The Presence Calculus

A Gentle Introduction

Dr. Krishna Kumar
The Polaris Advisor Program

1 What is The Presence Calculus?

The Presence Calculus is a quantitative model for reasoning about signal dynamics in a domain.

Its purpose is to support principled modeling and rigorous decision-making using operational data and signals in business-critical contexts—ensuring that such decisions rest on a mathematically precise foundation.

Figure 1: The Presence Calculus - Key Concepts

The presence calculus emerged from a search for better tools to reason about operations management in software product development and engineering—domains where operational measurement techniques lack a formal mathematical foundation and rely on ad hoc applications of principles imported from other domains.

At a minimum, its foundational constructs bring mathematical precision and clarity to widely used—but poorly defined—concepts such as flow, stability, equilibrium, and coherence for measurable signals in a domain.

More importantly, it offers a uniform set of abstractions and computational tools that connect path-dependent domain signals to business-relevant measures such as delay, cost, revenue, and user experience.

For software development, this enables the construction of bespoke, context-specific measurement models for operational improvement—a powerful alternative to one-size-fits-all metrics frameworks and visibility tools, which remain the dominant option today.

1.1 What is it?

The presence calculus is a novel modeling and measurement substrate with measure theory, a branch of real analysis, as its mathematical foundation. It treats time as a first-class concept, making it well suited for analyzing continuous, time-dependent behaviors of systems.

It complements classical statistical and probabilistic analysis, and is particularly well suited to domains like software development where state, history, and path dependence complicate traditional inference techniques 1.

On its own, the calculus provides a precisely defined set of modeling and computational primitives (shown in Figure 1) for analyzing the history and evolution of time-varying signal systems—structures about which we can make mathematically provable claims.

We can then use these primitives to construct richer, more expressive measurement models to reason about time varying behavior in real-world domains.

The presence calculus is not a modeling framework, a measurement system, or a methodology. It is a mathematical and analytical substrate—a foundation on which such frameworks or methodologies can be constructed.

It is not a replacement for existing mathematical tools. Rather, it is a tool that can operate alongside techniques such as statistical and probabilistic analyses. Its particular strength lies in analyzing a class of problems that are not easily addressed using those techniques.

Our goal in this document to present the foundational ideas of the presence calculus as a coherent whole. We will use examples to motivate key concepts but the focus is here is not on domain modeling or specific applications but on presenting the core theory, tools and techniques of the calculus and how they fit together.

As we’ll see, however, the core concepts are broadly applicable—well beyond the software domain where they originated.

1.2 The pitch

We introduce the simple but powerful concept of a presence.

This lets us reason about the history and evolution of a set of signals that measure time-varying, path-dependent properties of elements that are present in defined boundaries in a domain using techniques from measure theory, topology and complex analysis.

Classical statistics and probability theory often struggle here.

History—the sequence and structure of changes in the domain over time— is usually fenced off under assumptions like ergodicity, stationarity, and independence.

Our thesis is that to complement and extend statistical or probabilistic inference to reason effectively about global and long run behavior of many real world systems, especially those that arise commonly in software development, we need new analytical techniques that treat time and the history of signal interactions as first-class concepts we can model and calculate with.

The Presence Calculus is a novel, constructive approach to this problem—an analytical framework for modeling observed behavior in systems ranging from simple, linear, and ordered to non-linear, stochastic, adaptive, and complex, all based on a small, uniform set of underlying concepts rooted in a mathematically precise primitive called presence.

1.3 Why should I care?

Intuitively, the presence calculus is useful when a system signal has ” time-value” - i.e., its significance lies in how the underlying value of a property accumulates over time, not just its instantaneous values or their statistical properties.

We can categorize such signals into those we want to accumulate (e.g., revenue, profits, satisfied customers) and those we want to constrain (e.g., costs, risks, debts, delays). The presence calculus offers tools to frame decisions about such signals in terms of directly observing and actively managing a property’s accumulation over time and timescales.

Specifically we focus on systems properties and signals satisfying the following criteria.

For practical applications, we also require:

While this may sound restrictive, a surprisingly large class of operational and business-critical questions—especially those directly related to economic outcomes—fall squarely within this domain. Because these questions often underpin sound decision-making in business contexts, the presence calculus has the potential to significantly improve the quality of those decisions.

Modern technology makes is possible to directly instrument many such signals. The presence calculus thus opens up many more avenues for building operational systems to ask and answer such economically significant questions in a robust and principled way, and often in real-time.

For the questions that fall in this sweet spot, the calculus provides better tools to accurately measure and reason about signal dynamics - how economically significant properties of a system accumulates time-value across time-scales. These are questions that are hard to answer with purely statistical or probabilistic techniques.

As we will see, this lets us create some really novel tools for operational analysis of messy real-world systems like the ones we often encounter in software product development and engineering.

1.4 Learning about the presence calculus

In this document, we’ll motivate and introduce the intuitions and key ideas in the calculus with lots of evocative examples and mathematical simplifications to illustrate core concepts 3.

While the calculus was developed with mathematical rigor, an equally important goal was not to let mathematics get in the way of understanding the simple but very powerful ideas the calculus embodies.

However, in order to maintain precision, we dont shy away from mathematics where it is needed. We augment these with examples to build intuition throughout. However, given the nature of the material we have opted to stay on the side of rigor rather than dilute the concepts, even in this “gentle” introduction.

If you are inclined to skim over anything with mathematical notation in it, working through the examples alongside the math should be sufficient to grasp the key ideas and claims. However, for those who are comfortable with it, the mathematics should be easy to understand.

The presence calculus is constructive - specifically everything in the calculus has a computational aspect, and this document is designed to lay the framework for describing how to perform those computations in a step by step manner.

This means that it is best to read the sections in order, as each section builds on the concepts from the previous sections systematically. The concepts are not particularly difficult to grasp, but they will make more sense if you take time to understand how they fit together in the order they are presented here.

So even though it is a gentle introduction, it is not the kind of introduction that you will get as much value from if you simply skim this document superficially.

If you are not into reading long technical documents and would like to listen to a podcast that explains the key ideas, you can find a remarkably astute summary that was produced by Google’s NotebookLM here

Listen to it (if you have 30 minutes) and then come back and read this document if it sounds interesting!

If that deeper dive is not your cup of tea, we’ll continue with ongoing informal exposition on our blog The Polaris Flow Dispatch, where we will focus mostly on motivations, short pieces talking about specific areas, and applications of the concepts.

If you are interested in working with the calculus, we recommend reading and understanding all the main ideas in this document before jumping deeper into the rest of the documentation at this site, which get a fair bit more dense and technical.

This document can be thought as the middle ground: detailed enough to understand the concepts and even implement and extend them yourself if you are so inclined, but just a starting point if you want to really dig deeper. The background and context from this document will be important if you want to apply the concepts from first principles and develop new applications.

That next level of detail is in the API docs for The Presence Calculus
Toolkit.

The toolkit is an open source reference implementation of the core concepts in the presence calculus. It is currently situated 4 as a middleware layer suitable for interfacing real world operational systems and complex system simulation, to the analytical machinery of the presence calculus.

In the API documentation, we go into the concepts at a level of rigor that you’ll need to compute and build applications based on the presence calculus.

Finally, for those who want to dive deeper into the formal mathematical
underpinnings of the calculus, we have The Theory Track These are terse, technical documents that go into more detail than the average practitioner will need to read or understand, but should be straightforward for a reader with a background in real analysis to clearly grasp the mathematical definitions and claims behind the calculus. In the footnotes and references, we link to specific documents in this track for deeper mathematical treatments of the concepts we discuss here.

This document, and all related documentation can be found at the Presence Calculus Documentation Site

With all that as preamble, let’s jump in…

2 Why presence?

Presence is what we observe in the world.

We dont experience reality as a sequence of discrete events in time, but as a continuous unfolding—things are present, or they appear, endure for a while, and then slip away.

Permanence is simply a form of lasting presence. What we call change is the movement of presences into and out of awareness, often set against that permanence.

The sense that something is present—or no longer present—is our most immediate way of detecting change. This applies equally to the tangible—people, places, and things—and the intangible—emotions, feelings, and experiences.

Either way, reasoning about the presences and absences in our environment over time is key to understanding the dynamics of the world around us.

The Presence Calculus begins here.

Before we count, measure, compare, or optimize, we observe what is.

And what we observe is presence.

2.1 An example

Imagine you see a dollar bill on the sidewalk on your way to get coffee.
Later, on your way back home, you see it again—still lying in the same spot. It would be reasonable for you to assume that the dollar bill was present there the whole time.

Of course, that may not be true. Someone might have picked it up in the
meantime, felt guilty, and quietly returned it. But in the absence of other
information, your assumption holds: it was there before, and it’s there now, so it must have been there in between.

This simple act of inference is something we do all the time. We fill in gaps, assume continuity, and reason about what must have been present based on what we know from partial glimpses of the world.

The presence calculus gives formal shape to this kind of inference about things, places and time—and shows how we can build upon it to reason about presence and measure its effects in an environment.

2.2 A software example

Since the ideas here emerged from the software world, let’s begin with a
mundane, but familiar example: task work in a software team.

We usually reason about task work using events and snapshots of the state
of a process in time. A task “starts” when it enters development, and
“finishes” when it’s marked complete. We track “cycle time” by measuring the elapsed time between events, “throughput” by counting finish events, and ” work-in-process” by counting tasks that have started but not yet finished.

When we look at a Kanban board, we see a point-in-time snapshot of where tasks are at that moment—but not how they got there. And by the time we read a summary report of how many tasks were finished and how long they took to go through the process, much of the history of the system that produced those measurements has been lost. They become mere descriptive statistics about the system at a point in time. That makes it hard to reason about why those measurements are the way they are.

In software development, workflows are path-dependent: each task often has a distinct history—different from other tasks present at the same time. Losing history makes it hard to reason about the interactions between tasks and how they impact the global behavior of the process.

This problem is not unique to task work. Similar problems exist in almost all areas of business analysis that rely primarily on event models and descriptive statistics derived from events as the primary measurement tool for analyzing system behavior.

We are reduced to trying to make inferences from local descriptive
statistics —things like cycle times, throughput, and work-in-process levels- over highly irregular, path-dependent processes.

We try to reason about a process which is shaped by its history, whose behavior emerges from non-uniform interactions of individual tasks that have impacts at different timescales, with measurement techniques that lack the ability to represent or reason about that history or the interactions.

This is difficult to do, and we have no good tools right now that are fit for this purpose. So we either try to force fit our processes so that we can model and measure them more easily, or we simply make invalid inferences about them using the techniques that are not designed to operate accurately in these kinds of domains.

This is where the presence calculus begins.

By looking closely at how we reason about time varying quantities in the presence calculus, we can see how a subtle shift from an event-centered to a presence-centered perspective changes not just what we observe, but what we measure, and thus can reason about.

In this particular case, the calculus focuses on the time in between snapshots of history: when a task was present, where it was present, for how long, and whether its presence shifted across people, tools, or systems.

The connective tissue is no longer the task itself, or the process steps it
followed, or who was working on it, but a continuous, observable thread of
presence—through all of them, moving through time, interacting, crossing boundaries—a mathematical representation of history.

With the presence calculus, these threads and their interactions across time and space can now be measured directly, dissected, composed, and analyzed as first-class constructs—built on a remarkably simple yet general primitive—the presence.

In the above example, the calculus exploits the difference between the two independent statements—“The task started development on Monday” and “The task completed development on Friday”—and a single, unified assertion: “The task was present in development from Monday through Friday.”

The latter is called a presence, and it is the foundational building block
of the calculus. At first glance, this might not seem like a meaningful difference.

But treating the presence as the primary object of reasoning—as a first-class construct—opens up an entirely new space of possibilities.

Specifically, it allows us to apply powerful mathematical tools that exploit the topology of time and the algebra of time intervals to reason about the interactions and configurations of presences in a rigorous and structured, and more importantly, computable way.

3 What is a presence?

Let’s start by building intuition for the concept. Consider the statement: “The task \(X\) was in Development from Monday to Friday.”

In the presence calculus, this would be expressed as a statement of the form: “The element \(X\) was in boundary \(Y\) from \(t_0\) to \(t_1\) with mass 1.”

Presences are statements about elements (from some domain) being present in a boundary (from a defined set of boundaries) over a continuous period of time, measured using some timescale.

So why do we say “with mass 1”?

The presence calculus treats time as a physical dimension, much like space. Just as matter occupies space, presences occupy time. Just as mass quantifies how matter occupies space, the mass of a presence quantifies how a presence occupies time.

The statement “The task \(X\) was in Development from Monday through Friday” is a binary presence with a uniform mass of 1 over the entire duration. The units of this mass are element-time—in this case, task-days.

Binary presences are sufficient to describe the fact of presence or absence
of things in places in a domain. These presences always have mass 1 in whatever units we use for elements and time.

Typically, but not always, these represent the presence or absence of activity in a domain and can also be considered activity signals.

3.1 Presence mass: the manifestation of presence

Let’s consider a different set of statements:

“Task \(X\) had 2 developers working on it from Monday to Wednesday,
3 developers on Thursday, and 1 developer on Friday.”

These are no longer about just presence, but about the effects of presence.
They describe the load that task \(X\) placed on the Development boundary
over time.

The units of this presence are developer-days - potentially in a completely different dimension from the task, but grounded over the same time interval as the task.

Here we are saying: “the task being in this boundary over this time period, had this effect in a related dimension.”

We will describe this using a presence with an arbitrary real valued mass. We will assume that there is some function, that computes this mass. In our example, let’s call this function \(\mathsf{load}.\)

The presence can then be described as

Here, \(\mathsf{load}(e, b, t)\) is a time-varying function that takes an
element \(e\), a boundary \(b\), and a time \(t\), and returns a real-valued number
describing how much presence is concentrated at that point in time.

The presence mass of such a presence is the total presence over the
interval \([t_0, t_1]\), defined as:

\[ \text{mass} = \int_{t_0}^{t_1} \mathsf{load}(e, b, t)\, dt \]

where \(\mathsf{load}\) is called a presence density function 5.

Presence mass is composite that measures some real valued quantity and the amount of time it was present. Think of it as measuring an area in a two-dimensional space with time as one of the dimensions.

Binary presences are much easier to understand intuitively, but the real power of the presence calculus comes from generalizing to presence density functions with arbitrary mass.

3.2 Presence density functions aka Signals

Figure 2: Signals, Presence and Presence Mass

We can extrapolate from the example of the load function and think of defining a presence over an arbitrary time varying function with real numbers as values. We will call these presence density functions.

Such functions represent an underlying signal from the domain that we are interested in measuring. In what follows, we will use the terms signal and presence density functions interchangeably, opting for the latter only those cases where we want to focus specifically on the fact that what we are representing about the signal is the “amount” of the signal (its presence) over time.

As shown in Figure 2, the mass of a presence density function, over any given time interval \([t_0, t1)\) is the integral of the function over the interval, which is also the area under the signal over that interval6.

The only requirement for a function to be a presence density function (signal) is that it is measurable, and that you can interpret presence mass—defined as the integral of the function over a finite interval—as a meaningful measure of the effect of presence in your domain.

This is where measure theory enters the picture. It’s not essential to understand the full technical details, but at its core, measure theory tells us which kinds of functions are measurable—in other words, which functions can support meaningful accumulation, comparison, and composition of values via integration.

When a presence density functions (signal) is measurable, it gives us the confidence to do things like compute statistics, aggregate over elements or boundaries, and compose presence effects—while preserving the semantics of the domain.

From our perspective, a presence density function is a domain signal whose value that can be accumulated across time and across presences.

This lets us reason mathematically about presences with confidence—and since most of this reasoning will be performed by algorithms, we need technical constraints that ensure those calculations are both mathematically valid and semantically sound.

For a mathematically precise definition of these concepts please see our theory track document Presence: A measure-theoretic-definition.

3.3 More examples

Let’s firm up our intuition about what signals in the presence calculus can describe with a few more examples of presence density functions.

3.3.1 “Work” in software

If you’ve ever written a line of code in your life, you’ve heard the question:
“When will it be done?” Work in software can be a slippery, fungible concept— and the presence calculus offers a useful way to describe it.

We can express the work on a task using a presence density function whose value at time \(t\) is the remaining work on the task at \(t\).

This lets us model tasks whose duration is uncertain in general, but whose
remaining duration can be estimated, subject to revision, at any given time—a common scenario in software contexts.

A series of presences, where the (non-zero) mass of each presence corresponds to the total remaining work over its interval (interpreting the integral as a sum), gives us a way to represent work as presence.

Such presences can represent estimates, forecasts, or confidence-weighted
projections—and as we’ll see, they can be reasoned about and computed with just like any other kind of presence.

3.3.2 “Batch size” in software

Let’s consider another, more familiar example. There are well-understood benefits to delivering software more frequently, in smaller batches, and modern software engineering practices encourage teams to adopt this delivery model instead of relying on infrequent, large batch deliverables.

For companies with legacy processes built around large batch delivery, and looking to measure progress on the journey toward smaller batches, the state-of-the-art metrics used to guide the transition are deployment frequency and lead time for changes — two of the four key DORA metrics considered gold standards in the industry [5].

However, both deployment frequency and lead time for changes are proxies for batch size. And like all proxies, they are imperfect measurements of the true quantity of interest, often providing little insight into the causal drivers that shape the proxies themselves.

With the presence calculus, we have a different way to model this problem.
We can define a system of presence density functions, where:

Under this model, batch size becomes the total number of unmerged lines of code across the system at any given point in time — a quantity that is directly measurable using the machinery of presence calculus.

With modern tooling, it is entirely feasible to instrument such a model in a real-world delivery system and keep it updated in real time.

This kind of real-time, direct model has several important advantages over measuring the proxy metrics deployment frequency and lead time for changes after the fact.

Here are some benefits of this approach 7:

These are all substantial improvements over the status quo measurement techniques in DevOps and it all falls out of the representation and computation machinery of the presence calculus we describe in this document. This also means the presence calculus gives an precise mathematical model that lets us directly observe the relationship between batch size and the two DORA metrics in any given delivery context in real time.

3.3.3 The effects of interruptions

Another useful example from the software world illustrates a different
application of a presence. Let’s assume the boundary in this case is a
developer, the element is an interruption (defined appropriately in the
domain), and the presence density function captures the context switching
cost—measured in lost time—associated with that interruption.

The key insight here is that the effects of the presence extend beyond the
interval of the interruption itself. This is a classic case of a delayed or
decaying effect—a pattern that appears frequently in real-world systems.

The presence density function can be modeled in different ways:

This approach gives us a precise way to model and reason about the
aftereffects of events—effects that outlast the events themselves and
accumulate in subtle but measurable ways over longer timeframes.

In this case, we measured an effect that decayed from a peak, but a similar
approach can be taken, for example, with a presence density function that
grows from zero and plateaus over the duration of the presence—such as the net increase in revenue due to a released feature.

Used this way, presence density functions give us a powerful tool for modeling the impact of binary presences—capturing their downstream or distributed effects over time, and reasoning about their relationship over a shared timeline.

Another important use case in the same vein is modeling the cost of delay for a portfolio-level element—and analyzing its cascading impact across a portfolio.

These use cases show that it is possible to analyze not just binary presences, but entire chains of influence they exert across a timeline—a key prerequisite for causal reasoning.

3.3.4 Self-reported developer productivity

Imagine a developer filling out a simple daily check-in: “How productive did you feel today?”—scored from 1 to 5, or sketched out as a rough curve over the day8.

Over a week, this forms a presence density function—not of the developer in a place, but of their sense of productivity over time.

These types of presences, representing perceptions, are powerful—helping
teams track experience, spot early signs of burnout, or correlate perceived
productivity with meetings, environment changes, build failures, or interruptions.

Now, let’s look at some examples outside software development.

3.3.5 Browsing behavior on an e-commerce site

Imagine a shopper visiting an online store. They spend 90 seconds browsing kitchen gadgets, then linger for five full minutes comparing high-end headphones, before briefly glancing at a discounted blender.

Each of these interactions can be modeled as a presence: the shopper’s
(element) attention occupying different parts of the site (boundaries) over
time. The varying durations reflect interest, and the shifting presence reveals patterns of engagement in a population of visitors in a boundary (an area of the site).

By analyzing these presences across visitors to the site —where and for how long attention dwells—we can begin to understand population level preferences, intent, and even the likelihood of conversion (modeled as a different presence density function).

3.3.6 Patient movement in a hospital

Consider a patient navigating a hospital stay. They spend the morning in
radiology, move to a recovery ward for several hours, then are briefly
transferred to the ICU overnight.

Each location records a presence—when and where the patient was, and for how long. These presences can reveal bottlenecks, resource utilization, and potential risks.

Over time, analyzing patient presences helps surface patterns in care
delivery, delays in treatment, and opportunities for improving patient flow.

These are examples of classic operations management problems expressed in the language of the presence calculus. The calculus is well-suited to modeling scenarios like these as a base case.

3.4 Presence: a summary

Let’s summarize what we’ve described so far.

With signals and presences, we now have a framework for describing and measuring the behavior of time-varying domain signals—each representing how a specific element behaves within a given boundary.

The key feature of a presence is that it abstracts these behaviors into a uniform representation—one we can reason about and compute with.

Formally, a general presence is defined by:

Its mass is the integral of \(f\) over the interval:

\[ \text{mass}(e, b, [t_0, t_1]) = \int_{t_0}^{t_1} f(e, b, t)\, dt \]

This mass captures both that the element was present, and how it was
present—uniformly, variably, or intermittently—over the interval.

Intuitively, you can think of integration as the mathematical process by which we construct a continuous temporal model from discrete events in a domain.

Sensors in the real world often generate both discrete event streams and continuous signals. Modeling all signals uniformly as presences with temporal mass is the first step toward analyzing interactions and dynamics across heterogeneous signals within a domain.

4 Systems of presence

In this section, we move from individual presence density functions to systems of signals—each capturing the presence behavior of multiple elements across multiple boundaries within a domain.

A core aspect of the Presence Calculus is its treatment of fine-grained signals: each domain element is associated with its own presence density function, reflecting its path-dependent trajectory through one or more boundaries in the system.

We study how the presence masses of these signals interact over time to produce cumulative effects—observable outcomes that carry semantic meaning in the domain.

Each signal represents the continuous presence behavior of a specific element within a boundary over time. The calculus emphasizes that each signal traces a distinct trajectory, and seeks to explain how their interactions over shared intervals shape system-level behavior.

Figure 3: A System of Presences

From the standpoint of the calculus itself, it does not matter what the signals represent—we treat them uniformly as real-valued presence masses and define a standard set of mathematical operations over them.

From the standpoint of semantics, however, the meaningfulness of these operations depends on the domain. What it means to combine signals, which ones to combine, and how to interpret the result—all of this is context-dependent and crucial for producing useful insights.

This aspect—modeling and model development—is essential for applying the machinery effectively. However, our focus here is on the calculus itself: what it can say in general about arbitrary systems of signals and presence density functions that interact in time.

In what follows, we will treat signals and presence density functions as abstract mathematical objects and describe the properties and operations of such systems. We will build some domain-specific intuition through examples, but will not focus on applications.

Those will be the subject of future posts—an area that is both rich and complex, and which will depend heavily on the foundational machinery developed in this document.

4.1 Presence as a sample of a signal

A signal, in general, describes the continuous behavior of an element within a boundary over time. This continuous signal may have one or more disjoint periods where its value is non-zero. These non-zero periods are called the support of the signal.

As we see in Figure 4, a single signal (for a given element and boundary) might have multiple support intervals. These may correspond to episodic behavior in the underlying domain, for example, user sessions in an e-commerce context, or rework loops in software development when a task “returns” to development many times over its lifecycle.

Figure 4: A presence mass as a sample of a signal over an interval

A presence (the 5-tuple \(p = (e, b, t_0, t_1, m)\) that we work with in the calculus) is generated by taking an observation of this underlying signal over a specific time interval \([t_0, t_1)\), and computing its mass. This interval can be chosen in many ways:

All we require is that the interval chosen for the presence calculation intersects a region of non-zero area from the signal that can be reduced to a non-zero presence mass.

4.2 Presence assertions

Thus presence is best thought of as a sampled measurement of the underlying signal, taken by an observer over a specific time interval, which yields a non-zero mass for that interval.

A given observer may not “see” the underlying signal—only the mass of the presence they sampled over an observation window.

Different observers may observe different intervals of the same signal and derive different presence masses for the same signal.

So here is the first key assumption of the presence calculus:

A system of presences refers to a collection of discrete observations drawn from an underlying set of presence density functions. In general, we do not assume we have access to the “true” underlying functions. Instead, we reason about these functions based sampled presence masses and treat these presences as the basis of representation and inference within the calculus.

This brings us to the concept of presence assertions, which formalize the act of recording a presence based on an observer’s local “view” of the underlying density function.

A presence assertion is simply a presence as defined in the previous section augmented with additional attributes:

4.2.1 The open world assumption

Time in the presence calculus is explicitly defined to be over extended reals \(\overline{\mathbb{R}}\): the real line \(\mathbb{R}\) extended with the symbols \(-\infty\) and \(+\infty\).

This is a mathematical representation of an open world assumption, which holds that the history of a system of presences extends indefinitely into the past and future.

An observer will typically only see a finite portion of this history and has to make inferences on the basis of those observations, but in general, we need to make inferences with partial information about the past and contingent assumptions about the future.

A presence with \(t_0 = -\infty\) represents a presence whose beginning is unknown, and \(t_1 = +\infty\) represents a presence whose end is unknown.

Presences with both start and end unknown are valid constructs and represent eternal presences.

Many of the most interesting questions in the presence calculus involve reasoning about the signal dynamics of a domain under the epistemic uncertainty introduced by such presences.

Further we will note that the assertion time in a presence assertion doesn’t need to align with the time interval of the presence. This allows assertions to refer to the past, reflect the present, or even anticipate the future behavior of a signal.

4.2.2 A note on epistemology

Presence assertions give us the ability to assign provenance to a presence— not just what we know, but how we know it. This is essential in
representing contexts where the observer and the act of observation are first-class concerns.

Further, since reasoning about time and history is a primary focus of the calculus, presences with unknown beginnings or endings provide a way to explicitly model what is known—and unknown—about that history. This will prove more valuable than it might initially seem.

We won’t go too deeply into the epistemological aspects of the presence
calculus in this document—this remains an active and open area of research and development and complements much of what we discuss here.

But it’s important to acknowledge that this layer exists, that modeling and interpreting the output of the presence calculus requires an explicit treatment of how observations are made and by whom, and the fact that this has a huge impact on the validity of the inferences one makes using the machinery of the calculus.

With this caveat in place, once we’ve represented a problem domain as a
system of presences, much of the machinery of the presence calculus (which we’ll introduce in the next few sections) can be applied uniformly.

4.2.3 A note on path dependence

By representing a presence at the granularity of an element in a boundary, we explicitly recognize the path dependent nature of the domain signals.

In the presence calculus, signals represent some behavior of domain elements in boundaries. The calculus itself is agnostic to what counts as an “element” or a “boundary”— this is a domain modeling decision.

Even if they represent the same underlying quantity, we recognize that system behavior emerges from the interactions between individual signals at the element-boundary granularity—each potentially with distinct presence density functions. We are interested in studying how system-level behavior arises from the interactions between these signals over time.

Modeling boundaries are crucial because, for a given domain element, the boundary typically determines both which signals are relevant and how we wish to analyze system behavior using them.

For example, a software feature (an element) may be modeled using one set of signals during development (one boundary), another during production (a second boundary), and yet another when customers begin using it (a third boundary).

All three signals are part of the feature’s history. Each feature follows a unique path through these boundaries, producing its own independent set of signals with a distinct history and evolution. These signals interact in complex ways—over time, within boundaries, and with each other. The boundary is what brings coherence to the analysis—it defines which signals and interactions we choose to focus on, and why.

The boundary allows us to bring a coherent set of elements and related signals together for analysis. That analysis focuses on how these signals interact in time.

Constructing an appropriate set of element-boundary signals is the key modeling decision. But once these are defined, much of the machinery of the presence calculus can be applied without regard to the semantics of the specific element-boundaries involved.

Just as important as modeling time, is the explicit modeling of timescales. In software development the effects of presence often manifest across boundaries and across timescales.

We will see how the machinery of the presence calculus enables us to reason precisely and deterministically about presence at different timescales.

Semantics are, of course, crucial in interpreting the inferences one draws
using the machinery of the calculus.

When we refer to a “system” in the presence calculus we are explicitly defining it as an evolving set of presence assertions—i.e., the “system” is what we can assert about a domain using presence assertions at a given time.

In summary, this is what we refer to as a “system of presences” - a time-indexed collection of presence assertions derived from an underlying set of path dependent element-boundary signals.

5 Co-presence and the presence invariant

Figure 5 illustrates an example of a system of presences, where we focus on the subset of presences observed over a common interval.

Figure 5: Co-Presence and The Presence Invariant

These presences are called co-present—they represent an observer making simultaneous measurements of presence mass across multiple signals over a common interval of time.

Co-presence is a necessary (but not sufficient) condition for interaction between one or more signals. This section introduces a key construct: the presence invariant. It expresses a general and powerful relationship that holds for any co-present subset of presences within a finite observation window, and it is a fundamental relationship that governs how the masses of co-present signals interact.

Let’s establish this relationship.

Given, any finite observation interval, we’ve already shown that each presence density function has a presence mass, defined as the integral of the density over the observation interval.

These can be thought of as mass contributions from those presences to that interval. The sum of these individual mass contributions gives the total presence mass observed across the system in that interval.

In our example from Figure 5,

\[ A = M_0 + M_1 + M_3 \]

is the cumulative mass contribution from the signals that have non-zero mass over the interval \([t_0, t_1)\). The length of this interval is \(T = t_1 - t_0\).

Let \(N\) be the number of active signals: distinct signals with a presence in the observation window9.

Now let’s consider the quantity \[ \delta = \frac{A}{T} \]

Since the mass comes from integrating a density function over time, the quantity \(\frac{A}{T}\) represents the presence density over the observation interval \(T\) 10.

We can now decompose this as:

\[ \delta = \frac{A}{T} = \frac{A}{N} \times \frac{N}{T} \]

This separates the presence density into two interpretable components:

This leads to the presence invariant:

\[ \text{Presence Density} = \text{Signal Incidence Rate} \times \text{Mass Contribution per Signal} \] or in our notation

\[ \delta = \iota \cdot \bar{m} \]

This identity holds for any co-present subset of signals over any finite time interval.

The key insight here is that the presence invariant establishes a fundamental relationship between the individual mass contributions of signals and their cumulative, observable effect over a shared time interval—that is, the presence density they induce together over time.

That this identity holds for any co-present subset over any finite observation window makes it a powerful constraint. It connects local behaviors of individual signals to a global, observable quantity 11.

As we’ll see, this constraint—linking local contributions to global measurable structure—is central to how the calculus enables reasoning about the dynamics of a system of presences.

5.1 An example

To build intuition for these abstract terms, let’s look at a practical example.

For example, suppose our signals represent revenues from customer purchase over some time period. If we look at a system of presences, across an interval of time, say a week, the total presence mass \(A\) represents the total revenues across all customers who contributed to that revenue. \(T\) is the time period measured in some unit of time (say days) and \(N\) is the number of paying customers in that period.

The presence density is the daily revenue rate, the signal mass contribution for each signal is revenue for each customer, the signal mass contribution is the revenue per customer for that week, and the incidence rate represents the daily rate of active customers over the week.

So the presence invariant is stating that the revenue rate for the week is the product of the revenue per customer and the number of active customers over the week.

5.2 Why it matters

While algebraically, the presence invariant is a tautology, it imposes a powerful constraint on system behavior—one that is independent of the specific system, semantics, or timescale. Think of it as the generalization of our intuitive revenue example to any arbitrary system of presences.

The presence invariant is a foundational conservation law of the presence calculus: the conservation of mass (contribution).

Just as the conservation of energy or mass constrains the evolution of physical systems—regardless of the specific materials or forces involved—the conservation of presence mass constrains how observable mass is distributed over time in a system of presences.

The version of the invariant described here is a special case of an even more general measure theoretic construction that we derive in the document The Presence Invariant. We show there that the presence invariant is not just a mathematical trick, but a deep property of product measure spaces.

Thus, the conservation of presence mass plays a role in the presence calculus similar to that of other conservation laws in physics: it constrains the behavior of three key observable, measurable parameters of any system of presences.

While independent of the semantics of what is being observed, like energy, presence mass can shift, accumulate, or redistribute, but its total balance across presences within a finite time interval remains invariant.

Exploiting this constraint allows us to study and characterize the long-run behavior of a system.

5.3 Binary presences and Little’s Law

Let’s make things a bit more concrete by interpreting this identity in the special case of binary presences.

Recall that a binary signal is a function whose density is either \(0\) or \(1\). That is, we are modeling the presence or absence of an underlying signal in the domain.

In this case, the mass contribution of a signal becomes an element-time duration. For example, if the signal represents the time during which a task is present in development, the mass contribution of that task over an observation interval is the portion of its duration that intersects the interval. This is also called the residence time 12 for the task in the observation window.

Figure 6: The Presence Invariant for binary signals

Figure 6 shows possible configurations of binary signals intersecting a finite observation interval. Suppose the unit of time is days.

The total presence mass accumulation \(A\) is \(11\) task-days. The number \(N\) of tasks that intersect the observation interval is \(4\). The length of the observation window is \(T = 4\) days. It is straightforward to verify that the presence invariant holds.

Now, let’s interpret its meaning.

Since each task contributes \(1\) unit of mass for each unit of time it is present, the presence density \(\delta=\frac{A}{T}\) represents the * number of tasks* present per unit time in the interval—denoted \(L\).

Conversely, since each unit of mass corresponds to a unit of time associated with a task, the mass per active signal, \(\bar{m} = \frac{A}{N}\), is the time a task spends in the observation window. This value is typically called the residence time \(w\) of a task in the observation window.

The incidence rate \(\iota = \frac{N}{T}\) may be interpreted as the activation rate of tasks in the interval—a proxy for the rate at which tasks start (onset) or finish (reset) within the window.

For example, \(N\) may be counted as the number of tasks that start inside the interval, plus the number that started before but are still active. Thus, \(\frac{N}{T}\) is a cumulative onset rate \(\Lambda\).

The presence invariant can now be rewritten as:

\[ L = \Lambda \times w \]

which you may recognize as Little’s Law in its finite-window form.

Thus, the presence invariant serves as a generalization of Little’s Law—extending it to arbitrary systems of presence density functions (signals) measured over finite observation windows.

It is important to note that we are referring to Little’s Law over a finite observation window, rather than the more familiar steady-state, equilibrium form of Little’s Law 13.

Unlike the equilibrium form of the law, this version holds unconditionally. The key is that the quantities involved are observer-relative: the time signals spend within a finite observation window, and the incidence rate of signals over the window, rather than the signal-relative durations or long-run onset/reset rates assumed in the equilibrium form.

Indeed, the difference between these two forms of the identity will serve as the basis for how we define whether a system of presences is in equilibrium or not. The idea is that the system of presences is at equilibrium when observed over sufficiently long observation windows such that the observer-relative and task-relative values of presence density, incidence rate and presence mass converge.

Since real-world systems often operate far from equilibrium—and since the presence invariant holds regardless of equilibrium—the finite-window form becomes far more valuable for analyzing the long-run behavior of such systems as they move into and between equilibrium states.

All this is the focus of section 6, where we will formally make the connections between the presence invariant and equilibrium states in systems of presence with the help of a very general form of Little’s Law originally proven by Brumelle [8], and Heyman and Stidham [9].

5.4 Signal dynamics

Now that we’ve defined how to observe a system of presences over a single finite interval, we turn to what happens when we observe the system continuously over time—computing the parameters of the presence invariant across consecutive, equal-sized, half-open intervals.

Figure 7: Sampling a system of presences across uniform intervals

This step is fundamental to the presence calculus: it allows us to study how presence density evolves across consecutive observations—the signal dynamics of the system.

The presence invariant holds at each interval and defines a constraint among the three key parameters: presence density, signal incidence rate, and mass contribution per signal.

Given any two of these, the third is completely determined.
Among them, presence density is the output; incidence rate and mass contribution are the inputs.

At each interval, presence density can be directly observed—but the invariant requires that it always equal the product of incidence rate and mass contribution:

Any change in presence density must result from a change in incidence rate, mass contribution, or both.

In other words, the system has only two degrees of freedom among three interdependent variables.

Because the invariant holds across any finite interval, tracking how these parameters shift reveals how a particular system of presences evolves.

In Section 7, we’ll explore a geometric view of this idea. If we treat these three parameters as coordinates of the system’s state in each interval, we can trace its evolution as a trajectory through time.

This makes the presence invariant a powerful tool for causal reasoning—one that helps explain why presence density changes the way it does.

In the next section, we’ll introduce the presence matrix—a compact representation of the sampled signals shown in Figure 7. It is a key building block in the machinery for computing these trajectories.

6 The presence matrix

A presence matrix is a data structure that records the presence mass values resulting from the sampling process described in Figure 7.

The length of each sampling interval is called the sampling granularity. This defines the smallest time resolution at which the matrix represents presence 14.

The union of these intervals is called the observation window of the matrix.

Choosing the right sampling granularity is a key modeling decision. It directly affects the kinds of insights we can extract from the presence matrix using the machinery we develop later.

Given \(M\) presence density functions and an observation window consisting of \(N\) intervals at some fixed sampling granularity, the presence matrix \(P\) is an \(M \times N\) matrix where:

The presence matrix \(P\) is a discrete, time-aligned representation of presence mass for a given system of presences.

Since we are accumulating presence masses over an interval, the value of presence mass in a matrix entry is always a a real number. Figure 8 shows the presence matrix for the system in Figure 7.

Figure 8: Presence Matrix for a system of presences

In Figure 8,

The alert reader will note the difference between the first two rows in the matrix. Even though the underlying signals both have two distinct support intervals, the first signal is represented by a single presence in the matrix, while the second is broken up into two disjoint presences.

This is entirely an artifact of the granularity at which the signal is sampled. At a suitably fine sampling granularity, the first signal could also be represented by two presences. However, we are not losing any information - the total presence mass is simply distributed across different observation windows during the integration step that computes the presence mass.

In other words, this has no real impact on the behavior of the invariant 15 provided we cover the timeline with our observations.

The presence matrix encodes deep structural properties of a system of presences. Many of key concepts we want to highlight are easier to define and understand in terms of this representation.

6.1 The presence invariant and the presence matrix

Let’s revisit Figure 4, reproduced below, which introduced the idea of interpreting presence mass as a sample from an underlying signal.

When we observe a system of presences across a finite observation window—as we do when deriving the presence invariant—we are looking at presence mass across a “vertical” slice of time, spanning many signals.

E 1 2 3 4 5 6 7 8 9 10
e1_b1 0.3 2.3 3.4 1.1 2.9 3.2 1.1 0.0 0.0 0.0
e1_b2 0.3 2.3 3.4 1.1 0.0 1.1 2.2 2.4 2.3 0.8
e2_b2 0.0 0.0 0.0 0.0 0.0 0.0 0.9 1.8 3.2 0.9
e2_b1 0.8 1.3 2.4 2.8 3.0 3.2 3.4 2.4 0.0 0.0

Figure 9: Mass contributions by signal.

In the presence matrix from Figure 8, reproduced in Figure 9, each entry represents the presence mass obtained by sampling a signal over a particular interval. Rows correspond to distinct signals, and columns represent sampling intervals.

We can construct a presence matrix from any subset of the rows of another presence matrix—and it would still be a valid presence matrix for the underlying system. Similarly, any consecutive sub-sequence of columns within such a matrix also constitutes a presence matrix over a sub-interval of the system’s history.

Let’s see what this means in terms of the presence invariant.

Figure 10: Computing the invariant from the matrix

In Figure 10 we interpret a consecutive sequence of columns in the matrix as an observation window. Given such a window and the submatrix it induces:

Because \(A\), \(N\), and \(T\) are directly computable from the submatrix, we can derive the three parameters of the presence invariant:

  • presence density: \(\delta = \frac{A}{T}\)
  • signal incidence rate: \(\iota = \frac{N}{T}\)
  • mass contribution per signal: \(\bar{m} = \frac{A}{N}\)

Thus the parameters of the presence invariant are well defined for any observation window over a presence matrix.

Next, we introduce a structure that allows us to compute and analyze these parameters across multiple observation windows—the final piece we need to compute signal dynamics for the system.

7 The presence accumulation matrix

Signal dynamics requires us to reason about both the micro and macro behaviors of a system of presences. To do this efficiently we will derive a data structure called the Presence Accumulation Matrix.

This matrix compresses information about the interaction between micro and macro behavior of a system of presences across time windows and time scales. Let’s see how it is constructed.

We will start with the presence matrix of Figure 10. Recall that this is an \(M \times N\) matrix of \(M\) signal sampled at \(N\) consecutive intervals, and the value at row \(i\) and column \(j\) represents the sampled presence mass of signal \(i\) over the observation window \(j\).

Now, consider the matrix \(A\) that accumulates presences masses by applying the windowing operation of Figure 10 across all possible observation windows over the original matrix.

So, for each pair of columns \(i,j,\) the entry in \(A[i,j]\) equals the value of the total presence mass \(A\) for the submatrix induced by the columns \([i,..,j]\) in the original presence matrix.

As shown in Figure 11, the diagonal of the matrix contains the accumulated presence mass across signals at the sampling granularity. Each entry here is the sum of a single column in the presence matrix.

i\j 1 2 3 4 5 6 7 8 9 10
1 1.4
2 5.9
3 9.2
4 5.0
5 5.9
6 7.5
7 7.6
8 6.6
9 5.5
10 1.7

Figure 11: Presence mass for single observation window..

The next diagonal contains the cumulative presence mass for intervals of length 2 - ie \(A(1,2)\) contains the sum of all entries in columns 1 and 2. \(A(2,3)\) contains the sum of all entries column 2 and 3 and so on..

i\j 1 2 3 4 5 6 7 8 9 10
1 1.4 7.3
2 5.9 15.1
3 9.2 14.2
4 5.0 10.9
5 5.9 13.4
6 7.5 15.1
7 7.6 14.2
8 6.6 12.1
9 5.5 7.2
10 1.7

Figure 12: Cumulative presence mass along intervals of length 2.

We can continue filling the matrix out in diagonal order this way until we get the presence accumulation matrix shown below.

i\j 1 2 3 4 5 6 7 8 9 10
1 1.4 7.3 16.5 21.5 27.4 34.9 42.5 49.1 54.6 56.3
2 5.9 15.1 20.1 26.0 33.5 41.1 47.7 53.2 54.9
3 9.2 14.2 20.1 27.6 35.2 41.8 47.3 49.0
4 5.0 10.9 18.4 26.0 32.6 38.1 39.8
5 5.9 13.4 21.0 27.6 33.1 34.8
6 7.5 15.1 21.7 27.2 28.9
7 7.6 14.2 19.7 21.4
8 6.6 12.1 13.8
9 5.5 7.2
10 1.7

Figure 13: Final accumulation matrix.

Figure 14, shows this computation.

Figure 14: Accumulation Matrix Construction

At the observation granularity we have windows of length 1, next we have accumulated presence from two consecutive observation windows, next we have accumulated presence from three consecutive windows, and so on.

We can see that in practice, this matrix compresses a large amount of information in a very compact form.

For example if the columns represent weekly samples of signals a 52x52 matrix allows us to analyze a whole years worth of presence accumulation across every time-scale ranging from a single week to a whole year in one compact structure.

7.0.1 Presence Accumulation Matrix - Definition

To summarize lets define the general structure of the presence accumulation matrix, formally.

Let \(P \in \mathbb{R}^{M \times N}\) be the presence matrix of \(M\) signals sampled at \(N\) consecutive time intervals. The Presence Accumulation Matrix \(A \in \mathbb{R}^{N \times N}\) is defined as:

\[ A(i, j) = \sum_{k=1}^{M} \sum_{\ell=i}^{j} P(k, \ell) \quad \text{for all } 1 \leq i \leq j \leq N \]

That is, \(A(i,j)\) gives the total presence mass of all signals across the interval \([i,j]\).

As we will see below, this matrix compactly encodes multi-scale information about system behavior and supports the analysis of both micro and macro scale behavior of a system of presences.

7.1 The presence accumulation recurrence

In this section, we demonstrate a key implication of modeling signals as sampled presences over time: that the relationship between presence mass accumulation at micro and macro timescales can be described deterministically.

This powerful property arises directly from the fact that presence masses are measures over time intervals—and that such measures satisfy a mathematical property called finite additivity.

To explain this, we reproduce the presence accumulation matrix from Figure 12 below.
i\j 1 2 3 4 5 6 7 8 9 10
1 1.4 7.3 16.5 21.5 27.4 34.9 42.5 49.1 54.6 56.3
2 5.9 15.1 20.1 26.0 33.5 41.1 47.7 53.2 54.9
3 9.2 14.2 20.1 27.6 35.2 41.8 47.3 49.0
4 5.0 10.9 18.4 26.0 32.6 38.1 39.8
5 5.9 13.4 21.0 27.6 33.1 34.8
6 7.5 15.1 21.7 27.2 28.9
7 7.6 14.2 19.7 21.4
8 6.6 12.1 13.8
9 5.5 7.2
10 1.7

As we will see in the next section, the diagonal of the matrix represents a sample path through the system’s history, and the top row records cumulative presence mass over the entire observed history. The diagonal shows accumulation at the finest (micro) timescale; the top row reflects accumulation at the coarsest (macro) timescale. Each diagonal represents accumulation across intervals of increasingly coarse granularity.

In essence, the accumulation matrix is a compact encoding of how presence mass builds up across timescales.

But there’s more: the matrix entries obey a local recurrence relationship, which allows us to reconstruct the entire matrix from its diagonal:

\[ A[i,j] = A[i,j-1] + A[i+1,j] - A[i+1,j-1] \quad \text{for } 1 \le i < j \le N \]

It tells us that the cumulative mass from column \(i\) to column \(j\) can be
computed from three nearby entries:

This equation reflects the finite additivity of cumulative presence mass over rectangular regions in the presence matrix 17.

So, the entire matrix—and thus, the system’s accumulation dynamics—is governed by a simple, local rule. Given the values on the diagonal, the rest of the matrix is completely determined.

The physical meaning is this: if we know the first \(N{-}1\) values along a sample path, then observing the \(N^\text{th}\) value allows us to explain how the macro behavior of the system evolved across all timescales up to that point.

What’s remarkable is that this determinism holds regardless of the nature of the underlying processes. The signals might come from deterministic, stochastic, linear, non-linear, or even chaotic processes. As long as the signals they generate are measurable they satisfy the finite additivity property and when we sample their observed presence masses, this local recurrence always applies 18.

Combined with the presence invariant—which also holds at every level of this accumulation—this gives us a powerful framework for dissecting the dynamics of a system of presences.

8 Computing signal dynamics

Presence calculus concepts—such as presence mass, incidence rate, and density— are not unlike abstract physical notions like force, mass, and acceleration 19. In principle, these are measurable quantities constrained by nature to behave in prescribed ways at a micro scale.

Once we understand the rules governing their micro-scale behavior, we gain the ability to measure, reason about, and explain a wide range of macro-scale phenomena. Much of physics is built on this principle.

In a similar vein, the presence calculus—and especially the presence invariant— provides a foundational constraint that governs the local, time-based behavior of any system described by measurable, time-varying signals and the measures they induce: presence masses.

Recognizing that such a constraint exists allows us to construct tools that describe, interpret, and explain macro-scale system behavior.

Newtonian mechanics, for example, enables us to describe and predict the motion of certain physical systems—such as planetary orbits or the trajectories of falling objects—with remarkable precision. Yet even within this well-established framework, limitations persist: the general three-body problem, for example, has no closed-form solution, and systems like the double pendulum exhibit chaotic behavior that defies exact prediction.

Still, such systems can be represented and observed as deterministic trajectories through a parameter space. Even when precise long-term behavior is inaccessible, we can often uncover structure and explain observed behavior.

In much the same way, by explicitly modeling signal histories and representing system trajectories in a parameter space—the parameters of the presence invariant—the presence calculus provides powerful descriptive tools for explaining how systems of presence evolve.

Within this context, convergence and divergence of presence density are the two most important macro-scale behaviors we can study. These allow us to define—formally and operationally—what it means for the accumulation of presence mass in a system to be in equilibrium.

It’s important to note that the presence calculus is, at its core, a tool for explanation, not prediction. However, when supplemented with additional domain knowledge and assumptions, it can provide a distinct, non-statistical substrate on which to base predictive reasoning.

8.1 Sample paths

Consider the highlighted portions of the accumulation matrix \(A\) in Figure 15.

i\j 1 2 3 4 5 6 7 8 9 10
1 1.4 7.3 16.5 21.5 27.4 34.9 42.5 49.1 54.6 56.3
2 5.9
3 9.2
4 5.0
5 5.9
6 7.5
7 7.6
8 6.6
9 5.5
10 1.7

Figure 15: Diagonal and top row of the accumulation matrix.

Each diagonal entry represents the total presence mass observed across all signals at the sampling granularity. Thus, the diagonal traces the discrete-time evolution of the system’s directly observed signals—one sample at a time.

We will call the sequence of values on the diagonal a sample path for the system of presences 20.

Figure 16: Computing the accumulated values on the sample path

By contrast, each entry on the top row represents the accumulated presence mass along a prefix of a sample path. In other words, the top row represents the accumulated presence over the observed history of the system.

Figure 17: The top row: accumuleted presences over system history

In other words, the diagonal and top row represent presence mass accumulations in the system at fundamentally different timescales: the diagonal captures the micro-level behavior— presence mass across signals in each interval at the sampling granularity—while the top row encodes the macro-level behavior—the cumulative effect of those presences over the observed history of the system.

Both views are compactly encoded in the structure of the
accumulation matrix, as are every timescale in between.

As shown in Figure 18, the following geometric relationship holds between the values on the diagonal and the top row:

Figure 18: Top row as the area under the sample path

Each entry on the top row is an integral 21 and equals the area under a prefix of the sample path represented by the diagonal.

8.2 Convergence and divergence of presence density

If we divide each of the entries in the accumulation matrix by the length of the time interval it covers, we get the presence density for each interval. For the diagonal interval has length 1 (time unit at sampling granularity) and for the top row the lengths range from 1 to \(N-1.\)

i\j 1 2 3 4 5 6 7 8 9 10
1 1.4 3.6 5.5 5.4 5.5 5.8 6.1 6.1 6.1 5.6
2 5.9
3 9.2
4 5.0
5 5.9
6 7.5
7 7.6
8 6.6
9 5.5
10 1.7

Figure 19: Presence density: diagonal and top row..

So now we have the left-hand side of the presence invariant encoded in matrix form for every pair of continuous intervals in the system.

Let’s chart the values on the diagonal and the top row in the matrix.

Figure 20: Convergence of long-run presence density.

We can see from Figure 20 that while the values on the sample path are volatile, the values on the top row converge towards a finite value.

We can define this notion of convergence of the values on the top row precisely using the mathematical concept of a limit.

Let:

\[ \Delta = \lim_{T \to \infty} \frac{1}{T} \int_0^T \delta(t) \, dt \]

Here, \(\delta(t)\) is the presence density across all signals, measured at the base sampling granularity \(t\)—that is, the total presence mass per unit time over a single co-presence interval of width \(t\).

The quantity \(\Delta\) represents the presence density over the system’s sample path, that is, its observed history.

To make this correspondence explicit in terms of the accumulation matrix, we can express \(\Delta\) as the limiting value of the presence density along the top row - the value towards which the value on the top row in

\[ \Delta = \lim_{j \to \infty} \frac{A(1,j)}{j} \]

where \(A(1,j)\) is the total presence mass accumulated from time 0 through interval \(j\).

Not every system of presences has such a limit.

We call a system convergent if \(\Delta\) exists and is finite, and divergent otherwise.

To summarize: in a fully convergent system, the presence density over time converges toward a finite value and stays there. Intuitively, this means that after observing enough of the system’s history, additional observation does not significantly alter our understanding of its long-term behavior.

Some systems converge rapidly to a single stable limit. Others may not settle at all, but instead move among a small number of such limits. These
represent dominant behavioral modes — quasi-equilibrium states that the system can enter and sustain for extended periods. Such behavior is called
metastable or multi-modal.

It is also possible for the presence system to exhibit multiple recurring modes, but never remain in any one of them for any meaningful length of time. This is characteristic of a chaotic operating regime. At the macro level, the presence density may appear to trace out a pattern across a limited set of values—known as attractors—but the micro-scale trajectory is deterministic yet highly sensitive to initial conditions, making long-term behavior effectively unpredictable. In such cases, the attractor is not a fixed point but a complex structure that the system continuously explores. Such systems are convergent yet unpredictable and impossible to steer.

Divergence, by contrast, implies the absence of such limiting behavior. The
presence density in divergent systems continues to grow without bound,
indicating that the dominant behavior of the system is an unbounded accumulation of presence.

Figure 21 shows examples of each of these behaviors.

Figure 21: Convergent, divergent, and metastable behavior in systems of presences.

If a limit \(\Delta\) exists and is sustained over time, it signifies a stable long-run presence density for the system. This value represents a specific pattern of behavior toward which the system’s observable presence density gravitates over extended periods, regardless of short-term fluctuations.

This concept aligns with the broader notion of attractors in dynamical systems. While a system’s full, high-dimensional state might exhibit complex dynamics, the long-run presence density can itself stabilize around a particular value or set of values. When the presence density consistently settles around such a limit, it indicates that the system’s observable behavior has entered a stable regime.

This provides a powerful way to characterize the system’s overall operational modes in the long term.

It is important to emphasize that convergence and divergence are properties of the observed long-run behavior of a system of presences — not intrinsic properties of the underlying system.

We cannot infer the system’s nature (whether it is deterministic, stochastic, linear, non-linear, chaotic etc) solely from whether it appears convergent or divergent at any given time. Any of these types of systems may be convergent or divergent at different points in time. We can only observe the dynamics of presence accumulation over time and assess whether they exhibit convergence over an observation interval.

The key difference between convergence and the other two modes is that a
fully convergent system can effectively forget its history beyond the point of convergence. Its future behavior becomes representative of its past, allowing the system to be characterized by a stable long-run value.

Such systems are relatively rare in the real world. This is where much of the utility of the presence calculus lies. It shines when analyzing the behavior of systems of presence when they operate in those liminal phases between convergence and divergence - the states where most real-world systems spend most of their time.

Among other things, the presence calculus equips us with the computational tools needed to identify convergent, divergent and metastable states of a system of presences and monitor how they move in between these states over time.

8.2.1 The semantics of convergence

An important point to emphasize is that, depending on how presence density is interpreted in a given domain, any of the three behavioral modes — convergent, divergent, or metastable — may be desirable. Convergence is not inherently ” good,” nor is divergence necessarily “bad.”

For example, in repetitive manufacturing or many customer service domains, convergence is often desired. In these contexts, presence typically represents demand on a constrained resource, and managing the demand is essential for ensuring consistent service times, throughput,
and resource utilization.

Traditional operations management and queueing theory therefore seek out — and emphasize — stability and convergence in key operational signals.

By contrast, if presence represents a company’s customer base or market
share, we want the long-run presence density to look like the chart on
the right: up and to the right — that is, divergent.

Metastable modes are common in presence signals generated in complex
systems, for example market facing software delivery teams, where teams must shift between modes of operation in response to external demands or changing market conditions. Indeed, the ability to transition between such modes effectively is often a hallmark of a high-functioning, adaptive software organization — provided it is done intentionally and with awareness.

One of the major practical applications of the presence calculus is to
bring new analytical tools to observe, categorize, and steer the
behavior of such systems — aligning low level presence signals with the desired modes of operation in a given domain, before critical tipping points are reached.

8.3 Detecting convergence

In the last section, we defined convergent behavior in terms of the
existence of the limit \(\Delta\), the long-run presence density
of the system.

Now we ask: under what observable conditions does such a limit exist?

If we can identify these conditions, we gain levers to begin steering
systems toward desired modes of operation.

It turns out the answer is hiding in plain sight — in the presence
invariant, which, as we’ve seen, holds for any finite observation
window. The limit \(\Delta\) represents the asymptotic value of \(\delta(t)\),
the left-hand side of the invariant, measured over a sequence of consecutive overlapping intervals, each one a prefix of the sample path.

For each such prefix interval \(t\) , the presence invariant gives us:

\[ \delta(t) = \iota(t) \times \bar{m}(t) \]

This tells us that each value of \(\delta(t)\) is determined by the
product of two measurable quantities: \(\iota(t)\), the incidence rate,
and \(\bar{m}(t)\), the mass contribution per signal.

To understand when the long-run value of \(\delta(t)\) converges, we can ask
a simpler question: do the corresponding long-run s of \(\iota(t)\)
and \(\bar{m}(t)\) converge? If both do, we should expect that their product —
and hence \(\Delta\) — converges as well,and it does, with some technical conditions in place22.

So, let’s write down precise definitions for the limits of \(\iota(t)\) and
\(\bar{m}(t)\) and examine how these limits behave.

8.3.1 Convergence of mass contribution per signal

We will derive the limit for \(\bar{m}\). We will denote this by \(\bar{M}\).

\[ \bar{M} = \lim_{T \to \infty} \frac{1}{N(0,T)} \sum_{(e,b)} \int_0^T P_{(e,b)}(t) \, dt \]

This expression means: for each signal \((e,b)\), accumulate its total presence mass over time, then sum across all signals, and divide by the total number of signals active during that window. Each integral in the sum is a row sum in the original presence matrix - the mass contribution of an individual signal over the interval.

Thus we can also write this as

\[ \bar{M} = \lim_{j \to \infty} \frac{1}{N(1,j)} \sum_{(e,b)} \sum_{k=1}^j P_{(e,b)}(k) \]

\(\bar{M}\) is the limit of mass contribution per signal over a sufficiently long observation interval. Here, \(P_{(e,b)}(t)\) is the presence density function for signal \((e,b)\), and \(N(0,T)\) is the total number of signals observed during the interval \([0,T]\).

Let’s work this out for our running example and see what it means. We’ll reproduce Figure 9, our starting presence matrix, here for easy reference.

E 1 2 3 4 5 6 7 8 9 10
e1_b1 0.3 2.3 3.4 1.1 2.9 3.2 1.1 0.0 0.0 0.0
e1_b2 0.3 2.3 3.4 1.1 0.0 1.1 2.2 2.4 2.3 0.8
e2_b2 0.0 0.0 0.0 0.0 0.0 0.0 0.9 1.8 3.2 0.9
e2_b1 0.8 1.3 2.4 2.8 3.0 3.2 3.4 2.4 0.0 0.0

The cumulative mass per signal over each interval \([1, j], j \le 10\) is shown below. Each value in this matrix is the sum of all the values in that row to the left (inclusive) of the value.

E 1 2 3 4 5 6 7 8 9 10
e1_b1 0.3 2.6 6.0 7.1 10.0 13.2 14.3 14.3 14.3 14.3
e1_b2 0.3 2.6 6.0 7.1 7.1 8.2 10.4 12.8 15.1 15.9
e2_b2 0.0 0.0 0.0 0.0 0.0 0.0 0.9 2.7 5.9 6.8
e2_b1 0.8 2.1 4.5 7.3 10.3 13.5 16.9 19.3 19.3 19.3

Figure 22: Signal mass contribution matrix.

Now lets chart each row of this matrix to see how the mass contribution for each signal grows over time. In Figure 23 we are showing each row in the matrix as a line in the chart.

Figure 23: Mass contributions of each signal over time

Finally Figure 24 shows the how mass contribution per signal grows over time. Each point in this chart represents the mass contributions per signal for the window \([1,j]\) which is the sum of the value in column j divided by the number of non-zero rows in the sub-matrix spanned by the columns in \([1,j]\).

As we can see, this curve converges to a limit.

Figure 24: Convergence of mass contribution per signal

So lets ask, what would make the mass contribution per signal not converge to a finite value?

Figure 23 suggests that the mass contribution of every individual signal is monotonically non-decreasing and it increases continuously over every non-zero support interval of the signal and flattens out over every interval where the underlying signal is zero.

Suppose when measured over a sufficient long interval, each signal remains bounded, that is every onset is matched with a corresponding reset, then each individual signal contributes a finite mass to the cumulative value.

Thus, the only way the cumulative mass can grow without limit is if some signal grows without limit.

Figure 25: Onset-reset-patterns

For example, in Figure 25 we show some onset-reset patterns for signals and the signal for Element-4, which has an onset but no apparent reset within the observation window, would grow without limit in Figure 23 assuming there was no reset.

This gives us the first condition for convergence of \(\Delta\) :

Boundedness of Signal Mass

In a convergent system of presences, every signal onset is eventually followed by a corresponding reset, when observed over a sufficiently long time interval.

We’ll note, once again, that depending upon the semantics of the domain, we may or may not want to have this condition hold depending upon whether we are looking to steer the system towards convergence or towards divergence.

For example, if a signal represents a new revenue source, a mass contribution represents incremental revenues and ideally we want many onsets without matching resets: every reset corresponds to a lost revenue stream.

If, on the other hand, a signal onset represents a new unfinished task on your to-do list, then a reset marks its completion — and convergence becomes desirable, as it indicates tasks are being completed in a timely manner and that your todo list is not growing without limit.

8.3.2 Convergence of Incidence Rate

We now define the long-run incidence rate, denoted \(I\), in exact analogy to how we defined the mass contribution per signal \(\bar{M}\). Recall that \(\iota(t)\) is the incidence rate observed over the interval \([0, t]\), defined as the number of signals observed in that interval divided by its duration. Then we define the long-run incidence rate as the following limit:

\[ I = \lim_{T \to \infty} \iota(T) = \lim_{T \to \infty} \frac{N(0, T)}{T} \]

where \(N(0, T)\) is the number of signals observed over the interval \([0, T]\)— that is, the number of distinct element-boundary signals that are active at some point during the interval. The incidence rate measures how many such signals are activated, per unit time.

This limit \(I\), when it exists, represents the asymptotic rate at which signals appear in the system. It plays a symmetric role to \(\bar{M}\) in the convergence of the presence density, and its existence is the second key condition we will examine next.

To better understand the behavior of the incidence rate \(\iota(T)\), let’s now examine the cumulative signal count \(N(0,T)\) over the interval \([0,T]\) for increasing values of \(T\). This is directly analogous to how we analyzed the growth of cumulative mass contributions per signal when analyzing \(\bar{M}\).

Recall that for a given observation window \([0,T]\), \(N(0,T)\) counts the number of element-boundary signals that are active at some point within the interval. We can compute this by scanning across the presence matrix and, for each column (from \(t=1\) to \(t=T\)), counting how many new signals appear—that is, how many unique element-boundary rows have non-zero values in that column.

The result is a sequence of signal counts, which we can arrange as a \(1 \times T\) row vector:

Time 1 2 3 4 5 6 7 8 9 10
Signal Count 3 3 3 3 3 3 4 4 4 4

Figure 26: Signal incidence counts over the observation window..

where \(n_j\) is the total number of distinct signals that have appeared at or before time \(j\). Each \(n_j\) counts the number of signals with support intersecting the interval \([0, j]\).

We can chart this row to visualize how the cumulative number of observed signals grows over time. If \(N(0,T)\) grows linearly in \(T\), then the incidence rate \(\iota(T) = N(0,T)/T\) should converge to a finite value \(I\). On the other hand, if \(N(0,T)\) grows faster than linearly, the incidence rate will diverge—and if it grows sublinearly, the rate will decay toward zero.

Figure 27: Signal incidence rate

Figure 27 shows the incidence rate \(\iota(T) = N(0,T)/T\) over time. In this example, the rate initially decreases and then stabilizes, since the number of distinct signals \(N(0,T)\) stops increasing after a point. In general, the incidence rate will converge to a finite limit if \(N(0,T)\) grows no faster than linearly with \(T\). If \(N(0,T)\) grows faster than \(T\), the ratio \(\iota(T)\) will diverge—indicating that signals are being activated at an unbounded rate. Conversely, if \(N(0,T)\) grows slower than \(T\), the incidence rate will decay toward zero. Thus, convergence of \(\iota(T)\) requires that \(N(0,T)\) grows approximately linearly in \(T\).

This kind of divergence typically arises in systems where the onset rate— the rate at which new signals are activated—exceeds the reset rate, which closes those signals. While transient imbalances between onsets and resets are common during transitions between equilibrium states, divergence only occurs if this imbalance is sustained indefinitely. In that case, \(N(0,T)\) grows without bound relative to \(T\), and the system exhibits an asymptotically increasing incidence rate. So, divergence of \(\iota(T)\) directly reflects a persistent structural imbalance between signal onsets and resets over time.

Boundedness of incidence rate

In a convergent system of presences, the long-run rate of signal onsets does not exceed the long-run rate of resets, when observed over a sufficiently long time interval.

8.3.3 Recap

We began by defining convergence in terms of the existence of a long-run limit for presence density.

We then showed how the existence of this global limit depends on the existence of two other measurable limits: the incidence rate of signals and their mass contribution.

Next, we traced each of these limits back to the local behavior of individual signals—specifically, the presence or absence of well-behaved signal onsets and resets.

With this connection in place, we now have a principled way to reason about the global convergence or divergence of a system by analyzing the patterns of local signal behavior over time.

8.3.4 Formal proof of convergence and Little’s Law

In this document, we have presented a somewhat simplified— account of the criteria required to ensure that a system of presences is convergent. Specifically, based on the definitions above, we assert that for a given system of presences, if the limits \(I\) and \(\bar{M}\) exist and are finite, then the limit \(\Delta\) also exists and is finite. Furthermore, we claim that

\[ \Delta = I \times \bar{M} \]

Technically, this relationship does not follow automatically from the arguments we have presented so far. In fact, the statement above is a restatement of a generalized form of Little’s Law originally proven by Brumelle [8], and later by using sample path techniques by Heyman and Stidham [9]. The full proof—along with the additional technical conditions required to ensure that the limit of the product equals the product of the limits—is a discussed in a separate document on our theory track.

For our purposes, it is safe to state that this relationship, and the conditions under which it holds, constitute a general form of Little’s Law for a system of presences.

This general form of Little’s Law is usually presented in the form \(H = \lambda \cdot G.\) In our notation \(H = \Delta\), \(\lambda = I\) and \(\bar{M} = G\) We have chosen to develop a new consistent notation to describe these terms as limiting values of measure-theoretic parameters \(\delta, \iota, \text{ and } \bar{m}\) of the presence invariant, but the underlying terms can be shown to be equivalent to each other.

For a more detailed explanation of the correspondence between the two, please see the document Convergence of systems of presence on our Theory Track.

8.3.5 Convergence, coherence, and Little’s Law

One important point to note is that “Little’s Law” is not a single law, but rather a family of related laws that apply at different time scales, in different forms, and with different interpretations. The presence invariant is the most general version of this law, as it holds at all time scales.

In this document, we stated it as a relationship between presence density, signal incidence rate, and mass contribution per signal over any finite observation window. This relationship holds at all scales, including those sufficiently long windows where the parameters approach their limits: \(\Delta\), \(I\), and \(\bar{M}\).

It is natural to ask: what is special, if anything, about those limiting values?

Without going too deeply into technical arguments here, we note that the limits are indeed meaningful. When a system is observed over a non-convergent interval, the quantities in the presence invariant are dominated by partial mass contributions from signals. The first interval Figure 28 shows an example of this behavior.

Figure 28: Convergent and non-convergent intervals

The system may appear convergent when the window is long enough for complete signals to dominate the presence density. For example, the interval \(T'\) in Figure 28 includes full support for nearly all the signals, and in this case the system would appear convergent over that interval.

The main difference between the two intervals in Figure 28 is that in the latter, the mass contributions from the signals include nearly the entire presence mass of each each signal.

In other words, if the observation window is long enough that most signal contributions equal the masses of the signals in the window, the system will appear convergent over the interval and \(\bar{m}\) as measured for that interval will be close to its limit value \(\bar{M}\) for the system as whole.

The key point here is that in such situations, the presence invariant is not just a relationship among mass contributions, but also implicitly a relationship among the masses of the signals involved. This distinction has direct operational implications.

For instance, if the signals in Figure 28 represent customer service times, then over a convergent interval, mass contributions equal signal masses and reflect what the customer actually experiences. But over shorter, non-convergent intervals, those same contributions primarily reflect partial masses—what a system operator might observe on a day-to-day basis.

In Figure 28, the second interval represents the state where the mass contribution per signal is near its limit value \(\bar{M}\). When we measure presence density, signal masses, and incidence rates over this longer window, we are implicitly aligning the customer’s perspective with the operator’s.

In this way, convergence brings these two perspectives—the customer’s and the operator’s—into alignment. More generally, if a system is convergent over a long enough observation window, the invariant over those intervals expresses a relationship between presence density in the time domain and mass per signal ( which also equals mass contribution per signal) in the element-boundary domain.

We will also note that in these cases, there is a parallel argument that can be made about the incidence rate \(I\) and signal onset rates and reset rates. When the observation window is long enough that it includes complete signal masses, this implies will also have fewer signals where onsets are not matched with resets.

So over those intervals the incidence rate, onset rates and reset rates will all converge to the same limiting value \(I\) over the interval. These are the well known equilibrium conditions we call the “conservation of flow” under the classic treatments of Little’s Law, but now generalized to arbitrary systems of presences.

When the parameters of the presence invariant over an interval are at or close to their limit values, the system is in a state of epistemic coherence: multiple observers, using different vantage points, arrive at a consistent interpretation of parameters in the presence invariant.

This coherent state occurs only when the system is operating at or near equilibrium.

We will return to this idea in future posts, particularly in the context of flow measurement in systems that operate far from equilibrium. But for now, it is enough to recognize that identifying whether a system is operating in a convergent or divergent mode is fundamental to making meaningful decisions when reasoning about a system of presences.

8.3.6 A note on determinism

A final point we emphasize in this section is that the form of Little’s Law derived here is entirely deterministic. It does not depend on any probabilistic or stochastic assumptions about the behavior of the system. In fact, the notion of a sample path used here originates in a deterministic proof of Little’s Law by Stidham in 1972. The presence invariant, as we have introduced it, is a direct analogue of the finite-window constructs used in that proof.

It is important to recognize that Little’s Law is not a statistical artifact. It is a structural property deeply woven into the behavior of of any system of presences. Convergence and divergence are deterministic features of how signals evolve and interact over time—regardless of whether the underlying signals are random or not.

Even when the signals have randomness, the observed evolution of presence density is deterministic23 and governed by the law of conservation of presence mass—that is, the presence invariant. This determinism extends to any functional quantity that depends on presence density. As we will see, a large and operationally useful class of system behaviors can be characterized in terms of the presence density of domain signals. That is why the machinery developed here is more than just a theoretical curiosity.

8.4 The presence invariant and rate conservation laws

This is our main point of departure in the presence calculus: we treat equilibrium not as a precondition for Little’s Law, but as a special case of a more general principle.

We place the finite version of Little’s Law—in the general form of the presence invariant—at the center of our analysis, because it continues to hold and yield meaningful insight even when the system is far from equilibrium. These are precisely the operating modes that traditional queueing theory and other classical approaches de-emphasize—but where most real-world systems actually live.

In fact, a broader principle is at work. Miyazawa [10] was among the first to identify a general class of rate conservation laws that allow relationships like the generalized Little’s Law to be derived in wider settings. Sigman [11] showed that the generalized form of Little’s Law can be seen as an instance of such rate conservation principles.

The document The Presence Invariant and Rate Conservation Laws in our theory track explores this connection in detail.

From this perspective, the presence calculus offers constructive tools to discover and formalize conservation laws within a given domain.

Every system of presences may be considered to generate a rate conservation law when presence is interpreted according to the semantics of the domain.

The interactions of element-boundary signals in any system of presences naturally give rise to rate conservation laws based on the principle of conserved signal mass. Mapping these back into the language of the domain appears to be a fruitful path for uncovering the mechanisms by which systems in that domain evolve.

9 Visualizing signal dynamics

Convergence, as discussed in the last section, is a fundamental concept in the presence calculus. We now have the tools to detect convergence or divergence in the long-run behavior of a system of presences—specifically, the evolution of presence density and its underlying drivers: signal mass contributions and signal incidence rates.

As noted earlier, whenever we can model the meaningful behaviors of a system as interactions between element-boundary signals within a system of presences, the presence calculus gives a constructive analytical framework for studying signal dynamics of the system.

This framework applies to a wide range of operational problems across many domains, and should be viewed as an alternate analytical lens to statistical analysis of operational data.

We also observed that much of the structure in the presence calculus is
deterministic. That is, given the observed behavior of a system of presences, we can deterministically explain the evolution of presence density in terms of its drivers, signal mass and incidence rate—across time and across different timescales.

In this section, we elaborate on this deterministic structure and introduce
new tools that help us analyze how local behaviors evolve into global patterns, and help us understand the dynamics of a system of presence.

First we will develop fundamental tools that help us establish a uniform sense of place and direction of the system evolution, that scales from the local to global timescales.

While convergence and divergence establish this at the macro scale, these concepts are somewhat unwieldy and inapplicable when the system is operating far from equilibrium, and we need some better tools to navigate in this complementary region.

We will show how to detect emerging patterns in this evolution that reveal the direction in which the system is moving—and just as importantly, how to detect when that direction is changing.

These are essential capabilities for building mechanisms that can steer system behavior in a desired direction—toward convergence or divergence of presence density.

9.1 The presence invariant in phase space

In the previous section, we showed that the accumulation of presence density across time and timescales follows a deterministic rule—the presence accumulation recurrence. That rule described how presence density evolves, but only in terms of magnitude and scale.

We now introduce machinery that allows us to understand the direction in which the system is evolving. To do this, we recast the entries in the accumulation matrix using our central tool: the presence invariant.

Since the presence invariant holds at each cell of the accumulation matrix, we can write:

\[ \delta_{i,j} = \iota_{i,j} \cdot \bar{m}_{i,j} \]

This is just the familiar identity \(\delta = \iota \cdot \bar{m}\) applied at each interval \((i,j)\) in the matrix.

Changes in presence density are thus driven by changes in the product of incidence rate and signal mass contribution.

Because products are more difficult to reason about directly than sums, we analyze the system in logarithmic space, where the invariant becomes additive:

\[ \log \delta = \log \iota + \log \bar{m} \]

This additive form reveals how changes in rate and mass contribute linearly (in log space) to changes in presence density.

We now introduce a compact coordinate system for visualizing these dynamics by embedding the two terms into the complex plane:

\[ z = \log \iota + i \log \bar{m} \]

Here, \(\Re(z) = \log \iota\) and \(\Im(z) = \log \bar{m}\). This maps each interval to a point in \(\mathbb{C}\) representing the logarithmic decomposition of the observed presence density.

This representation has powerful interpretive value:

Let’s define these explicitly:

Together, the polar representation

\[ z = \|z\| \cdot e^{i\theta} \]

allows us to interpret each interval’s presence dynamics in terms of both intensity and direction.

We now have a formal notion of flow for presence in log-space—one that combines incidence and mass into a single complex number that represents intensity and direction of presence accumulations.

Figure 29 shows this mapping of the presence accumulation matrix into polar co-ordinates in the complex plane. We will call this the presence accumulation dual.

i 1 2 3 4 5 6 7 8 9 10
1 1.34 ∠ -0.61 0.98 ∠ 1.14 1.70 ∠ 1.57 1.99 ∠ 1.72 2.27 ∠ 1.80 2.55 ∠ 1.85 2.43 ∠ 1.80 2.60 ∠ 1.84 2.74 ∠ 1.87 2.80 ∠ 1.90
2 1.29 ∠ 0.55 1.67 ∠ 1.32 1.90 ∠ 1.57 2.18 ∠ 1.70 2.47 ∠ 1.78 2.36 ∠ 1.74 2.54 ∠ 1.79 2.68 ∠ 1.83 2.74 ∠ 1.87
3 1.57 ∠ 0.80 1.61 ∠ 1.32 1.90 ∠ 1.57 2.24 ∠ 1.70 2.19 ∠ 1.67 2.38 ∠ 1.74 2.53 ∠ 1.79 2.60 ∠ 1.84
4 1.21 ∠ 0.44 1.35 ∠ 1.27 1.81 ∠ 1.57 1.87 ∠ 1.57 2.11 ∠ 1.68 2.29 ∠ 1.75 2.36 ∠ 1.81
5 1.28 ∠ 1.00 1.55 ∠ 1.31 1.68 ∠ 1.40 1.93 ∠ 1.57 2.12 ∠ 1.68 2.20 ∠ 1.76
6 1.43 ∠ 0.70 1.50 ∠ 1.09 1.72 ∠ 1.40 1.92 ∠ 1.57 1.99 ∠ 1.68
7 1.53 ∠ 0.43 1.44 ∠ 1.07 1.62 ∠ 1.39 1.68 ∠ 1.57
8 1.35 ∠ 0.62 1.45 ∠ 1.29 1.53 ∠ 1.57
9 1.23 ∠ 0.97 1.28 ∠ 1.57
10 0.71 ∠ -0.23

Figure 29: The presence accumulation dual: the presence accumulation matrix in polar co-ordinates on the complex plane..

9.2 Flow fields

Figure 29 is not particularly insightful, so let’s visualize this as a field of vectors as in Figure 30. We will call this the flow field for the system.

Figure 30: Flow field visualization

To understand its construction, let’s go back and start with Figure 29. The grid represents rows and columns of the accumulation matrix and represents timescales.

Since the matrix in Figure 29 is upper triangular, there is no significant information encoded in the bottom half of the matrix. So as a convention, when we visualize it we will drop the bottom half of the matrix and only show the entries in the upper diagonal.

Now imagine this matrix rotated by 90 degrees so that entries on the diagonal are on the bottom row, the entries on the next diagonal are laid out above it, and so on until the we get to the top left entry in the matrix. The result in the pyramid shape of the flow field diagram in Figure 30.

We can also think of this as a compact visualization of the pyramid in Figure 14 represented in polar-coordinates. Each vector in the flow field is drawn with a proportional magnitude and theta.

Figure 30 compresses a tremendous amount information about the dynamics of a system into a very compact representation that is easy to scan and interpret visually as well as analytically.

9.3 Interpreting flow fields

When interpreting flow fields, we are not interested in absolute magnitudes or angles for the most part.

Rather we focus on the relative change in magnitude and direction of these vectors as we sweep left to right in time along a row and from bottom to top in time scales across the rows.

That is, the flow field encodes the dynamics of the system across time and time scales.

The phase angle \(\theta\), provides a concise, single-value representation of the system’s directional flow. Its values range from \(-\pi\) to \(\pi\) radians, spanning all four quadrants of the complex log-space. In this log-space, a positive value signifies the original quantity is greater than 1, a negative value indicates it’s less than 1, and a zero value means it’s exactly 1.

The table in Figure 31 interprets that range, showing how the combination of incidence rate and mass per signal contributes to the system’s overall dynamics, indicating whether growth or decline is primarily driven by rate, mass, or a balanced interplay between them.

θ (rad) Dir. Dynamics Conditions Interpretation
\(\approx \pi/2\) Mass-Driven Growth \(\log \iota \approx 0\), \(\log \bar{m} > 0\) \(\iota \approx 1\), \(\bar{m} > 1\). Density increases from growing mass per signal; incidence stable.
\(\in (0, \pi/2)\) General Growth/Expansion \(\log \iota > 0\), \(\log \bar{m} > 0\) \(\iota > 1\), \(\bar{m} > 1\). Both signal incidence rate and mass per signal are increasing. (\(\theta = \pi/4\) for balanced growth).
\(\approx 0\) Rate-Driven Growth (Mass Stable) \(\log \iota > 0\), \(\log \bar{m} \approx 0\) \(\iota > 1\), \(\bar{m} \approx 1\). Density increases from growing signal incidence; mass per signal stable.
\(\in (-\pi/2, 0)\) Mass Dilution \(\log \iota > 0\), \(\log \bar{m} < 0\) \(\iota > 1\), \(\bar{m} < 1\). Many signals entering, but each contributes less mass. (\(\theta = -\pi/4\) for proportional change).
\(\approx -\pi/2\) Mass-Driven Decline \(\log \iota \approx 0\), \(\log \bar{m} < 0\) \(\iota \approx 1\), \(\bar{m} < 1\). Density declines due to decreasing mass per signal; incidence stable.
\(\in (-\pi, -\pi/2)\) General Decline/Contraction \(\log \iota < 0\), \(\log \bar{m} < 0\) \(\iota < 1\), \(\bar{m} < 1\). Both signal incidence rate and mass per signal are decreasing. (\(\theta = -3\pi/4\) for balanced proportional decline).
\(\approx \pm \pi\) Rate-Driven Decline (Mass Stable) \(\log \iota < 0\), \(\log \bar{m} \approx 0\) \(\iota < 1\), \(\bar{m} \approx 1\). Density declines primarily due to decreasing signal incidence; mass per signal stable.

Figure 31: Interpretation of θ as directional flow in log-space between incidence rate and signal mass..

Let’s review the flow field in Figure 30 with these interpretations in mind. While this example is somewhat simplified, many of the key features of a flow field are already observable.

Here are some important observations from the flow field in Figure 30:

9.4 Sample path trajectories and attractors

In the last section, we showed how we could mathematically define the concept of flow in a system of presences via a mapping of the right-hand product in the presence invariant to a complex number and then visualizing the entries in the accumulation matrix as a vector field using the polar coordinates of the corresponding complex numbers.

There is much more that can be done with this complex plane mapping besides visualization, but these applications are beyond the scope of this gentle introduction.

While the flow field representation focused on the evolution of incidence rate and presence mass, an equally important set of visualizations directly examines the presence density matrix. Recall that this matrix represents the presence density for each entry in the accumulation matrix. There are many straightforward and direct visualizations of the matrix using heatmaps and contour plots, which we will describe later elsewhere, but in this document, we want to describe a less obvious visualization that is very useful for studying
the dynamics of the system when it operates far from equilibrium.

In dynamical systems theory, a standard object of study is the path trajectory of a system through some parameter space. For a system of presences, the key input parameters that drive the system’s dynamics are the incidence rate and mass contribution per signal and their changes over time.

Since the product of these values determines presence density over an interval, a sample path in the accumulation matrix is a good candidate to study the evolution of the system in this parameter space. Figure 32 shows this matrix for our running example.

i\j 1 2 3 4 5 6 7 8 9 10
1 1.40 3.65 5.50 5.38 5.48 5.82 6.07 6.14 6.07 5.63
2 5.90 7.55 6.70 6.50 6.70 6.85 6.81 6.65 6.10
3 9.20 7.10 6.70 6.90 7.04 6.97 6.76 6.13
4 5.00 5.45 6.13 6.50 6.52 6.35 5.69
5 5.90 6.70 7.00 6.90 6.62 5.80
6 7.50 7.55 7.23 6.80 5.78
7 7.60 7.10 6.57 5.35
8 6.60 6.05 4.60
9 5.50 3.60
10 1.70

Figure 32: Presence Density Matrix for our running example.

This matrix encodes presence density along sample paths at different observation time scales and each one can be plotted as a trajectory.

In visualizing these trajectories, we are focusing on the values of the presence density that the system inhabits—how close they are to each other, how often the system returns to the same values, etc.

These are useful in identifying values to which the system converges when the behavior is not fully convergent or divergent, and help identify attractors of the system as well as the movement of the system between them.

We can construct this visualization as follows:

The resulting diagram is shown in Figure 33.

Figure 33: Sample path trajectories

While this example is again fairly simple, here are some common characteristics we can see:

Overall, the diagram is complementary to the information shown in the flow field: how presence densities evolve and concentrate across temporal scales, revealing both volatility at finer scales and convergence toward attractor bands at coarser scales.

9.5 Feedback loops and steering

We should also note that while flow fields and trajectory maps are useful to a get a quick high level signature of flow dynamics in a system of presences as above, they shine when we use them reason closely about the local and global changes in the field, and trace these back to the specific drivers of the change - incidence rate, mass contributions, or both.

The flow field representation, in particular, lets us reliably attribute a proximate cause to a local change in presence density, and this is the key to being able to build feedback loops into the system to steer it in a desired direction.

However, to act on this insight effectively, we need to drop back down into the domain and what presence density, incidence rate, and mass contribution mean for a specific system of presences. It also requires careful choices about timescale at which we sample presence density, the timescales at which we sense changes, and timescales at which we intervene in the system to change direction.

What is important to note here, is that underlying machinery to detect and respond to change is signal and timescale agnostic and this is the key contribution of the presence calculus in this regard.

We will have much more say about using the tools of the presence calculus to create such feedback loops and steer systems in future posts.

10 Taking stock

The last few chapters have introduced many concepts that build on the primitive idea of presence and systems of presence. Let’s pause and take stock of these ideas to build some intuition for what the presence calculus is and what it tells us.

10.1 A summary

Figure 34: The presence calculus machinery

The machinery of the presence calculus is shown in Figure 34

All of this are significantly more powerful tools than analyzing isolated point samples using standard statistical or probabilistic techniques. The key advantage is that it allows us to manage a system of presences even when the underlying signals lack a stable statistical or probability distribution and the system is far from equilibrium 24.

So for this very broad category of signals, the presence calculus is a much more powerful mathematical substrate on which to build decision support systems.

10.2 Towards systems of systems

When managing a single operational property—like cost, risk, delay, or a market-facing reward such as revenue—a single presence system is often sufficient to model the relevant variables. Many useful operational questions can be answered with such models, though these are typically local optimizations.

Real-world decision-making environments, however, involve many such systems, each representing distinct tradeoffs between aspects of work, interaction, or constraint.

A general theory for how these systems of presence interact would be valuable but quite challenging to develop. Much work remains in understanding how to combine individual presence systems into larger, interacting systems of systems, and it seems naive to hope this can be done for arbitrarily complex systems.

The mathematics of the presence calculus are well-suited for modeling local interactions. When tackling measurement in a larger complex system, I believe their power lies in modeling the interactions within and between smaller subsystems and assemblages. This remains an active and largely unexplored application area for the calculus.

Still, as we’ve seen in concrete applications, modeling within specific domains already offers substantial value. Domain assumptions often allow us to prove properties that might not hold in the general case.

For instance, in the domain where the presence calculus was first developed—operations management in software product development—systems of presence are a powerful unifying concept.

Today, operational improvement in software largely remains in a pre-scientific phase. It relies on a fragmented mix of metrics, developed using gut instinct and folk theories, and awkwardly transplanted models from unrelated domains, many designed under assumptions that provably don’t hold in the software context.

The result is a patchwork of measurement systems—DORA metrics, Flow Metrics, project metrics, customer experience measures—each operating independently and largely without an underlying theory that is provably correct in the software domain. Yet each can be reframed as a system of presences and analyzed uniformly with the tools developed here.

We will finally have measurement systems that start with software product development concepts and operating principles, and yet have mathematically provable relationships between key measurements. This is what the presence calculus brings to the table. Our first set of application articles will explore what this looks like in practice.

The next step is to construct structural causal models that link these systems to understand their local interactions in a specific company context. The presence calculus gives us tools to steer presence density in individual systems of presence and explain observed behavior in a linked system. This means we can make a local change to one system and explain the changes in a linked system. The interactions are local, but now they span directly linked systems.

This gives us the ability to build such causal models iteratively through experimentation on the system. A value stream is a promising example of such a system of systems where an approach like this appears tractable.

This idea is very similar to statistical causal modeling, but with a crucial difference: we’re not inferring causal relationships through statistical correlations between proxy variables on large heterogeneous datasets. Instead, we’re proving causal relationships between local interactions on directly observed presence in a specific complex system.

In my view, the most productive path forward lies not in seeking a fully general theory of interacting presence systems, but in building domain-specific measurement frameworks grounded in presence systems as primitives that can be effectively tuned through experimentation for a specific complex system, like a single value stream in a company. This approach is both more tractable and more likely to yield practical insights.

That’s where we’ll begin—focusing on specific, bounded, real-world applications. Perhaps a larger theory will emerge from this, perhaps not, but in either case, we expect to see tremendous value created from taking a careful, measured mathematically rigorous approach to understanding the capabilities and limits of this formalism.

11 A personal note

This work represents the culmination of nearly ten years of research, product development, and field work in operations management in software product development in The Polaris Advisor Program.

The foundations were laid between 2017 and 2022 while I was developing Polaris, a proprietary measurement platform for software product development that powers our program. But—as is the nature of evolving software platforms—ideas, theories, and practice all blended together over the years into a somewhat incoherent jumble, making it hard to distinguish what was fundamental from what was anecdotal practice.

I was also deeply frustrated with the inadequacy of existing measurement techniques to realistically model the systems we actually work in. Personally, as someone who has been a software developer for nearly 30 years, building measurements systems in software over the last decade has highlighted, in vivid detail, the shortcomings in our ability to measure anything meaningful and actionable about software development work.

The presence calculus is a foundational step in addressing this problem, but it is only a starting point. It makes no grand claims beyond what is stated mathematically here. Much work still remains to explore its applications, and its practical limits will become clear as we test its applicability in various contexts.

That said, in my view, the calculus represents a fundamental departure in how we approach measurement in software, distinguishing it both from the purely empirical and statistical techniques that are in the mainstream today and also from ideas imported from other domains like manufacturing—another significant branch.

It allows us to model and measure software development as it is— bespoke, messy, irregular, highly variable, deeply time- and history-dependent work involving humans, machines, and codified knowledge, forming complex, interconnected systems—where being able to move beyond proxies and observe what is in rich detail is a prerequisite to reasoning quantitatively about the system.

It formally identifies the mathematical foundations behind how we can do this—how to better measure properties of such real-world systems without compromise, adapt the ideas that are fundamental to this messy world, and build measurement models that reflect the reality of software development.

In developing the presence calculus, I drew upon and re-contextualized many existing ideas in the field. For example, Little’s Law plays a foundational role in the presence calculus. But this is because, mathematically, Little’s Law encodes some deep and general structural properties of arbitrary time-varying signals in any domain.

The version we use commonly in the software industry is a rather trivial special case of the underlying mathematical concepts. This version is based on nearly 50-year-old ideas developed in the context of studying queueing systems and their application to manufacturing. It does not reflect the modern mathematical developments in the theories underpinning the law. In fact, those developments are materially important to apply it correctly in the software development context.

The presence calculus starts from this up-to-date view but goes beyond this to position this law within a more general, measure-theoretic mathematical framework.

When we start from the presence calculus framing, we can derive a provably correct version of Little’s Law from first principles for any measurement context involving time-varying signals. It is no longer simply a formula copied from Lean manufacturing.

It may be surprising to many that such a derivation is even possible—much less in a mathematically provable way. It certainly was, to me!

This has fundamental implications on our ability to reason about these measurements—well beyond what is possible with statistical or probabilistic techniques alone.

Those statistical and probabilistic techniques, too, require much more careful treatment when modeling the non-stationary, time-varying behaviors that are the default in real-world software systems. However, that care is very often missing in the mainstream applications of these techniques in industry—for example, in the field of developer experience and in mainstream developer productivity measurement products, where statistical measures are applied to time-varying signals with nary a concern about such technicalities.

The presence calculus offers a much more careful and rigorous foundation for bridging this gap—quite different from standard statistical approaches to dealing with time-varying data.

In this way, I believe the presence calculus has deep and fundamental applications for the software domain and beyond.

Even though the concepts require a bit of effort to understand and apply, the good news is that there are relatively straightforward points of departure between how we would model problems using the calculus and how we would typically model them using conventional techniques.

By comparing and contrasting these approaches, we can better understand where it brings new benefits.

11.1 The road ahead

The presence calculus formalizes and generalizes the theory behind many of the ideas developed and remixed from existing practices and techniques in the industry and that we have used in The Polaris Advisor Program over the years.

I am now embarking on a new phase: rebuilding a mix of open-source and proprietary tooling grounded in the principles laid out here.

If the concepts in this document and the project are of interest, I welcome collaborators who can help pressure test and apply these ideas—and explore and understand what their limits are.

This document, along with others on this site, are intended to provide a broad and deep foundation to support anyone who finds this prospect intriguing or exciting.

My objective here is to develop a robust substrate for many common modeling and measurement problems in software, so that more people can extend and apply these ideas in real-world environments with greater precision, and with confidence in the mathematical validity of their measurement systems.

I welcome constructive feedback and thoughtful skepticism—especially from those who can help surface areas where the approach needs refinement or may even be the wrong fit. That kind of scrutiny is essential if we want these ideas to be broadly useful.

References

[1]
S. Stidham, “A last word on \(L=\lambda W\),” Operations Research, vol. 22, no. 1–4, pp. 417–421, 1972.
[2]
M. El-Taha and S. Jr. Stidham, Sample-path analysis of queueing systems, vol. 11. in International series in operations research & management science, vol. 11. Boston, MA: Springer Science & Business Media, 1999, p. 295.
[3]
S.-H. Kim and W. Whitt, “Statistical analysis with little’s law,” Operations Research, vol. 61, no. 4, pp. 1030–1045, 2013, doi: 10.1287/opre.2013.1193.
[4]
P. J. Denning and J. P. Buzen, “Operational analysis of queueing networks,” in Measuring, modelling and evaluating computer systems, H. Beilner and E. Gelenbe, Eds., North-Holland Publishing Company, 1977.
[5]
N. Forsgren, J. Humble, and G. Kim, Accelerate: The science of lean software and DevOps: Building and scaling high performing technology organizations. Portland, OR: IT Revolution Press, 2018.
[6]
J. Z. Muller, The tyranny of metrics. Princeton, NJ: Princeton University Press, 2018.
[7]
J. Little, “Little’s law as viewed on its 50th anniversary,” Operations Research, vol. 59, no. 3, pp. 536–549, 2011.
[8]
S. Bruemelle, “On the relation between customer and time averages in queues.” Journal of Applied Probability, vol. 8, no. 1–4, pp. 508–520, 1971.
[9]
D. Heyman and S. Stidham, “The relation between customer and time averages in queues,” Operations Research, vol. 28, no. 4, pp. 983–994, 1980.
[10]
M. Miyazawa, “Rate conservation laws: A survey,” Queueing Systems, vol. 15, no. 1–4, pp. 1–58, 1994.
[11]
K. Sigman, “A note on a sample path rate conservation law and it’s relationship to \(H=\lambda G\),” Advanced App. Probability, vol. 23, no. 1–4, pp. 662–665, 1991.

  1. Mathematically, the presence calculus is a measure-theoretic framing of sample path analysis—a technique originally developed by Dr. Shaler Stidham to analyze stochastic processes and used to provide the first deterministic proof of Little’s Law and its generalized form. [1].

    Together with colleagues, he developed it into a powerful tool for establishing general properties of stochastic processes without relying on probabilistic assumptions—a major breakthrough. Its theory and applications to queueing systems are detailed in [2].

    Although these techniques were always applicable beyond stochastic process theory, development remained largely within queueing theory adjacent domains.

    Presence calculus makes this separation explicit. It begins with measure theory and develops sample path tools for observable signals in a general domain. It relies on established results from both areas to justify the correctness of its claims. This situates it firmly within operational analysis [4] as both a practical modeling and measurement substrate where sample path reasoning arises naturally.

    Crucially, like sample path analysis, it is a deterministic paradigm, but one that is applicable to both stochastic and non-stochastic systems.

    The history of Little’s Law and its relationship to the presence calculus is discussed in detail in our theory track post A Deep Dive into Little’s Law.↩︎

  2. Direct observation of the quantity under analysis is a core tenet of the presence calculus. When a quantity is not directly observable, or hard to instrument, the most common practice is to rely on statistical correlations and proxy variables (often pulled out of thin air) to measure the quantity indirectly.

    The presence calculus gives us better techniques to derive provable mathematical relationships between a hard-to-observe quantity and an easier-to-observe one and using these it becomes possible to construct a rigorous model to measure the quantity from directly observable signals.

    The presence calculus thus expands the domain of “directly observable” signals, reducing the reliance on proxies and statistcal correlations in many practical measurement scenarios.

    While often considered “difficult,” this approach is usually tractable—even when the path isn’t immediately obvious. This is where modeling skills and a solid mathematical foundation for identifying such indirect relationships prove essential.

    The presence calculus provides tools that make this easier to operationalize in real-world settings.↩︎

  3. Much of the mathematical theory behind the Presence Calculus is derived from the concepts and techniques that were used to proved Little’s Law. You might be familiar with Little’s Law from its applications in Lean software development. If so, one possible path to understanding the material here is to start with the post A Deep Dive into Little’s Law But this is not required. The material here is designed to stand alone and refer back to its origins in context.↩︎

  4. As of writing, the toolkit is still in an alpha state, and is under active development and evolution. It is not ready for production use, but can be used as reference to understand how the concepts we define are implemented in code.↩︎

  5. If integration signs in a “gentle” introduction feels like a bait-and-switch, rest assured, for the purposes of this document you just need to think of them as a way to add up presence masses, in a way that the ideas we use for binary presences will generalize when we apply them to arbitrary functions.↩︎

  6. The way we’ve defined signals and mass is directly analogous to how mass is defined for matter occupying space in physics.

    A binary signal can be thought of as defining a one-dimensional interval over time. For a fixed element and boundary, this gives us an area under the curve in two dimensions: time vs. amount of signal.

    If we treat elements and boundaries as additional independent dimensions, then the signal defines a volume in three dimensions, with time as one axis.

    This interpretation—presence as a physical manifestation of density over time—is a powerful way to reason intuitively and computationally about duration, overlap, and accumulation in time.

    And when we allow multiple signals to interact over the same time periods, we begin to model complex, higher-dimensional effects of presence—exactly the kind of generality we’ll need when we move beyond simple binary presences.↩︎

  7. The details here depend on ideas we will develop later in the document. The key point is that the presence calculus gives us tools to accurately represent the time-varying behavior of the actual system property we care about, rather than relying on statistical proxies. This also gives us stronger protection against Goodhart’s Law [6].↩︎

  8. The use of a rough curve here is an example of how presences can encode continuous inputs more effectively than discrete techniques, thanks to their explicit model of time. Forcing a developer to rank their productivity on a Likert scale often loses valuable nuance—whereas a fine-grained presence captures temporal variation with ease, making it available for downstream analysis.↩︎

  9. It is equally valid to define \(N\) as the number of distinct presences in the observation window. For example for the signal \(P2\) in Figure 5, this corresponds to asking if \(N=5\) (if we count the disjoint presences individually) or \(N=3\) (if we count the signals). These give different values for \(\bar{m}\) and \(\iota\) but their product still equals \(\delta\), as long as a single consistent definition of N is used.

    As we show in The Generalized Presence Invariant \(N\) can be a dicrete counting measure on the signal domain.

    This is ultimately a modeling decision that depends on what you are trying to measure. By default we will assume that \(N\) is measured at the signal granularity.↩︎

  10. \(\delta\) may also be considered the rate at which presence density accumulates over the interval. This latter view will be useful when interpreting the dynamics of the system.↩︎

  11. As discussed in The Presence Invariant – a Measure-Theoretic Generalization, the presence invariant admits a formal generalization: whenever two orthogonal measure spaces are involved, and accumulated presence mass resides in their product space, a similar invariant emerges. This reveals that the presence invariant is not a superficial constraint, but a deep structural property rooted in the measurability of systems of presence.↩︎

  12. We note that the residence time represents only the portion of the duration of the task in some arbitrary observation window. This is a different quantity from the overall duration of the task from start to finish (or from signal onset to reset in our terminology). This is the more familiar metric typically called the cycle time.↩︎

  13. We introduce Little’s Law here as a special case of the presence invariant. This is a deliberate decision: we aim to contextualize classical treatments of Little’s Law through the lens of what we consider the more foundational, measure-theoretic constructs of the presence calculus.

    In fact, the ideas in presence calculus may be considered generalizations of the concepts developed to prove Little’s Law. For an excellent overview of those techniques, see [7]. Many of the concepts surveyed there will reappear in this document—transformed, but recognizable—under the measure-theoretic definitions of the presence calculus.

    If you are not familiar with Little’s Law and wish to understand it better, Dr. Little’s survey paper remains the best starting point.↩︎

  14. The sampling granularity is typically coarser than time resolution of the underlying signals. For example, the signals themselves may be timestamped at millisecond granularity, while the presence matrix may be constructed by sampling at hourly, daily, or weekly intervals. Many different presence matrices can be constructed from the same underlying set of signals. Depending on the observation window and sampling granularity, we may arrive at very different matrices. A presence matrix is therefore an observer-relative analytical construct, derived from a system of presences—not a direct representation of the underlying signals.↩︎

  15. It is equally valid to define \(N\) as the number of distinct presences in the observation window. For example for the signal \(P2\) in Figure 5, this corresponds to asking if \(N=5\) (if we count the disjoint presences individually) or \(N=3\) (if we count the signals). These give different values for \(\bar{m}\) and \(\iota\) but their product still equals \(\delta\), as long as a single consistent definition of N is used.

    As we show in The Generalized Presence Invariant \(N\) can be a dicrete counting measure on the signal domain.

    This is ultimately a modeling decision that depends on what you are trying to measure. By default we will assume that \(N\) is measured at the signal granularity.↩︎

  16. Assuming \(N\) and \(T\) can be mapped to row and column counts of the presence matrix is a simplification. It relies on the assumption that each sampling interval is equal sized, and also that \(N\) represents signal counts. It is entirely possible to apply the machinery and techniques here to irregular observation windows, as well as for other measures for N on the signal dimension. These generalizations merely lead more complicated mappings to compute \(N\) and \(T\) from the presence matrix structure, and need to be specified as part of the modeling process. To avoid complicating matters, we will use this simple mapping, which covers a large number of practical use cases, as the default.↩︎

  17. Recall that we required signals to be measurable functions. This implies presence masses are measures over time intervals. Given intervals \(A\) and \(B\), a measure \(\mu\) satisfies the property of finite additivity \[\mu(A \cup B) = \mu(A) + \mu(B) - \mu(A \cap B),\] \(A[i,j]\) is a measure over the union of two time intervals, so the
    recurrence follows from this property of finite additivity where \[A = [i, j{-}1] \text{ and } B = [i{+}1, j].\] When \(A\) and \(B\) intersect, the subtraction removes the presence mass of overlap \([i{+}1, j{-}1]\) from the sum to avoid double-counting it.↩︎

  18. We’ll note at this point, that the determinism is retrospective. Since it is based on observed presence, this recurrence in no way implies we can predict how the presences will evolve in the future.↩︎

  19. Though we should hasten to add that this is just a loose analogy—we do not imply any conceptual equivalence.↩︎

  20. This concept is equivalent to the concept of a sample path developed by Stidham in [1] to provide the first deterministic proof of Little’s Law. This concept has been extensively studied in the context queueing systems with [2] being the comprehensive reference for applications of sample path techniques.↩︎

  21. Specifically, the Riemann sum approximation of the integral \[ A(1, j) = \sum_{k=1}^j A(k,k) \approx \int_0^j \left( \sum_{(e,b)} P_{(e,b)}(t) \right) dt \] where each \(P_{(e,b)}(t)\) represents the presence density function of an underlying element-boundary signal.↩︎

  22. While it’s tempting to assume that the limit of a product is simply the product of the limits, this doesn’t automatically hold here. The long-run value of presence density, \(\Delta\), is defined as a time-based density, while the signal mass contribution, \(\bar{M}\), is defined over the number of signals. Since these limits are taken over different denominators, additional technical conditions are required to ensure that their product equals the limit of the product.↩︎

  23. It is worth emphasizing the word observed in this statement. Even though the evolution is deterministic, this does not mean the future behavior of the system is predictable based on past behavior. That depends entirely on the nature of the signals involved, which may or may not be predictable. What we can say is that, given a sufficiently complete history of the system, we can deterministically reconstruct the current state of presence density from any starting point. This explanatory power is useful in its own right, as we will soon see.↩︎

  24. This is a key differentiation between the presence calculus and standard statistical and probabilistic techniques. The latter struggle with non-stationary systems—i.e., systems where the underlying distributions of values continuously change over time.↩︎