A measure-theoretic generalization
Dr. Krishna Kumar The Polaris Advisor Program
© 2025 Krishna Kumar. All rights reserved.
The Presence Invariant fundamentally expresses the conservation of presence mass across distinct, interacting domains: time and signals. This measure-theoretic framing closely parallels the structure of Rate Conservation Laws (RCL) in stochastic processes, particularly as formulated by Masakiyo Miyazawa [1].
This document explores this connection, showing that the Presence Invariant provides a generalized, deterministic conservation law for any measurable presence distribution over joint domains. We demonstrate that Miyazawa’s RCL emerges as a specific instance of the Presence Invariant under the additional assumptions of stochastic stationarity and the interpretation of quantities as expectations.
The power of the generalized Presence Invariant, however, is that it requires no such assumptions—as long as presence is defined over a product measure space, the invariant holds by construction.
This perspective echoes Little’s observation [2] that Sigman [3] has shown the sample path proofs of the general form of Little’s Law, \(H = \lambda G\), are equivalent to the rate conservation laws of Miyazawa.
Our formulation of the Presence Invariant makes this correspondence explicit by deriving the conservation law directly from measure-theoretic assumptions, without requiring stochastic stationarity or expectation-based interpretations.
It generalizes both Little’s Law and Miyazawa’s Rate Conservation Law. Rather than indexing presence by items or arrival processes, we define it over an arbitrary product measure space and use Fubini’s Theorem to show that all quantities and relationships in both laws arise as consequences of marginalizing an integrable density over this space.
This removes the need for stationarity, stochasticity, or item-based indexing. The invariant holds by construction whenever presence is represented as an integrable density over a product of measurable domains.
In the context of stationary queueing systems and marked point processes, Miyazawa’s Rate Conservation Law (RCL) establishes a fundamental balance principle. It asserts that:
The *rate at which a measurable event (e.g., arrivals, transitions, completions) occurs at a boundary is equal to the rate at which it leaves**, under stationary expectations.
Formally, for a counting process ( N(t) ), the rate is often defined as ( r( t) = _{} [N(t + ) - N(t)] ). In a stable system, this incoming rate is matched by an outgoing rate: ( [] = [] ).
Miyazawa’s framework, often leveraging Palm calculus, provides a rigorous foundation for invariants such as Little’s Law, typically expressed in the form: \[\mathbb{E}[H] = \lambda \cdot \mathbb{E}[G],\] where:
A key aspect of this formulation is its reliance on expected values and stochastic assumptions, particularly stationarity**, which implies that the statistical properties of the process do not change over time.
The Presence Calculus adopts a measure-theoretic approach to define and quantify presence, independent of stochastic assumptions. We model presence as a quantity distributed over a product space formed from two orthogonal measurable domains:
This leads to the product measurable space \((T \times S, \mathcal{F}_T \otimes \mathcal{F}_S)\) and the product measure \(\mu = \mu_T \otimes \mu_S\).
Let \(f: T \times S \to \mathbb{R}_{\geq 0}\) be a measurable function representing the presence density over this joint domain. The function \(f\) is assumed to be integrable under the product measure \(\mu\).
The total presence mass (\(A\)) accumulated across the full product space ( e.g., over a finite observation window) is defined as the joint integral of the presence density: \[A = \iint_{T \times S} f(t, s)\, d\mu_T(t)\, d\mu_S(s).\]
We define marginal densities by integrating out one of the variables:
By defining average presence densities over their respective domains: \[\bar{D}_T = \frac{A}{\mu_T(T)}, \quad \bar{D}_S = \frac{A}{\mu_S(S)},\] where \(\mu_T(T)\) and \(\mu_S(S)\) are the finite measures (extents) of the time and signal domains, respectively.
The generalized presence invariant then states the proportional relationship between these average densities: \[\frac{\bar{D}_S}{\bar{D}_T} = \frac{\mu_T(T)}{\mu_S(S)}.\] This fundamental relationship reflects the conservation of total presence mass, ensuring that marginal densities are scaled by the ratio of the measures of their support domains.
The connection between the measure-theoretic Presence Invariant and Miyazawa’s stochastic RCL is established by recognizing how the components of the Presence Invariant map to, and generalize, the terms in \(\mathbb{E}[H] = \lambda \cdot \mathbb{E}[G]\).
Let’s define the co-presence rate as the ratio of the “size” of the signal domain to the “size” of the time domain within an observation window: \[\text{Co-Presence Rate} = \frac{\mu_S(S)}{\mu_T(T)}.\] In many practical scenarios, where \(\mu_S\) is a counting measure over \(N\) active signals and \(\mu_T\) represents a time duration \(T\), this becomes the familiar \(\frac{N}{T}\) ratio. This ratio quantifies the “density of signals per unit time” induced by the co-presence.
Rearranging the generalized presence invariant, we get: \[\bar{D}_T = \frac{\mu_S(S)}{\mu_T(T)} \cdot \bar{D}_S.\] This form explicitly highlights the structural analogy:
Thus, the Presence Invariant, expressed as \(\bar{D}_T = \left(\frac{\mu_S(S)}{\mu_T(T)}\right) \cdot \bar{D}_S\), structurally mirrors Miyazawa’s \(\mathbb{E}[H] = \lambda \cdot \mathbb{E}[G]\).
The fundamental link here is that measure theory provides the bedrock for probability theory and, consequently, stochastic process theory.
The Presence Calculus leverages general Lebesgue measures (or other \(\sigma\) -finite measures) over arbitrary time and signal domains. This means its invariants hold for any measurable function \(f(t,s)\), regardless of whether \(f\) arises from a stochastic process or exhibits stationary behavior.
Miyazawa’s RCL becomes a special case of the Presence Invariant when:
Fubini’s Theorem is central to the consistency of this framework. It ensures that the total presence mass can be computed by integrating over time first and then over signals, or vice versa, without altering the result: \[ \iint_{T \times S} f(t, s)\, d\mu_T(t)\, d\mu_S(s) = \int_T \left( \int_S f(t, s)\, d\mu_S(s) \right) d\mu_T(t) = \int_S \left( \int_T f(t, s)\, d\mu_T(t) \right) d\mu_S(s). \] This integral symmetry precisely supports the structure of the invariant, allowing for coherent “marginalization” of presence mass and confirming the internal consistency of flow conservation principles in both measure theory and, by extension, RCL theory.
The*Presence Invariant** is a powerful*generalized, measure-theoretic conservation law** that applies to any measurable distribution of presence across time and signal domains.
Thus, the Presence Calculus offers a broader and more fundamental lens for analyzing flow and conservation, applicable even in complex, non-stationary systems where traditional stochastic models face significant limitations.