A consequence of Little’s Law
Dr. Krishna Kumar
The Polaris Advisor Program
In this technical note, we formalize the relationship between convergence in the presence calculus and the generalized form of Little’s Law, as originally proven by Brumelle (1971) and refined through sample path techniques by Heyman and Stidham (1980).
This addresses a key gap in our informal exposition: while we introduced the convergence conditions, and appealed to the general form of Little’s Law for proof, we did not make the connections explicit.
Here, we explicitly bridge that gap by mapping the concepts in the presence calculus to those in the general form of Little’s Law and make clear the relationships between the two.
In the rest of this document, when we say Little’s Law it should be assumed that we are referring to the general version discussed here.
The generalized form of Little’s Law is concerned with a countable collection of time-varying functions that represent the activity or occupancy of individual items in a system over time. Let each item \(i\) have an associated function \(f_i(t)\) defined for \(t \geq 0\), such that:
\[ \int_0^\infty |f_i(t)| \, dt < \infty \]
and for some finite \(l_i > 0\),
\[ f_i(t) = 0 \quad \text{for} \quad t \notin [t_i, t_i + l_i] \]
In other words, each \(f_i(t)\) is a function of compact support with a finite integral. These functions describe the temporal footprint or participation profile of item \(i\), capturing its influence on the system while it is “active”— for example, time spent in a queue, time holding a resource, or time contributing to some measurable system state.
This formulation originated as a generalization of classical queueing models: rather than simply recording the arrival and departure times of items, it allows us to associate an arbitrary time-varying function with each item’s time in the system. This enables a more flexible description of occupancy-like behavior, including partial participation, weighted influence, or overlapping resource use.
It is not hard to see that the notion of presence in the presence calculus is a close generalization of this same idea. The key difference lies in emphasis: while the classical formulation aims to derive steady-state relationships like \(H = \lambda \cdot G\), the presence calculus takes presence mass—the values of these integrals over finite time windows—as a foundational primitive of the framework.
From this base, we derive not only long-run behavior but also a rich vocabulary for describing the dynamics of systems in non-equilibrium states. Convergence is just one property of interest; divergence and metastability are equally important in understanding the real-world behavior of complex systems.
Nevertheless, when the system does converge, the identity \(\Delta = I \cdot \bar{M}\) stated in our earlier exposition is not an empirical observation—it is a direct consequence of the generalized Little’s Law presented here.
To prove this we need to show that our definitions in the presence calculus are isomorphic to those in this theorem
Lets start with the definitions of the key quantities in Little’s Law.
\[ G_i = \int_0^\infty f_i(t) \, dt, \quad i \geq 1 \]
\[ H(t) = \sum_{i=1}^\infty f_i(t), \quad t \geq 0 \]
From these, define the limiting quantities:
\[ G = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n G_i, \quad H = \lim_{T \to \infty} \frac{1}{T} \int_0^T H(t) \, dt, \quad \lambda = \lim_{n \to \infty} \frac{n}{t_n} \]
Theorem:
If \(\lambda\) and \(G\) exist and are finite, and if \(l_i / t_i \to 0\) as \(i \to \infty\), then \(H\) exists and:
\[ H = \lambda \cdot G \]
This is the general form of Little’s Law [1], [2], [3]. Please see Dr. Little’s survey article [3] for an extensive discussion of this law, and its history.
We now map each element of the theorem above to its counterpart in the presence calculus.
In the presence calculus, the system is composed of a (possibly infinite) set of signals indexed by \((e, b)\) — each representing the interaction of an element \(e\) with a boundary \(b\). Each such signal is associated with a presence function \(P_{(e,b)}(t)\), which records its temporal contribution at time \(t\).
This maps directly to the \(f_i(t)\) functions in the classical theorem: each \(f_i(t)\) corresponds to a presence function \(P_{(e,b)}(t)\).
The condition \(\int_0^\infty |f_i(t)| \, dt < \infty\) ensures that each item contributes a finite total presence over its lifespan. In the presence calculus, we refer to this as bounded signal mass. This is a technical requirement only for convergence. The presence calculus allows signals with unbounded mass in general; we only enforce integrability when analyzing asymptotic behavior like long-run presence density.
Thus, in the presence calculus:
The function \(H(t)\) in the theorem aggregates the instantaneous contributions of all \(f_i(t)\). In the presence calculus, the analog is:
\[ \delta(t) = \sum_{(e,b)} P_{(e,b)}(t) \]
We define long-run presence density as:
\[ \Delta = \lim_{T \to \infty} \frac{1}{T} \int_0^T \delta(t) \, dt \]
So:
\[ \Delta = H \]
Define:
\[ \bar{M} = \lim_{T \to \infty} \frac{1}{N(0,T)} \sum_{(e,b)} \int_0^T P_{(e,b)}(t) \, dt \]
This matches the limiting average of \(G_i\):
\[ G = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n G_i \]
So:
\[ \bar{M} = G \]
The classical limit \(\lambda = \lim_{n \to \infty} n / t_n\) expresses the asymptotic rate at which functions \(f_i\) arrive in time. In the presence calculus, the equivalent is:
\[ I = \lim_{T \to \infty} \frac{N(0, T)}{T} \]
This is the incidence rate of signals — the average number of signal onsets per unit time.
Thus:
\[ I = \lambda \]
The theorem requires that \(l_i / t_i \to 0\) as \(i \to \infty\). This means that the duration of each function \(f_i\) must become small relative to its arrival time.
In presence calculus terms, this ensures that signals do not overlap indefinitely with later-arriving signals, which could distort the time-based averages.
Equivalently, it ensures that:
\[ \lim_{i \to \infty} \frac{l_i}{t_i} = 0 \quad\Rightarrow\quad \text{bounded overlap and finite cumulative presence mass.} \]
Importance: This condition ensures that the support of each signal becomes negligible in the long run, preventing cumulative mass from diverging. Intuitively it formalizes the intuition that over sufficienctly long intervals the onset and reset rates of signals converge. Without this, presence can accumulate indefinitely.
Substituting these mappings:
\[ \Delta = I \cdot \bar{M} \]
This is the presence calculus analog of the generalized Little’s Law.
We restate a general convergence law for systems of presence:
Theorem (Presence Calculus version of Little’s
Law):*
Suppose the following limits exist and are finite:
Long-run incidence rate:
\[
I = \lim_{T \to \infty} \frac{N(0,T)}{T}
\]
Average presence per signal:
\[
\bar{M} = \lim_{T \to \infty} \frac{1}{N(0,T)}
\sum_{(e,b)} \int_0^T P_{(e,b)}(t) \, dt
\]
and assume the technical condition that presence durations become vanishingly small relative to arrival time:
\[ \text{duration}_{(e,b)} / \text{onset}_{(e,b)} \to 0 \]
Then the long-run presence density exists:
\[ \Delta = \lim_{T \to \infty} \frac{1}{T} \int_0^T \delta(t) \, dt \]
and:
\[ \Delta = I \cdot \bar{M} \]
This makes precise the claim used in our introductory material and affirms that the convergence of long-run presence density is a direct generalization of Little’s Law. More importantly, it highlights when such a relationship does not hold — namely, when the relevant limits do not exist, or the technical conditions fail.
While the presence calculus derives much of its analytical machinery from the tools used to prove Little’s Law, it focuses on very complementary applications enabled by generalizing some of the underlying proof techniques.