A first passage problem for a Poisson counting process with a linear moving boundary

TL;DR

This work addresses the first-passage problem for a Poisson counting process across a linearly moving boundary with slope $\alpha$ and offset $\beta$. It develops and juxtaposes two frameworks—direct time-domain path decomposition and a Laplace-domain mapping to an effective random walk with Pollaczek-Spitzer analysis—to derive exact results and verify their equivalence. The authors deliver new analytical expressions, including a large-deviation form of the first-passage time distribution in the subcritical regime, closed-form conditional mean first-passage times for arbitrary offset, and a universal relaxation timescale $\xi(\alpha)=1/(1-\alpha+\alpha\log\alpha)$ governing survival decay, together with a detailed study of critical behavior at $\alpha=1$. These results yield a complete, exactly solvable portrait of a classic stochastic-crossing problem with rich phase-transition-like structure, and illuminate potential generalizations to broader moving-boundary problems and renewal processes.

Abstract

The time to first crossing for the Poisson counting process with respect to a linear moving barrier with offset is a classic problem, although key results remain scattered across the literature and their equivalence is often unclear. Here we present a unified and pedagogical treatment of two approaches: the direct time-domain approach based on path-decomposition techniques and the Laplace-domain approach based on the Pollaczek-Spitzer formula. Beyond streamlining existing derivations and establishing their consistency, we leverage the complementary nature of the two methods to obtain new exact analytical results. Specifically, we derive an explicit large deviation function for the first-passage time distribution in the subcritical regime and closed-form expressions for the conditional mean first-passage time for arbitrary offset. Despite its simplicity, this first crossing process exhibits non-trivial critical behavior and provides a rare example where all the main results of interest can be derived exactly.

Figures (15)

• Figure 1: Schematic of a sample path of the Poisson process showing its evolution in relation to the linear moving boundary with positive offset $B(t) = \alpha t + \beta$. In this example, the path crosses the boundary for the first time on the fifth jump at time $\tau$. The time intervals $t_i$ between jumps are independent random variables drawn from the exponential distribution with unit rate.
• Figure 2: Schematic representation of the D/M/1 queue with $k=2$ initial customers. The Poisson process (blue) counts the number of customers served by time $t$. The piecewise constant function (brown) represents the total number of customers who have arrived by time $t$. The linear boundary $B(t) = \alpha t + \beta$ with $\beta=k-1$ serves as the lower enveloping curve for this piecewise constant function. The boundary crossing corresponds to the moment when the number of served customers equals the number of customers who have arrived, hence the queue is depleted.
• Figure 3: Path decomposition corresponding to the recurrence relation \ref{['eq:Pn=recurrence_0']} for $n=5$. There are 4 jumps without crossing in $(0,t')$, the fifth jump occurs at $t'$, and no jumps occur in the time interval $(t',t)$. The instances at which the boundary hits an integer value $n$, i.e., the earliest times at which the $n$-th jump may occur without causing a crossing, are given by $T_n$ as in \ref{['eq:Tn=def']}. In this example, the parameters are $\beta=2{.}2$, $\alpha=0.5$ and hence $T_1=T_2=0$, $T_3=1.6$, $T_4=3.6$, and $T_5=5.6$.
• Figure 4: First-passage time density $\mathbb{P}\left[\, \tau\,\vert\, \beta\,\right]$ as a function of time for $\beta = 1.5$ and $\alpha = 2$. The red solid line shows the analytical result \ref{['eq:P=result_time_domain']}, while black circles correspond to direct numerical simulations obtained by generating $10^{7}$ independent trajectories of the Poisson process. The density exhibits characteristic discontinuities at $T_n = (n-\beta)/\alpha$ for $n > \lfloor\beta\rfloor$.
• Figure 5: An example of a configuration with the crossing at $\tau=T_3-\epsilon$ (left) and at $\tau=T_3+\epsilon$ (right). A crossing at $\tau=T_3+\epsilon$ would require two jumps within the interval $(T_3,T_3+\epsilon)$, which becomes impossible as $\epsilon\to0$, hence the different limits in \ref{['eq:P[tn-epsilon]=']} and \ref{['eq:P[tn+epsilon]=']}.