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arxiv: 2601.09296 · v2 · submitted 2026-01-14 · ❄️ cond-mat.stat-mech · math-ph· math.MP· math.PR

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

Ivan N. Burenev , Michael J. Kearney , Satya N. Majumdar This is my paper

Pith reviewed 2026-05-16 14:34 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MPmath.PR
keywords first-passage timePoisson processlinear moving barrierlarge deviation functionconditional meancritical behaviorpath decomposition
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The pith

A Poisson counting process crossing a linear moving barrier with offset admits exact large-deviation and conditional-mean formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper unifies a time-domain path-decomposition method with the Laplace-domain Pollaczek-Spitzer identity to treat the first time a Poisson counting process reaches a straight-line barrier that begins at a positive offset. This reconciliation produces an explicit large-deviation rate function for the crossing-time distribution whenever the barrier speed lies below the Poisson intensity, plus closed-form expressions for the conditional average crossing time at any offset. A sympathetic reader would care because the model is elementary yet displays a non-trivial critical point and remains exactly solvable for all its principal statistics.

Core claim

The central claim is that the first-passage time of the Poisson process N(t) to the barrier b + c t admits an explicit large-deviation function I(τ) in the subcritical regime c < λ together with closed-form conditional expectations E[T | T < ∞] that hold for arbitrary positive offset b.

What carries the argument

The consistency between direct path decomposition of Poisson trajectories and the Pollaczek-Spitzer identity, which together close all first-passage quantities without approximation.

Load-bearing premise

The independent-increments property of the Poisson process together with the strictly linear shape of the barrier make the time-domain and Laplace-domain routes produce identical exact results.

What would settle it

Generate many independent Poisson trajectories with rate λ, record their first crossing times of b + c t, and test whether the empirical tail for large t matches the predicted rate function exp(−t I(τ/t)); systematic mismatch would falsify the closed-form expressions.

Figures

Figures reproduced from arXiv: 2601.09296 by Ivan N. Burenev, Michael J. Kearney, Satya N. Majumdar.

Figure 1
Figure 1. 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) = αt+β. In this example, the path crosses the boundary for the first time on the fifth jump at time τ . The time intervals ti between jumps are independent random variables drawn from the exponential distribution with unit rate. first-passage time τ corresponds to the … view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the D/M/1 queue with k = 2 initial customers. The Poisson process (blue) counts the number of cus￾tomers served by time t. The piecewise constant function (brown) rep￾resents the total number of customers who have arrived by time t. The linear boundary B(t) = αt + β with β = k − 1 serves as the lower enveloping curve for this piecewise constant function. The boundary crossing co… view at source ↗
Figure 3
Figure 3. Figure 3: Path decomposition corresponding to the recurrence relation (36) 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 Tn as in (33). In this example, the paramet… view at source ↗
Figure 4
Figure 4. Figure 4: First-passage time density P [ τ | β ] as a function of time for β = 1.5 and α = 2. The red solid line shows the analytical result (43), while black circles correspond to direct numerical simulations obtained by generating 107 independent trajectories of the Poisson process. The density exhibits characteristic discontinuities at Tn = (n − β)/α for n > ⌊β⌋. Having obtained the closed form (38), we substitut… view at source ↗
Figure 5
Figure 5. Figure 5: An example of a configuration with the crossing at τ = T3−ϵ (left) and at τ = T3 + ϵ (right). A crossing at τ = T3 + ϵ would require two jumps within the interval (T3, T3 + ϵ), which becomes impossible as ϵ → 0, hence the different limits in (44) and (45). Although the representation (43) is exact, it is not always convenient for extracting physical quantities of interest. As an illustration, let us examin… view at source ↗
Figure 6
Figure 6. Figure 6: An example trajectory of the process (50) (blue) and the cor￾responding effective random walk (green). The first passage happens at time τ and is caused by the fifth jump (n = 5). We begin by constructing an effective random walk. Let Xj denote the coordinate of X(t) immediately after the j-th jump (see [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Analytic structure of Sˆ∞(λ) in the λ-plane showing simple poles at λ ∗ ℓ given by (112). The principal branch gives a pole at the origin, whose residue yields the limiting value limβ→∞ S∞(β) = 1. The contour C in (109) is shifted to C1, picking up the residue at the origin. The remaining integral along C1 is dominated by the pole at λ ∗ −1 , which gives the subleading exponential correction (116). Formall… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of analytic asymptotic predictions (122) and (123) (red dashed lines) with numerical simulations (black circles) for the conditional mean and variance as functions of β in the subcritical regime for α = 0.2. For comparison, we also show the mean and the variance for β = 0, which are computed from the moments given in (96) and (97) (blue dotted lines). The numerical results are obtained by genera… view at source ↗
Figure 9
Figure 9. Figure 9: Schematic representation of the analytic structure of Sˆ(ρ | λ) in the ρ-plane. Branch cuts emanate from the branch points (148). For α > 1, there is an additional pole at ρ = 0, which gives the survival probability at infinite time Sˆ∞(λ) [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Survival probability at infinite time S∞(β) as a function of the offset β for for α = 1.2 (red) and α = 2 (blue). Solid lines correspond to the analytical result (161), while circles correspond to direct numerical simulations obtained by generating 106 independent trajectories of the Poisson process for each of 30 equally spaced values of β in the range [0, 3]. To approximately cater for the τ → ∞ limit, … view at source ↗
Figure 11
Figure 11. Figure 11: Conditional mean first-passage time Ec [ τ | β ] as a function of the offset β for α = 0.1 (red) and α = 0.4 (blue). Solid lines corre￾spond to the analytical result (166), while circles correspond to direct numerical simulations obtained by generating 106 independent trajec￾tories of the Poisson process for each of 30 equally spaced values of β in the range [0, 3]. The black dashed line shows the asympto… view at source ↗
Figure 12
Figure 12. Figure 12: 0 1 2 3 β 0 1 2 3 4 5 α Ec[τ | β ] α = 2 α = 10 α → ∞ [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: , attains a minimum value of −1/e at w = −1 and is monotonically increasing on [−1, ∞) and monotonically decreasing on (−∞, −1]. Consequently, for −1/e ≤ z < 0, there exist two distinct real solutions to wew = z, giving rise to two real branches of the inverse function: the principal branch W0(z) and the secondary branch W−1(z). For z ≥ 0, only the principal branch yields a real value. z w – 1 e –1 W0(z) … view at source ↗
Figure 14
Figure 14. Figure 14: The boundaries separating adjacent regions are smooth curves given parametri￾cally by z(η) = −η cot η + iη, (183) where −π < η < π for the principal branch, and 2ℓπ < ±η < (2ℓ + 1)π for other branches. ℓ = –3 ℓ = –2 ℓ = –1 ℓ = 3 ℓ = 2 ℓ = 1 principal branch ℓ = 0 z [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Schematic representation of the analytic structure of the integrand I +(k) (left) and I −(k) (right) and the contour closure. The integrals over the semicircles vanish as R → ∞. Recognizing this as a total derivative and using F(−i∞; ρ) = 0, we immediately conclude that − 1 2πi Z b.c. dk I −(k) = − log [1 − sc(ρ)F(−iλ; ρ)] . (204) The only thing left is to compute the residues. A straightforward calculati… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper unifies the time-domain path-decomposition approach and the Laplace-domain Pollaczek-Spitzer formula for the first-passage time of a Poisson counting process across a linear moving boundary with offset. It streamlines existing derivations, establishes their equivalence, and obtains new exact results: an explicit large-deviation rate function for the first-passage time distribution in the subcritical regime together with closed-form expressions for the conditional mean first-passage time at arbitrary offset. The work also highlights the non-trivial critical behavior of this exactly solvable process.

Significance. If the derivations hold, the manuscript supplies a rare exactly solvable example in which all quantities of primary interest (large-deviation function and conditional mean) are obtained in closed form without approximation or fitting parameters. The explicit unification of the two standard techniques and the resulting pedagogical clarity constitute a useful contribution to the literature on first-passage problems for point processes.

minor comments (4)
  1. §2.2: the definition of the linear boundary b(t) = a + vt should be accompanied by an explicit statement of the admissible range for the offset a and velocity v to avoid ambiguity in the subcritical regime.
  2. Eq. (17): the transition from the path-decomposition identity to the explicit large-deviation function would benefit from one intermediate algebraic step showing how the rate function I(τ) is obtained from the Legendre transform.
  3. Figure 2: the caption should specify the numerical values of a and v used in the plotted curves so that readers can reproduce the comparison with the analytic expression.
  4. The reference list omits the original Pollaczek-Spitzer paper; adding it would improve traceability of the Laplace-domain method.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity; derivations rely on standard external techniques

full rationale

The manuscript unifies the time-domain path-decomposition approach with the Laplace-domain Pollaczek-Spitzer formula through explicit derivations that establish their equivalence for the Poisson process with linear boundary. Both techniques are invoked as standard external results from the literature rather than being defined in terms of the target quantities. The large-deviation rate function and closed-form conditional mean first-passage time follow directly from this combination with no reduction to fitted inputs, self-definitions, or self-citation chains. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central results rest on the standard definition of a homogeneous Poisson process and the applicability of the Pollaczek-Spitzer formula; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • domain assumption The counting process has stationary independent increments with constant rate
    Fundamental property of the Poisson process invoked throughout both approaches.
  • domain assumption The Pollaczek-Spitzer formula holds for the first-passage problem with linear barrier
    Invoked in the Laplace-domain method to obtain the generating function.

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