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arxiv: 2602.05009 · v2 · pith:3OHNMBEQnew · submitted 2026-02-04 · ❄️ cond-mat.stat-mech

Resetting-induced instability in queues fed by a search process in an interval

Jos\'e Giral-Barajas , Paul C. Bressloff This is my paper

Pith reviewed 2026-05-16 06:31 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords stochastic resettingqueuing theorysearch processesbounded domainsteady-state convergencemulti-server queuesthreshold rateresource management
0
0 comments X

The pith

Stochastic resetting in search processes feeding queues exhibits a threshold rate that reverses its impact on convergence to steady state, growing exponentially with the number of servers.

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

This paper examines queues where resource arrivals are governed by a search process with stochastic resetting inside a bounded interval. It combines standard results from multi-server queuing theory with calculations of mean search times under resetting to map out the regions of search parameters that allow the queue to reach a steady-state distribution. A key finding is the existence of a critical resetting rate: below it, higher resetting rates shrink the stable parameter region, while above it they expand it. This threshold itself increases exponentially as the number of servers grows, implying that resetting becomes less effective at promoting stability in larger systems.

Core claim

By merging the steady-state analysis of an M/M/c queue with the mean first-passage time for a resetting search process in an interval, the authors identify parameter regions ensuring queue convergence and locate a resetting-rate threshold at which the effect of resetting on those regions switches from contraction to expansion; this threshold scales exponentially with server number.

What carries the argument

The threshold resetting rate separating regimes of reduced versus expanded convergence regions in the search-parameter space, obtained by matching the effective arrival rate from the resetting search to the service capacity of the multi-server queue.

If this is right

  • Below the threshold, stochastic resetting contracts the region of parameter space where the queue converges to steady state.
  • Above the threshold, stochastic resetting expands that convergence region.
  • The value of the threshold resetting rate increases exponentially with the number of servers in the system.
  • For systems with many servers, stochastic resetting is less likely to improve convergence compared with small-server cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exponential scaling suggests a practical limit on using resetting to stabilize large-scale resource queues, such as in distributed computing or biological transport networks.
  • If correlations between the search position and current queue length are strong, they could shift the location of the threshold or introduce new instability boundaries not captured by the decoupled analysis.
  • Extensions to non-Markovian search or time-varying resetting rates would likely alter the functional form of the exponential growth with server number.
  • In ecological models of foraging, the same threshold mechanism could determine when environmental resetting helps or harms population-level resource buffering.

Load-bearing premise

The steady-state queue length distribution can be accurately determined by treating the mean search time from the resetting process as an effective arrival rate without additional interaction terms between the search and queue dynamics.

What would settle it

Direct simulation of the combined stochastic process for a small number of servers that tracks whether the boundary of convergence regions in search-parameter space shifts exactly at the analytically predicted threshold resetting rate.

Figures

Figures reproduced from arXiv: 2602.05009 by Jos\'e Giral-Barajas, Paul C. Bressloff.

Figure 1
Figure 1. Figure 1: FIG. 1. Sequence of search-and-capture events for a diffusin [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Critical spatial configurations for the existence of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sample path of a search-and-capture process with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Critical, convergence-inducing initial condition [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Different scenarios of interest for the exploration o [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Contour plots for the traffic intensity in the ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Critical, convergence-inducing starting position [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: (b). Therefore, we finally obtain that the subre￾gion of [0, L] for which an initial position allows stochastic resetting to expedite the search process is X0 =  0,  1 − 1 √ 5  · L  . (A7) Appendix B: Partial derivative of the critical starting position with resetting For the sake of completeness, we include the resulting expression for the partial derivative with respect to the resetting rate of the c… view at source ↗
read the original abstract

Proper management of resources whose arrival and consumption are subject to environmental randomness is an intrinsic process in both natural and artificial systems. This phenomenon can be modeled as a queuing process whose arrival distribution is determined by a search process with stochastic resetting. When the queuing system has a limited number of servers and the search process occurs within a bounded domain, the dynamics of expediting or delaying the search through stochastic resetting interact with the long-term dynamics of the number of resources in the queue. We combine results from queuing theory with those from search processes with stochastic resetting in a bounded domain to obtain regions of the parameter space of the search process that ensure convergence of the number of resources in the queue to a steady state. Furthermore, we find a threshold resetting rate at which the effects of stochastic resetting shift from reducing convergence regions to expanding them. Finally, we demonstrate that this threshold value grows exponentially with the number of servers, making it harder for stochastic resetting to improve the convergence of the queueing system.

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

1 major / 0 minor

Summary. The manuscript models a multi-server queue (M/M/c) whose arrivals are supplied by a diffusive search process with stochastic resetting inside a bounded interval. Combining the standard M/M/c stability criterion (arrival rate set by the reciprocal of the mean search time) with the known mean first-passage time under resetting, the authors map regions of parameter space in which the queue length converges to a steady state. They report a threshold resetting rate r* that separates regimes in which increasing the resetting rate shrinks versus expands the convergence region, and show that r* grows exponentially with the number of servers c.

Significance. If the factorization assumption is justified, the work supplies a concrete, quantitative connection between stochastic resetting and the stability boundary of multi-server queues, together with a striking exponential dependence of the critical rate on system size. This could be relevant for resource-management models in fluctuating environments (e.g., biological transport or distributed computing) and offers a falsifiable prediction for the location of the transition.

major comments (1)
  1. [Main derivation of the threshold (following the abstract)] The threshold r* is obtained by intersecting the M/M/c stability condition with the mean first-passage time of the resetting search process. Because the instantaneous arrival rate is a functional of the searcher's position (which is itself modulated by resetting), the joint Markov chain on (queue length, searcher coordinate) is not separable. The manuscript supplies neither a master equation for the combined state space nor a perturbative estimate of the correlations between queue state and arrival rate; consequently the reported location of r* and its exponential scaling with c rest on an unverified factorization assumption.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: The threshold r* is obtained by intersecting the M/M/c stability condition with the mean first-passage time of the resetting search process. Because the instantaneous arrival rate is a functional of the searcher's position (which is itself modulated by resetting), the joint Markov chain on (queue length, searcher coordinate) is not separable. The manuscript supplies neither a master equation for the combined state space nor a perturbative estimate of the correlations between queue state and arrival rate; consequently the reported location of r* and its exponential scaling with c rest on an unverified factorization assumption.

    Authors: We thank the referee for highlighting this important point. The search process evolves independently of the queue length, as there is no feedback from the queue state to the searcher's position in the interval. Thus, the stationary distribution of the searcher can be determined separately, and the long-term arrival rate is the expectation of the position-dependent rate under this distribution, equivalent to the reciprocal of the mean first-passage time. In queueing theory, for a multi-server queue driven by a stationary arrival process, the stability condition is determined solely by whether this average arrival rate is below the total service capacity cμ. The correlations between the queue length and the searcher position affect the detailed dynamics and the queue length distribution but do not alter the stability threshold. We will add a clarification in the revised version explaining this separation and citing appropriate references from queueing theory on stability for general stationary inputs. The reported exponential scaling of r* with c follows from the c-dependence in the M/M/c stability criterion combined with the r-dependence of the mean first-passage time. revision: partial

Circularity Check

0 steps flagged

No circularity: threshold emerges from intersection of independent steady-state expressions

full rationale

The derivation obtains the convergence region by intersecting the stability condition of an M/M/c queue (arrival rate equal to the reciprocal of the mean search time) with the mean first-passage time under resetting in a bounded interval. Both expressions are taken from prior, externally established results in queuing theory and resetting search theory; the threshold resetting rate r* is the explicit solution to that intersection and is not fitted to data, defined in terms of itself, or obtained via a self-citation chain. The paper states the combination explicitly as a modeling step rather than deriving one quantity from the other by algebraic identity. No load-bearing ansatz is smuggled through citation, and the exponential scaling with server number c follows directly from the closed-form expressions without renormalization or redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Because only the abstract is available, the ledger is necessarily incomplete. The central claim rests on the unstated assumption that the two bodies of theory combine without residual interaction terms.

axioms (1)
  • domain assumption Steady-state convergence regions of the queue can be obtained by direct superposition of results from queuing theory and resetting search theory.
    Invoked when the authors state they combine the two literatures to obtain the parameter regions.

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Works this paper leans on

78 extracted references · 78 canonical work pages

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    Another topic of current interest is search processes with stochastic resetting [ 26]

    Therefore, starting the process closer than x∗ 0 to the target is detrimental for the long- term stability of the number of resources. Another topic of current interest is search processes with stochastic resetting [ 26]. A typical motivating ex- ample is a diffusing particle searching for a specific tar- get U in an unbounded domain Ω. The introduction of ...

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    (5) Therefore, all the quantities of interest for the search process can be described via the Laplace transform of the probability flux into the target, ˜J0(x0,s )

    we directly have that ˜f0(x0,s ) = ˜J0(x0,s ), from where we obtain that T0(x0) := E[T0(x0)] = − ∂ ∂s ˜J0(x0,s ) ⏐ ⏐ ⏐ s=0 . (5) Therefore, all the quantities of interest for the search process can be described via the Laplace transform of the probability flux into the target, ˜J0(x0,s ). An explicit expression for ˜J0(x0,s ) can be found using the Green’s...

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    yields ˜F0(s) = ˜f0(s) ˜ϕ (s). (10) B. Convergence zones for the G/M/c without resetting As highlighted in the introduction, the accumulation of resources after several rounds of the search-and-capture process can be mapped onto a queueing process. To do so, we need to determine the departure process to coun- terbalance the continual arrival coming from t...

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    ( 7b) to derive the follow- ing condition for convergence to a steady-state number of resources [ 20]: x0 >x ∗ 0 where x∗ 0 =L − √ L2 − 2D cµ + 2Dτcap

    with the explicit expression of the MFPT in Eq. ( 7b) to derive the follow- ing condition for convergence to a steady-state number of resources [ 20]: x0 >x ∗ 0 where x∗ 0 =L − √ L2 − 2D cµ + 2Dτcap. (12) We also obtained a minimal interval length that ensures the existence of regions in the search process’s parame- ter space that allow the G/M/c system t...

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    Stochastic resetting has been 5 FIG. 2. Critical spatial configurations for the existence of a steady-state distribution with instantaneous refractor y periods. (a) The threshold starting position of the search process en sures that the number of resources in the G/M/c system converges to a steady state. Each curve is determined by Eq. ( 12) as a function ...

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    (19) Substituting this expression into the convergence con- dition in Eq

    to com- pute the arrival rate as λ r = r cosh [ √ r D (L − x0) ] cosh [ √ r DL ] + (rτcap − 1) cosh [ √ r D (L − x0) ] . (19) Substituting this expression into the convergence con- dition in Eq. ( 11) and solving the inequality for the initial position, x0, we obtain the following critical starting po- sition x∗ r =L − √ D r arccosh [ cµ cosh [ √ r DL ] r...

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    We obtain that L∗ r = √ D r arccosh [ r +cµ − rcµτcap cµ ]

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    ( 20) and solving for L

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    ( 22) and solving for L, ob- taining Leq = √ D r 2r + 2rcµτcap +cµ log ( r+cµ − rcµτ cap cµ ) 2 2cµ log ( r+cµ − rcµτ cap cµ )

    to Eq. ( 22) and solving for L, ob- taining Leq = √ D r 2r + 2rcµτcap +cµ log ( r+cµ − rcµτ cap cµ ) 2 2cµ log ( r+cµ − rcµτ cap cµ ) . (25) To determine the validity of this explicit expression, we explore the rate of convergence of x∗ r to xhl r , as L → ∞ , for different values of the resetting rate. Let r >0, and define dr(L) := x∗ r(L) − xhl r . Numeri...

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    Furthermore, the threshold between convergence and blow up can be determined by the critical starting position x∗ r, given in Eq

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