Instability of G/M/c queues under stochastic resetting in the interval

TL;DR

This work studies how stochastic resetting intersects with the long-time behavior of a target-centric search driving a $G/M/c$ queue in a bounded interval. By merging first-passage time analysis for a diffusive search with queueing theory and renewal resetting, it derives explicit critical quantities that separate convergence to a steady state from unbounded growth. Key findings include a resetting-induced shift of the convergence boundary, characterized by the interval-length threshold $L_{eq}$ and resetting-rate threshold $r_{eq}$, and the existence of a rate $r_{max}$ that minimizes the convergence region. The results reveal that, as the number of servers $c$ grows, the threshold resetting rate grows (as a power law $r_{eq}\sim \alpha c^{\beta}$ with $\beta>2$, while $r_{max}\sim \gamma c$ with $\gamma\approx3.93$), making resetting less beneficial in larger multi-server queues. Overall, the paper links renewal-resetting search dynamics to queue stability, with implications for designing resetting-based control in bounded domains.

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.

Figures (9)

• Figure 1: Sequence of search-and-capture events for a diffusing particle in the interval with an absorbing boundary at the origin and a reflecting boundary at the endpoint mapped into a $G/M/c$ queueing process. (a) The particle follows a one-dimensional (1D) Brownian motion starting from $x_{0}\in[0,L]$ and searching for the target at $x=0$. Once the particle finds the target, it delivers a resource, returns to its starting position, and restarts the process. In each round, the delivery, return, and reload take a random time $\hat{\tau}$. Repeating this process produces a sequence of delivery times, known as the sequence of bursts. (b) Under the additional assumption that the random consumption times are exponentially distributed, the sequence of bursts can be fed into a $G/M/c$ queueing process to track the accumulation of resources within the target.
• Figure 2: Critical spatial configurations for the existence of a steady-state distribution with instantaneous refractory periods. (a) The threshold starting position of the search process ensures that the number of resources in the $G/M/c$ system converges to a steady state. Each curve is determined by Eq. (\ref{['eq:CriticalX0']}) as a function of $L$, fixing $D=1$, $\mu=1$, and $\tau_{\mathrm{cap}}=0$ and varying the number of servers in the system. The green dots represent the threshold interval length, $L^{*}$, determined by Eq. (\ref{['eq:CriticalL']}). (b) Critical initial position and convergence and blow up zones for an interval of length $L=2$ and different numbers of servers. This representation identifies with the endpoints of the curves in panel (a), when $L=2$. Whenever $x_{0}>x_{0}^{*}$, the number of resources converges to a steady state, and whenever $x_{0}\leq x_{0}^{*}$, the number of resources blows up.
• Figure 3: Sample path of a search-and-capture process with instantaneous resetting and instantaneous refractory periods. The resetting position is assumed to be the same as the initial position, i.e., $x_{0}=x_{r}$. The dynamics introduced by stochastic resetting are shown in orange, straight arrows, and the restart dynamics of the search-and-capture process are shown in dotted lines.
• Figure 4: Critical, convergence-inducing initial conditions without stochastic resetting and with fixed resetting rates of (a) $r=1$, (b) $r=10$, and (c) $r=100$. The light curve corresponds to the scenario without stochastic resetting, determined by Eq. (\ref{['eq:CriticalX0']}), and the dark curve corresponds to the scenario with stochastic resetting, determined by Eq. (\ref{['eq:CriticalStartingResetting']}). The green-shaded area represents the resetting-induced spatial configurations that ensure convergence, compared to the case without resetting, while the red-shaded area represents the lost spatial configurations that ensure convergence when stochastic resetting is introduced. The dotted line corresponds to the critical starting position in the half-line, $x_{r}^{\text{hl}}$, shown in Eq. (\ref{['eq:CriticalLimitResetting']}). Panels (d-f) show the threshold quantities $L_{r}^{*}$, $L^{*}$, and $L_{\text{eq}}$ corresponding to the panel directly above. In every case, we fix $D=1$, $\tau_{\mathrm{cap}}=0$, $\mu=1$, and $c=1$.
• Figure 5: Different scenarios of interest for the exploration of the effects of stochastic resetting in the long-term behavior of the $G/M/c$ queueing system. (a) When $x_{0}^{*}>(1-1/\sqrt{5})L$, the zone of interest for the initial position of the search process is $((1-1/\sqrt{5})L,x_{0}^{*})$. (b) When $x_{0}^{*}<(1-1/\sqrt{5})L$, the zone of interest for the initial position of the search process is $(x_{0}^{*},(1-1/\sqrt{5})L)$.