Queueing models with random resetting
Dongzhou Huang, Guodong Pang, Izabella Stuhl, Yuri Suhov
TL;DR
This paper develops a unified analytical framework for queues with random resetting to zero, spanning Markovian (M/M/∞, M/M/$r$, M/M/1+M) and non-Markovian (GI/GI/1, GI/GI/$r$, GI/GI/∞) models. It derives explicit stationary distributions using partial balance equations, generating functions, and Wiener–Hopf equations, and introduces modified Lindley and Kiefer–Wolfowitz recursions together with an operator calculus that expresses stationary laws as convergent series. The resetting mechanism yields positive recurrence and tractable, interpretable structures (Poisson-like below a threshold, geometric above it) and offers new results for non-Markovian queues under arrival-time resets. By connecting clearing/disaster-queue literature with modern recurrence and operator techniques, the work provides versatile tools for performance analysis under random disruptions and suggests pathways for extensions to general reset rules and diffusion approximations.
Abstract
We introduce and study some queueing models with random resetting, including Markovian and non--Markovian models. The Markovian models include M/M/$\infty$, M/M/r and M/M/1+M queues with random resetting, in which a continuous-time Markov chain is formulated, with transitions including a resetting to state zero in addition to arrivals and services. We explicitly characterize the stationary distributions of the queueing processes in these models by using parting balance equations. Although the stationary distribution for M/M/$\infty$ queue with resetting has been previously derived in the literature, we obtain an alternative and more interpretable expression by a different approach. That provides useful insights for the analysis of M/M/r and M/M/1+M queues with resetting under the first-come first-served (FCFS) discipline. The non--Markovian models include GI/GI/1, GI/GI/$r$ and GI/GI/$\infty$ queues with random resetting to state zero at arrival times. For GI/GI/1 and GI/GI/$r$ queues under the FCFS discipline, we introduce modified Lindley and Kiefer--Wolfowitz recursions, respectively. Using an operator representation for these recursions, we characterize the stationary distributions via convergent series, as solutions to the modified Wiener--Hopf equations. For GI/GI/$\infty$ queues with resettings, we utilize a version of the Kiefer--Wolfowitz recursion, and also characterize the corresponding stationary distribution.