Queueing models with random resetting

Abstract

We introduce and study some queueing models with random resetting, including Markovian and non--Markovian models under the first-come first-served (FCFS) discipline. The Markovian models include 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. We derive expressions for standard performance measures such as the delay probability, expected queue length and waiting time, as well as probability of a customer completing service before resetting in the M/M/$r$ model, and probability of abandonment before service or resetting in the M/M/1+M model. 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.