The M/G/1 retrial queue with event-dependent arrivals

TL;DR

This paper introduces a versatile M/G/1 retrial queue model with event-dependent arrivals, analyzing stability, stationary distribution, asymptotics, and optimization to better understand complex service system dynamics.

Contribution

It presents a novel queue model with event-dependent arrivals and retrial times, providing stability conditions, explicit performance measures, and optimization methods.

Findings

Derived stability conditions for the model

Explicit stationary distribution at different epochs

Insights into the impact of event-dependency on performance

Abstract

We introduce a novel single-server queue with general retrial times and event-dependent arrivals. This is a versatile model for the study of service systems, in which the server needs a non-negligible time to retrieve waiting customers upon a service completion, while future arrivals depend on the last realized event. Such a model is motivated by the customers' behaviour in service systems where they decide to join based on the last realized event. We investigate the necessary and sufficient stability condition and derive the stationary distribution both at service completion epochs, and at an arbitrary epoch using the supplementary variable technique. We also study the asymptotic behaviour under high rate of retrials. Performance measures are explicitly derived and extensive numerical examples are performed to investigate the impact of event-dependency. Moreover, constrained optimisation problems are formulated and solved with ultimate goal to investigate the admission control problem.