Performance Analysis of an Unreliable $M/G/1$ Retrial Queue with Two-way Communication

TL;DR

This paper analyzes the stationary behavior of an unreliable M/G/1 retrial queue with two-way communication, considering server failures, and derives stability conditions and queue length distributions.

Contribution

It introduces a novel analysis of an M/G/1 retrial queue with server failures and two-way communication, providing stability conditions and distribution formulas.

Findings

Derived stability condition for the queue.

Obtained probability generating functions for queue length.

Numerical evaluation of system size and server availability.

Abstract

Efficient use of call center operators through technological innovations more often come at the expense of added operation management issues. In this paper, the stationary characteristics of an $M/G/1$ retrial queue is investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. The incoming calls arriving at the server follow a Poisson arrival process, while outgoing calls are made in an exponentially distributed time. On finding the server unavailable (either busy or temporarily broken down), incoming calls intrinsically join the virtual orbit from which they re-attempt for service at exponentially distributed time intervals. The system stability condition along with probability generating functions for the joint queue length distribution of the number of calls in the orbit and the state of the server are derived and evaluated numerically in the context of mean system size, server availability, failure frequency and orbit waiting time.