Performance Analysis of a Two-Server Heterogeneous Retrial Queue with Threshold Policy

Abstract

In the paper we deal with a Markovian queueing system with two heterogeneous servers and constant retrial rate. The system operates under a threshold policy which prescribes the activation of the faster server whenever it is idle and a customer tries to occupy it. The slower server can be activated only when the number of waiting customers exceeds a threshold level. The dynamic behaviour of the system is described by a two-dimensional Markov process that can be seen as a quasi-birth-and-death process with infinitesimal matrix depending on the threshold. Using a matrix-geometric approach we perform a stationary analysis of the system and derive expressions for the Laplace transforms of the waiting time as well as arbitrary moments. Illustrative numerical results are presented for the threshold policy that minimizes the mean number of customers in the system and are compared with other heuristic control policies.

Key Words:

• Controllable queueing system
• retrial queue
• sojourn time distribution
• steady-state probabilities
• threshold control policy
• waiting time distribution

Additional information

Notes on contributors

Dmitry Efrosinin

Dmitry Efrosinin is University-Assistant for Research and Teaching in the Institute for Stochastics, Johannes Kepler University of Linz, Austria. He is a Research consultant at the Department of Probability theory and Math. Statistics, Peoples Friendship University, Russia. His main scientific interests are in queueing and reliability systems, optimization problems and structural properties of optimal control policies.

Janos Sztrik

Janos Sztrik is Professor in the Department of Informatics Systems and Networks, University of Debrecen, Hungary. He is a Member of J. Bolyai Mathematical Society, Budapest and London Mathematical Society. His research interests are in mathematical statistics, queueing theory and computer performance evaluation.