The multi-server retrial system with Bernoulli feedback and starting failures

Abstract

In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probability p or to join the retrial orbit again for another service with complementary probability 1−p. We analyse this queueing system as a quasi-birth–death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model.

Keywords:

• Bernoulli feedback
• cost
• matrix-geometric method
• retrial queue
• starting failure

2010 AMS Subject Classifications::

• 60K25
• 68M20
• 90B22
• 90C53

Acknowledgements

This work was partly supported by the National Science Council of Republic of China under Contract No. NSC 99-2221-E-025-002-MY3.