Retrial queue with unreliable server and second optional service under <i>K</i> types of Bernoulli vacations

Abstract

This paper discusses a single server retrial queue with two phases of service of which the second phase is optional. Further, the server is subjected to starting failures and takes K types of Bernoulli vacations. A necessary and sufficient condition for the system to be stable is investigated. The joint probability generating function of the system size and the server states are obtained using supplementary variable technique. Some of the existing results are deduced as special cases. Further, significant system performance measures such as mean system size, mean orbit size, server utilisation and probability of the orbit being empty are derived. The stochastic decomposition law is shown to hold good for this model also. Finally, extensive numerical illustrations are provided to showcase the effect of the parameters on the system performance measures.