Markovian retrial queues with two way communication

Abstract

In this paper, we first consider single server retrial queues with two way communication. Ingoing calls arrive at the server according to a Poisson process. Service times of these calls follow an exponential distribution. If the server is idle, it starts making an outgoing call in an exponentially distributed time. The duration of outgoing calls follows another exponential distribution. An ingoing arriving call that finds the server being busy joins an orbit and retries to enter the server after some exponentially distributed time. For this model, we present an extensive study in which we derive explicit expressions for the joint stationary distribution of the number of ingoing calls in the orbit and the state of the server, the partial factorial moments as well as their generating functions. Furthermore, we obtain asymptotic formulae for the joint stationary distribution and the factorial moments. We then extend the study to multiserver retrial queues with two way communication for which a necessary and sufficient condition for the stability, an explicit formula for average number of ingoing calls in the servers and a level-dependent quasi-birth-and-death process are derived.