Markovian single server retrial queues with two way communication

Abstract

In this paper, we 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 derive explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server as well as for the partial generating functions and the partial factorial moments. We also derive recursive formulae for the stationary distribution and the partial factorial moments for which both symbolic and numerical algorithms can be implemented. We further present a cost model from which the optimal rate of outgoing calls is explicitly obtained.