Single Server Markovian Retrial Queues with Multiple Types of Outgoing Calls

We consider single server Markovian retrial queues with mul-tiple types of outgoing calls. Incoming calls arrive at the server according to a Poisson process. The service time of an incoming call follows the exponential distribution. In-coming calls that nd the server busy upon arrival join an orbit and retry after some exponentially distributed time. On the other hand, the server makes an outgoing call after some exponentially distributed idle time. We assume that there are multiple types of outgoing calls whose durations follow the exponential distribution with dierent parame-ters. For this model, we obtain explicit expressions for the joint stationary distribution of the number of calls in the or-bit and the state of the server using the generating function approach. We also obtain simple asymptotic and recursive formulae for the joint stationary distribution. We show a stochastic decomposition property where we prove that the total number of incoming calls in the system (server and orbit) can be expressed by the sum of three independent random variables which have a clear physical meaning. 1