M(1,2)/G(1,2)/1/(L1, L2) queue with Markov dependent shift in service

We consider a queueing process with a single server where the inputs form two different streams. Inputs to these two streams are independent Poisson processes of rates λ i and λ 2 respectively. Service times for the two streams have distributions G 1 and G 2 respectively. There is only one server. He selects units to be served according to a Markov Chain Rule from the two streams, only one customer is served at a time. We assume that the system can accommodate a maximum of L 1 of Type-1 and L 2 of Type-2 in the respective waiting lines. The time dependent and steady state system state probabilities are obtained. In addition from a given set of Markov chains determining the type of customer to be served the one that minimizes the total cost is identified. This model generalizes the alternating and priority queues.