A Finite Capacity Queueing Model with Primary and Secondary Jobs

We consider a queueing model consisting of a primary buffer of capacity K and a secondary buffer of capacity N. Upon completion of the primary service, customers enter the secondary buffer for one additional service with probability p, 0 ⩽ p ⩽ 1, or leave the system with probability q = 1 − p. The server exhaustively processes the jobs from the secondary buffer whenever the primary buffer becomes empty or the secondary buffer fills up. The service times of the primary customers are assumed to follow a phase type distribution and those of the secondary jobs follow an exponential distribution with parameter μ2. Customers arrive to the system according to a homogeneous Poisson process of rate λ. Using Markov chain theory, we discuss numerically stable algorithms to compute various steady-state system performance measures such as throughput, the fraction of time the server is busy, the fraction of time a primary customer is blocked, the fraction of time new customers are lost, expected numbers of customers waiting in the primary and secondary buffers. A number of numerical examples are presented