A Two-Stage Tandem Queue with Specialist Servers

Abstract

The queueing system considered consist of two multi-server stations in series. Customers arrive according to a Markovian Arrival Process to an infinite capacity queue at the first station. There are c servers who provide identical exponentially distributed service at the first station. A customer at the head of the queue can enter into service if any one of the servers at the first stage is idle. At the second station there are N identical servers called specialist servers . The service time distribution of specialist severs is phase type. There is a finite buffer in between the two stations. On completion of service at first stage, a customer needs service at the second station with probability p or leaves the system with probability 1 − p. In the former case, the customer joins the second station for service in case the waiting room is not full, else he is lost to the system. A customer in the finite buffer can enter into service if at least one of these servers is free. Stability of the system is established and stationary distribution is obtained using Matrix Analytic Methods. We compute distribution of waiting time of customers in the first queue, the mean number of customers lost due to capacity restriction of the waiting space of the second station and the mean waiting time of customers who get into service at the second station. An optimization problem on the capacity of second waiting station is also analyzed.