A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example

In this study a two stage queueing model is analyzed. At first stage there is a single server having exponential service time with parameter μ_1 and no waiting is allowed in front of this server. There are two parallel phase-type servers at second stage and these parallel servers have exponential service time with parameter μ_2. Arrivals to this system is Poisson with parameter λ. An arriving customer to this system has service if the server at first stage is available or leaves the system if the server is busy where the first loss occurs. After having service in first stage the customer proceeds to the second stage, if both of the phase-type parallel servers in second stage are available the customer chooses one of these servers with probability 0.50 or leaves the system if any of these servers in second stage is busy so the second loss occurs. A customer who has service at both stages leaves the system. The number of customers in this model is represented by a 3-diamensional Markov chain and Kolmogorov differential equations are obtained. After that mean number of customers and mean waiting time in the system is obtained by limit probabilities. We have shown that the customer numbers at first and second stages are dependent to each other. The numerical analysis of obtained performance measures are shown by a numeric example. Finally the graphs of loss probabilities and measure of performances given for some values of arrival rate λ and the service parameters.