Optimization of loss probability in the GI/M/n/0 queueing model with heterogeneous servers

This study is mainly concerned with the finite-capacity queueing system with recurrent input, n heterogeneous servers, and no waiting line represented by GI/M/n/0. The service discipline is addressed in two different ways. Firstly, customers choose only one server from the empty servers with equal probability. Secondly, customers choose the server with the lowest index number among the empty servers with probability 1. In both cases, when all servers are busy, customers depart from the system without taking any service. These customers are called `lost customers? and the flows of lost customers are called `stream of overflows?.The queueing model GI/M/n/0 with heterogeneous servers is analyzed using semi-Markov process. The semi-Markov process representation of the system is described and the kernel functions of semi-Markov process are derived. An implementation of this formula is performed for the queueing model GI/M/3/0 with heterogeneous servers. Using the kernels of semi-Markov process, one-step transition probabilities, and steady-state probabilities are obtained for the related queueing model.The stream of overflows is analyzed for the queueing model GI/M/n/0 with heterogeneous servers, the Laplace-Stieltjes transform of the distribution of the time between overflows is obtained and the loss probability of customers is formulated. An implementation of this formula is performed for the queueing model GI/M/2/0 with heterogeneous servers, and the loss probability of customers is computed.It becomes computationally intractable to compute the exact solution of loss probability, besides it is impossible to minimize the loss probability according to distribution of arrival process as the number of servers increases. In this respect a quite extensive simulation study is performed and the loss probability is computed for different distributions of interarrival times and different service disciplines. The conditions in which the loss probability is minimum are determined by simulation optimization.Keywords: Semi-Markov process, Laplace-Stieltjes transform, loss probability, stream of overflows, queueing, simulation.