Implementation of Markovian Queueing Network Model with Multiple Closed Chains

Abstract

Mathematical strategy portrays the performance evaluation of computer and communication system and it deals with the stochastic properties of the multiclass Markovian queueing system with class-dependent and server-dependent service times. An algorithm is designed where the job transitions are characterized by more than one closed Markov chain. Generating functions are implemented to derive closed form of solutions and product form solution with the parameters such as stability, normalizations constant and marginal distributions. For such a system with N servers and L chains, the solutions are considerably more complicated than those for the systems with one sub-chain only. In Multi-class queueing network, a job moves from a queue to another queue with some probability after getting a service. A multiple class of customer could be open or closed where each class has its own set of queueing parameters. These parameters are obtained by analyzing each station in isolation under the assumption that the arrival process of each class is a state-dependent Markovian process along with different service time distributions. An algorithmic approach is implemented from the generating function representation for the general class of Networks. Based on the algorithmic approach it is proved that how open and closed sub-chain interact with each other in such system. Specifically, computation techniques are provided for the calculation of the Markovian model for multiple chains and it is shown that these algorithms converge exponentially fast.