Performance Measures Analysis of M/M/m/K/N Systems with Finite Customer Population

Abstract

This article provides an analytic solution for performance measures of finite channel capacity m queuing system M/M/m/K/N. Queuing system with a finite number of call sources size N and restricted storage K is investigated. Markov chains are used for the performance measures evaluation. The solution to our queuing system with losses, which is the birth-death system, was obtained by applying the finite Markov chains. There is used system with primitive call flow model. Interarrival time and service time represented using exponential distribution. Packets arriving to find K+m already in the system are lost and return immediately to the arriving state as if they had just completed service. The most important performance measures of queuing system are defined in an analytic form, such as delay probability P(W>0), mean value of waiting time mean value of queue length E[Nq], call loss probability B, service channel utilization ym, and etc. Taken queuing models for M/M/m/K/N systems are very elegant in analysis and provide good closed results. Some of these models were extensively used in designing telephone networks, but however they are very useful in modeling of packets or messages services in data networks. We describe the Markov birth-death model used for calculating characteristics of asymmetric queuing system M/M/2/K/N, when the fastest free channel is selected. So we examine the system linked to two channels with different packets transmission rates μ1 > μ2. Some calculation results are presented in diagram form. Ill. 15, bibl. 7 (in English; summaries in English, Russian and Lithuanian).