Scheme for determining the jump priorities in queuing systems with heteregeneous servers

Abstract

In this paper, we propose the mathematical model of a queuing system with heteregenous servers, calls of different types and jump priorities. Both type of calls are formed Poisson flows and they are waits in finite separate buffers at heterogeneous servers. Calls of high priority are served by fast server while calls of low priority are servered in slow server. Jump priorities are defined rules for transfer of low priority calls to buffer of high priority calls. If upon arrival of a high priority call (low priority call) there is one free position in appropriate buffer, then it occupies it; otherwise, the call is lost. The distribution functions of channel occupation time by heterogeneous calls are exponential with different average values. It is shown that the mathematical model of the system is a certain two-dimensional Markov chain with a finite set of states. An algorithm is proposed for constructing the generating matrix of this chain and it is proved that this chain is irreducible and therefore there exists stationary probability distribution of the states of this Markov chain. An explicit form of the balance equations is obtained. Explicit formulas have been developed for calculating the characteristics of the system under study. Main characteristics are call loss probabilities of each type flow, average length of the both queue of calls different types and their average waiting times in queue. The developed formulas allow us to conduct numerical experiments in order to study the characteristics of the system relative to changes in its parameters, as well as solve the problems of their optimization with respect to the selected quality criterion for the functioning of the system.