Analysis of queuing system with calls of various width

Abstract

In this paper, we study the mathematical model of a multi-channel queuing system in which Poisson flows of voice and data calls are served. It is assumed that narrow-band voice calls require only one free channel to service, while at the same time, number of free channels that required to service broadband data calls is b, b>1. The system adopts a widely used in practice, fully accessible access strategy to the common channel pool, according to which there are no differences between different types of speech and data calls. If upon arrival of a voice call (data call) there is one free channel (b>1 free channels), then it occupies free channel(s) of the system; 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 from any state of this chain in a finite number of steps it is possible to go to any other state. Using the Kolmogorov theorem, it is proved that the constructed two-dimensional Markov chain is reversible, and therefore the stationary probability distribution of the states of this Markov chain has a multiplicative form. An explicit form of the multiplicative representation is obtained. Explicit formulas have been developed for calculating the probabilities of the states of the Markov chain under study, and using these probabilities of states, explicit formulas have been proposed for calculating the indicators of service quality — call probabilities of each type calls and channel utilization. The developed formulas allow us to conduct numerical experiments in order to study the behavior of indicators of the quality of service 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.