Performance Analysis of Multi-Server Queueing System Operating under Control of a Random Environment

Abstract

Since the early 1900th the Erlang multi-server queueing systems with losses ( B -model or 0 M/M/N/ system) and with an infinite size buffer ( C -model or M/M/N system) provided good mathematical tools for capacity planning and performance evaluation in the classic telephone networks for many years. Good quality of the loss probability forecasting in real world networks based on the formulas obtained for the 0 M/M/N/ system and the delay prediction based on the formula obtained for the M/M/N system was a rather surprising because the requirement that inter-arrival and service times have an exponential distribution, which is imposed in the 0 M/M/N/ and M/M/N models, seems to be too strict. The interest of mathematicians to the fact of good matching of the calculated under debatable assumptions characteristics to their measured value in real world systems have lead to the following two results. By efforts of many mathematicians (A. Ya. Khinchin and B. I. Grigelionis first of all), it was proved that the superposition of a large number of independent flows having uniformly small intensity approaches to the stationary Poisson input when the number of the superposed inputs tends to infinity. It explains the fact that the flows in classic telephone networks (where flows are composed by small individual flows from independent subscribers) have the exponentially distributed inter-arrival times. Concerning the service time distribution, situation was more complicated. The real-life measurements have shown that the service (conversation) time can not be well approximated by means of the exponentially distributed random variable. So, due to the good matching of results obtained for the 0 M/M/N/ queue performance characteristics to characteristics of real systems modelled by such a queue, the hypothesis has arisen that the stationary state distribution in the 0 M/M/N/ queue is the same as the one in the 0 M/G/N/ 15