Development of an Interval Method for Queue Analysis in Queueing Systems

The article considers the peculiarities of traffic of multiservice communication networks, its processing in queuing systems and interval methods of traffic analysis. As an analyzed random variable, the number of applications arriving in the system at constant time intervals equal to the mathematical expectation of application processing times is considered. The generalized Hinchin-Pollachek formula for average queue values occurring when processing flows of common kind requests is given. The generalized formula specifies the dependence of average queue values on the variance and correlation properties of a given random variable, at different values of the system load factor. Group Poisson flows are given as an example of a flow for which there is no correlation component. The formula for group Poisson flows of requests is considered. For such flows, the average queue value, for a given system load, is completely determined by the variance of the number of applications at their service intervals. It is shown that the processing of applications is cyclic, and there is always no queue at the last interval of the active part of each cycle. For a stationary flow of applications with finite mathematical expectation and variance, formulas are given from which it follows that when incoming applications are placed at time intervals distant from the beginning of each cycle, their contribution to the average queue value decreases. The queue reaches its maximum value if all applications are concentrated at the beginning of cycles. The development of the interval method of queue analysis was the determination of the number of requests in the time intervals between neighboring served requests leaving the queue. The obtained relations are fundamental because they show that if the conditions of stationarity, ergodicity and convergence of the covariance function are satisfied, the conditional average values of queues in a single-channel queuing system are completely determined by the variance and the sum of covariances of the number of applications received in the time intervals between neighboring serviced applications.