TY - JOUR
AU - Tarasov, V. N.
AU - Bakhareva, N. F.
PY - 2019/11/25
Y2 - 2022/05/26
TI - COMPARATIVE ANALYSIS OF TWO QUEUING SYSTEMS M/HE2/1 WITH ORDINARY AND WITH THE SHIFTED INPUT DISTRIBUTIONS
JF - Radio Electronics, Computer Science, Control
JA - RIC
VL - 0
IS - 4
SE - Mathematical and computer modelling
DO - 10.15588/1607-3274-2019-4-5
UR - http://ric.zntu.edu.ua/article/view/192089
SP - 50 - 58
AB - Context. In the queueing theory, studies of particular systems of the M/G/1 type are relevant in that they are still actively used in<br />the modern theory of teletraffic. The problem of finding a solution for the mean waiting time in a queue in the closed form of two<br />systems with ordinary and shifted exponential and hypererlangian input distributions is considered.<br />Objective. Obtaining a solution for the main system characteristic – for the average waiting time in a queue for two queuing systems<br />of type M/G/1 and G/G/1 with conventional and offset exponential and hypererlangian input distributions.<br />Method. To solve this problem, we use the classical method of spectral decomposition of the solution of the Lindley integral<br />equation. This method allows to obtain a solution for the average waiting time for the systems under consideration in closed form.<br />The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems<br />G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used.<br />Results. Spectral decompositions of the solution of an integral equation of Lindley for couple of systems by means of which<br />formulas for the average time of waiting in queue in the closed form are received. The shifted exponential distribution transforms the<br />system M/G/1 into the system G/G/1.<br />Conclusions. The spectral decompositions of the solution of the Lindley integral equation for the systems under consideration<br />are obtained and with their help, the formulas for the average waiting time in the queue for these systems in a closed form are derived.<br />These expressions expand and complement the known queuing theory formulas for the average waiting time for M/G/1 and<br />G/G/1 systems with arbitrary laws of input flow and service time distributions. This approach allows us to calculate the average latency<br />for these systems in mathematical packages for a wide range of traffic parameters. All other characteristics of the systems are<br />derived from the waiting time. In addition to the average waiting time, such an approach makes it possible to determine also moments<br />of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the<br />spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time. The method<br />of spectral decomposition of the solution of the Lindley integral equation for the systems under consideration makes it possible to<br />obtain a solution in a closed form and these solutions are published for the first time.
ER -