TY - JOUR
AU - Tarasov, V.N.
AU - Bakhareva, N. F.
PY - 2019/04/16
Y2 - 2022/05/21
TI - RESEARCH OF QUEUEING SYSTEMS WITH SHIFTED ERLANGIAN AND EXPONENTIAL INPUT DISTRIBUTIONS
JF - Radio Electronics, Computer Science, Control
JA - RIC
VL - 0
IS - 1
SE - Mathematical and computer modelling
DO - 10.15588/1607-3274-2019-1-7
UR - http://ric.zntu.edu.ua/article/view/163430
SP -
AB - <p>Context. In queuing theory, the study of G/G/1 systems is particularly relevant due to the fact that until now there is no solution in the final form<br />in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary systems with<br />erlangian and exponential input distributions and for the same systems with shifted distributions is considered.<br />Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing<br />systems of type G/G/1 with conventional and shifted erlangian and exponential input distributions.<br />Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows<br />one to obtain a solution for average the waiting time for systems under consideration in a closed form. The method of spectral decomposition of<br />the solution of Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained,<br />the well-known method of moments of probability theory is used.<br />Results. The spectral decompositions of the solution of the Lindley integral equation for the three kinds of systems were first obtained with the<br />help of which the calculated expressions for the average waiting time in the queue for the above systems in a closed form were derived.<br />Conclusions. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration<br />turns them into systems with a delay with a shorter waiting time. This is due to the fact that the time shift operation reduces the coefficient of<br />variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time<br />of requirements is related to these coefficients of variation by a quadratic dependence. The system with erlangian input distributions of the second<br />order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service<br />time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope<br />of these systems. Similarly the situation and with the shifted exponential distributions is. In addition, the shifted exponential distribution contains two<br />parameters and allows one to approximate arbitrary distribution laws using the first two moments. This approach allows us to calculate the average<br />latency for these systems in mathematical packages for a wide range of traffic parameters. All other characteristics of the systems are derived from the<br />waiting time. The method of spectral decomposition of the solution of the Lindley integral equation for the systems under consideration makes it<br />possible to obtain a solution in a closed form and these solutions are published for the first time.</p>
ER -