Comparative Analysis of the Queueing Systems E2/M/1 and M/E2/1 with Shifted Distribution Laws

Abstract

The objective of this work is to the output the solution in closed form for the average waiting time in a queue for queueing systems with shifted Erlangian and exponential input distributions and comparative analysis of their results with the results for conventional systems. As is known from queuing theory, the average waiting time for requirements in a queue is the main characteristic of a queuing system. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows one to obtain a solution for average the waiting time for systems under consideration in a closed form. The method of spectral decomposition of 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, the well-known method of moments of probability theory is used. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration turns them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the coefficient of 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 of requirements is related to these coefficients of variation by a quadratic dependence. The obtained results expand and supplement the well-known incomplete formula of the queuing theory for the average waiting time of requirements in the queue for G/G/1 systems.