Nonstationary moments for queuing systems

TL;DR

This paper proves the exponential convergence of the non-stationary moments of virtual waiting time in M/G/1 and M/M/1 queueing systems to their stationary values, under certain conditions, using regenerative process analysis.

Contribution

It establishes the exponential convergence of virtual waiting time moments in non-stationary queueing systems, extending results to M/G/1 and M/M/1 models with explicit formulas.

Findings

Exponential convergence of virtual waiting time moments proven.

Results applicable to M/G/1 and M/M/1 queueing systems.

Uses properties of Bessel functions for analysis.

Abstract

In this paper, we prove the exponential convergence of the non-stationary moment of a random variable that defines the virtual waiting time in the mass service system M\G\1\ $\infty $, where the distribution of the service time satisfies the Kramer condition and system load factor $\rho < 1$, to the value of the virtual waiting time in the stationary mode in this system. We consider the virtual waiting time as a regenerative process, which regenerative periods is the sum of busy period and period of time, when the system is free. We show, that the busy period in such system satisfies the Kramer condition, so, period of the virtual waiting time too. Using this, we prof the the exponential convergence of the non-stationary moment of the virtual waiting time distribution $ \phi (t)$ to its value in the stationary mode. Also, we show, that the results from the general case is still true for the M\M\1\ $\infty $ system as a special case of the M\G\1\ $\infty$. Here we use explicit expression for the virtual waiting time distribution in such system and some Bessel functions properties to get the same convergence.