The System Point Method in Exponential Queues: A Level Crossing Approach

Abstract

The purpose of this paper is to describe the new System Point method for analyzing queues. It considers the stationary probability distribution of the waiting time in variations of the M/M/R queue with first come first served discipline for a large class of service mechanisms. It is shown that the stationary probability density function of the waiting time evaluated at w > 0 can be interpreted as the long run average of the number of times that the virtual wait becomes less than w, per unit time. Theorems are presented which establish this interpretation of the probability density function of the virtual waiting time in terms of point processes generated by level crossings in the state space. These theorems, in combination with the principle of stationary set balance, generate a system of model equations that can be written directly. In addition, the forms of these equations are often linear Volterra integral equations of the second kind with parameter, which yield direct analytical solutions. An analogous theorem is proved for a variant of M/G/1. Two illustrative examples are presented.

A continuous-review inventory system with lost sales and variable lead time

“…In this regard, we apply a system-point (SP) method of level-crossing theory (Brill and Posner [3,4]) to develop an inventory model in which the demand process is compound Poisson, and lead times are assumed to be independent and identically distributed (i.i.d.) random variables following Erlang and hyperexponential distributions.…”

Analysis Of A Production-Inventory System With Unreliable Production Facility

“…Using the generalized technique of the 'system point' in levelcrossings analysis (Brill and Posner, 1977;Brill and Posner, 1981), Posner and Berg derive an analytical formula from which a number of performance measures can be calculated.…”

Alternative Analysis of Finite-Time Probability Distributions of Renewal Theory

“…It provides a completely different perspective of the time-t quantities and suggests new avenues of research. References for level-crossing methods are Brill and Posner [16,17], Cohen [18,19], Brill [20,21] and others. Regenerative processes are discussed in Smith [5], Cohen [18], Sigman and Wolff [11], Asmussen [12], Tijms [13], and others.…”