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

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.