The Output of a Queueing System

Abstract

The output of a queueing system is defined as the number of elements leaving the service facility during a given interval of time after completion of their service. This random variable is written σ(t, t + z) if (t, t + z] is the relevant interval of time, and $$\begin{array}{*{20}{c}} {{{q}_{m}}(z|t) = Prob\{ \sigma (t,t + z) = m\} ,} & {m \in \{ 0,1,2, \ldots \} } \\ \end{array}$$ (1.01) denotes its distribution. A knowledge of this distribution can claim an autonomous interest; besides (1.01) is of considerable importance in the discussion of decision problems which arise typically in queueing situations. The length of the queue and thereby the waiting time of an element depend on the capacity of the service facility. The capacity of the service facility is dependent on the number of channels and on the service process. For the purposes of the present discussion it is sufficient to consider a service facility which consists of exactly one channel. If the elements and the service facility belong to the same organization costs may be used as an appropriate performance criterion. Let K(t, t + z) denote the total cost of the queueing system in the relevant interval, then $$\kappa (t,t + z) = {\text{cost}} {\text{of}} {\text{service}} {\text{ + }} {\text{waiting}} {\text{costs}}{\text{.}}$$ (1.02)