In Section 3.1 we use the arrival times \(\{A_{k}\}\) and the service times \(\{S_k\}\) to construct the waiting times \(\{\W_k\}\), the sojourn times \(\{\J_k\}\), the arrival process \(\{A(t)\}\), the departure process \(\{D(t)\}\), and the number in the system \(\{\L(t)\}\). If the queueing system is rate stable, several long-run average performance measures¹¹ Also known as `steady-state' or `stationary' limits. along sample paths exist almost surely.²² Proving the existence of these limits below requires substantial mathematics; see S. Asmussen (2003), Applied Probability and Queues for details. Here we assume these limits exist.

As the next example demonstrates, we need to distinguish between sampling the system at arrival times and sampling it over time. Suppose that jobs arrive exactly at the start of an hour and require 59 minutes of service time. If we sample the server occupancy at arrival epochs, the server is always idle, whereas over time it is almost always busy. Thus, what jobs see upon arrival generally differs from what the server perceives.

One set of measures focuses on what customers perceive at the moment of arrival.³³ Such statistics are often described as `as seen by arrivals'. The expectation and distribution function⁴⁴ This is simply counting. of the waiting time are obtained from, respectively,

\begin{align*} \E{\W} &= \lim_{n\to\infty} \frac 1 n\sum_{k=1}^n \W_{k}, & \P{\W \leq x} &= \lim_{n\to\infty} \frac 1n\sum_{k=1}^n \1{\W_k\leq x}. \tag{4.2.1} \end{align*}

For the sojourn time \(\J\), we use similar definitions.

Since \(L(A_k-)\) denotes the number of jobs in the system just before the \(k\)th arrival⁵⁵ Note the \(A_k-\); since \(L(t)\) is right-continuous, we focus on what an arrival sees just before its arrival. , the average and distribution function of the number of jobs in the system as seen by arrivals are given by

\begin{align*} \E\L &= \lim_{n\to\infty}\frac 1 n \sum_{k=1}^n L(A_k-), & \P{\L \leq m} &= \lim_{n\to\infty} \frac 1n\sum_{k=1}^n \1{\L(A_{k}-)\leq m}. \tag{4.2.2} \end{align*}

A second set of performance measures follows by tracking the system's behavior over time and taking the time-average.⁶⁶ These performance measures are `as seen by the server'. Since at time \(s\) the number in the system is \(L(s) = A(s) - D(s)\), we define the time-average number of jobs⁷⁷ Although the symbol is the same, this expectation need not be the same as Eq. (4.2.2). and the time-average fraction of time the system contains at most \(m\) jobs, respectively, as

\begin{align*} \E\L &= \lim_{t\to\infty} \frac 1 t\int_0^t L(s) \d s, & \P{\L\leq m} &=\lim_{t\to\infty} \frac 1 t\int_0^t \1{\L(s)\leq m} \d s. \tag{4.2.3} \end{align*}