In this section we define the quantities and introduce the notation that will be used in the rest of the chapter. We do this by focusing on a specific queueing system and beginning to count at some instant t = 0 the number of users who arrive at the system.
    Let us concentrate on the ith user to arrive at that system after we begin our counting process. Three important "events" can be identified with respect to this ith user: the arrival of the user at the queueing system, the beginning of service to the user, and the completion of service to the user. We shall denote the instants when these three events occur as ta(i), tb(i), and tc(i), respectively (with a standing for "arrival," b for "beginning" of service, and c for "completion" of service).

We can now define the following quantities:

x(i) ta(i) - ta(i - 1) = ith interarrival time

s(i) ta(i) - ta(i) = service time for the ith (in terms of order of arrival to the system) user

Wq(i) tb(i) - ta(i) = waiting time in the queue for the ith user

W(i) tc(i) - ta(i) = total time spent in the queueing system by the ith user ("system occupancy time")

Obviously, from the foregoing definitions we also have

W(i) = Wq(i) + s(i)     (4.1)

In general, the interarrival and service times are random variables, X(i) and S(i), with pdf's fX(i)(x) and fS(t)(s), respectively.

[The use of the term "probability density function" is made here in a general sense. The random variables X(i) and S(i), i = 1, 2, 3... can be discrete, continuous, or mixed.]

We shall assume from now on that, unless otherwise stated, the interarrival-time random variables X(i) are independent and identically distributed [i.e., fX(1)(x) = fX(2)(x) = ... = fX(x)]. Similarly, we shall assume that the service times S(i) are independent and identically distributed with fS(1)(s) = fS(2)(s) = ... = fS(s). The expected values of random variables X and S appear so frequently in the analysis of queueing systems that special symbols have been adopted for them:


(1/ ) E[X]             (4.2)

(1/) E[X]              (4.3)

    In words, represents the expected number (or the "rate") of user arrivals at the queueing system per unit of time. Similarly, ju is the expected number of service completions per unit of time when a server is working continuously. Note that when all m parallel and identical servers in a queueing system are working simultaneously, the rate of service completions is equal to mp. 
    Now if the queueing system is allowed to operate for a long time, it can be expected, under certain conditions, to reach an equilibrium ("steady state"). Without specifying what these conditions are, it is reasonable to assume that the system occupancy times, w(i), and waiting times, wq(i), for large values of i, will tend to become samples of two random variables W and W, respectively, whose pdf's fW(w) and fq w(Wq) are independent of the order, i, of a user's arrival. We shall refer to fw(w) and fW q(wq) as the steadystate probability density functions for the system occupancy times and the waiting times of users, respectively. We shall then define the quantities

E[W] = E[W[i]] = expected system occupancy time
for a user under steady-state conditions
q E[Wq] = E[Wq[i]] = expected waiting time in queue for a user
under steady-state conditions

Rather than focus on user-related events at the queueing system [such as the ta(i), tb(i), tc(i), etc.], we also could have looked at the system at random points in time and defined the quantities

N (t) total number of users (including those in service) who are in the queueing system at time t

Nq(t) number of users waiting in the queue at time t

For large values of t and under the (yet unspecified) proper conditions, we can expect the distributions of variables N(t) and N,(t) to approach equilibrium (steady-state) pmf's pN(n) and pNq(n).

We shall then use the symboIs

We shall also define here the quantity

From the definition of the utilization ratio (the reason for its name will become obvious later), it is clear that, for a single-server queueing system,

whereas for a m-server queueing system,

1 The use of the term "probability density function" is made here in a general sense. The random variables X(i) and S(i), i = 1,2,3, ... can be discrete, continuous, or mixed.