4.3 DEFINING THE QUANTITIES OF INTEREST
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.
We can now define the following quantities:
Obviously, from the foregoing definitions we also have
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]
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.
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
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,