Page 348 - A First Course In Stochastic Models

BASIC CONCEPTS 343

The random variable S denotes the service time of a customer. Note that λ =
1/E(A), where the random variable A denotes the interarrival time. An important
quantity is the offered load, which is defined as λE(S). This dimensionless quan-
tity indicates the average amount of work that is offered to the system per time
unit. In the GI/G/c queue (N = ∞) the offered load should be less than the
maximum load the system can handle, otherwise infinitely long queues ultimately
build up. Letting
λE(S)
ρ = ,
c
the following assumption is made.

Assumption 9.1.1 For the GI/G/c queue the load factor ρ is below 1.
It will be seen below that in the GI/G/c queue the quantity ρ can be interpreted
as the long-run fraction of time that a given server is busy. This explains why ρ is
called the server utilization in the GI/G/c queue. In addition to Assumption 9.1.1
we make the following technical assumption.
Assumption 9.1.2 (a) The interarrival-time distribution A(t) or the service-time
distribution B(t) has a positive density on some interval.
(b) The probability that the interarrival time A is larger than the service time S
is positive.

Define a cycle as the time elapsed between two consecutive arrivals that find
the system empty. Then, under Assumptions 9.1.1 and 9.1.2, it can be shown that
the expected value of the cycle length is always finite. The proof of this result is
quite deep and is not given here; see Wolff (1989). Let us now define the following
random variables:
L(t) = the number of customers in the system at time t (including those
in service),
L q (t) = the number of customers in the queue at time t (excluding those
in service),
D n = the amount of time spent by the nth accepted customer in the
queue (excluding service time),
U n = the amount of time spent by the nth accepted customer in the
system (including service time).

The continuous-time stochastic process {L(t)} and {L q (t)} and the discrete-time
stochastic processes {D n } and {U n } are all regenerative. The regeneration epochs are
the epochs at which an arriving customer finds the system empty. The regeneration
cycles have finite means. Thus the following long-run averages exist:
1 t
L = lim L(u) du (the long-run average number in system)
t→∞ t 0