Page 61 - A First Course In Stochastic Models

52 RENEWAL-REWARD PROCESSES

formula of Little is easiest understood (and reconstructed) when imagining that each
customer pays money to the system manager according to some non-discrimination
rule. Then it is intuitively obvious that

the long-run average reward per time unit earned by the system
= (the long-run average arrival rate of paying customers) (2.3.3)
× (the long-run average amount received per paying customer).

In regenerative queueing processes this relation can often be directly proved by
using the renewal-reward theorem; see Exercise 2.26. Taking the ‘money principle’
(2.3.3) as starting point, it is easy to reproduce various representations of Little’s
law. To obtain (2.3.1), imagine that each customer pays $1 per time unit while
waiting in queue. Then the long-run average amount received per customer equals
the long-run average time in queue per customer (= W q ). On the other hand, the
system manager receives $j for each time unit that there are j customers waiting
in queue. Hence the long-run average reward earned per time unit by the system
manager equals the long-run average number of customers waiting in queue (= L q ).
The average arrival rate of paying customers is obviously given by λ. Applying the
relation (2.3.3) gives next the formula (2.3.1). The formula (2.3.2) can be seen by
a very similar reasoning: imagine that each customer pays $1 per time unit while
in the system. Another interesting relation arises by imagining that each customer
pays $1 per time unit while in service. Denoting by E(S) the long-run average
service time per customer, it follows that
the long-run average number of customers in service = λE(S). (2.3.4)

If each customer requires only one server and each server can handle only one
customer at a time, this relation leads to

the long-run average number of busy servers = λE(S). (2.3.5)

Finite-capacity queues
Assume now there is a maximum on the number of customers allowed in the
system. In other words, there are only a finite number of waiting places and each
arriving customer finding all waiting places occupied is turned away. It is assumed
that a rejected customer has no further influence on the system. Let the rejection
probability P rej be defined by

P rej = the long-run fraction of customers who are turned away,
assuming that this long-run fraction is well defined. The random variables L(t),
L q (t), D n and U n are defined as before, except that D n and U n now refer to the
queueing time and sojourn time of the nth accepted customer. The constants W q
and W now represent the long-run average queueing time per accepted customer