Page 59 - A First Course In Stochastic Models

50 RENEWAL-REWARD PROCESSES

n
Assuming that the failure probability q is close to 0, the approximations (1−q) ≈
1 − nq and e −nq ≈ 1 − nq apply. Thus we find that
P {U > t} ≈ e −tq/T , t ≥ 0.
In other words, the time until the first system failure is approximately exponentially
distributed.

2.3 THE FORMULA OF LITTLE

To introduce the formula of Little, we consider first two illustrative examples. In
the first example a hospital admits on average 25 new patients per day. A patient
stays on average 3 days in the hospital. What is the average number of occupied
beds? Let λ = 25 denote the average number of new patients who are admitted
per day, W = 3 the average number of days a patient stays in the hospital and L
the average number of occupied beds. Then L = λW = 25 × 3 = 75 beds. In the
second example a specialist shop sells on average 100 bottles of a famous Mexican
premium beer per week. The shop has on average 250 bottles in inventory. What is
the average number of weeks that a bottle is kept in inventory? Let λ = 100 denote
the average demand per week, L = 250 the average number of bottles kept in stock
and W the average number of weeks that a bottle is kept in stock. Then the answer
is W = L/λ = 250/100 = 2.5 weeks. These examples illustrate Little’s formula
L = λW. The formula of Little is a ‘law of nature’ that applies to almost any
type of queueing system. It relates long-run averages such as the long-run average
number of customers in a queue (system) and the long-run average amount of time
spent per customer in the queue (system). A queueing system is described by the
arrival process of customers, the service facility and the service discipline, to name
the most important elements. In formulating the law of Little, there is no need to
specify those basic elements. For didactical reasons, however, it is convenient to
distinguish between queueing systems with infinite queue capacity and queueing
systems with finite queue capacity.

Infinite-capacity queues
Consider a queueing system with infinite queue capacity, that is, every arriving
customer is allowed to wait until service can be provided. Define the following
random variables:

L q (t) = the number of customers in the queue at time t
(excluding those in service),
L(t) = the number of customers in the system at time t
(including those in service),