Page 290 - Probability, Random Variables and Random Processes

QUEUEING THEORY [CHAP. 9

L: the average number of customers in the system
L,: the average number of customers waiting in the queue
L,: the average number of customers in service
W: the average amount of time that a customer spends in the system
W,: the average amount of time that a customer spends waiting in the queue
W,: the average amount of time that a customer spends in service
Many useful relationships between the above and other quantities of interest can be obtained by
using the following basic cost identity:
Assume that entering customers are required to pay an entrance fee (according to some rule) to
the system. Then we have
Average rate at which the system earns = A, x average amount an entering customer (9.1)
pays where 1, is the average arrival rate of entering customers
X(t)
A, = lim 7
and X(t) denotes the number of customer arrivals by time t.
If we assume that each customer pays $1 per unit time while in the system, Eq. (9.1) yields
L=R,W (9.4
Equation (9.2) is sometimes known as Little's formula.
Similarly, if we assume that each customer pays $1 per unit time while in the queue, Eq. (9.1)
yields
Lq = A, wq (9.3)

If we assume that each customer pays $1 per unit time while in service, Eq. (9.1) yields

Note that Eqs. (9.2) to (9.4) are valid for almost all queueing systems, regardless of the arrival process,
the number of servers, or queueing discipline.

9.3 BIRTH-DEATH PROCESS
We say that the queueing system is in state S, if there are n customers in the system, including
those being served. Let N(t) be the Markov process that takes on the value n when the queueing
system is in state S, with the following assumptions:
1. If the system is in state S,, it can make transitions only to S,-, or S,+ , , n 2 1 ; that is, either a
customer completes service and leaves the system or, while the present customer is still being
serviced, another customer arrives at the system ; from So , the next state can only be S, .
2. If the system is in state S, at time t, the probability of a transition to Sn+, in the time interval
(t, t + At) is an At. We refer to a, as the arrival parameter (or the birth parameter).
3. If the system is in state S, at time t, the probability of a transition to S,-, in the time interval
(t, t + At) is dn At. We refer to d, as the departure parameter (or the death parameter).
The process N(t) is sometimes referred to as the birth-death process.
Let p,(t) be the probability that the queueing system is in state S, at time t; that is,

Then we have the following fundamental recursive equations for N(t) (Prob. 9.2):