Page 229 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes
Did u know? Different models in queueing theory are classified by using special notations
described initially in 1953 in the form (a/b/b). Later A.M. Lee in 1966 added the symbols
d and c to the Kendall notation. Now in the literature of queuing theory the standard
format used to describe the main characteristics of parallel queues is as follows:
{(a/b/c) = (d/c) }
Where a = arrivals distribution

b = service time
c = number of service channels (servers)
d = maximum number of customers allowed in the system
e = queue discipline

11.5 Model 1: (MM1) : ( / FIFO)

Assumptions

This model is based on the following assumptions:

1. The arrivals follow Poisson distribution, with a mean arrival rate .
2. The service time has exponential distribution, average service rate .
3. Arrivals are infinite population .
4. Customers are served on a First-in, First-out basis (FIFO).

5. There is only a single server.
System of Steady-state Equations

In this method, the question arises whether the service can meet the customer demand. This
depends on the values of  and .
If   , i.e., if arrival rate is greater than or equal to the service rate, the waiting line would
increase without limit. Therefore for a system to work, it is necessary that  < .
As indicated earlier, traffic intensity  =  / . This refers to the probability of time. The service
station is busy. We can say that, the probability that the system is idle or there are no customers
in the system, P = 1 – .
0
From this, the probability of having exactly one customer in the system is P =  P .
1 0
Likewise, the probability of having exactly 2 customers in the system would be
P =  P =  P
2
3 1 0
The probability of having exactly n customers in the system is
P =  P =  n(1-) = ( / ) P
n
n
n 0 0

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