Page 226 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 11: Queuing Theory

1. Probability that an arrival is observed during a small time interval (say of length v) is Notes
proportional to the length of interval. Let the proportionality constant be , so that the
probability is  v.
2. Probability of two or more arrivals in such a small interval is zero.
3. Number of arrivals in any time interval is independent of the number in non-
overlapping time interval.
These assumptions may be combined to yield what probability distributions are likely to be,
under Poisson distribution with exactly n customers in the system.

Suppose function P is defined as follows:
P (n customers during period t) = the probability that n arrivals will be observed in a time
interval of length t

  

then, P (n, t) = (n = 0, 1, 2,……………) (1)
This is the Poisson probability distribution for the discrete random variable n, the number of
arrivals, where the length of time interval, t is assumed to be given. This situation in queuing
theory is called Poisson arrivals. Since the arrivals alone are considered (not departures), it is
called a pure birth process.
The time between successive arrivals is called inter-arrival time. In the case where the number
of arrivals in a given time interval has Poisson distribution, inter-arrival times can be shown to
have the exponential distribution. If the inter-arrival times are independent random variables,
they must follow an exponential distribution with density f(t) where,

f (t) = e (t > 0) (2)
–t
Thus for Poisson arrivals at the constant rate  per unit, the time between successive arrivals
(inter-arrival time) has the exponential distribution. The average Inter - arrival time is denoted
by .
By integration, it can be shown that E(t) =  (3)
If the arrival rate  = 30/hour, the average time between two successive arrivals are 1/30 hour
or 2 minutes.
For example, in the following arrival situations, the average arrival rate per hour,  and the
average inter arrival time in hour, are determined.
1. One arrival comes every 15 minutes.

Average arrival rate, l = = 4 arrivals per hour.

Average inter arrival time = 15 minutes = ¼ or 0.25 hour.
2. Three arrivals occur every 6 minutes.
Average arrival rate, l = 30 arrivals per hour.

Average Inter-arrival time, = = 2 minutes = or 0.33 hr.

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