Page 228 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 11: Queuing Theory

Solution: Notes
Given, l = 6 customers / hour
t = 30 Minutes = 0.5 hour
n = 2

  

we know, P(n, t) =
   
P(6,2) = = 0.22404

Similarly, when the time taken to serve different customers are independent, the probability
that no more than t periods would be required to serve a customer is given by exponential
distribution as follows:
– t
p(not more than t time period) = 1 – e where  = average service rate

Task A manager of a fast food restaurant observes that, an average of 9 customers are
served by a waiter in a one-hour time period. Assuming that the service time has an
exponential distribution, what is the probability that
1. A customer shall be free within 12 minutes.
2. A customer shall be serviced in more than 25 minutes.

11.4 Symbols and Notations

The symbols and notations used in queuing system are as follows:

n = Number of customers in the system (both waiting and in service).
 = Average number of customers arriving per unit of time.

 = Average number of customers being served per unit of time.
 /  = , traffic intensity.

C = Number of parallel service channels (i.e., servers).
L = Average or expected number of customers in the system (both waiting and in
s
service).
L = Average or expected number of customers in the queue.
q
W = Average waiting time in the system (both waiting and in service).
s
W = Average waiting time of a customer in the queue.
q
P = Time independent probability that there are n customers in the system (both
n
waiting and in service).
P (t) = Probability that there are n customers in the system at any time t (both waiting and
n
in service).

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