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It is a queuing model where the arrivals follow a Poisson process, service times are exponentially
distributed and there is only one server. In other words, it is a system with Poisson input, exponential
waiting time and Poisson output with single channel
Queue capacity of the system is infinite with first in first out mode. The first M in the notation stands
for Poisson input, second M for Poisson output. 1 for the number of servers and © for infinite
capacity of the system.
Formulas
Probability of zero unit in the queue (P.) = ae weee
Average queue length (Ly ) =
Average number of units in the system (L.) =
Average waiting time of an arrival (Wa) =
Average waiting time of an arrival in the
system (WW) =
Example 1
Students arrive at the head office of Universal Teacher Publications according to a Poisson input
process with a mean rate of 40 per hour. The time required to serve a student has an exponential.
distribution with a mean of 50 per hour. Assume that the students are served by a single individual,
find the average waiting time of a student.
Solution.
Given
A= 40/hour, u = 50/hour
Average waiting time of a student
po nsann = 48 minutes
before receiving service (W/q) = 4
50(50 - 40)
ED sample 2
New Delhi Railway Station has a single ticket counter. During the rush hours, customers arrive at the
rate of 10 per hour. The average number of customers that can be served is 12 per hour. Find out the
following:
* Probability that the ticket counter is free.
* Average number of customers in the queue.
Solution.
Given]
I. 10/hour, p = 12/hour
10
Probability that the counter is free = d- 0 =1/6
12
(101
secsee = 25/6
12 (42 - 10)
At Bharat petrol pump. customers arrive according to a Poisson process with an average time of 5
minutes between arrivals. The service time is exponentially distributed with mean time = 2 minutes.
On the basis of this information, find out
1 What would be the average queue length?
2. What would be the average number of customers in the queuing system?
3. What is the average time spent by a car in the petrol pump?
4. What is the average waiting time of a car before receiving petrol?
Solution.
1 1
Average inter arrival time - — =5minutes- — --- hour
A 12
A= 12/hour
1 1
Average service time - — =2minutes = --- hour
u 30
= 30/hour
