M/M/1 Model
Formulas and Practice Questions
Examples-
1. In a self service store with one cashier, 8 customers arrive on an average of every 5 mins. and the
cashier can serve 10 in 5 mins. If both arrival and service time are exponentially distributed, then
determine a) Average number of customer waiting in the queue for average. b) Expected waiting
time in the queue c) What is the probability of having more than 6 customers In the system .
Ans. a. 3.2
b. 0.033 ; c. 0.209
2.
• Consider a box office ticket window being manned by a single server. Customer
arrives to purchase ticket according to Poisson input process with a mean rate of
30/hr. the time required to serve a customer has an ED with a mean of 90 seconds
determine:(a) Mean queue length (2.25) (b) Mean waiting time in the system (0.1)
c. The fraction of the time for which the server is busy (0.75).
• Note- mean arrival rate is inverse of mean arrival time similarly mean service rate
is inverse of mean service time.
3.
• There is congestion on the platform of a railway station. The trains arrive at a rate
of 30/days. The service time for any train is ED with an average of 36mins.
Calculate: (a) Mean queue size (2.25) (b) Probability that there are more than 10
trains in the system. (0.0422)
• 4. Arrivals at a telephone both are considered to be Poisson at an
average time of 8 min between our arrival and the next. The length
of the phone call is distributed exponentially, with a mean of 4 min.
Determine
(a) Expected fraction of the day that the phone will be in use (0.5).
(b) Expected number of units in the queue Expected waiting time in
the queue (0.5; 0.066).
(c) Expected number of units in the system (1).
(e) Expected waiting time in the system (.133)
(f) Expected number of units in queue that from time to time (2).
(g) What is the probability that an arrival will have to wait in queue
for service (0.5)?
(h) What is the probability that exactly 3 units are in system (0.0625)
(i) What is the probability that an arrival will not have to wait in
• 5. A T.V repairman repair the sets in the order in which they arrive and expects that
the time required to repair a set has an ED with mean 30mins. The sets arrive in a
Poisson fashion at an average rate of 10/8 hrs a day.
(a) What is the expected idle time / day for the repairman? (0.375x8)
b) How many TV sets will be there awaiting for the repair? (1.04)
• 6. The arrival rate for a waiting line system obeys a P.D with a mean of 0.5 units/hr. it
is
• required that the probability of one or more units in the system does not exceed 0.25.
what is
• the minimum service rate that must be provided if the service duration will be
distributed
• exponentially? (02/hr)
7. Jobs arrive at an inspection station according to Poisson process at a mean rate of
2/hr and are inspect one at a time on a FIFO basis. The quality control engineer both
inspects and makes minor adjustments. The total service time for the job appears to
be ED with a mean of 25mins. Jobs that arrive but cannot be inspected immediately
by the engineer must be stored until the engineer is free to take them. Each job
requires 1 sq mts space determine
a)The waiting line length [4.16]
b) The waiting time [2.08]
c) % of idle time of the engineer [16.66%]
d) The floor space to be provided in the quality control room. [5]
8. A mechanic is to hired to repair a machine which breaks down at an average
rate of 3/hr. breakdowns are distributed in time in a manner that may be regarded
as Poisson. The non productive time on any machine is considered to cost the
Company. Rs. 5/ hr the choice to 2 mechanics A & B. The mechanic A repairs
the machines at an average rate of 4/hr and he will demand Rs. 3/hr. The
mechanic B costs Rs. 5/hr and can repair the machines exponentially at an
average rate of 6/hr. Decide which mechanic should be hired.
