Group-B
1. A T.V repairman finds that the time spent on his jobs has an exponential distribution
with mean 30 minutes. If he repairs sets in the order in which they came in, and if the
arrival of sets is approximately Poisson with an average rate of 10 per 8-hour day, what
is repairman’s expected idle time each day? How many jobs are ahead of the average
set just brought in?
λ
Ans- ρ = μ = 0.625, P0 = 1 − ρ
Expected idle time each day = 8 × P0 = 3 hrs
ρ 5
Average no. of T.V E(n) = 1−ρ = 3 = 2 approx.
2. Customers arrive at a sales counter manned by a single person according to a Poisson
process with a mean rate of 20 per hour. The time required to serve a customer has an
exponential distribution with a mean of 100 seconds. Find the average waiting time of
a customer.
Ans- Average waiting time in queue =125 seconds.
Average waiting time in system =225 seconds.
3. A petrol station has two pumps. The service time follows the exponential distribution
with mean 4 minutes and cars arrive for service in a Poisson process at the rate of 10
cars per hour. Find the probability that a customer has to wait for service. What
proportion of time do the pumps remain idle?
Ans- Customer has to wait for service =0.167 , The pumps remain idle = 67%
Group-C
4. The rate of arrival of customers at a public telephone booth follows Poisson distribution,
with an average time of 10 minutes between one customer and the next. The duration of
a phone call is assumed to follow exponential distribution, mean time of 3 minutes.
(i) What is the probability that a person arriving at the booth will have to wait?
(ii) What is the average length of the non-empty queues that form from time to time?
(iii) Estimate the fraction of a day that the phone will be in use.
(iv) What is the probability that it will take him more than 10 minutes altogether to
wait for phone and complete his call?
Ans- i) 0.3 ii) 1.43 iii) 0.3 iv) 0.03
5. At a railway station, only one train is handled at a time. The railway yard is sufficient
only for two trains to wait while other is given signal to leave the station. Trains arrive
at the station at an average rate of 6 per hour and the railway station can handle them on
an average of 12 per hour. Assuming Poisson arrivals and exponential service
distribution, find the steady-state probabilities for the various number of trains in the
system. Also find the average waiting time of a new train coming into the yard.
Ans- Expected number of train 𝐸(𝑛) = 0.74,
Average number of trains in the system = 0.74
Average service time = 0.08 hours.
Expected waiting time 𝐸(𝑤) = 3.5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
6. If the arrival rate is 𝜆 and service rate is 𝜇 then prove that the expected queue length is
𝜆2
.
𝜇(𝜇−𝜆)
7. The arrival patterns of awaiting line system follow a Poisson distribution and the service
completion follow an exponential distribution with service rate 1 unit per period. If the
average length of the waiting line be 0.5 unit find the arrival rate per period.
𝜆2
Ans- = 0.5 , 𝜆 = 0.5 = arrival rate
𝜇(𝜇−𝜆)
8. Customers arrive at a post office manned by a single person, according to a Poisson
input process with a mean rate of 10 per hour. The time required to serve a customer
has an exponential distribution with a mean of 4 minutes. Find (i) The average number
in the system. (ii) The probability that there would be 2 customers in the queue.
2
Ans- 𝜌 = traffic intensity = 3
𝜌
(i) The average number in the system = 1−𝜌 = 2
4
(ii) The probability that there would be 2 customers in the queue = (1 − 𝜌)𝜌2 = 27
9. The tool room company’s quality control department is manned by a single clerk who
takes an average of 5 minutes in checking parts of each of the machine coming for
inspection. The machine arrives once in every 8 minutes on the average. One hour of
the machine is valued at Rs. 15 and the clerk’s time is valued at Rs. 4 per hour. What
is the average hourly queuing system costs associated with the quality control
department?
1 2
Ans- 𝜆 = 7.5/ℎ𝑟 , 𝜇 = 12/ℎ𝑟 𝑊𝑠 = 𝜇−𝜆 = 9
10
Average queuing cost/machine = 𝑅𝑠. 3
Average queuing cost/hr = 𝑅𝑠. 25
Average cost of the clerk/hr = 𝑅𝑠. 4
Total cost for the deptt./hr = 𝑅𝑠. 29
