1. What do you understand by a queue? Give some important applications of queuing theory.

2. Explain the basic queuing process.

3. What do you understand by queue discipline and input process

4. Explain the constituents of a queuing model.

5. State some of the important distributions of arrival interval and service times.

6. Give the essential characteristics of the queuing process.

1. Students arrive at the head office of Universal Teacher Publications according to a Poisson input process with a mean rate of 30 per hour. The time required to serve a student has an exponential distribution with a mean of 40 per hour. Assume that the students are served by a single individual, find the average waiting time of a student.

2. A barber with a one-man shop takes exactly 30 minutes to complete one haircut. If customers arrive according to a Poisson process at a rate of one every 40 minutes, how long on the average must a customer wait for service?

3. In PNB there is only one window, a solitary employee performs all the service required and the window remains continuously open from 10.00 A.M. to 5.00 P.M. It has been discovered that the average number of clients are 54 during the day and that the average services time is of five minutes per person. Calculate:

- the average number of clients in the system
- the average number of client in the waiting line
- the average waiting time,

4. At the PVR Cinema Hall, customers arrive to purchase tickets according
to a Poisson process with a mean rate of 30 per hour. The time required
to serve a customer has an exponential distribution with a mean of 90
seconds. Find the value of P_{o}, L_{s} and W_{s}.

5. Patients arrive at the Lifeline Hospital according to a Poisson distribution at the rate of 35 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 25 per hour.

- Find the effective arrival rate at the clinic.
- What is the probability that an arriving patient will not wait?
- What is the expected waiting time until a patient is discharged from the clinic?

6. A booking counter at New Delhi Railway Station takes 10 minutes
to book a ticket for each customer. If the customers are arriving according
to a Poisson process with a rate of 4 per hour, find out

- Expected queue length.
- Expected waiting time of a customer in the queue.
- Expected time a customer spends in the system.

7. Universal Bank is considering opening a drive in window for customer service. Management estimates that customers will arrive at the rate of 12 per hour. The teller whom it is considering to staff the window can service customers at the rate of one every three minutes.

Assuming Poisson arrivals and exponential service find

- Average number in the waiting line.
- Average number in the system.
- Average waiting time in line.
- Average waiting time in the system.

8. Himachal fertilizers Ltd. distributes its products' by trucks loaded
at its loading station. Both company trucks and contractors' trucks
are used for this purpose. Trucks arrive at a rate of 10 per minute
and the average loading time is 6 minutes.

You are required to determine

- The probability that a truck has to wait.
- The waiting time of a truck that waits.

9. The Janta transport company has one reservation clerk on duty at
a time. He handles information of bus schedules and makes reservations.
Customers arrive at a rate of 6 per hour and the clerk can service 10
customers on the average per hour. You are required to answer the following.

- What is the average number of customers waiting for the service of the clerk?
- What is the average time a customer has to wait before getting service?
- What is the average waiting time of a customer in the system?

10. The Priya Cinema has recently released a famous movie (Dinosaur Park). Customers arrive at the box office window, being managed by a single individual according to a Poisson input process with a mean rate of 20 per hour. The time required to serve a customer has an exponential distribution with a mean of 90 seconds. Find the average waiting time of a customer.