UNIT – II

NON MARKOVIAN QUEUES &QUEUE NETWORKS

1. Derive Pollaczek-Khintchine formula.

2. A patient goes to a single doctor clinic for a general checkup with 4 phases. The doctor takes on

average 4 minutes for each phase of the check up and time taken for each phase is exponentially

distributed. If the arrivals of the patient at the clinic are approximately Poisson at the average of 3 per

hour. What is the average time spent by the patient i) in the examination ii) waiting in the clinic?

3. An automatic car washing facility operates with only one bay. Cars according to a Poisson distribution

with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. If service time

for all cars is constant and equal to 10 minutes determine.

4. An automatic car washing facility operates with only one bay. Cars according to a Poisson distribution

with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. The parking lot

is large enough to accommodate any number of cars. Find average number of cars waiting in the

parking lot, the time for washing and cleaning a car follows

a) Uniform distribution between 8 and 12 minute

b) A normal distribution with mean 12 minutes and S.D of 3 minutes

c) A discrete distribution with values equal to 4,8,15 minutes and corresponding probabilities 0.2,

0.6and 0.2.

5. A one man barber shop exactly 25 minutes to complete one haircut. If customer arrive at the barber

shop in a Poisson fashion at an average rate of one every 40 minutes, how long on the average a

customer spends in the shop? Also find the average time customer must wait for service?

6. In a heavy machine factory, the overhead crane is utilized 75%. The study of time an average service

time as 10.5 minutes with a standard deviation of 7.5 minutes. Find

a) The average rate of the crane for service

b) The average delay for getting service.

PROBLEMS UNDER SERIES QUEUE WITH BLOCKING

1. For a 2-stage sequential queue model with blockage, compute the average number of customer in the

system and the average time that a customer has to spend in the system if

2. There are 2 chairs in a barber shop, each of them managed by a barber.B1 is a specialist in a hair –

cutting and B2 is a specialist in shaving and washing. B1 and B2 do their jobs according to exponential

distribution with parameters 1 and 3 respectively. Customers who require both hair –cutting and

shaving enter the shop only if B1 is free. If a customer finished his job with B1 chair until B2 becomes

free. Find the average number of customer in the shop and average waiting time spends in the shop.

3. There are 2 salesman in a ration shop , one in charge of billing and receiving payment and other is

charge of weighing and delivering the items. Due to limited availability of space, only one customer is

allowed to enter in to the shop, that too when the billing clerk is free. The customer who finished his

billing job has to wait there until the delivery section becomes free. If customer arrive in accordance

with a Poisson fashion at the rate of 1 and service time of two clerks are independent and have

exponential rates of 3, 2 ,find

a) The proportion of customer who enter into ration shop

b) The average number of customer in the shop

c) The average amount of time that entering customer spends in the shop.

PROBLEMS UNDER TWO STAGE TANDEM QUEUE

1. For a walk an interview conducted by a company, candidates arrive at the interview hall at a Poisson

rate of 6/hour. In the interview hall, verification of certificates is done by a clerk and the personal

interview is conducted by an officer in different rooms. Both of them do the job in an exponential

manner each taking 6 minutes on the average. If no queue is allowed to form in front of the rooms and

the candidates has to wait in the clerk room if the officer is busy. Find the numbers of candidates

present in the interview hall and the average time spend by a candidate in the hall.

2. On the first day of admission of fresher’s in engineering college , fresher’s , after surrendering the

original certificates , join the queue in front of the principals chamber for getting actual admission at

the rate of 18/hour in a Poisson manner. After getting the nod from the principal, they join the queue

in front of the cash collection counter to make a payment of fees. If the principal and the cashier do the

job in an exponential manner taking 2.5, 3 minutes respectively. Find the average number of fresher’s

inside the college office and the average time spends by a fresher’s inside the office. Find also the

probability that they are 3 students in front of principle chamber and 4 students in front of cash

counter.

3. In an ophthalmic clinic, there are two sections one section for assessing power approximately and

other for final assessment and prescription of glasses. Patients arrive at the clinic in a Poisson fashion

at the rate of 3 per hour. The assistant in the first section takes nearly 15 minutes per patient of power

and the doctor in the second section takes nearly 6 minutes per patient for final prescription. If

the service time in the two sections is approximately exponential, find probability that there are

3 patients in the first section, and2 patients in the second section. Find also average waiting time,

average number of customer in the clinic. Assume that enough space is available for the patient to

wait in front of both sections.

4. Arepairfacilitysharedbyalargenumberofmachineshassequentialstations

with respective service rateof 2perhour and3perhour.Thecumulativefailure rate of all machines is 1 per

hour. Assuming that the systembehaviormaybe approximated by 2-stage tandem queue, find

a) The average repair time including the waiting time

b) The probability that both the service stations are idle

c) The bottle neck of the system.

5. In a busy medical shop located in a heart of the city, there 3 salesmen who receive the customer,

supply the drugs and prepare the bills. On finishing the job with any one of the salesmen , the

customer goes to the payment counter, manned by the owner himself and leave the shop after

paying the bill. If customer enter in to the medical shop in a Poisson rate of 30 per hour each

salesmen takes on an average 5 minutes to serve a customer in an exponential manner and the

owner takes on an average 1 minute per customer to check the bill and receive the payment in an

exponential fashion. Find the average number of customer in the shop. If the owner wishes that no

customer should stay in shop more than 8 minutes and the number of customer in the shop should

exceed 5 and the same time he wishes to take on the average 1.5 minutes per customer for his job, find

if these can be accomplished by appointing one more salesman of the same caliber as others.

6. In the railway reservation section of a city junction, there is enough space for the customer to

assemble, form a queue and fill up the reservation forms. There are 5 reservation counters in front of

which also there is enough space for the customer to wait. Customer arrive at number the

reservation section at the rate of 50/hour according to Poisson process, take 1 minute each on the

average to fill up forms and then move to the reservation counter section. Each fill up the forms and

then move to the reservation clerks takes 5 minutes on the average to complete the business customer

in an exponential manner. i) Find the probability that a customer has to wait to get the service in the

reservation counter. Ii) Find the total number of customer in the entire section. Assume that only

those who have filled up reservation forms will be allowed into counter section.

7. In a festival season cracker shop, there are 4 sales men, and 1 manager. Customers arrive at the cracker

shop according to Poisson process at the rate of 15/hour. On entry the customer stands in the queue,

goes to any one of the sales man and gets his requirement and also the bill. After his finishing the job

with the salesman the customer goes to the manager counter and joins the queue there. The

manager checks the bill and receives the payment. If the service time of each sales man is exponential

distributed with mean 6 minutes and the same for the manager is 3 minutes. Find the average number

of customer in the cracker shop and average waiting time for a customer in it.

PROBLEMS UNDER OPEN JACKSON NETWORK

1. In a open Jackson network the following information are given

Station j i=2 i=3

1 1 10 1 0 0.1 0.4

2 2 10 4 0.6 0 0.4

3 1 10 3 0.3 0.3 0

Find i) the joint probability for the the number of customer in 1st, 2nd and 3rd stations are 3,2,4

respectively.ii) the expected number of customer in the system iii) the expected total waiting in the

system.

2. In a book shop there 2 section, one for text book and other for note book. Customer from outside

arrive at the text book section at a Poisson at the rate of 4 per hour and the note book section at a

Poisson at the rate of 3 per hour. The service rate of the T.B section and N.B section are respectively 8,

10 per hour. A customer upon completion of service at T.B is equally likely to go to N.B section or

leave the book shop, whereas a customer upon completion of service at N.B section will go to the

T.B section with the probability is 1/3 and will leave the book otherwise. Find the joint steady-state

probability that there are 4 customer in the T.B section and 2 customer in the N.B section . Find also

the average number of customer in the shop and the average waiting time of customer in the shop.

Assume there is only one salesman in this counter.

3. In a network of 3 service station 1,2,3 customer arrive at 1,2,3 from outside , in accordance with

Poisson process having the rate 5,10,15 respectively.. The service time at the 3 section are exponential

with respective rate 10, 50,100 respectively. A customer completing service at the station 1 is equally

to i) go to station 2 ii) go to the station 3 or iii) leave the system. A customer departing from service at

the station 2 always goes to station 3. A departure from service at the station 3 is equally likely to go to

station 2 or leave the system.

a) What is the average number of customer in the system, consisting all the 3 station?

b) What is the average time a customer spends in the system?

4. Customer arrives at a service center consisting of 2 service points at a Poisson rate of 35/hour and

form a queue at the entrance. .on studying the situation at the center, they decide to go to either

or . The decision making takes on the average 30 seconds in an exponential fashion .Nearly 55%

of the customer go to , that consists of 3 parallel servers and rest go to that consists 7 parallel

servers. The service time at are exponential with a mean rate of 6 minutes and those of with a

mean of 20 minutes. About 2% of customer finishing service at go to . Find the average queue

size in front of each node and the total average time customer spends in the service centre.

PROBLEMS UNDER CLOSED JACKSON NETWORK

1. There are 2 clerks in a bank, one processing housing application and other processing agricultural loan

applications. While processing, they get doubts according to an exponential distribution each with a

mean of ½. To get clarifications, clerk goes to the deputy manager, with a probability ¾ and to the

senior manager with probability ¼ . After completing the job with the D.M, a clerk goes to S.M with

probability 1/3 and returns to his seat otherwise. Completing the job with S.M, a clerk always returns

to his seat. If the D.M clarifies the doubts and advises a clerk according to an exponential distribution

with parameter 1 and S.M with parameter 3

Find

a) The steady-state probabilities P ( ) for all possible values of .

b) The probability that both the manager is idle

c) The probability that at least at least one manager are idle.

2. In a factory there are 2 machines which are expected to operational at all times and 2 service man ,

one of them will rectify ordinary defects and other will do the service in respect of serious defects. The

machine breaks down according to an exponential distribution with parameter 2. When a machines breaks

down with the probability 0.8 of being served for ordinary defect and a probability 0.2 being served for

serious defect. After service for ordinary defect, a machine will require for service for serious defect with

probability 0.4 and return to the operation otherwise. After service for serious defect, the machine will

always become operational. Treating the operational status of the machine as node 1,

a) Find the probability that both the serviceman is idle.

b) At least one of them is operational.

Assume that the service time for of the repairman is exponential distributed with parameter 3 and 4

respectively.

PROBLEMS UNDER MEAN VALUE ANALYSIS

1. Assuming that there is only one clerk (N=1) in the loan section in the previous example, compute the

probabilities that he is nodes 1, 2, 3. Verify that these are the same as the average length of the queue

at the nodes. Checks the correctness of these values by comparing with the probabilities obtain by

using mean-value analysis.

2. Use M.V.A algorithms for i) single server ii) multiple server to compute .Assuming there is N=2 clerks

in the system.