DEPARTMENT OF MATHEMATICS

PROBABILITY AND QUEUEING THEORY (MA 2262)

UNIT – I RANDOM VARIABLESPART-A

1. The probability density function of a continuous random variable is given by

. Find and C.D.F of X

2. and are independent random variables with variance 2 and 3. Find the variance of.

3. A Continuous random variable has a probability density function

Find ‘a’ such that

4. A random variable has the p.d.f given by Find the value

of and cumulative density function of .

5. If a random variable has the p.d.f , find the mean and

variance of .

6. A random variable has density function given by . Find the

moment generating function.

7. If is a Poisson variate such that , find the

variance.8. Comment the following: “The mean of a binomial distribution is 3 and variance is 49. If and are independent binomial variates

and find

10. If is uniformly distributed with Mean and Variance , find

11. If is ind where

12. If the probability is that a man will hit a target, what is the chance that he will hit the

target for the first time in the 7th trial?

PART-B 1. A random variable has the following probability function:Values of

Find (i) , (ii) Evaluate and

(iii). Determine the distribution function of . 2. If the probability distribution of is given as

Find

(ii). If

find (a) (b)

3. is a continuous random variable with pdf given by

Find the value of and also the cdf .

4. A random variable has the P.d.f

Find (i) (ii) (iii)

5. The elementary probability law of a continuous random variable is

where a, b and are constants. Find , the rth moment about the point and also find the mean and variance.6. The first four moments of a distribution about are 1,4,10 and 45 respectively. Show that the mean is 5, variance is 3, and

7. A continuous random variable X has the p.d.f Find the rth moment of X

about the origin. Hence find mean and variance of X.8.Find the moment generating function of the random variable X, with probability density

function . Also find , .

9. Let the random variable have the p.d.f

Find the moment generating function, mean & variance of .10. a) A die is cast until 6 appears. What is the probability that it must cast more than five times?

b) Suppose that a trainee soldier shoots a target in an independent fashion. If the probability that the target is shot on any one shot is 0.8.

(i) What is the probability that the target would be hit on 6th attempt? (ii) What is the probability that it takes him less than 5 shots?11. a) Find the m.g.f. of the geometric distribution and hence find its mean and variance. b) Six coins are tossed 6400 times. Using the Poisson distribution, what is the approximate probability of getting six heads times?12. a) Define Binomial distribution Obtain its m.g.f., mean and variance. b) Six dice are thrown 729 times. How many times do you expect at least 3 dice show 5 or 6 ?13. a) If are two independent random variables each flowing negative binomial

distribution with parameters and , show that the sum also follows negative binomial

distribution b) If a boy is throwing stones at a target, what is the probability that his 10 th throw is his 5th hit, if the probability of hitting the target at any trial is 0.5?14. a) State and prove the memoryless property of exponential distribution. b) A component has an exponential time to failure distribution with mean of 10,000 hours. (i) The component has already been in operation for its mean life. What is the probability that it will fail by 15,000 hours? (ii) At 15,000 hours the component is still in operation. What is the probability that it will operate for another 5000 hours.15. The Daily consumption of milk in a city in excess of 20,000 gallons is approximately

distributed as a Gamma variate with parameters and . The city has a daily

stock of 30,000 gallons. What is the probability that the stock is insufficient on a particular day?16. a) suppose that the lifetime of a certain kind of an emergency backup battery (in hours) is a r.v. , having the Weibull distribution with parameter and .

(i) Find the mean life time of these batteries (ii)The probability that such a battery will last more than 800 hours. b) Each of the 6 tubes of a radio set has the life length (in years) which may be considered as a r.v that follows a Weibull distribution with parameter and .If these tubes function independently of one another, what is the probability that no tube will have to be replaced during the first 2 months of service?

UNIT-II

1. Let and have joint density function .Find the marginal density

functions and the conditional density function given .

2. Two random variables X and Y have the joint p.d.f . Find A.

3. Verify whether X and Y are independent if

4. Two random variables X and Y have the joint p.d.f

Determine the marginal distributions of X and Y.5. Suppose that the joint density function of X and Y is

Determine .

6. Examine whether the variables and are independent, whose joint density function is .

7. If has an exponential distribution with parameter 1. Find the pdf of

8. If is uniformly distributed random variable in , Find the probability density

function of

9. If the Joint probability density function of is given by

Find .

10. If and are random Variables, Prove that

11. If and are independent random variables prove that

12. Write any two properties of regression coefficients.13. Write the angle between the regression lines.14. State central limit theorem

PART-B

1. The joint probability density function of a bivariate random variable is

where is a constant.

i. Find .ii. Find the marginal density function of and .

iii. Are and independent?

iv. Find and .

2.a) If and are two random variables having joint probability density function

Find (i)

(ii) (iii) .

b). Three balls are drawn at random without replacement from a box containing 2 white, 3 red and 4 black balls. If denotes the number of white balls drawn and denotes the number of

red balls drawn find the joint probability distribution of .

3.a) Two fair dice are tossed simultaneously. Let denotes the number on the first die and denotes the number on the second die. Find the following probabilities.

(i) , (ii) , (iii) and (iv) .

b) The joint probability mass function of a bivariate discrete random variable in given

by the table. 1 2 3

1 0.1 0.1 0.2

2 0.2 0.3 0.1

Find i. The marginal probability mass function of and .

ii. The conditional distribution of given .

iii.

4.a) If and are two random variables having the joint density function

where and can assume only integer values 0, 1 and 2, find the

conditional distribution of for .

b) The joint probability density function of is given by

. Find (i) , (ii) and

(iii)

5.a) If the joint distribution functions of and is given by

i. Find the marginal density of and .ii. Are and independent.

iii. .

b) The joint probability distribution of and is given by

. Find .

6.a) Two random variables and have the following joint probability density function

. Find the marginal probability density function of

and . Also find the covariance between and .

b) If for a bivariate , find the correlation

coefficient

7.a) Let the random variables and have pdf .

Compute the correlation coefficient. b) Let and be two independent random variables with means 5 and 10 and standard

devotions 2 and 3 respectively. Obtain the correlation coefficient of where

and .

8.a) Let the random variable has the marginal density function and let

the conditional density of be . Prove that the variables

and are uncorrelated.

b) Given . Find the regression curve of on .

9.a) Given , obtain the regression of on and on

. b) Distinguish between correlation and regression Analysis

10.a) any are two random variables with variances and respectively and is the

coefficient of correlation between them. If and , find the value of

so that and are uncorrelated. b) Find the regression lines:

6 8 10 18 20 2340 36 20 14 10 2

11.a) Using the given information given below compute and . Also compute when

and . b) The joint pdf of and is

-1 1

0

1

Find the correlation coefficient of and .

`12.a) Calculate the correlation coefficient for the following heights (in inches) of fathers and their sons .

65 66 67 67 68 69 70 7267 68 65 68 72 72 69 71

b) If and are independent exponential variates with parameters 1, find the pdf of .

13.The joint pdf of and is given by . Find the pdf of

.

b) If and are independent random variables each following , find the pdf of

. If and are independent rectangular variates on find the distribution of

.

14.a) If are Poisson variates with parameter . Use the central limit theorem

to estimate where and .

b) A random sample of size 100 is taken from a population whose mean is 60 and variance is 400. Using central limit theorem, with what probability can we assent that the mean of the sample will not differ from by more than 4.

15.a) If the variable are independent uniform variates in the interval ,

find using central limit theorem.

b) A distribution with unknown mean has a variance equal to 1.5. Use central limit theorem to find how large a sample should be taken from the distribution in order that the probability will be at least 0.95 that the sample mean will be within 0.5 of the population mean.

UNIT– III – MARKOV PROCESSES AND MARKOV CHAINS

PART -A1. Distinguish between wide sense stationary (WSS) and strict sense stationary (SSS) random processes.

2 Find the invariant probabilities for the Markov chain with state space and

one step tpm .

3 What is a Markov process?4 Define irreducible Markov chain.5 State four properties of a Poisson process.

6 Consider the random process where is uniformly distributed in the

interval . Check whether {X (t)} is stationary or not?7 Prove that a first order stationary process has a constant mean.

8 If patients arrive at a clinic according to poisson process with mean rate of 2 per minutes, find the probabilities that during a 1-minute interval, no patient arrives.

9 Define Poisson random process. Is it a stationary process? Justify the answer.10 What is meant by steady state distribution of a Markov chain?11 What is the super position of n independent Poisson processes with respective average rate

?12 What is a stochastic matrix? When is it said to be regular?

PART –A1.Two random processes and are defined by and

show that and are jointly wide sense stationary if A and B

are uncorrelated random variables with zero means and the same variances and is a constant.2.Find the mean and auto correlation of the Poisson process

3.Given a random variable Y with characteristic function and a random process

defined by ; show that is stationary in the wide sense if

.

4.If is a Gaussian Process with and Find the probability

that i) ii) .

5.A man either drives a car or catches a train to go to office each day. He never goes 2 days in a row by trains but if he drives one day, then the next day he is just as likely to drive again he is to travel by train. Now suppose that on the first day of the week, the man tossed a fair die and drove to work if and only if a ‘6’ appeared. Find i) The probability that he takes a train on the third day. ii) The probability that he drives to work in the long ran.6.Prove that the difference of two independent Poisson processes is not a Poisson process.

7.Show that the random process is wide sense stationary if and are

constant and is uniformly distributed random variable

8.Given a random process where is uniformly distributed over

, prove that the process is correlation ergodic.

9.Define a random process (stochastic process). Explain the classification of random processes. Give an example to each class.

10.If the process is a Poisson process with parameter , obtain

and

11.(a).Let where A and B are independent normally distributed

random variables . Obtain the covariance function of .

(b). The auto correlation function for a stationary process is given by .

Find the mean value of the R.V. and variance of .

(c). If is a WSS process with auto correlation function and if

.Show that

12.Consider two random processes

and where is a random variable uniformly

distribution in . Prove that .

13.Define Markov chain and explain how you would classify the states and identify different classes of a Markov chain. Given an example to each class14.State the postulates of a Poisson process. State its properties and establish the additive property for the Poisson process.15.Let X be the random variable which gives the intervals between two successive occurrences of a Poisson process with parameter . Find out the distribution of X.

16.Find the nature of the states of the Markov chain with the TPM and the state

space .

17.The one-step T.P.M of a Markov chain having state space is

and the initial distribution is . Find (i)

(ii) (iii) .

18.Let be a Markov chain with state space and 1 – step

Transition probability matrix (i) Is the chain ergodic? Explain (ii) Find the

invariant probabilities.

19.Define autocorrelation function and state its properties. Given that the autocorrelation

function for a stationary process,

. Find the mean and variance of

the process .

20.Three are 2 white marbles in urn A and 3 red marbles in urn B. At each step of the process, a marble is selected from each urn and the 2 marbles selected are inter changed. Let the state of the system be the number of red marbles in A after changes. What is the probability that there are 2 red marbles in A after 3 steps? In the long run, what is the probability that there are 2 red marbles in urn A?

UNIT– IV – QUEUEING THEORY

1.For model, write down the Little’s Formula.

2.For model, write down the Formula for

a) Average number of customers in the queue.b) Average waiting time in the system.

3.What is the probability that a customer has to wait more than 15 minutes to get his service

completed in a queuing system, if per hour per hour?

4.What is the probability that an arrival to an infinite capacity 3 server Poisson queueing system

with and entries the service without waiting.

5.In a given queue, , What is the probability that the queue

contain 5 or more students?

6.What is the effective arrival rate for queuing model when and

7.a) In write down the expression for P0.

b) In Calculate P0

8.In Calculate P0

9.In a two server system with infinite capacity and and , find P0.

10.Write the steady states equations in where

11.In , what is the probability that the server

will be idle.

12.In write the expression for P0.

13.What is effective arrival rate in

14. with compute where

.

PART-B

1.A repairman is to be hired to repair machines which break down at an average rate of 3 per hour. The breakdown follows Poisson distribution. Non- productive time of machine is considered to cost Rs.16/hr. Two repairmen have been invited. One is slow but cheap while the other is fast and expensive. The slow repairman charges Rs.8 per hour and he services machines at rate of 4 per hour. The fast repairman demands Rs.10 per hour and service at the average rate of 6 per hour. Which repair man should be hired?

2.A two person barber shop has 5 chairs to accommodate waiting customers. Customers who arrive when all 5 chairs are full leave without entering barber shop. Customers arrive at the average rate of 4 per hour and spend an average of 12 minutes in the barber’s chair. Compute

and average number of customers in the queue3.On average 96 patients over 24 hour day require the service of an emergency classic. Also on average a patient requires 10 minutes of active attention. Assume that the facility can handle any one emergency at a time. Suppose that it costs the clinic Rs.100 per patient treated to obtain an average servicing time of 10 minutes and that each minute of decrease in this average time would cost Rs.10 per patient treated. How much would have to be budgeted by the clinic to decrease the

average size of the queue from patient to patient?

4.Obtain the expressions for steady state probability of a M/M/C queuing system

5.Arrivals at a telephone booth are considered to be Poisson with an average time of 12 mins. The length of a phone call is assumed to be distributed exponentially with mean 4 min. Find the average number of persons waiting in the system. What is the probability that a person arriving at the booth will have to wait in the queue? Also estimate the traction of the day. Where phone will be in use

6.There are three typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour, what fraction of time all the typists per hour will be busy? What is the average number of letters waiting to be types?

7.A bank has two tellers working are savings account. The first teller handles with drawals only the second teller handles deposits only. It has been found that the service time distributions for both deposits and withdrawals are exponential with mean service time of 3 minutes per customer. Depositors are found to arrive in a poisson fashion through out the day with mean arrival rate of 16 per hour withdrawals also arrive in a poisson fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting time for the customers if each teller could handle both withdrawals and deposits?

8.In a heavy machine shop, the overhead crane is 75% utilized. Time study observations gave the average slinging time as 10.5 minutes with a SD of 8.8 minutes. What is the average calling rate

for the service of the crane and what is the average delay in getting service? If the average service time is cut to 8.0 minutes, with a standard deviation of 6.0 minutes, how much reduction will occur on average in the delay of getting service?9.A Super market has two girls ringing up sales at the customers. If the service time for each customer is exponential with mean 4 minutes and if the people arrive in a Poisson fashion at the rate of 10 per hour

a) What is the probability of having to wait for service?b) What is the expected percentage if idle time for each girl?Hence expected percentage of idle time each girl is 67%.

10.At a one man barber shop, the customers arrive following poisson process at an average rate of 5 per hour and they are serviced according to exponential distribution with an average service. Rate of 10 minutes assuming that only 5 seats are available for waiting customers find the average time a customer spends in the system.

11.At a railway station, only one train is handled at a time. The railway yard is sufficient only for two trains to wait while the other is given signal to leave the station. Trains arrive at the station at an average of 12 per hour. Assuming poisson arrivals and exponential service distribution, find the steady state probabilities for the number of trains in the system also find the average waiting time of a new train coming into the yard. If the handling rate is reduced to half, what is the effect of the above results?

12.At a port there are 6 unloading berths and 4 unloading crews. When all the berths are full, arriving ships are diverted to an overflow facility 20 kms down the river. Tankers arrive according to a poisson process with a mean of 1 for every 2 hrs. If it takes for an unloading crew on the average, 10 hrs to unload a tanker, the unloading time follows an exponential distribution Determine.

a) How many tankers one of the port on the average?b) How long does a tanker spend at the port as the average?

13.A petrol pump has 2 pumps. The service time follows the exponential distribution with a mean of 4 minutes and ears arrive for service in a poisson process at the rate of 10 cars per hour. Find the probability that a customer has to want for service. What proportion of time the pumps remain idle?

14.A supermarket has two girls serving at the customers. The customers arrive in a Poisson fashion at the rate of 12 per hour. The service time for each customer is exponential with mean 6 minutes. Find

i) The probability that an arriving customer has to want for service.ii) The average number of customers in the system, and iii)The average time spent by a customer in the supermarket.

15.Four counters are being run on the frontier of a country to check the passports of the tourists. The tourists choose a counter at random. If the arrival at the frontier is poisson at the rate and

the service time is exponential with parameter , find the steady average queue at each counter.

16.In a given M/M/1 queuing system the average arrival is 4 customers per minutes what are

1) Mean number of customers in the system

2) Mean number of customers in the queue.3) Probability that the service is idle.4) Mean waiting time in the system.

17.A TV repairman finds the time spend on his job has an exponential distribution with mean 30’.If he repairs sets in the model in which they come in and of the arrival of sets is approximately poisson with an average rate of 10 sets per 8 hours day, what is the repairman’s expected idle time each day ? How many jobs ahead of the average set just brought in?

18.If for a period of 2 hours in the day trains arrive at the yard every 20 minutes but the service time continuous to remain 36 minutes. Calculate the following for the above period.

i) The Probability that the yard is empty ii) The average number of trains on the assumption that the line capacity of the yard is limited to 4 trains only.

19.Suppose there are 3 typists is a typing pool. Each typist can type an average of 6 letters / hr. If letters arrive to be typed at the rate of 15 letters/hr

a) What is the probabilities that there is only one letter in the systemb) What is the average number of letters waiting to be typed?c) What is the average time a letter spends in the system?

20.An automobile inspection station has 3 inspection stalls. Assume that cans wait in such a way that when a stall becomes vacancy the car at the head of the time pulls up to it. The Station can accommodate at most 4 cars waiting at one time. The arrival pattern is poisson with a mean of I car every minute during the peak hours. The service time is exponential with mean 6 min. Find the average number of customers is the queue during peak hours and the average waiting time in the queue.

21.Patients arrive at a clinic according to Poisson distribution at a rate of 60 patients per hour. The waiting room does not accommodate more than 14 patients Investigation time per patient in exponential with mean rate of 40 per hour.

a) Determine the effective arrival rate at the clinic.b) What is the probability that an arriving patient will not wait?c) What is the expected time (waiting) until the patient is discharged from the clinic?

22.A group of users in a computer browsing centre has 2 terminals. The average computing job requires 20 min of terminal time and each user requires some computation about once every half an hour .Assume that the arrival rate is poisson and service rate is exponential and the group contains 6 users. Calculate

a) The average number of users waiting to use one of the terminals and in the computing job.

b) The total time lost by all users per day when the centre is opened 12 hrs/day.23.The railway marshalling yard is sufficient only for trains (there are 11 lines one of which is earmarked for the shutting engine to reverse itself from the crest of the hump to the rate of the train). Train arrive at the rate of 25 trains per day, inter-arrival time and service time follow exponential distribution with an average of 30 minutes Determine.

a) The probability that the yard is empty.b) The average queue length.

UNIT– V – NON-MARKOVIAN QUEUES AND QUEUE NETWORKS

PART-A1.Write down Pollaczek - Khinchine formula. 2.Write the steady-state equations (flow balance equations) for a two-station sequential Queue with blocking.3.Write the flow-balance equations of open Jackson networks.4.Write the flow-balance equations of closed Jackson networks.5.State the characteristics of Jackson networks. 6.Distinguish between open Jackson networks and closed Jackson networks

PART-B

1.Derive Pollaczek - Khinchine relation for

2.A one man barber shop takes exactly 25 minutes to complete one haircut. If customers 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 a customer must wait for service.

3.Suppose a one person tailor shop is in business of making men’s suits. Each suit requires four district tasks to be performed before it is completed. Assume all four tasks must be completed on each suit before another is started. The time to perform each task has an exponential distribution with a mean of 2 hr. If orders for a suit come at the average rate 5.5 per week (assume an 8hr day, 6 day week), how long can a customer expect to wait to have a suit made?4.In a heavy machine shop, the overhead crane is 75% utilized. Time study observations gave the average slinging time as 10.5 minutes with a standard deviation of 8.8 minutes. What is the average calling rate for the services of the crane and what is the average delay in getting service? If the average service time is cut to 8.0 minutes, with a standard deviation of 6.0 minutes, how much reduction will occur, on average, in the delay of getting served?5.A car manufacturing plant uses one big crane for loading cars into a truck. Cars arrive for loading by the crane according to a Poisson distribution with a mean of 5 cars per hour. Given that the service time for all cars is constant and equal to 6 minutes determine.6.A car wish facility operates with only one bay. Cars arrive according to a Poisson distribution with a 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 the average number of cars waiting in the packing lot, if the time for washing and cleaning a car follows. (a) Uniform distribution between 8 and 12 minutes (b) A normal distribution with mean 12 minutes and S.D.3minutes.

(c) A discrete distribution with values equal to 4,8 and 15 minutes and corresponding probabilities 0.2,0.6 and 0.2.7.An automatic car wash facility operates with only one bay; Cars arrive according to a Poisson process with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. If the service times for all cars is constant and equal to 10 minutes. Determine .

8.In a big factory, there are a large number of operating machines and two sequential repair shops, which do the service of the damaged machines exponentially with respective rates of 1/hour and 2/hour. If the cumulative failure rate of all the machines in the factory is 0.5 / hour, find (i) the probability that both repair shops are idle, (ii) the average number of machines in the service section of the factory and (ii) The average repair time of a machine.

9.A TVS company in Chennai containing a repair section shared by a large number of machines has 2 sequential stations with respective service rates of 3 per hour and 4 per hour. The cumulative failure rate of all the machines is 1 per hour. Assuming that the system behavior can be approximated by the above 2-stage tendon queue, find

(i) the probability that booth the service stations are idle (free)(ii) the average repair time including the waiting time.(iii) the bottleneck of the repair facility.

10.In the Airport reservation section of a city junction, there is enough space for the customers 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 customers to wait. Customers arrive at the reservation counter section at the rate of 40 per hour and takes one minute on the average to fill up the forms. Each reservation clerk takes 5 minutes on the average to complete the business of a customer in an exponential manner.(i) Find the probability that a customer has to wait to get the service in the reservation counter section (ii) Find the total waiting time for a customer in the entire reservation section. Assume that only those who have the filled up reservation forms will be allowed into the counter section.

11.The last two things that are done to a car before its manufacture is complete are installing the engine and putting on the tires. An average of 54 cars per hour arrives, requiring these two tasks. One worker is available to install the engine and can service an average of 60 cars per hour. After the engine is installed, the car goes to the tire station and waits for its tires to be attached. Three workers serve at the tire station. Each works on one car at a time and can put tires on a car in an average of 3 minutes. Both inter arrival times and service times are exponential.(i) Determine the mean queue length at each work station.(ii) Determine the total expected time a car spends waiting for service.

12.For a 2- stage (service point) sequential queue model with blockage, compute the average number of customers in system and the average time that a customer has to spend in the system, if ,

13.For a 2-stage (service point) sequential queue model with blockage, compute and

.14.There are two salesmen in a shop, one in charge of receiving payment and the other in charge of delivering the items. Due to limited availability of space, only one customer is allowed to

enter the shop, that too when the clerk is free. The customer who has finished his job has to wait there until the delivery section becomes free. It customers arrive in accordance with a Poisson process at rate 1 and the service times to two clerks are independent and have exponential rates of 1 and 3, find(i) the proportion of customers who enter the ration shop(ii) the average number of customers in the shop and(iii) the average amount of time that an entering customer spends in the shop.15.Consider a system of two servers where customers from outside the system arrive at server 1 at a Poisson rate 4 and at server 2 at a Poisson rate 5. The service rates 1 and 2 are respectively 8 and 10. A customer upon completion of service at server 1 is equally likely to go to server 2 or to leave the system (i.e.,P11=0,P12=1/2); whereas a departure from server 2 will go 25 percent of the time to server 1 and will depart the system otherwise (i.e.,P21=1/4,P22=0). Determine the limiting probabilities, L and W.

16.In a departmental store there are 2 sections, namely, grocery section and perishable (vegetables and fruits) section. Customers from outside arrive at the G-Section according to a Poisson process at a mean rate of 10/hour and they reach the P-section at a mean rate of 2/hour. The service times at both the sections are exponentially distributed with parameters 15 and 12 respectively. On finishing the job in the G-section, a customer is equally likely to go the P-section or to leave the store, whereas a customer on finishing his job in the P-section will go to the G-section with probability 0.25 and leave the store otherwise. Assuming that there is only one salesman in each section, find the probability that there are 3 customers in the G-section and 2 customers in the P-section. Find also the average number of customers in the store and the average waiting time of a customer in the store.17.Consider two serves. Ana average of 8 customers per hour arrive from outside at server 1 and an average of 17 customers per hour arrive from outside at server 2. Inter arrival times are exponential. Server 1 can serve at an exponential rate of 20 customers per hour and server 2 can serve at an exponential rate of 30 customers per hour. After completing service at server 1, half of the customers leave the system, and half go to server 2. After completing service at server 2,

of the customers complete service , and return 1.

(i) What fraction of the time is server 1 idle?(ii) Find the expected number of customers at each server.(iii) Find the average time a customers spends in the system.(iv) How would the answers to parts (i) –(iii) change if server 2 could server only an average of 20 customers per hour?