CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY

Transcription

1 CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY (Approved by AICTE New Delhi, Affiliated to Anna University Chennai. Zamin Endathur Village, Madurntakam Taluk, Kancheepuram Dist ) PROBABILITY AND QUEUEING THEORY QUESTION BANK UNIT-1 RANDOM VARIABLES PART-A 1. Define Random Variable 2. If X and Y are independent random variables with variance 2 and 3. Find the variance of 3X+4Y. 3. If the range of X is the set {0,1,2,3,4} and P[X=x]=0.2, determine the mean and variance of the random variable. 4. A continuous random variable X has probability density function given by Find K such that P(X>K)= Find the value of C given the pdf of a random variable X as 6. Is the cumulative distribution function F(x) of a random variable X always continuous? Justify your answer. 7. A continuous random variable X that can assume any value between x=2 and x=5 has a density function given by f(x)=k(1+x), Find P[X<4] 8. For a binomial distribution mean is 6 and S.D is.find the first two terms of the distribution. 9. What is the relationship between Weibull and Exponential distribution? 10. Obtain the mean for a Geometric random variable. 11. What is meant by memoryless property? Which continuous distribution follows this property? 12. If a random variable X has the distribution function Where is the parameter, then find P( Every week the average number of wrong-number phone calls received by a certain mail order house is seven. What is the probability that they will receive two wrong calls tomorrow? 14. Give the probability law of poisson distribution and also its mean and variance. 15. State memory loss property of exponential distribution 16. A coin is tossed 2 times, if X denotes the number of heads, find the probability distribution of X. 17. If the probability that a target is destroyed on any one shot is 0.5, find the probability that it would be destroyed on 6 th attempt. 18. Let the random variable X denote the sum of obtaqined in rolling a pair of fair dice. Determine the probability mass function of X 19. Check whether the following is a probability density function or not. 20. If the random variable has the moment generating function given by, determine the variance of X.

2 21. Identify the random variable and name the distribution it follows, from the following statement: A realtor claims that only 30% of the houses in a certain neighbourhood, are appraised at less than Rs.20 lakhs. A random sample of 10 houses from that neighbourhood is selected and appraised to check the realtor s claim is acceptable are not. 22. Suppose X has an exponential distribution with mean equal to 10.Determine the value of X 23. The probability that a candidate can pass in an exam is 0.6. (a) What is the probability that he pass in the third trial? (b) What is the probability that he pass before the 3 rd trial? 24. Define moment generating function 25. Define gamma distribution (or) General gamma distribution. PART-B 1. Find the MGF of the random variable X having the pdf 2. A manufacturer of pins knows that 2% of his products are defective. If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective, what is the probability that a box fail to meet the guaranteed quality? 3. Six dice are thrown 729 times. How many times do you expect atleast three dice to show a five (or) a six? 4. If a continuous RV, X follows uniform distribution in the interval (0,2) and a continuous RV, Y follows exponential distribution with parameter λ. Find λ such that. 5. A continuous random variable has the pdf Find the value of k and also 6. Find the moment generating function of Uniform distribution. Hence find its mean and variance. 7. Find the moment generating function and rth moment for the distribution whose pdf is Hence find the mean and variance. 8. In a large consignment of electric bulbs, 10 percent are defective. A random sample of 20 is taken for inspection. Find the probability that (1) All are good bulbs (2) A t most there are 3 defective bulbs (3) Exactly there are 3 defective bulbs 9. If the random variable x takes the value3s 1,2,3 and 4 such that find the probability distribution and cumulative distribution function of X. 10. Find the moment generating function of Binomial distribution. Hence find its mean 11. If the probability that an applicant for a driver s license will pass the road test on any given trial is 0.8, what is the probability that he will finally pass the test (1) on the 4 th trial (2) in fewer than 4 trials? 12. The number of monthly breakdown of a computer is a random variable having a pois son distribution with mean equal to 1.8.Find the probability that this computer will function fir a month (1) without a breakdown (2) with only one breakdown

3 13. A random variable X has the following probability distribution. X: P(x): 0 k 2k 2k 3k 2k 2 7 k 2 +k (1) Find the value of k (2) Evaluate P(X ),P(X (3) If, find the minimum value of C 14. Find the moment generating function of an exponential distribution. Hence find its mean and variance. 15. If X is a poisson variate such that P(X=2)=9P(X=4)+90P(X=6) Find (1) Mean and E(X 2 ) (2) P(X 2) 16. In a certain city, the daily consumption of electric power in millions of kilowatt-hours can be treated as a random variable having Gamma distribution with parameters and. If the power plant in this city has a daily capacity of 12 millions kilowatt-hours, what is the probability that this supply of power will be inadequate on any given day? 17. Find the moment generating function of Binomial random variable with parameters m and and hence find its mean and variance. 18. Define Weibull distribution and write its mean and variance. 19. Derive mean and variance of a Geometric distribution. Also establish the forgetfulness property of the Geometric distribution. 20. Suppose that telephone calls arriving at a particular switchboard follow a Poisson process with an average of 5 calls coming per minute. What is the probability that up to a minute will elapse unit 2 calls have come into the switch board? 21. The DF of a continuous random variable X is given by Find the pdf of X and evaluate P( and P( using both the pdf and cdf. 22. A random variable X has the following probability distribution X: P(x): 0.1 k 0.2 2k 0.3 3k (1) Find k, (2) Evaluate P(X<2) and P(-2<X<2), (2) Find the pdf of X and (4) Evaluate the mean of X. 23. The probability function of an infinite discrete distribution is given by P(X=j)= ; j=1,2,.. Verify that the total probability is 1 and find the mean and variance of the distribution.find also P(X is even), P(X and P(X is divisible by 3). 24. Define Gamma distribution and find its mean and variance. 25. The random variable X has the probability density function Find the probability density function of 26. If the mgf of X is given by.find the variance X mgf of

4 27. If the RVs X and Y have the joint pdf Check whether they are independent. 28. If X and Y are independent RVs uniformly distributed in [0,1] obtain the distribution of XY. 29. Two RVs X and Y have the following joint pdf Find Var(X), Var(Y) and Co-Variance between X and Y 30. Given the prob. distribution X P(x) Find P(X and P(X<3 X>1) UNIT-II TWO DIMENSIONAL RANDOM VARIABLES PART-A 1. The joint Probability of the RV (X,Y) is given by Find the value of K and prove also that X and Y independent 2. State central limit theorem 3. If the joint pdf of (X,Y) is given by f(x,y)=2, find E(X) 4. When will the two regression lines be (a) a right angles (b) coincident? 5. A small college has 90 male and 30 female professors. An ad-hoc committee of 5 is selected at random to unite the vision and mission of the college. If X and Y are the number of men and women in the committee, respectively, what is the joint probability mass function of X and Y? 6. The regression equation of X and Y and Y on X are respectively and.find the means of X and Y. 7. Given two regression lines, find the coefficient of correlation between X and Y. 8. The joint Probability of the RV (X,Y) is given by Find the value of K 9. Given the RV X with density function Find the pdf of Y=8 10. If there is no linear correlation between teo random variables X and Y, then what can you say about the regression lines? (Nov/Dec 2010) 11. Let the joint pdf of the random variable (X,Y) be given by ; x>0 and y>0. Are X and Y independent? Why or why not?(nov/dec 2010) 12. Give a real life example each for positive correlation and negative correlation..(apr/may 2010) 13. State central limit theorem for independent and identically distributed random variables. (Apr/May 2010) 14. Find the marginal and conditional densities if (May/June 2009)

5 15. Define covariance and correlation between the random variables X and Y (Nov/Dec 2009) 16. Find the correlation coefficient for the following data: X Y For 10 observation in price X and Supply Y the following data were obtained. 220, 2=2288, 2=5506 and =3467. Obtain the line of regression of Y on X and estimate the supply when the price is 16 units. (May/June 2009) 18. If two dimensional random variable X and Y are uniforemly distributed in find (i) Correlation coefficient rxy (ii) Regression equations(nov/dec 2009) 19. Write the applications of central limit theorem 20. The lifetime of a TV tube (in years) is an exponential random variable with mean 10. What is the probability that the average lifetime of a random sample of 36 TV tube is atleast 10.5? 21. A coin is tossed 10 times. What is the probability of getting 3 or 4 or 5 heads. Use central limit theorem. 22.If X 1 and X 2 are independent uniform variates on [0,1], find the distribution of X 1 /X 2 and X 1 X Define Regression 24. What are regression lines? 25. Find the acute angle between the two lines of regression. PART-B 1. Let X and Y be random variables having joint density function (Nov/Dec 2013) 2. The joint distribution of X and Y is given by Find the marginal distributions. (Nov/Dec 2013) 3. If the pdf of X is find the pdf of y=3x+1(nov/dec 2013) 4.The life time of a certain band of an electric bulb may be considered as a RV with mean 1200 h and SD 250 h. Using central limit theorem, find the probability that the average life time of 60 bulbs exceeds 1250 h. (Nov/Dec 2013) 5. The joint probability density function of a two-dimensional random variable (X,Y) is Find(1) P(X 1 Y<3) (2)P(X+Y<3) (3)P(X<1/Y<3) (May/June 2013) 6. If X and Y each follow an exponential distribution with parameter 1 and are independent, find the pdf of U=X-Y(May/June 2013)

6 7. The marks obtained by 10 students in Mathematics and statistics are given below.find the correlation coefficient between the two subjects. Marks in Mathematics Marks in statistics (May/June 2013) 8. A distribution with unknown mean has 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 atleast 0.95 that the sample mean will be within 0.5 of the population mean. (May/June 2013) 9. Obtain the equations of the lines of regression from the following data: (Nov/Dec 2012) X Y The joint pdf of randomvariable X and Y is given by (1) Determine the value of (2) Find the marginal probability density function of X (Nov/Dec 2012) 11. Let X 1,X 2,X 100 be independent identically distributed random variables with. Find P(192< X 1 +X 2 + +X 100 <210) (Nov/Dec 2012) 12. The regression equation of X and Y is 3y-5x+108=0. If the mean value of Y is 44 and the variance of X were th of the variance of Y. Find the mean value of X and the correlation coefficient. (Nov/Dec 2012) 13. Let X and Y be two random variables having the joint probability function, where X and Y can assume only the integer values 0,1 and 2. Find the marginal and conditional distributions. (May/June 2012) 14. Two random variables X and Y have joint probability density function Find cov(x,y) (May/June 2012) 15. Two dimensional random variable (X,Y) have the joint probability density function (1) Find (2) Find the marginal and conditional distributions. (3) Are X and Y independent? (May/June 2012) 16. Suppose that in a certain circuit, 20 resistors are connected in series. The mean and variance of each resistor are 5 and 0.20 respectively. Using central limit theorem, find the probability that the total resistance of the circuit will exceed 98 ohms, assuming independence. (May/June 2012)

7 17. Given the joint density function Find marginal densities g(x),h(y) and the conditional density f(x/y) and evaluate (Apr/May 2011) 18. Determine whether the random variable X and Y are independent, given their joint probability density function as (Apr/May 2011) 19. If X and Y are independent random variables having density functions and respectively, find the density functions of Z=X-Y (Apr/May 2011) 20. The joint probability mass function of (X,Y) is given by P(X,Y)=K(2X+3Y), X=0,1,2;Y=1,2,3. Find all the marginal and conditional distributions. (Nov/Dec 2011) 21. State and Prove Central Limit Theorem(Nov/Dec 2011) 22. If X and Y are independent RVs with pdf s and, respectively, find the density function of and V=X+Y. Are U and V independent? (Nov/Dec 2011) 23. Given and 0 elsewhere. Evaluate c and find f x (x) and f y (y). (Nov/Dec 2011) 24. Compute the coefficient of correlation between X and Y using the following data: X (Nov/Dec 2010) Y For two random variables X and Y with the same mean, the two regression equations are y=ax+b and x=cy+d. Find the common mean, ratio of the standard deviations and also show that (Nov/Dec 2010) 26. If X 1,X 2,X n are poisson variates with parameter Use the central limit theorem to estimate P(120< S n <160), where S n = X 1 +X 2 +X n and n=75. (Nov/Dec 2010) 27. If the correlation coefficient is 0, then can we conclude that they are independent? Justify your answer, through an example. What about the converse? (Apr/May 2010) 28. Let X and Y be independent random variables both uniformly distributed in [0,1]. Calculate the probability density of X+Y (Apr/May 2010) 29. Suppose X and Y are independent non negative continuous random variables having densities f x (x) and f y (y) respectively. Compute P[X<Y] (Nov/Dec 2009) 30. A coin is tossed 10 times, What is the probability of getting 3 or 4 or 5 heads. Use central limit theorem. (Nov/Dec 2009)

8 UNIT-III MARKOV PROCESS AND MARKOV CHAINS PART-A 1. Prove that first order stationary random process has a constant mean. (Nov/Dec 2013) 2. Prove that poisson process is a markov process. (Nov/Dec 2013) 3. Define Wide sense stationary process. (May/June 2013) 4. If the transition probability matrix (tpm) of a markov chain is, find the steady state distribution of the chain. (May/June 2013) 5. If N(t) is the poisson process, then what can you say about the time we will wait for the first event to occur? And the time we will wait for the nth event to occur? (Nov/Dec 2012) 6. Is poisson process is stationary? Justify. (Nov/Dec 2012) 7. If the initial state probability distribution of a markov chain is P (0) = and the transition probability matrix of the chain is, find the probability distribution of the chain after 2 steps. (Apr/May 2011) 8. Define continuous time random process(apr/may 2011) 9. Define Discrete state random process(apr/may 2011) 10. Find the transition probability matrix of the process represented by the state transition diagram(apr/may 2011) 11.Define transition probability matrix(nov/dec 2011) 12. Define Markov Process(Nov/Dec 2011) 13. Examine whether the Poisson process {x(t)} is stationary or not? (Nov/Dec 2010) 14. When is a Markov Chain, called homogeneous? (Nov/Dec 2010) 15. Is a poisson process a continuous time Markov Chain? Justify your answer(apr/may 2010) 16. Consider the markov chain consisting of the three states 0,1,2 and transition probability matrix P= it irreducible? Justify. (Apr/May 2010) 17. Define strict sense and wide sense stationary process. 18. A gambler has Rs.2.He bets Re.1 at a time andf wins Re.1 with probability ½.He stops playing if he loses Rs.2 or wins Rs.4.What is the transition probability matrix of the related Markov Chain? 19. What do you understand by stationary process? 20. Give an example of Ergodic process 21. State Chapman-Kolmogorov Theorem 22. Explain any two applications of a Binomial Process. 23. Define Binomial Process.Give an example for its sample function 24. State any four properties of poisson process. 25. State the properties of an ergodic process 26. If the transition probability matrix of a Markov Chain is, find the limiting distribution of the chain.

9 27. Determine whether the given matrix is irreducible or not P= (Nov/Dec 2009) 28. The one-step transition probability matrix of a Markov Chain with states (0,1) is given by P=.Is it irreducible Markov chain? (Nov/Dec 2009) 29. What will be the superposition of n independent poisson processes with respective average rates 1, 2,. n? 30. Define Markov process and Markov Chain PART-B 1. Let {x n } be a markov chain with states space {0,1,2} with initial probability vector p (0) =(0.7,0.2,0.1) and the one step transition probability matrix p=. Compute P(x 2 =3) and P(x 3 =2,x 2 =3,x 1 =3, x 0 =2).(May/June 2014) 2. If {x 1 (t)} and {x 2 (t)} are two independent poisson processes with parameters 1and 2 respectively, show that the process {x 1 (t)+x 2 (t)}is also a Poisson process..(may/june 2014) 3. Show that the random process X(t)=A is a WSS if A and are constants and is a uniformly distributed random variable in (0,2 ).(May/June 2014) 4. Consider a Markov chain with transition probability matrix P=.Find the limiting probabilities of the system.(may/june 2014) 5. The tpm of a markov chain {x n },n=1,2,3, having three states 1,2 and 3 is P= and the initial distribution is P (0) ={ } Find (1)P[X 2 =3] (2)P[X 3 =2, X 2 =3, X 1 =3, X 0 =2] (Nov/Dec 2013) 6. A salesman territory consista of three cities A,B and C. He never sells in the same city on successive days.if he sells in city A, then the next day he sells in city B. however if he sells in either B or C the next day he is twice as likely as to sell in city A as in other city.in the long run how often does he sell in each the cities? (Nov/Dec 2013) 7. Suppose that customers arrive at a bank according to a poisson process with mean rate of 3 per minute. Find the probability that during a time interval of two minutes (1) Exactly 4 customers arrive (2)More than 4 customers arrive (3)Less than 4 customers arrive (Nov/Dec 2013) 8. Prove that process {x(t)} whose probability distribution given by P[X(t)=n]= is not stationary. (Nov/Dec 2013) 9. Show that the process x(t)=a is WSS, if E(A)=E(B)=0, E(A 2 )=E(B 2 ) and E(AB)=0, where A and B are random variables. (May/June 2013)

10 11. A gambler has Rs.2 he bets Re.1 at a time and wins Re.1 with probability ½.He stops playing if he lose Rs.2 or wins Rs.4 (1) What is the tpm of the related markov chain? (2)what is the probability that he has lost his money at the end of 5 plays? (May/June 2013) 11. Find the nature of the states of the markov chain with the tpm P= (May/June 2013) 12. Prove that the difference of two independent poisson process is not a poisson process 13. Prove that the poisson process is a markov process(may/june 2013) 14. A fair die is tossed repeatedly. The maximum of the first n outcomes is denoted by X n. Is {X n :n=1,2,.} a markov chain? Why or why not? If it is a Markov chain, calculate its transition probability matrix. Specify the classes (Nov/Dec 2012) 15. An observer at a lake notices that when fish are caught, only 1 out of 9 trout is caught after another trout, with no other fish between, whereas 10 out of 11 non-trout are caught following non-trout, with no trout between.assuming that all fish are equally likely to be caught, what fraction of fish in the lake is trout? (Nov/Dec 2012) 16. The following is the tpm of markov chain with atate space {1,2,3,4,5}. Specify the classes and determine which classes are transient and which are recurrent. (Nov/Dec 2012) 17. For an English course there are four popular textbooks dominating the market. The English department of an institution allows its faculty to teach only from these 4 textbooks. Each year Prof. Rose Mary o donoghue adopts the same book she was using the previous year with probability The probabilities of her changing to any of the other 3 books are equal. Find the proportion of years Prof. o donoghue uses each book. (Nov/Dec 2012) 18. Show that random wide sense stationary process where A and process {X(t)} = A, -, is a B are independent random variables each of which has a value -2 with probability 1/3 and a value 1 with probability 2/3 (Apr/May 2011) 19. Derive probability distribution of poisson process and hence find its auto correlation function. (Apr/May 2011)

11 20. Find the limiting state probabilities associated with the following transition probability matrix. (Apr/May 2011) 21. Show that the difference of two independent poisson processes is not a poisson process. (Apr/May 2011) 22. Define poisson process and derive the poisson probability law. (Apr/May 2011) 23. A man either drives a car (or) catches a train to go to office each day. He never goes two days in a row by train but if he drives one day, then the next day he is just as likely to drive again as 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 (1) The probability that he takes a train on the third day and(2) The probability that he drives to work in the long run. (Nov/Dec 2011) 24. Show that the random process X(t)=A is a WSS if A and are constants and is a uniformly distributed random variable in (0,2 ). (Nov/Dec 2011) 25. If customers arrive at a counter in accordance with a poisson processes with a mean rate of 2 per minute, find the probability that the interval between 2 consecutive arrivals is (1) more than 1 min. (2) between 1 min and 2 min and (3) 4 min (or) less (Nov/Dec 2011) 26. An engineer analyzing a series of digital signals generated by a testing system observes that only 1 out of 15 highly distorted signals follow a highly distorted signal, with no recognizable signal between, whereas 20 out of 23 recognizable signals follow recognizable signals, with no highly distorted signal between. Given that only highly distorted signals are not recognizable, find the fraction of signals that are highly distorted. (Nov/Dec 2010) 27. Suppose that a mouse is moving inside the maze shown in the adjacent figure from one cell to another, in search of food. When at a cell,the mouse will move to one of the adjoining cells randomly. For n X n be the cell number the mouse will visit after having changed cells n times. Is { X n : n=0,1, } a markov chain? If so, write its state space and tpm. (Nov/Dec 2010) 28. Suppose that whether or not it rains today depends on previous weather conditions through the last two days, Show how this system may be analyzed by using a Markov Chain. How many states are needed?.(apr/may 2010) 29. Derive Chapman-Kolmogorov equations.(apr/may 2010) 30. Three out of every four trucks on the road are followed by a car, while only one out of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?.(apr/may 2010) UNIT-IV QUEUEING THEORY PART- A 1. A super market has a single cashier. During peak hours, customers arrive at a rate of 20 per hour.the average number of customers that can be serviced by the cashier is 24 per hour. Calculate the probability that the cashier is idle.(m/j2014) 2. State the steady state probabilities of the finite source queueing model represented by (M/M/R):(GD/K/K) (M/J2014) 3. Define series queues(n/d 2013) 4. Define open network(n/d 2013) 5. State Pollaczek-Khinchine formula(m/j 2013)

12 6. Define closed network of a queuing system. (M/J 2013) 7. Consider a random queue with 2 independent markovian servers. The situation at server 1 is just as in an M/M/1 model. What will be the type of queue in server 2? Why?(N/D 2012) 8. State jackson s theorem for an open network(n/d 2012) 9. State little s formula for a (M/M/1):(GD/N/ ) Queuing model.(m/j 2012) 10. Define steady state state and transient state in Queuing theory(m/j 2012) 11. When a M/G/1 queuing model will become a classic M/M/1 Queuing model. (M/J 2012) 12. State Pollazcek-Khintchine formula for the average number of customers in a M/G/1 queuing model(m/j 2012) 13. Arrival rate of telephone calls at a telephone booth is according to poisson distribution with no average time of 9 minutes between two consecutive arrivals. The length of a telephone call is assumed to be exponentially distributed with mean 3 minutes. Determine the probability that a person arriving at the both will have to wait. (A/M 2011) 14. Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity of the yard is limited to 4 trains find the probability that yard is empty. (A/M 2011) 15. Write the P-K formula(n/d 2011) 16. State the various discipline in queuinf model(n/d 2010) 17. Define various distribution in arrival at service mechanism. (N/D 2010) 18. Explain effective arrival rate in a finite queue capacity model(n/d 2010) 19. Draw the state transition rate diagram of an M/M/C queuing model(n/d 2010) 20. Suppose that customers arrive at a poisson rate of one per every 12 minutes and that the service time is exponential at a rate of one service per 8 minutes. What is the average number of customers in the system?(a/m 2010) 21. Define M/M/2 queueing model. Why the notation M is used? (A/M 2010) 22. What are the basic characteristics of a queueing system? 23. What is the probability that a customer has to wait more than 15 minutes to get his service completed in (M/M/1) : ( /FIFO) queue system if λ=6 per hour and µ=10 per hour? 24. What are the basic characteristics of Queueing process? 25. Give any two examples for series queueing situations. PART-B 1. Derive the governing equations for the (M/M/1):(GD/N/ ) queueing model and hence obtain the expression for the steady state probabilities and the average number of customers in the system.(m/j 2014) 2. Four customers are being run on the frontiers of the 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 /2. Find the average queue length at each counter (M/J 2014) 3. Derive the governing equations for the (M/M/c):(GD/ / ) queueing model and hence obtain the expression for the steady state probabilities and the average number of customers in the queue (M/J 2014)

13 4. Customers arrive at the express checkout lane in a super market in a poisson process with a rate of 15 per hour. The time to check out a customer is an exponential random variable with mean of 2 minutes. Find the average number of customers present. What is the expected waiting time for a customer in the system? (M/J 2014) 5. A tv repair man finds that the time spend on his job has an exponential distribution with mean 30 minutes. If he repair 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 the repairman s expected idle time each-day? How many are ahead of average set just brought?(n/d 2013) 6. Consider a single server queueing system with poisson input, exponential service times. Suppose the mean arrival rate is 3 calling units per hour, the expected service time is 0.24 hours and the maximum permissible number calling units in the system is two. Find the steady probability distribution of the number of calling units in the system and the expected number of calling units in the system. (N/D 2013) 7. A telephone exchange has two long distance operators. It is observed that, during the peak load distance calls arrive in a poisson fashion at an average rate of 15 per hour. The length of service on these calls is approximately exponentially distributed with mean length 5 minutes. Find (1) The probability a subscriber will have to wait for long distance call during the peak hours of the day. (2) If the subscribers will wait and are serviced in turn, what is the expected waiting time? (N/D 2013) 8. Customers arrive at a sales counter manned by a single person according to a poisson process with 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. (N/D 2013) 9. Derive (1) L S, average number of customers in the system (2) L q, average number of customers in the queue for the queueing model (M/M/1):(N/FIFO) (M/J 2013) 10. Customers arrive at oneman barber shop according to a poisson process with a mean inter arrival time of 20 minutes. Customers spend an average of 15 minutes in the barber chair. The service time is exponentially distributed. If an hour is used as a unit of time, then (1) What is the probability that a customer need not wait for a hair cut? (2) What is the expected number of customer in the barber shop and in the queue? (3) How much time can a customer expect to spend in the barber shop? (4) Find the average time that a customer spend in the queue. (5) Estimate the fraction of the day that the customer will be idle?(6) What is the probability that three will be 6 or more customer? (7) Estimate the percentage of customers who have to wait prior to getting into the barber s chair.(16 marks) (M/J 2013) 12. 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 will be busy? What is the average number of letters waiting to be typed? (Assume poisson arrivals and exponential service times) (M/J 2013) 12.Obtain the steady state probabilities of birth-death process. Also draw the transition graph (16 marks) (N/D 2012)

14 13.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 poisson process with a mean of 1 every 2 hrs. It takes for an unloading crew, on the average, 10 hrs to unload a tanker, the unloading time following an exponential distribution. Find (i) how many tankers are at the port on the average? (ii) how long does a tanker spend at the port on the average? (iii) what is the average arrival rate at the overflow facility?(16) (N/D 2012) 13. A supermarket has 2 girls running up sales at the counters. If the service time for each customer is exponential with mean 4 minutes and if people arrive in poisson fashion at the rate of 10 per hour, find the following: (1) What is the probability of having to wait for service? (2) What is the expected percentage of idle time for each girl? (3) What is the expected length of customer s waiting time? (M/J 2012) 15.Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacitu of the yard is limited to 5 trains, find the probability that the yard is empty and the average number of trains in the system, given that the inter arrival time and service time are following exponential distribution. (M/J 2012) 16. Customers arrive at a one window drive-in bank according to poisson distribution with mean 10 per hour. Service time per customer is exponential with mean 5 minutes. The space is front of window, including that for the serviced car can accommodate a maximum of three cars. Others cars can wait outside this space.(1) What is the probability that an arriving customer can drive directly to the space in front of the window? (2) What is the probability that an arriving customer will have to wait outside the indirected space? (3) How long is an arriving customer expected to wait before being served? (16 MARKS) (A/M 2011) 17. Show that for the (M/M/1):(FCFS/ / ), the distribution of waiting time in the system is w(t)=(µ- t>0 (A/M 2011) 18. Find the steady state solution for the multiserver M/M/C model and hence find L q,w q,w s and L S by using Little s formula. (16 marks) (A/M 2011) 19. Find the mean number of customers in the queue, system average waiting time in the queue and system of M/M/1 queueing model (16 MARKS) (N/D 2011) 20. 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 will be busy? What is the average number of letters waiting to be typed?(3) What is the average time a letter has to spend for waiting and for being typed?(4) what is the probability that a letter will take longer than 20 min. waiting to be typed and being typed?(16 marks) (N/D 2011,2010) 21. Customers arrive at a watch repair shop in a poisson process at a rate of one per every 10 minutes and the service time is exponential with mean 8 minutes. Find the average number of customer L S average waiting time a customer spends in the shop W S and the average time a customer spends in the watching for service W Q

15 22.If people arrive to purchase cinema tickets at the average rate of 6 per minute, it takes an average of 7.5 seconds to purchase a ticket. If a person arrives 2 min before the picture starts and it takes exactly 1.5 min to reach the correct seat after purchasing the ticket, (i) Can he expect to be seated for the start of the picture? (ii) what is the probability that her will be seated for the start of the picture? (iii) How early must he arrive in order to be 99% sure of being seated for the start of the picture?(16 marks)(n/d 2010) 23. Calculate any four measures of effectiveness of M/M/1 queueing model.(16 marks) (A/M 2010) UNIT-V NON-MARKOVIAN QUEUES AND QUEUE NETWORKS PART-A 1. State Pollazcek-Khincthine formula for the average number in the system in a M/G/1 queueing model and hence derive the same when the service time is constant with mean 1/µ ( M/ J 2014) 2. Distinguish between open and closed queueing network ( M/ J 2014) 3.Define markovian queueing models (N/D 2013) 4. Define series queues (N/D 2013) 5. What are the basic characteristics of a queueing system?(m/j 2013) 6. Give any two examples for series queueing situations 7. 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 L q,w q,w s 8. Classify the queueing networks 9. Define Arrival theorem 10. Write closed Jackson network 11. State transition diagram 12. Define tandem queue 13. Define Mean value Analysis 14. M/G/1 queueing system is Markovian. Comment on this statement (A/M 2010) 15. Define M/M/2 queueing model. Why the notation Mis used? (A/M 2010) 16. What do you mean by bottleneck of a network? 17. Consider a service facility with two sequential stations with respective service rates of 3/min and 4/min. The arrival rate is 2/ min. What is the average service time of the system, if the system could be approximated by a two stage tandem queue? 18. Explain effective arrival rate in a finite queue capacity model 19. Define various distribution in arrival at service mechanism 20. 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? PART-B 1. An automatic car wash facility operates with only one bay. Cars arrive according to a poisson distribution with a mean of 4 cars/hour and may wait in the facility s parking lot if the bay is busy. Find the average number of customers in the system in the service time is (1) constant and is equal to 10 minutes (2) uniformly distributed between 8 and 12 minutes.(m/j 2014)

16 2. An average of 120 students arrive each hour times are at the controller office to get their hall tickets.. To complete this process, a candidate must pass through three counters. Each counter consists of a single server. Service times at each counter are exponential with the following mean times: counter1,20 seconds; counter 2,15 seconds and counter 3,12 seconds. On the average, how many students will be present in the controller s office. 3. Consider a open queueing network with parameter values shown below: (1) Find the steady state distribution of the number of customers at facility 1, facility 2 and facility 3. (2) Find the expected total number of customers in the system. (3) Find the expected total waiting time for a customer (16 marks) 4. 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 (N/D 2013) 5. Derive Pollazcek-Khincthine formula (N/D 2013) 6. There are two salesman in a ration shop one incharge of billing and receiving payment and the other incharge of weighing and delievering the items. Due to limited availability of space, only one customer is allowed to enter the shop, that too when the billing clerk is free. The customer who has finished his billing job has to wait there until the delivery section becomes free. If customers arrive in accordance with a poisson process at rate 1 and the service times of two clerks are independent and have exponential rate of 3 and 2 find (1) the proportion of customers who enter the ration shop (2) the average number of customers in the shop. (3) the average amount of time that an entering customer spends in the shop (N/D 2013) 7. An automatic car wash facility operates with only one bay. Cars arrive according to a poisson distribution with a mean of 4 cars/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. If the service time for a car has uniform distribution between 8 and 12 minutes find: (1) The average number of cars waiting in the parking lot and (2) The average waiting time of a car in the parking lot. (N/D 2013) 8. An automatic car wash facility operates with only one bay. Cars arrive according to a poisson distribution with a mean of 4 cars/hour and may wait in the facility s parking lot if the bay is busy. The service time for all cars is constant and equal to 10 minutes. Determine L q,w q,w s (16 marks) (M/J 2013) 9. Consider a system of two servers where customers from outside the system arrive at sever 1 at a poisson rate 4 and at server 2 at a poisson rate 5. The service rates for server 1 and 2 are 8 and 10 respectively. A customer upon completion of service at server 1 is likely to go to server 2 or leave the system; whereas a departure from server 2 will go 25 percent of the time to server 1 and will depart the system otherwise. Determine the limiting probabilities W s, L s (16 marks) (M/J 2013) 10. Derive Pollazcek-Khincthine formula for M/G/1 queue(16 marks) (N/D 2012)

17 11. Consider two servers. An 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 station 1, half the customers leave the system and half go to server 2. After completing service at station 2, 3/4 of the customer complete service and ¼ return to server 1. Find the average time a customer spends in the system. (N/D 2012) 12. Consider a two stage twndem queue with external arrival rate to node 0. Let µ 0 and µ 1 be the service rates of the exponential servers at node 0 and 1 respectively. Arrival process is poisson. Model this system using a Markov chain and obtain the balance equations(n/d 2012) 13.An automatic car wash facility operates with only one bay. Cars arrive according to a poisson distribution with a mean of 4 cars/hour and may wait in the facility s parking lot if the bay is busy. The service time for all cars is constant and equal to 10 minutes. Determine (1) mean number of customers in the system L s (2) mean number of customers in the queue L q (3) mean waiting time of a customer in the system W s (4) mean waiting time of a customer in the queue W q (M/J 2012) 14. Derive the P-K formula for the (M/G/1) : (GD/ / ) queueing model and hence deduce that with the constant service time the P-K formula reduces to where and (M/J 2012) 15. For a open queueing network with three nodes 1,2 and 3, let customers arrive from outside the system to node j according to a poisson input process with parameters r j and let P ij denote the proportion of customers departing from facility i to facility j.given (r 1, r 2, r 3 )=(1,4,3) and P ij = (M/J 2012) 16. Derive the expected steady state system size for the single server queues with poisson input and general service (16 marks) ( A/M 2011) 17.Write short notes on : (i) Series Queues (ii) Open and Closed Queue networks (16 marks ) ( A/M 2011) ALL THE BEST