MA2262 Probability and Queuing theory(Common to Fourth Semester B.Tech IT)Time: Three Hours Maximum Marks: 100Answer all the questions
PART A-(10 X 2 = 20 marks)
1. Given the probability density function f (x) = k / (1 + x2), - ∞ <x <∞, Find k
and C.D.F. F (x).
2. If the probability is 0.10 that a certain kind of measuring device will show
excessive drift, what is the probability that the fifth measuring device
tested will be the first show excessive drift? Find its expected value also.
3. If X has mean 4 and variance 9, while Y has mean -2 and variance 5, and
the two are independent, find (a). E (XY) (b). E (XY2)
4.Let X and Y be continuous RVs with J.p.d.f
f (x, y) = 2xy +3 / 2y2, 0 <1 0 = "0" br = "br" f = "f" find = "find" otherwise = "otherwise" p = "p" x = "x "y =" y "> 5. Define (a). Markov chain (b). Wide-Sense stationary process.6. State any two properties of the Poisson process7. In the usual notation of an M / M / I queuing system, if λ = 3/hour andμ = 4/hour, find P (X = 5) where X is the number of customers in the system.8. Find P (X = c + n) for an M / M / C queuing system.9. Write the P - K Formula in M/G/1 Queuing Model10.Write the balance equation for the closed Jackson Network.
PART B-(5 X 16 = 80 marks)
11a (i). The time required to repair a machine is exponentially distributedwith mean 2. What is the probability that a repair takes at least 10hours given that its duration exceeds 9 hours? (8)(Ii). A discrete R.V. X has moment generating function MX (t) = (1/4 +3 / 4e t) 5Find E (X), Var (X) and P (X = 2). (8)(OR)11. (B). (I). Find the moment generating function of a poisson variable andhence obtain its mean and variance. (8)(Ii). A man draws 3 balls from an urn containing 5 white and and 7black balls. He gets Rs.10 for each white ball and Rs.5 for each blackball. Find his Expectation. (8)
12. (A). (I). (X, Y) is a two dimensional random variable uniformly distributedover the triangular region R bounded by y = 0, x = 3, y = 4/3 x. Findthe correlation coefficient (8)(Ii). Suppose that orders at a restaurant are iid random variables withmean μ = Rs. 8 and standard deviation s = Rs. 2. Estimate (1) theprobability that first 100 customers spend a total of more thanRs.840 (2). P (780 <x <820 br = "br"> (OR)12. (B). (I). Let X and Y be non-negative continuous random variables havingthe joint probability density functionf (x, y) = 4xy e-(x2 + y2), x> 0, y> 0Find the p.d.f. of U = √ (x2 + y2). (8)(Ii). If the joint p.d.f. of X and Y is given by X and Y is given byg (x, y) = e-(x + y), x> = 0, y> = 0then(1) find the m.p.d.f. of X(2) find the m.p.d.f of Y.(3) Are X and Y independent RVs? Explain?(4) Find P (X> 2, Y <4 br = "br"> (5) Find P (X> Y). (8)
13. (A). (I). Let {Xn; n = 1,2,3 ...} be a Markov chain on the spaceS = {1,2,3} with one step transition probability matrix0 1 0p = ½ 0 ½1 0 0(1). Sketch the transition diagram.(2). Is the chain irreducible? Explain(3). Is the chain Ergodic? Explain. (8)(Ii) If the customers arrive in accordance with Poisson process, with meanrate of 2 per minute, find the probability that the interval between 2consecutive arrivals is (1) more than 1 minute (2) between 1 and 2minutes (3) less than 4 minutes. (8)(OR)13. (B). (I). Consider a random process X (t) defined by X (t) = Ucost + (V +1) sint,where U and independent random variables for whichE (U) = E (V) = 0; E (U2) = E (V2) = 1.(1). Find the auto-covariance function of X (t)(2). Is X (t) wide-sense stationary? Explain your answer. (8)(Ii). Ther are 2 white marbles in urn A and 4 red marbles in urn B. At eachstep of the process, a marble is selected from each urn and the 2marbles selected are interchanged. The state of the relaxed Markovchain is the number of red balls in A after the interchange. What isthe probability that there are 2 red balls in urn A (i) after 3 steps and(Ii) in the long run? (8)
14. (A). (I). A concentrator receives messages from a group of terminals andtransmits them over a single transmission line. Suppose thatmessages arrives according to a Poisson process at a rate of onemessage every 4 milliseconds and suppose that message transmissiontimes are exponentially distributed with mean 3ms. Find the meannumber of messages in the system and the mean total delay in thesystem. What percentage increase in arrival rate results in a doublingof the above mean total delay? (8)(Ii). Discuss the M/M/1 queuing system finite capacity and obtain itssteady-state probabilities and the mean number of customers in thesystem. (8)(OR)
14. (B). (I). A petrol pump station has 2 pumps. The service times follow theexponential distribution with mean of 4 minutes and cars arrive forservice is a Poisson process at the rate of 10 cars per hour. Find theprobability that a customer has to wait for service. What is theprobability that the pumps remain idle? (8)(Ii) There are 3 typists in an office. Each typist can type an average of 6letters per hour. If letters arrive for being typed at the rate of 15 lettersper hour, what fraction of time all the typists will be busy? What isthe average number of letters waiting to be typed? (8)
15. (A) (i). Automatic car wash facility operates with only one bay. Carsarrive according to a Poisson process, with mean of 4 cars per hourand may wait in the facility parking lot if the bay is busy. If theservice time for the cars is constant and equal to 10 min, determine(1). Mean number of customers in the system, (2). Mean number ofcustomers in the queue (3). mean waiting time in the system (4). meanwaiting time in the queue. (8)(Ii) A repair facility shared by a large number of machines has 2sequential stations with respective service rates of 2 per hour and 3 perhour. The cumulative failure rate of all the machines is 1 per hour.Assuming that the system behavior may be approximated by the2-stage tandem queue, find(1) the average repair time including the waiting time.(2) the probability that both the service stations are idle and(3) the bottleneck of the repair facility. (8)(OR)15 (b). Customers arrive at a service centre consisting of 2 service points S1and S2 at a Poisson rate of 35/hour and form a queue at the entrance.On studying the situation at the centre, they decide to go to either S1 orS2. The decision making takes on the average 30 seconds in anexponential fashion. Nearly 55% of the customers go to S1, thatconsists of 3 parallel servers and the rest go to S2, that consist of 7parallel servers. The service times at S1, are exponential with a meanof 6 minutes and those at S2 with a mean of 20 minutes. About 2% ofcustomers, on finishing service at S1 go to S2 and about 1% ofcustomers, on finishing service at S2 go to S1. Find the average queuesizes in front of each node and the total average time a customerspends in the service centre. (16)
PART A-(10 X 2 = 20 marks)
1. Given the probability density function f (x) = k / (1 + x2), - ∞ <x <∞, Find k
and C.D.F. F (x).
2. If the probability is 0.10 that a certain kind of measuring device will show
excessive drift, what is the probability that the fifth measuring device
tested will be the first show excessive drift? Find its expected value also.
3. If X has mean 4 and variance 9, while Y has mean -2 and variance 5, and
the two are independent, find (a). E (XY) (b). E (XY2)
4.Let X and Y be continuous RVs with J.p.d.f
f (x, y) = 2xy +3 / 2y2, 0 <1 0 = "0" br = "br" f = "f" find = "find" otherwise = "otherwise" p = "p" x = "x "y =" y "> 5. Define (a). Markov chain (b). Wide-Sense stationary process.6. State any two properties of the Poisson process7. In the usual notation of an M / M / I queuing system, if λ = 3/hour andμ = 4/hour, find P (X = 5) where X is the number of customers in the system.8. Find P (X = c + n) for an M / M / C queuing system.9. Write the P - K Formula in M/G/1 Queuing Model10.Write the balance equation for the closed Jackson Network.
PART B-(5 X 16 = 80 marks)
11a (i). The time required to repair a machine is exponentially distributedwith mean 2. What is the probability that a repair takes at least 10hours given that its duration exceeds 9 hours? (8)(Ii). A discrete R.V. X has moment generating function MX (t) = (1/4 +3 / 4e t) 5Find E (X), Var (X) and P (X = 2). (8)(OR)11. (B). (I). Find the moment generating function of a poisson variable andhence obtain its mean and variance. (8)(Ii). A man draws 3 balls from an urn containing 5 white and and 7black balls. He gets Rs.10 for each white ball and Rs.5 for each blackball. Find his Expectation. (8)
12. (A). (I). (X, Y) is a two dimensional random variable uniformly distributedover the triangular region R bounded by y = 0, x = 3, y = 4/3 x. Findthe correlation coefficient (8)(Ii). Suppose that orders at a restaurant are iid random variables withmean μ = Rs. 8 and standard deviation s = Rs. 2. Estimate (1) theprobability that first 100 customers spend a total of more thanRs.840 (2). P (780 <x <820 br = "br"> (OR)12. (B). (I). Let X and Y be non-negative continuous random variables havingthe joint probability density functionf (x, y) = 4xy e-(x2 + y2), x> 0, y> 0Find the p.d.f. of U = √ (x2 + y2). (8)(Ii). If the joint p.d.f. of X and Y is given by X and Y is given byg (x, y) = e-(x + y), x> = 0, y> = 0then(1) find the m.p.d.f. of X(2) find the m.p.d.f of Y.(3) Are X and Y independent RVs? Explain?(4) Find P (X> 2, Y <4 br = "br"> (5) Find P (X> Y). (8)
13. (A). (I). Let {Xn; n = 1,2,3 ...} be a Markov chain on the spaceS = {1,2,3} with one step transition probability matrix0 1 0p = ½ 0 ½1 0 0(1). Sketch the transition diagram.(2). Is the chain irreducible? Explain(3). Is the chain Ergodic? Explain. (8)(Ii) If the customers arrive in accordance with Poisson process, with meanrate of 2 per minute, find the probability that the interval between 2consecutive arrivals is (1) more than 1 minute (2) between 1 and 2minutes (3) less than 4 minutes. (8)(OR)13. (B). (I). Consider a random process X (t) defined by X (t) = Ucost + (V +1) sint,where U and independent random variables for whichE (U) = E (V) = 0; E (U2) = E (V2) = 1.(1). Find the auto-covariance function of X (t)(2). Is X (t) wide-sense stationary? Explain your answer. (8)(Ii). Ther are 2 white marbles in urn A and 4 red marbles in urn B. At eachstep of the process, a marble is selected from each urn and the 2marbles selected are interchanged. The state of the relaxed Markovchain is the number of red balls in A after the interchange. What isthe probability that there are 2 red balls in urn A (i) after 3 steps and(Ii) in the long run? (8)
14. (A). (I). A concentrator receives messages from a group of terminals andtransmits them over a single transmission line. Suppose thatmessages arrives according to a Poisson process at a rate of onemessage every 4 milliseconds and suppose that message transmissiontimes are exponentially distributed with mean 3ms. Find the meannumber of messages in the system and the mean total delay in thesystem. What percentage increase in arrival rate results in a doublingof the above mean total delay? (8)(Ii). Discuss the M/M/1 queuing system finite capacity and obtain itssteady-state probabilities and the mean number of customers in thesystem. (8)(OR)
14. (B). (I). A petrol pump station has 2 pumps. The service times follow theexponential distribution with mean of 4 minutes and cars arrive forservice is a Poisson process at the rate of 10 cars per hour. Find theprobability that a customer has to wait for service. What is theprobability that the pumps remain idle? (8)(Ii) There are 3 typists in an office. Each typist can type an average of 6letters per hour. If letters arrive for being typed at the rate of 15 lettersper hour, what fraction of time all the typists will be busy? What isthe average number of letters waiting to be typed? (8)
15. (A) (i). Automatic car wash facility operates with only one bay. Carsarrive according to a Poisson process, with mean of 4 cars per hourand may wait in the facility parking lot if the bay is busy. If theservice time for the cars is constant and equal to 10 min, determine(1). Mean number of customers in the system, (2). Mean number ofcustomers in the queue (3). mean waiting time in the system (4). meanwaiting time in the queue. (8)(Ii) A repair facility shared by a large number of machines has 2sequential stations with respective service rates of 2 per hour and 3 perhour. The cumulative failure rate of all the machines is 1 per hour.Assuming that the system behavior may be approximated by the2-stage tandem queue, find(1) the average repair time including the waiting time.(2) the probability that both the service stations are idle and(3) the bottleneck of the repair facility. (8)(OR)15 (b). Customers arrive at a service centre consisting of 2 service points S1and S2 at a Poisson rate of 35/hour and form a queue at the entrance.On studying the situation at the centre, they decide to go to either S1 orS2. The decision making takes on the average 30 seconds in anexponential fashion. Nearly 55% of the customers go to S1, thatconsists of 3 parallel servers and the rest go to S2, that consist of 7parallel servers. The service times at S1, are exponential with a meanof 6 minutes and those at S2 with a mean of 20 minutes. About 2% ofcustomers, on finishing service at S1 go to S2 and about 1% ofcustomers, on finishing service at S2 go to S1. Find the average queuesizes in front of each node and the total average time a customerspends in the service centre. (16)
