Subject Code : MA2262
Subject name : Probability and Queuing theory
ANNA UNIVERSITY - April/MAY 2010
Fourth Semester
MA2262 Probability and Queuing theory
(Common to Fourth Semester B.Tech IT)
Regulation 2008
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 showexcessive 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<1 f(x,y)= 0 ,otherwise
Find P(X + Y<1)
5. Define (a).Markov chain (b).Wide-Sense stationary process.
6. State any two properties of the Poisson process
7. In the usual notation of an M/M/I queuing system, if λ = 3/hour and µ=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 Model
10.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 distributed with mean 2 . What is the probability that a repair takes at least 10 hours 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)5 Find E(X),Var(X) and P(X=2). (8)
(OR)
11.(b).(i).Find the moment generating function of a poisson variable and hence obtain its mean and variance. (8)
(ii). A man draws 3 balls from an urn containing 5 white and and 7 black balls. He gets Rs.10 for each white ball and Rs.5 for each black ball. Find his Expectation. (8)
12.(a).(i).(X, Y) is a two dimensional random variable uniformly distributed over the triangular region R bounded by y = 0 , x = 3 , y = 4/3 x. Find the correlation coefficient (8)
(ii). Suppose that orders at a restaurant are i.i.d random variables with mean µ=Rs. 8 and standard deviation s=Rs. 2. Estimate (1)the probability that first 100 customers spend a total of more than Rs.840 (2).P( 780 < x <820). (8)
(OR)
12.(b).(i). Let X and Y be non-negative continuous random variables having the joint probability density function f(x,y)=4xy e-(x2+y2) , x>0 , y>0 Find 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 by g(x,y) = e-(x+y) , x>=0, y>=0 then
(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)
(5) Find P(X>Y). (8)
13.(a).(i).Let {Xn;n=1,2,3…} be a Markov chain on the space S={1,2,3} with one step transition probability matrix
0 1 0
p = ½ 0 ½
1 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 mean rate of 2 per minute, find the probability that the interval between 2 consecutive arrivals is (1) more than 1 minute (2) between 1 and 2
minutes (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 which E(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 each step of the process, a marble is selected from each urn and the 2 marbles selected are interchanged. The state of the relaxed Markov chain is the number of red balls in A after the interchange. What is the 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 that messages arrives according to a Poisson process at a rate of one message every 4 milliseconds and suppose that message transmission times are exponentially distributed with mean 3ms. Find the mean number of messages in the system and the mean total delay in the system. What percentage increase in arrival rate results in a doubling of the above mean total delay? (8)
(ii). Discuss the M/M/1 queuing system finite capacity and obtain its steady-state probabilities and the mean number of customers in the system. (8)
(OR)
14.(b).(i).A petrol pump station has 2 pumps. The service times follow the exponential distribution with mean of 4 minutes and cars arrive for service is a Poisson process at the rate of 10 cars per hour. Find theprobability that a customer has to wait for service. What is the probability that the pumps remain idle? (8)
(ii) There are 3 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? (8)
15. (a)(i). 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 parking lot if the bay is busy. If the service time for the cars is constant and equal to 10 min, determine (1).mean number of customers in the system, (2).mean number of customers 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 2 sequential stations with respective service rates of 2 per hour and 3 per hour. The cumulative failure rate of all the machines is 1 per hour. Assuming that the system behavior may be approximated by the 2-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 S1 and 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 or S2 .The decision making takes on the average 30 seconds in an exponential fashion. Nearly 55% of the customers go to S1, that consists of 3 parallel servers and the rest go to S2, that consist of 7 parallel servers. The service times at S1, are exponential with a mean of 6 minutes and those at S2 with a mean of 20 minutes. About 2% of customers, on finishing service at S1 go to S2 and about 1% of customers, on finishing service at S2 go to S1. Find the average queuesizes in front of each node and the total average time a customer pends in the service centre. (16)
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