EE 5375/7375 Random Processes

[Pages:2]EE 5375/7375 Random Processes

Homework #8

Problem 1. Poisson process For a Poisson counting process N (t), find the joint probability P (N (t1) = i, N (t2) = j) for t2 > t1, j i.

Problem 2. textbook problem 6.31 Suppose that a secretary receives calls that arrive according to a Poisson process with a rate of 10 calls per hour. What is the probability that no calls go unanswered if the secretary is away from the office for the first and last 15 minutes of an hour?

Problem 3. textbook problem 6.32 Customers arrive at a soft drink dispensing machine according to a Poisson process with rate . Suppose that each time a customer deposits money, the machine dispenses a soft drink with probability p. Find the PMF for the number of soft drinks dispensed in time t. (Assume the machine holds an infinite number of soft drinks.)

Problem 4. textbook problem 6.33 Noise impulses occur on a telephone line according to a Poisson process of rate . (a) Find the probablity that no impulses occur during the transmission of a message that is t seconds long. (b) Suppose that the message is encoded so that the errors caused by a single impulse can be corrected. What is the probability that a t-second message is either error-free or correctable?

Problem 5. textbook problem 6.34

Messages arrive at a computer from 2 telephone lines according to independent Poisson

processes with rates 1 and 2, respectively. (a) Find the probability that a message arrives first on line 1. (b) Find the pdf for the time until a message arrives on either line. (c) Find

the PMF for N (t), the total number of messages that arrive in an interval of length t. (d)

Generalize the result of part (c) for the "merging" of k independent Poisson processes of rate

1, . . . , k, respectively:

N (t) = N1(t) + ? ? ? + Nk(t)

Problem 6. textbook problem 6.45 Messages arrive at a message center according to a Poisson process of rate . Every hour the messages that have arrived during the previous hour are forwarded to their destination. Find the mean of the total time waited by all the messages that arrive during the hour. Hint: condition on the number of arrivals.

Problem 7. textbook problem 8.15 A critical part of a machine has an exponentially distributed lifetime with parameter . Suppose that n spare parts are initially in stock, and let N (t) be the number of spares left

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at time t. (a) Find pij(t) = P (N (s + t) = j|N (s) = i). (b) Find the transition probability matrix. (c) Find pj(t) = P (N (t) = j). Problem 8. textbook problem 8.16 A machine shop initially has n identical machines in operation. Assume that the time until breakdown for each machine is an exponentially distributed random variable with parameter . Let N (t) denote the number of machines in working order at time t. Repeat parts (a), (b), and (c) of the previous problem and show that N (t) is a binomial random variable with parameter p = e-t. Problem 9. textbook problem 8.17 A shop has n machines and one technician to repair them. A machine remains in the working state for an exponentially distributed time with parameter ?. The technician works on one machine at a time, and it takes him an exponentially distributed time of rate to repair each machine. Let X(t) be the number of working machines at time t. (a) Show that if X(t) = k, then the time until the next machine breakdown is an exponentially distributed random variable with rate k?. (b) Find the transition rate matrix [ij] and sketch the transition rate diagram for X(t). (c) Write the global balance equations and find the steady-state probabilities for X(t). Problem 10. textbook problem 8.18 A speaker alternates between periods of speech activity and periods of silence. Suppose that the former are exponentially distributed with parameter and the latter exponentially distributed with parameter . Consider a group of n independent speakers and let N (t) denote the number of speakers in speech activity at time t. (a) Find the transition rate diagram and the transition rate matrix for this system. (b) Write the global balance equations and find the steady-state PMF.

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