ISyE 6650 Final ExamSpring 2005

NamePlease be neat and show all your work so that I can give you partial credit.GOOD LUCK AND HAVE A WONDERFUL SUMMER.

Question 1Question 2Question 3Question 4Question 5Total

1

(10) 1. a. Let {N(t) : t ≥ 0} be a Poisson process with rate λ. CalculateE[N(t)N(t + s)].

(10) b. Do interarrival times of a non-homogeneous (non-stationary) Poissonprocess have exponential distribution? Justify your answer.

2

(10) 2. a Let {N(t) : t ≥ 0} be a renewal process with Sn denoting the time ofthe nth renewal. Is it true that N(t) < n if and only if Sn > t? Justify youranswer.

(10) b. Consider a single server bank in which potential customers arrive withrespect to a Poisson process of rate λ. However, an arrival only enters the bankif the server is free when he arrives. Let G denote the service time distribution.At what rate do customers enter the bank?

3

(20) 3. Jobs arrive at a processing center according to a Poisson process ofrate λ/day. However, the center has waiting space for only N jobs and so anarriving job finding N others waiting goes away. At most 1 job per day canbe processed, and the processing of the job must start at the beginning of theday. Thus, if there are any jobs waiting for processing at the beginning of theday, then one of them is processed that day, and if no jobs are waiting at thebeginning of the day then no jobs are processed that day. Let Xn denote thenumber of jobs at the center at the beginning of day n.(10) a. Is {Xn : n ≥ 0} a Markov chain? If it is, write down the state spaceand the probability transition matrix.

(10) b. If {Xn : n ≥ 0} is a Markov chain, classify its states.

4

(20) 4. Consider a system having two independent M/M/1 queueing stations,each with its own waiting line. Let the arrival rate to each station be λ/2 andthe service rate of each server be µ. We now pool two stations together andthus form an M/M/2 queueing system with one waiting line. The arrival rateto the pooled system is therefore, λ, and the service rate of each server is µ.Assuming λ < 2µ, compare L and Lq for these two systems.

5

(20) 5. Dave repeatedly tosses three fair dice together. He stops only if thesame number appears on exactly two dice.

(5) a. What is the probability that he stops after the fifth toss?

(15) b. What is the expected number of tosses he will make?

6