Exercises

1. The time T required to repair a machine is an exponentially distributed random

variable with mean 1

2 (hours).

(a) What is the probability that a repair time exceeds 1

2 hour?

(b) What is the probability that a repair takes at least 121

2 hours given that

its duration exceeds 12 hours?

2. Suppose that you arrive at a single-teller bank to find five other customers in

the bank, one being served and the other four waiting in line. You join the

end of the line. If the service times are all exponential with rate μ, what is the

expected amount of time you will spend in the bank?

3. Let X be an exponential random variable. Without any computations, tell

which one of the following is correct. Explain your answer.

(a) E[X2|X>1] = E[(X +1)2] (b) E[X2|X>1] = E[X2] + 1

(c) E[X2|X>1] = (1 +E[X])2

4. Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously.

A and B go directly to the clerks, and C waits until either A or

B leaves before he begins service. What is the probability that A is still in the

post office after the other two have left when

(a) the service time for each clerk is exactly (nonrandom) ten minutes?

(b) the service times are i with probability 1

3 , i = 1, 2, 3?

(c) the service times are exponential with mean 1/μ?

*5. If X is exponential with rate !, show that Y = [X] + 1 is geometric with parameter

p = 1 −e−!, where [x] is the largest integer less than or equal to x.

6. In Example 5.3 if server i serves at an exponential rate !i, i = 1, 2, show that

P{Smith is not last} =

!

!1

!1 +!2

“2

+

!

!2

!1 + !2

“2

*7. If X1 and X2 are independent nonnegative continuous random variables, show

that

P{X1 <X2|min(X1,X2) = t} =

r1(t)

r1(t) +r2(t)

where ri (t) is the failure rate function of Xi .

8. If X and Y are independent exponential random variables with respective rates

! and μ, what is the conditional distribution of X given thatX<Y?

9. Machine 1 is currently working. Machine 2 will be put in use at a time t from

now. If the lifetime of machine i is exponential with rate !i, i = 1, 2, what is

the probability that machine 1 is the first machine to fail?

*10. Let X and Y be independent exponential random variables with respective

rates ! and μ. Let M = min(X, Y). Find

(a) E[MX|M = X],

358 Introduction to Probability Models

(b) E[MX|M = Y],

(c) Cov(X,M).

11. Let X, Y1, . . . , Yn be independent exponential random variables; X having rate

!, and Yi having rate μ. Let Aj be the event that the j th smallest of these

n + 1 random variables is one of the Yi. Find p = P{X >maxi Yi }, by using

the identity

p = P(A1 ···An) = P(A1)P (A2|A1) ···P(An|A1 ···An−1)

Verify your answer when n = 2 by conditioning on X to obtain p.

12. If Xi, i = 1, 2, 3, are independent exponential random variables with rates !i ,

i = 1, 2, 3, find

(a) P{X1 <X2 <X3},

(b) P{X1 <X2|max(X1, X2, X3) = X3},

(c) E[maxXi |X1 <X2 <X3],

(d) E[maxXi ].

13. Find, in Example 5.10, the expected time until the nth person on line leaves

the line (either by entering service or departing without service).

14. I am waiting for two friends to arrive at my house. The time until A arrives is

exponentially distributed with rate !a, and the time until B arrives is exponentially

distributed with rate !b. Once they arrive, both will spend exponentially

distributed times, with respective rates μa and μb at my home before departing.

The four exponential random variables are independent.

(a) What is the probability that A arrives before and departs after B?

(b) What is the expected time of the last departure?

15. One hundred items are simultaneously put on a life test. Suppose the lifetimes

of the individual items are independent exponential random variables

with mean 200 hours. The test will end when there have been a total of 5

failures. If T is the time at which the test ends, find E[T ] and Var(T ).

16. There are three jobs that need to be processed, with the processing time of

job i being exponential with rate μi . There are two processors available, so

processing on two of the jobs can immediately start, with processing on the

final job to start when one of the initial ones is finished.

(a) Let Ti denote the time at which the processing of job i is completed.

If the objective is to minimize E[T1 + T2 + T3], which jobs should be

initially processed if μ1 <μ2 <μ3?

(b) Let M, called the makespan, be the time until all three jobs have been

processed. With S equal to the time that there is only a single processor

working, show that

2E[M] = E[S] +

#3

i=1

1/μi

For the rest of this problem, suppose that μ1 = μ2 = μ, μ3 = !. Also,

let P(μ) be the probability that the last job to finish is either job 1 or job

The Exponential Distribution and the Poisson Process 359

2, and let P(!) = 1 − P(μ) be the probability that the last job to finish

is job 3.

(c) Express E[S] in terms of P(μ) and P(!).

Let Pi,j (μ) be the value of P(μ) when i and j are the jobs that are

initially started.

(d) Show that P1,2(μ) ” P1,3(μ).

(e) If μ>! show that E[M] is minimized when job 3 is one of the jobs that

is initially started.

(f) If μ < ! show that E[M] is minimized when processing is initially

started on jobs 1 and 2.

17. A set of n cities is to be connected via communication links. The cost to

construct a link between cities i and j is Cij , i #= j. Enough links should be

constructed so that for each pair of cities there is a path of links that connects

them. As a result, only n − 1 links need be constructed. A minimal cost algorithm

for solving this problem (known as the minimal spanning tree problem)

first constructs the cheapest of all the

$n2

%

links. Then, at each additional stage

it chooses the cheapest link that connects a city without any links to one with

links. That is, if the first link is between cities 1 and 2, then the second link

will either be between 1 and one of the links 3, . . . , n or between 2 and one of

the links 3, . . . , n. Suppose that all of the

$n2

%

costs Cij are independent exponential

random variables with mean 1. Find the expected cost of the preceding

algorithm if

(a) n = 3,

(b) n = 4.

*18. Let X1 and X2 be independent exponential random variables, each having rate

μ. Let

X(1) = minimum(X1,X2) and X(2) = maximum(X1,X2)

Find

(a) E[X(1)],

(b) Var[X(1)],

(c) E[X(2)],

(d) Var[X(2)].

19. In a mile race between A and B, the time it takes A to complete the mile is

an exponential random variable with rate !a and is independent of the time it

takes B to complete the mile, which is an exponential random variable with

rate !b. The one who finishes earliest is declared the winner and receives

Re−”t if the winning time is t, where R and ” are constants. If the loser

receives 0, find the expected amount that runner A wins.

20. Consider a two-server system in which a customer is served first by server 1,

then by server 2, and then departs. The service times at server i are exponential

random variables with rates μi, i = 1, 2. When you arrive, you find server 1

free and two customers at server 2—customer A in service and customer B

waiting in line.

360 Introduction to Probability Models

(a) Find PA, the probability that A is still in service when you move over to

server 2.

(b) Find PB, the probability that B is still in the system when you move over

to server 2.

(c) Find E[T ], where T is the time that you spend in the system.

Hint: Write

T = S1 +S2 +WA +WB

where Si is your service time at server i,WA is the amount of time you wait

in queue while A is being served, and WB is the amount of time you wait in

queue while B is being served.

21. In a certain system, a customer must first be served by server 1 and then by

server 2. The service times at server i are exponential with rate μi, i = 1, 2.

An arrival finding server 1 busy waits in line for that server. Upon completion

of service at server 1, a customer either enters service with server 2 if that

server is free or else remains with server 1 (blocking any other customer from

entering service) until server 2 is free. Customers depart the system after being

served by server 2. Suppose that when you arrive there is one customer in the

system and that customer is being served by server 1. What is the expected

total time you spend in the system?

22. Suppose in Exercise 21 you arrive to find two others in the system, one being

served by server 1 and one by server 2. What is the expected time you spend

in the system? Recall that if server 1 finishes before server 2, then server 1’s

customer will remain with him (thus blocking your entrance) until server 2

becomes free.

*23. A flashlight needs two batteries to be operational. Consider such a flashlight

along with a set of n functional batteries—battery 1, battery 2, . . . , battery n.

Initially, battery 1 and 2 are installed. Whenever a battery fails, it is immediately

replaced by the lowest numbered functional battery that has not yet been

put in use. Suppose that the lifetimes of the different batteries are independent

exponential random variables each having rate μ. At a randomtime, call it T ,

a battery will fail and our stockpilewill be empty.At that moment exactly one

of the batteries—which we call battery X—will not yet have failed.

(a) What is P{X = n}?

(b) What is P{X = 1}?

(c) What is P{X = i}?

(d) Find E[T ].

(e) What is the distribution of T ?

24. There are two servers available to process n jobs. Initially, each server begins

work on a job. Whenever a server completes work on a job, that job leaves

the system and the server begins processing a new job (provided there are still

jobs waiting to be processed). Let T denote the time until all jobs have been

processed. If the time that it takes server i to process a job is exponentially

distributed with rate μi, i = 1, 2, find E[T ] and Var(T ).

The

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