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theory of probability

Questions and Answers of Theory Of Probability

31. Consider a network of three stations. Customers arrive at stations 1, 2, 3 in accordance with Poisson processes having respective rates, 5, 10, 15. The service times at the three stations are
30. For the tandem queue model verify that Pn,m = (λ/μ1)n(1 − λ/μ1)(λ/μ2)m(1 − λ/μ2)satisfies the balance equation (8.15).
29. Potential customers arrive to a single server hair salon according to a Poisson process with rate λ. A potential customer who finds the server free enters the system; a potential customer who
28. Let D denote the time between successive departures in a stationary M/M/1 queue with λ
27. Consider a single-server exponential system in which ordinary customers arrive at a rate λ and have service rate μ. In addition, there is a special customer who has a service rate μ1. Whenever
26. In a queue with unlimited waiting space, arrivals are Poisson (parameter λ)and service times are exponentially distributed (parameter μ). However, the server waits until K people are present
25. Poisson (λ) arrivals join a queue in front of two parallel servers A and B, having exponential service rates μA and μB (see Figure 8.4). When the system is empty, arrivals go into server A
24. Reconsider Exercise 23, but this time suppose that a customer that is in the system when a breakdown occurs remains there while the server is being fixed.In addition, suppose that new arrivals
23. Consider the M/M/1 system in which customers arrive at rate λ and the server serves at rate μ. However, suppose that in any interval of length h in which the server is busy there is a
22. Customers arrive at a single-server station in accordance with a Poisson process with rate λ. All arrivals that find the server free immediately enter service. All service times are
21. Suppose in Exercise 20 we want to find out the proportion of time there is a type 1 customer with server 2. In terms of the long-run probabilities given in Exercise 20, what is(a) the rate at
20. There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate λ1 and λ2. There are two servers. A type 1 arrival will
19. The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate λ1, and during
18. Customers arrive at a two-server system at a Poisson rate λ. An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the
17. Customers arrive at a two-server station in accordance with a Poisson process with a rate of two per hour. Arrivals finding server 1 free begin service with that server. Arrivals finding server 1
16. Customers arrive at a two-server system according to a Poisson process having rate λ = 5. An arrival finding server 1 free will begin service with that server.An arrival finding server 1 busy
15. Consider a sequential-service system consisting of two servers, A and B.Arriving customers will enter this system only if server A is free. If a customer does enter, then he is immediately served
14. Customers arrive at a single-service facility at a Poisson rate of 40 per hour.When two or fewer customers are present, a single attendant operates the facility, and the service time for each
13. A supermarket has two exponential checkout counters, each operating at rate μ. Arrivals are Poisson at rate λ. The counters operate in the following way:(i) One queue feeds both counters.
12. Exponential queueing systems in which the state is the number of customers in the system are known as birth and death queueing systems. For such a system, let λn denote the rate at which a new
11. Consider a single-server queue with Poisson arrivals and exponential service times having the following variation: Whenever a service is completed a departure occurs only with probability α.
10. A group of m customers frequents a single-server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon
9. A facility produces items according to a Poisson process with rate λ. However, it has shelf space for only k items and so it shuts down production whenever k items are present. Customers arrive
8. A group of n customers moves around among two servers. Upon completion of service, the served customer then joins the queue (or enters service if the server is free) at the other server. All
7. Show that W is smaller in an M/M/1 model having arrivals at rate λ and service at rate 2μ than it is in a two-server M/M/2 model with arrivals at rateλ and with each server at rate μ. Can you
6. Two customers move about among three servers. Upon completion of service at server i, the customer leaves that server and enters service at whichever of the other two servers is free. (Therefore,
5. It follows from Exercise 4 that if, in the M/M/1 model, W∗Q is the amount of time that a customer spends waiting in queue, then W∗Q =0, with probability 1 − λ/μExp(μ − λ), with
4. Suppose that a customer of the M/M/1 system spends the amount of time x > 0 waiting in queue before entering service.(a) Show that, conditional on the preceding, the number of other customers that
3. The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of$3 per hour. Alice, who gives service at an
2. Machines in a factory break down at an exponential rate of six per hour.There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost
1. For the M/M/1 queue, compute(a) the expected number of arrivals during a service period and(b) the probability that no customers arrive during a service period.Hint: “Condition.”
57. Let h(x) = P{T i=1 Xi > x} where X1,X2,... are independent random variables having distribution function Fe and T is independent of the Xi and has probability mass function P{T = n} = ρn(1 −
56. Random digits, each of which is equally likely to be any of the digits 0 through 9, are observed in sequence.(a) Find the expected time until a run of 10 distinct values occurs.(b) Find the
55. A coin that comes up heads with probability 0.6 is continually flipped. Find the expected number of flips until either the sequence thht or the sequence ttt occurs, and find the probability that
54. Let Xi, i 1, be independent random variables with pj = P{X = j }, j 1. If pj = j/10, j = 1, 2, 3, 4, find the expected time and the variance of the number of variables that need be observed until
53. Write a program to approximate m(t) for the interarrival distribution F ∗ G, where F is exponential with mean 1 and G is exponential with mean 3.
52. Let Xi, i = 1, 2,..., be the interarrival times of the renewal process {N(t)}, and let Y , independent of the Xi, be exponential with rate λ.(a) Use the lack of memory property of the
51. In 1984 the country of Morocco in an attempt to determine the average amount of time that tourists spend in that country on a visit tried two different sampling procedures. In one, they
50. To prove Equation (7.24), define the following notation:Xj i ≡ time spent in state i on the j th visit to this state;Ni(m) ≡ number of visits to state i in the first m transitions In terms of
49. Consider a renewal process having the gamma (n,λ) interarrival distribution, and let Y(t) denote the time from t until the next renewal. Use the theory of semi-Markov processes to show that lim
48. A taxi alternates between three different locations. Whenever it reaches location i, it stops and spends a random time having mean ti before obtaining another passenger, i = 1, 2, 3. A passenger
47. In a semi-Markov process, let tij denote the conditional expected time that the process spends in state i given that the next state is j .
46. Consider a semi-Markov process in which the amount of time that the process spends in each state before making a transition into a different state is exponentially distributed. What kind of
45. Consider a system that can be in either state 1 or 2 or 3. Each time the system enters state i it remains there for a random amount of time having mean μi and then makes a transition into state
44. An airport shuttle bus picks up all passengers waiting at a bus stop and drops them off at the airport terminal; it then returns to the stop and repeats the process. The times between returns to
43. Consider a renewal process having interarrival distribution F such that F(x) ¯ = 1 2 e−x + 1 2 e−x/2, x> 0 That is, interarrivals are equally likely to be exponential with mean 1 or
42. For an interarrival distribution F having mean μ, we defined the equilibrium distribution of F, denoted Fe, by?
41. Each time a certain machine breaks down it is replaced by a new one of the same type. In the long run, what percentage of time is the machine in use less than one year old if the life
40. Three marksmen take turns shooting at a target. Marksman 1 shoots until he misses, then marksman 2 begins shooting until he misses, then marksman 3 until he misses, and then back to marksman 1,
39. A system consists of two independent machines that each functions for an exponential time with rate λ. There is a single repairperson. If the repairperson is idle when a machine fails, then
38. A truck driver regularly drives round trips from A to B and then back to A.Each time he drives from A to B, he drives at a fixed speed that (in miles per hour)is uniformly distributed between 40
37. There are three machines, all of which are needed for a system to work.Machine i functions for an exponential time with rate λi before it fails, i = 1, 2, 3.When a machine fails, the system is
36. Each of n skiers continually, and independently, climbs up and then skis down a particular slope. The time it takes skier i to climb up has distribution Fi, and it is independent of her time to
35. Satellites are launched according to a Poisson process with rate λ. Each satellite will, independently, orbit the earth for a random time having distribution F. Let X(t) denote the number of
34. An M/G/∞ queueing system is cleaned at the fixed times T, 2T, 3T,... .All customers in service when a cleaning begins are forced to leave early and a cost C1 is incurred for each customer.
33. In Example 7.14, find the long run proportion of time that the server is busy.
32. Determine the long-run proportion of time that XN(t)+1 < c.
31. If A(t) and Y(t) are, respectively, the age and the excess at time t of a renewal process having an interarrival distribution F, calculate P{Y(t) > x|A(t) = s}
30. For a renewal process, let A(t) be the age at time t. Prove that if μ < ∞, then with probability 1 A(t)t → 0 as t → ∞
29. Consider a single-server queueing system in which customers arrive in accordance with a renewal process. Each customer brings in a random amount of work, chosen independently according to the
28. In Example 7.15, what proportion of the defective items produced is discovered?
27. A machine consists of two independent components, the ith of which functions for an exponential time with rate λi. The machine functions as long as at least one of these components function.
26. Consider a train station to which customers arrive in accordance with a Poisson process having rate λ. A train is summoned whenever there are N customers waiting in the station, but it takes K
25. Suppose in Example 7.13 that the arrival process is a Poisson process and suppose that the policy employed is to dispatch the train every t time units.(a) Determine the average cost per unit
24. Wald’s equation can also be proved by using renewal reward processes. Let N be a stopping time for the sequence of independent and identically distributed random variables Xi, i 1.(a) Let N1 =
23. If H is the uniform distribution over (2, 8) and if C1 = 4, C2 = 1, and R(T ) = 4 − (T/2), then what value of T minimizes Ms. Jones’ long-run average cost in Exercise 22?
22. The lifetime of a car has a distribution H and probability density h.Ms. Jones buys a new car as soon as her old car either breaks down or reaches the age of T years. A new car costs C1 dollars
21. Consider a single-server bank for which customers arrive in accordance with a Poisson process with rate λ. If a customer will enter the bank only if the server is free when he arrives, and if
20. For a renewal reward process consider Wn = R1 + R2 +···+ Rn X1 + X2 +···+ Xn where Wn represents the average reward earned during the first n cycles. Show that Wn → E[R]/E[X] as n → ∞.
19. For the renewal process whose interarrival times are uniformly distributed over (0, 1), determine the expected time from t = 1 until the next renewal.
18. Compute the renewal function when the interarrival distribution F is such that 1 − F(t) = pe−μ1t + (1 − p)e−μ2t
17. In Example 7.7, suppose that potential customers arrive in accordance with a renewal process having interarrival distribution F. Would the number of events by time t constitute a (possibly
16. A deck of 52 playing cards is shuffled and the cards are then turned face up one at a time. Let Xi equal 1 if the ith card turned over is an ace, and let it be 0 otherwise, i = 1,..., 52. Also,
15. Consider a miner trapped in a room that contains three doors. Door 1 leads him to freedom after two days of travel; door 2 returns him to his room after a four-day journey; and door 3 returns him
14. Wald’s equation can be used as the basis of a proof of the elementary renewal theorem. Let X1,X2,... denote the interarrival times of a renewal process and let N(t) be the number of renewals by
13. Let X1, X2,... be a sequence of independent random variables. The nonnegative integer valued random variable N is said to be a stopping time for the sequence if the event {N = n} is independent
12. Events occur according to a Poisson process with rate λ. Any event that occurs within a time d of the event that immediately preceded it is called a d-event.For instance, if d = 1 and events
11. A renewal process for which the time until the initial renewal has a different distribution than the remaining interarrival times is called a delayed (or a general)renewal process. Prove that
10. Consider a renewal process with mean interarrival time μ. Suppose that each event of this process is independently “counted” with probability p. Let NC(t)denote the number of counted events
9. A worker sequentially works on jobs. Each time a job is completed, a new one is begun. Each job, independently, takes a random amount of time having distribution F to complete. However,
8. A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F
7. Mr. Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate
6. Consider a renewal process {N(t), t 0} having a gamma (r,λ) interarrival distribution. That is, the interarrival density is f (x) = λe−λx (λx)r−1(r − 1)! , x> 0
5. Let U1, U2,... be independent uniform (0, 1) random variables, and define N by N = min{n: U1 + U2 +···+ Un > 1}What is E[N]?
4. Let {N1(t), t 0} and {N2(t), t 0} be independent renewal processes. Let N(t) = N1(t) + N2(t).(a) Are the interarrival times of {N(t), t 0} independent?(b) Are they identically distributed?(c) Is
3. If the mean-value function of the renewal process {N(t), t 0} is given by m(t) = t/2, t 0, what is P{N(5) = 0}?
2. Suppose that the interarrival distribution for a renewal process is Poisson distributed with mean μ. That is, suppose P{Xn = k} = e−μ μk k!, k = 0, 1,...(a) Find the distribution of Sn.(b)
1. Is it true that(a) N(t) < n if and only if Sn > t?(b) N(t) n if and only if Sn t?(c) N(t) > n if and only if Sn < t?
42. (a) Show that Approximation 1 of Section 6.8 is equivalent to uniformizing the continuous-time Markov chain with a value v such that vt = n and then approximating Pij (t) by P∗n ij .(b) Explain
41. Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain {X(t)} and let P¯ij = P{X(Y) = j |X(0) = i}(a) Show that P¯ij = 1 vi + λk
40. Consider the two-state continuous-time Markov chain. Starting in state 0, find Cov[X(s),X(t)].
39. Let O(t) be the occupation time for state 0 in the two-state continuous-time Markov chain. Find E[O(t)|X(0) = 1].
38. In Example 6.20, we computed m(t) = E[O(t)], the expected occupation time in state 0 by time t for the two-state continuous-time Markov chain starting in state 0. Another way of obtaining this
37. For the continuous-time Markov chain of Exercise 3 present a uniformized version.
36. Consider a system of n components such that the working times of component i, i = 1,...,n, are exponentially distributed with rate λi. When failed, however, the repair rate of component i
35. Consider a time reversible continuous-time Markov chain having infinitesimal transition rates qij and limiting probabilities {Pi}. Let A denote a set of states for this chain, and consider a new
34. Four workers share an office that contains four telephones. At any time, each worker is either “working” or “on the phone.” Each “working” period of worker i lasts for an
33. Consider two M/M/1 queues with respective parameters λi,μi, i = 1, 2.Suppose they share a common waiting room that can hold at most three customers.That is, whenever an arrival finds her server
32. Customers arrive at a two-server station in accordance with a Poisson process having rate λ. Upon arriving, they join a single queue. Whenever a server completes a service, the person first in
31. A total of N customers move about among r servers in the following manner. When a customer is served by server i, he then goes over to server j , j = i, with probability 1/(r −1). If the

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