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introduction to probability statistics

Introduction To Probability Models 12th Edition Sheldon M Ross - Solutions

77. Suppose that customers arrive to a system according to a Poisson process with rate λ. There are an infinite number of servers in this system so a customer begins service upon arrival. The service times of the arrivals are independent exponential random variables with rate μ, and are
62. Suppose that the number of typographical errors in a new text is Poisson distributed with mean λ. Two proofreaders independently read the text. Suppose that each error is independently found by proofreader i with probability . Let denote the number of errors that are found by proofreader 1 but
61. A system has a random number of flaws that we will suppose is Poisson distributed with meanc. Each of these flaws will, independently, cause the system to fail at a random time having distribution G. When a system failure occurs, suppose that the flaw causing the failure is immediately located
46. LetI mage be a Poisson process with rate λ that is independent of the sequence of independent and identically distributed random variables with mean μ and variance . Find
47. Consider a two-server parallel queuing system where customers arrive according to a Poisson process with rate λ, and where the service times are exponential with rate μ. Moreover, suppose that arrivals finding both servers busy immediately depart without receiving any service (such a customer
48. Consider an n-server parallel queuing system where customers arrive according to a Poisson process with rate λ, where the service times are exponential random variables with rate μ, and where any arrival finding all servers busy immediately departs without receiving any service. If an arrival
49. Events occur according to a Poisson process with rate λ. Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time T, where . That is, if an event occurs at time, and we decide to stop, then we win
50. The number of hours between successive train arrivals at the station is uniformly distributed on (0, 1). Passengers arrive according to a Poisson process with rate 7 per hour. Suppose a train has just left the station. Let X denote the number of people who get on the next train. Find(a) ,(b) .
51. If an individual has never had a previous automobile accident,then the probability he or she has an accident in the next h time units is ; on the other hand, if he or she has ever had a previous accident, then the probability is . Find the expected number of accidents an individual has by time
52. Teams 1 and 2 are playing a match. The teams score points according to independent Poisson processes with respective rates and . If the match ends when one of the teams has scored k more points than the other, find the probability that team 1 wins.Hint: Relate this to the gambler's ruin problem.
53. The water level of a certain reservoir is depleted at a constant rate of 1000 units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson process with rate 0.2 per day. The amount of water added to the reservoir by a rainfall is 5000 units with
54. A viral linear DNA molecule of length, say, 1 is often known to contain a certain “marked position,” with the exact location of this mark being unknown. One approach to locating the marked position is to cut the molecule by agents that break it at points chosen according to a Poisson
55. Consider a single server queuing system where customers arrive according to a Poisson process with rate λ, service times are exponential with rate μ, and customers are served in the order of their arrival. Suppose that a customer arrives and finds others in the system. Let X denote the number
56. An event independently occurs on each day with probability p. Let denote the total number of events that occur on the first n days, and let denote the day on which the rth event occurs.(a) What is the distribution of ?(b) What is the distribution of Image?(c) What is the distribution of ?(d)
57. Each round played by a contestant is either a success with probability p or a failure with probability If the round is a success, then a random amount of money having an exponential distribution with rate λ is won. If the round is a failure, then the contestant loses everything that had been
58. There are two types of claims that are made to an insurance company. Let denote the number of type i claims made by time t, and suppose thatI mage andI mage are independent Poisson processes with rates and . The amounts of successive type 1 claims are independent exponential random variables
59. Cars pass an intersection according to a Poisson process with rateλ. There are 4 types of cars, and each passing car is, independently, type i with probability , .(a) Find the probability that at least one of each of car types but none of type 4 have passed by time t.(b) Given that exactly 6
60. People arrive according to a Poisson process with rate λ, with each person independently being equally likely to be either a man or a woman. If a woman (man) arrives when there is at least one man(woman) waiting, then the woman (man) departs with one of the waiting men (women). If there is no
78. A store opens at 8 a.m. From 8 until 10 a.m. customers arrive at a Poisson rate of four an hour. Between 10 a.m. and 12 p.m. they arrive at a Poisson rate of eight an hour. From 12 p.m. to 2 p.m. the arrival rate increases steadily from eight per hour at 12 p.m. to ten per hour at 2 p.m.; and
*79. Suppose that events occur according to a nonhomogeneous Poisson process with intensity function Further, suppose that an event that occurs at time s is a type 1 event with probability IfI mage is the number of type 1 events by time t, what type of process isI mage?
80. LetI mage denote the interarrival times of events of a nonhomogeneous Poisson process having intensity functionI mage.(a) Are the Image independent?(b) Are the Image identically distributed?(c) Find the distribution of Image.
98. Let in Example 5.21.(a) Show that(b) Use (a) to show that(c) Show that
99. Let X be the time between the first and the second event of a Hawkes process with mark distribution F. Find
1. A population of organisms consists of both male and female members. In a small colony any particular male is likely to mate with any particular female in any time interval of length h, with probabilityI mage. Each mating immediately produces one offspring, equally likely to be male or female.
*2. Suppose that a one-celled organism can be in one of two states—either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two new individuals of type A at an exponential rate β. Define an appropriate continuous-time Markov
3. Consider two machines that are maintained by a single repairman.Machine i functions for an exponential time with rate before breaking down, . The repair times (for either machine) are exponential with rate μ. Can we analyze this as a birth and death process? If so, what are the parameters? If
*4. Potential customers arrive at a single-server station in accordance with a Poisson process with rate λ. However, if the arrival finds n customers already in the station, then he will enter the system with probabilityI mage. Assuming an exponential service rate μ, set this up as a birth and
5. There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process having rate λ. When a contact occurs, it is equally likely to involve any of theI mage pairs of
6. Consider a birth and death process with birth ratesI mage, and death ratesI mage.(a) Determine the expected time to go from state 0 to state 4.(b) Determine the expected time to go from state 2 to state 5.(c) Determine the variances in parts (a) and (b).
*7. Individuals join a club in accordance with a Poisson process with rate λ. Each new member must pass through k consecutive stages to become a full member of the club. The time it takes to pass through each stage is exponentially distributed with rate μ. Let denote the number of club members at
8. Consider two machines, both of which have an exponential lifetime with mean . There is a single repairman that can service machines at an exponential rate μ. Set up the Kolmogorov backward equations; you need not solve them.
9. The birth and death process with parametersI mage andI mage is called a pure death process. Find .
10. Consider two machines. Machine i operates for an exponential time with rate and then fails; its repair time is exponential with rate . The machines act independently of each other.Define a four-state continuous-time Markov chain that jointly describes the condition of the two machines. Use the
*11. Consider a Yule process starting with a single individual—that is,supposeI mage. Let denote the time it takes the process to go from a population of size i to one of size .(a) Argue that Image, are independent exponentials with respective rates iλ.(b) Let Image denote independent
12. Each individual in a biological population is assumed to give birth at an exponential rate λ, and to die at an exponential rate μ. In addition, there is an exponential rate of increase θ due to immigration. However, immigration is not allowed when the population size is N or larger.(a) Set
13. A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with meanI mage hour.(a) What is the average number of customers in
97. Consider a conditional Poisson process in which the rate L is, as in Example 5.29, gamma distributed with parameters m and p. Find the conditional density function of L given thatI mage.
96. For the conditional Poisson process, let . In terms of and , find forI mage.
81.(a) Let Image be a nonhomogeneous Poisson process with mean value function Image. Given Image, show that the unordered set of arrival times has the same distribution as n independent and identically distributed random variables having distribution function(b) Suppose that workmen incur accidents
82. Let be independent positive continuous random variables with a common density functionf, and suppose this sequence is independent of N, a Poisson random variable with mean λ. Define Show thatI mage is a nonhomogeneous Poisson process with intensity function .
83. Prove Lemma 5.4.
*84. Let be independent and identically distributed nonnegative continuous random variables having density function Image. We say that a record occurs at time n if is larger than each of the previous values . (A record automatically occurs at time 1.) If a record occurs at time n, then is called a
85. Let where , are independent and identically distributed with mean , and are independent ofI mage, which is a Poisson process with rate λ. ForI mage, find(a) ;(b) ;(c) ;(d) .
86. In good years, storms occur according to a Poisson process with rate 3 per unit time, while in other years they occur according to a Poisson process with rate 5 per unit time. Suppose next year will be a good year with probability 0.3. LetI mage denote the number of storms during the first t
87. Determine whenI mage is a compound Poisson process.
88. Customers arrive at the automatic teller machine in accordance with a Poisson process with rate 12 per hour. The amount of money withdrawn on each transaction is a random variable with mean $30 and standard deviation $50. (A negative withdrawal means that money was deposited.) The machine is in
89. Some components of a two-component system fail after receiving a shock. Shocks of three types arrive independently and in accordance with Poisson processes. Shocks of the first type arrive at a Poisson rate and cause the first component to fail. Those of the second type arrive at a Poisson rate
90. In Exercise 89 show that and both have exponential distributions.
*91. Let be independent and identically distributed exponential random variables. Show that the probability that the largest of them is greater than the sum of the others is . That is, if then show Hint: What is ?
92. Prove Eq. (5.22).
93. Prove that(a) and, in general,(b)(c) Show by defining appropriate random variables ,, and by taking expectations in part (b) how to obtain the well-known formula(d) Consider n independent Poisson processes—the ith having rate . Derive an expression for the expected time until an event has
94. A two-dimensional Poisson process is a process of randomly occurring events in the plane such that(i) for any region of area A the number of events in that region has a Poisson distribution with mean λA, and(ii) the number of events in nonoverlapping regions are independent.For such a process,
95. LetI mage be a conditional Poisson process with a random rate L.(a) Derive an expression for .(b) Find, for Image, .(c) Find, for Image, .
14. Consider an irreducible continuous time Markov chain whose state space is the nonnegative integers, having instantaneous transition ratesI mage and stationary probabilities , . Let T be a given set of states, and letI mage be the state at the moment of the nth transition into a state in T.(a)
33. 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 finds the server busy goes away. Each potential customer is type i with probabilityI mage, whereI
51. In the k server Erlang loss model, suppose thatI mage andI mage.Find L ifI mage.
52. Verify the formula given for theI mage of the .
53. In the Erlang loss system suppose the Poisson arrival rate isI mage,and suppose there are three servers, each of whom has a service distribution that is uniformly distributed overI mage. What proportion of potential customers is lost?
54. In the system,(a) what is the probability that a customer will have to wait in queue?(b) determine L and W.
55. Verify the formula for the distribution of given for the model.
*56. Consider a system where the interarrival times have an arbitrary distribution F, and there is a single server whose service distribution is G. LetI mage denote the amount of time the nth customer spends waiting in queue. InterpretI mage so that
57. Consider a model in which the interarrival times have an arbitrary distribution F, and there are k servers each having service distribution G. What condition on F and G do you think would be necessary for there to exist limiting probabilities?
1. Prove that, for any structure function ϕ,where
2. Show that(a) if and , then(b)(c)
3. For any structure function ϕ, we define the dual structure by(a) Show that the dual of a parallel (series) system is a series (parallel) system.(b) Show that the dual of a dual structure is the original structure.(c) What is the dual of a k-out-of-n structure?(d) Show that a minimal path (cut)
*4. Write the structure function corresponding to the following:
5. Find the minimal path and minimal cut sets for:
50. In the model if G is exponential with rate λ show thatI mage.
*49. Calculate explicitly (not in terms of limiting probabilities) the average time a customer spends in the system in Exercise 28.
48. Consider the priority queueing model of Section 8.6.2 but now suppose that if a type 2 customer is being served when a type 1 arrives then the type 2 customer is bumped out of service. This is called the preemptive case. Suppose that when a bumped type 2 customer goes back in service his
35. Consider a network of three stations with a single server at each station. Customers arrive at stationsI mage in accordance with Poisson processes having respective ratesI mage, and 15. The service times at the three stations are exponential with respective ratesI mage, and 100. A customer
36. Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either server—that is,I mage . LetI mage denote the exponential service rate at serverI mage.(a) Determine the
37. Explain how a Markov chain Monte Carlo simulation using the Gibbs sampler can be utilized to estimate(a) the distribution of the amount of time spent at server j on a visit.Hint: Use the arrival theorem.(b) the proportion of time a customer is with server j (i.e., either in server j's queue or
38. For open queueing networks(a) state and prove the equivalent of the arrival theorem;(b) derive an expression for the average amount of time a customer spends waiting in queues.
39. Customers arrive at a single-server station in accordance with a Poisson process having rate λ. Each customer has a value. The successive values of customers are independent and come from a uniform distribution on . The service time of a customer having value x is a random variable with meanI
*40. Compare the system for first-come, first-served queue discipline with one of last-come, first-served (for instance, in which units for service are taken from the top of a stack). Would you think that the queue size, waiting time, and busy-period distribution differ? What about their means?
41. In an queue,(a) what proportion of departures leave behind 0 work?(b) what is the average work in the system as seen by a departure?
42. For the queue, let denote the number in the system left behind by the nth departure.(a) If Image what does represent?(b) Rewrite the preceding as where Image Take expectations and let in Eq. (8.64) to obtain(d) Argue that Image, the average number as seen by a departure, is equal to L.
*43. Consider an system in which the first customer in a busy period has the service distribution and all others have distribution . Let C denote the number of customers in a busy period, and let S denote the service time of a customer chosen at random.Argue that(a) Image.(b) Image where has
44. Consider a system with .(a) Suppose that service is about to begin at a moment when there are n customers in the system.(i) Argue that the additional time until there are only customers in the system has the same distribution as a busy period.(ii) What is the expected additional time until the
45. Carloads of customers arrive at a single-server station in accordance with a Poisson process with rate 4 per hour. The service times are exponentially distributed with rate 20 per hour. If each carload contains eitherI mage, or 3 customers with respective probabilitiesI mage, andI mage, compute
47. In a two-class priority queueing model suppose that a cost of per unit time is incurred for each type i customer that waits in queue, . Show that type 1 customers should be given priority over type 2 (as opposed to the reverse) if
*6. The minimal path sets are {1, 2, 4}, {1, 3, 5}, and {5, 6}. Give the minimal cut sets.
7. The minimal cut sets are {1, 2, 3}, {2, 3, 4}, and {3, 5}. What are the minimal path sets?
8. Give the minimal path sets and the minimal cut sets for the structure given by Fig. 9.21.
24. Show that if F is IFR, then it is also IFRA, and show by counterexample that the reverse is not true.
*25. We say that ζ is a p-percentile of the distribution F if .Show that if ζ is a p-percentile of the IFRA distribution F, then
26. Prove Lemma 9.3.Hint: Let . Note that is a concave function when, and use the fact that for a concave function is decreasing in t.
27. Let . Show that if , then
28. Find the mean lifetime of a series system of two components when the component lifetimes are respectively uniform on ( ) and uniform on ( ). Repeat for a parallel system.
29. Show that the mean lifetime of a parallel system of two components is when the first component is exponentially distributed with mean and the second is exponential with mean .
*30. Compute the expected system lifetime of a three-out-of-four system when the first two component lifetimes are uniform on ( )and the second two are uniform on ( ).
31. Show that the variance of the lifetime of a k-out-of-n system of components, each of whose lifetimes is exponential with mean θ, is given by
32. In Section 9.6.1 show that the expected number of that exceed is equal to 1.
33. Let be an exponential random variable with mean , for. Use the results of Section 9.6.1 to obtain an upper bound on , and then compare this with the exact result when the are independent.
34. For the model of Section 9.7, compute for a k-out-of-n structure(i) the average up time, (ii) the average down time, and (iii) the system failure rate.
35. Prove the combinatorial identity(a) by induction on i;(b) by a backwards induction argument on i—that is, prove it first for , then assume it for and show that this implies that it is true for .
23. Show that if each (independent) component of a series system has an IFR distribution, then the system lifetime is itself IFR by(a) showing that where is the failure rate function of the system; and the failure rate function of the lifetime of component i.(b) using the definition of IFR given in
*22. Let X denote the lifetime of an item. Suppose the item has reached the age of t. Let denote its remaining life and define In words, is the probability that a t-year-old item survives an additional timea. Show that (a) where F is the distribution function of X.(b) Another definition of IFR is
21. Consider the following four structures:(i) See Fig. 9.23:Let , and be the corresponding component failure distributions; each of which is assumed to be IFR (increasing failure rate). Let F be the system failure distribution. All components are independent.(a) For which structures is F

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