Markov Processes For Stochastic Modeling 2nd Edition Oliver Ibe - Solutions

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Mark and Kevin play a series of games of cards. During each game each player bets $1, and whoever wins the game gets the $2. Sometimes a game ends in a tie in which case neither player loses his money. Mark is a better player than Kevin and has a probability 0.5 of wining each game, a probability
A bag contains four red balls, three blue balls, and three green balls. Jim plays a game in which he bets $1 to draw a ball from the bag. If he draws a red ball, he wins $1; otherwise he loses $1. Assume that the balls are drawn with replacement and that Jim starts with $50 with the hope of
Consider an m-server queueing system that operates in the following manner. There are two types of customers: type 1 and type 2. Type 1 customers 214 Markov Processes for Stochastic Modeling arrive according to a Poisson process with rate λ1, and type 2 customers arrive according to a Poisson
Consider a queueing system in which the server is subject to breakdown and repair. When it is operational, the time until it fails is exponentially distributed with mean 1/η.When it breaks down, the time until it is repaired and brought back to service is also exponentially distributed with mean
Consider an MMBP(2)/Geo/1 queueing system, which is a single-server queueing system with a second-order Markov-modulated Bernoulli arrival process with external arrival parameters α and β and internal switching probabilities p and q, where q = 1 − p, and geometric service times with parameter
Consider the superposition of two identical interrupted Poisson processes with internal rates α and β and external arrival rate λ. Obtain the infinitesimal generator and arrival rate matrix for the superposed system.
Give the state-transition-rate diagram for the BMAP(2)/M/1 queue with internal rates α12 and α21, external arrival rates λ1 and λ2, and service rateμ. Specify the infinitesimal generator, Q, if the batch size is equally likely to be 1, 2, or 3.
Consider Problem 6.11. Assume that the time she spends with each of her children is exponentially distributed with the same means as specified.Obtain the transition probability functions {φij (t)} of the process.
In her retirement days, a mother of three grown-up children splits her time living with her three children who live in three different states. It has been found that her choice of where to spend her time next can be modeled by a semi-Markov chain. Thus, if the children are labeled by ages as child
A component is replaced every T time units and upon its failure. The lifetimes of successive components are independent and identically distributed random variables with PDF fX(x). A cost c1>0 is incurred for each planned replacement, and a fixed cost c2>c1 is incurred for each failure
Customers arrive at a taxi depot according to a Poisson process with rate λ.The dispatcher sends for a taxi when there are N customers waiting at the station. It takes M units of time for a taxi to arrive at the depot. When it arrives, the taxi picks up all waiting customers. The taxi company
A machine can be in one of three states: good, fair, and broken. When it is in a good condition, it will remain in this state for a time that is exponentially distributed with mean 1/μ1 before going to the fair state with probability 4/5 and to the broken state with probability 1/5. When it is in
Consider a Markov renewal process with the semi-Markov kernel Q given by Q =0.6(1 − e−5t) 0.4− 0.4e−2t 0.5− 0.2e−3t − 0.3e−5t 0.5− 0.5e−2t − te−2ta. Determine the state-transition probability matrix P for the Markov chain {Xn}.b. Determine the conditional distributions Gij
Larry is a student who cannot seem to make up his mind whether to live in the city or in the suburb. Every time he lives in the city, he moves to the suburb after one semester. Half of the time he lives in the suburb, he moves back to the city after one semester. The other half of Larry’s
A high school student has two favorite brands of bag pack labeled X and Y.She continuously chooses between these brands in the following manner.Given that she currently has brand X, the probability that she will buy brand X again is 0.8, and the probability that she will buy brand Y next is
A machine has three components labeled 1, 2, and 3, whose times between failure are exponentially distributed with mean 1/λ1, 1/λ2, and 1/λ3, respectively. The machine needs all three components to work, thus when a component fails the machine is shut down until the component is repaired and the
Victor is a student who is conducting experiments with a series of lightbulbs.He started with 10 identical lightbulbs, each of which has an exponentially distributed lifetime with a mean of 200 hours. Victor wants to know how long it will take until the last bulb burns out (or fails). At noontime
The Merrimack Airlines company runs a commuter air service between Manchester, New Hampshire, and Cape Cod, Massachusetts. Because the company is a small one, there is no set schedule for their flights, and no reservation is needed for the flights. However, it has been determined that their planes
Consider a machine that is subject to failure and repair. The time to repair the machine when it breaks down is exponentially distributed with mean 1/μ. The time the machine runs before breaking down is also exponentially distributed with mean 1/λ. When repaired the machine is considered to be as
Consider a queueing system in which customers arrive according to a Poisson process with rate λ. The time to serve a customer is a third-order Erlang random variable with parameter μ. What is the expected waiting time of a customer?
Consider a queueing system in which the interarrival times of customers are the third-order Erlang random variable with parameter λ. The time to serve a customer is exponentially distributed with mean 1/μ. What is the expected waiting time of a customer?
Consider a finite-capacity G/M/1 queueing that allows at most 3 customers in the system including the customer receiving service. The time to serve a customer is exponentially distributed with mean 1/μ. As usual, let rn denote the probability that n customers are served during an interarrival
Consider an M/G/1 queueing system where service is rendered in the following manner. Before a customer is served, a biased coin whose probability of heads is p is flipped. If it comes up heads, the service time is exponentially distributed with mean 1/μ1. If it comes up tails, the service time is
Consider an M/M/2 queueing system with hysteresis. Specifically, the system operates as follows. Customers arrive according to a Poisson process with rate λ customers per second. There are two identical servers, each of which serves at the rate of μ customers per second, but as long as the number
Consider an M/M/1 queueing system with mean arrival rate λ and mean service time 1/μ. The system provides bulk service in the following manner.When the server completes any service, the system returns to the empty state if there are no waiting customers. Customers who arrive while the server is
Consider an M/M/1 queueing system with mean arrival rate λ and mean service time 1/μ that operates in the following manner. When the number of customers in the system is greater than three, a newly arriving customer joins the queue with probability p and balks (i.e., leaves without joining the
Consider an M/M/1/5 queueing system with mean arrival rate λ and mean service time 1/μ that operates in the following manner. When any customer is in queue, the time until he defects (i.e., leaves the queue without receiving service) is exponentially distributed with a mean of 1/β. It is assumed
Customers arrive at a checkout counter in a grocery store according to a Poisson process with an average rate of 10 customers per hour. There are two clerks at the counter, and the time either clerk takes to serve each customer is exponentially distributed with an unspecified mean. If it is desired
Students arrive at a checkout counter in the college cafeteria according to a Poisson process with an average rate of 15 students per hour. There are three cashiers at the counter, and they provide identical service to students.The time to serve a student by any cashier is exponentially distributed
Consider a birth-and-death process representing a multi-server finite population system with the following birth-and-death rates:λk = (4 − k)λ k = 0, 1, 2, 3, 4μk = kμ k = 1, 2, 3, 4a. Find the pk, k = 0, 1, 2, 3, 4 in terms of λ and μ.b. Find the average number of customers in the system.
A cyber cafe has six PCs that customers can use for Internet access. These customers arrive according to a Poisson process with an average rate of six per hour. Customers who arrive when all six PCs are being used are blocked and have to go elsewhere for their Internet access. The time that a
A machine has four identical components that fail independently. When a component is operational, the time until it fails is exponentially distributed with a mean of 10 hours. There is one resident repairman at the site so that when a component fails he immediately swaps it out and commences 150
A small PBX serving a startup company can only support five lines for communication with the outside world. Thus, any employee who wants to place an outside call when all five lines are busy is blocked and will have to hang up. A blocked call is considered to be lost because the employee will not
minutes.a. How much capacity should the facility have to achieve this goal?b. With the capacity obtained in part (a), what is the probability that an arriving customer is lost?
A company is considering how much capacity K to provide in its new service facility. When the facility is completed, customers are expected to arrive at the facility according to a Poisson process with a mean rate of 10 customers per hour, and customers that arrive when the facility is full are
A clerk provides exponentially distributed service to customers who arrive according to a Poisson process with an average rate of 15 per hour. If the service facility has an infinite capacity, what is the mean service time that the clerk must provide in order that the mean waiting time shall be no
People arrive at a library to borrow books according to a Poisson process with a mean rate of 15 people per hour. There are two attendants at the library, and the time to serve each person by either attendant is exponentially distributed with a mean of 3 minutes.a. What is the probability that an
People arrive at a phone booth according to a Poisson process with a mean rate of five people per hour. The duration of calls made at the phone booth is exponentially distributed with a mean of 4 minutes.a. What is the probability that a person arriving at the phone booth will have to
A shop has five identical machines that break down independently of each other. The time until a machine breaks down is exponentially distributed with a mean of 10 hours. There are two repairmen who fix the machines when they fail. The time to fix a machine when it fails is exponentially
Cars arrive at a car wash according to a Poisson process with a mean rate of 8 cars per hour. The policy at the car wash is that the next car cannot pass through the wash procedure until the car in front of it is completely finished.The car wash has a capacity to hold 10 cars, including the car
People arrive to buy tickets at a movie theater according to a Poisson process with an average rate of 12 customers per hour. The time it takes to complete the sale of a ticket to each person is exponentially distributed with a mean of 3 minutes. There is only one cashier at the ticket window, and
Cars arrive at a parking lot according to a Poisson process with rate λ.There are only four parking spaces, and any car that arrives when all the spaces are occupied is lost. The parking duration of a car is exponentially distributed with mean 1/μ. Let pk(t) denote the probability that k cars are
Consider a system consisting of two birth and death processes labeled system 1 and system 2. Customers arrive at system 1 according to a Poisson process with rate λ1, and customers arrive at system 2 according to a Poisson process with rate λ2. Each system has two identical attendants. The time
Trucks bring crates of goods to a warehouse that has a single attendant.It is the responsibility of each truck driver to offload his truck, and the time that it takes to offload a truck is exponentially distributed with mean 1/μ1. When a truck is offloaded, it leaves the warehouse and takes a time
An assembly line consists of two stations in tandem. Each station can hold only one item at a time. When an item is completed in station 1, it moves into station 2 if the latter is empty; otherwise it remains in station 1 until station 2 is free. Items arrive at station 1 according to a Poisson
Consider a collection of particles that act independently in giving rise to succeeding generations of particles. Suppose that each particle, from the time it appears, waits a length of time that is exponentially distributed 102 Markov Processes for Stochastic Modeling with a mean of 1/λ and then
A taxicab company has a small fleet of three taxis that operate from the company’s station. The time it takes a taxi to take a customer to his or her location and return to the station is exponentially distributed with a mean of 1/μ hours. Customers arrive according to a Poisson process with
A service facility can hold up to six customers who arrive according to a Poisson process with a rate of λ customers per hour. Customers who arrive when the facility is full are lost and never make an attempt to return to the facility. Whenever there are two or fewer customers in the facility,
A switchboard has two outgoing lines serving four customers who never call each other. When a customer is not talking on the phone, he or she generates calls according to a Poisson process with rate λ calls per minute.Call lengths are exponentially distributed with a mean of 1/μ minutes.If a
Lazy Chris has three identical lightbulbs in his living room that he keeps on all the time. Because of his laziness, Chris does not replace a lightbulb when it fails. (Maybe Chris does not even notice that the bulb has failed!)However, when all three bulbs have failed, Chris replaces them at the
A small company has two PCs A and B. The time to failure for PC A is exponentially distributed with a mean of 1/λA hours, and the time to failure for PC B is exponentially distributed with a mean of 1/λB hours.The PCs also have different repair times. The time to repair PC A when it fails is
Customers arrive at Mike’s barber shop according to a Poisson process with rate λ customers per hour. Unfortunately Mike, the barber, has only five chairs in his shop for customers to wait when there is already a customer receiving a haircut. Customers who arrive when Mike is busy and all the
A small company has two identical PCs that are running at the same time.The time until either PC fails is exponentially distributed with a mean of 1/λ. When a PC fails, a technician starts repairing it immediately. The two PCs fail independently of each other. The time to repair a failed PC is
On a given day Mark is either cheerful, so-so, or glum. Given that he is cheerful on a given day, then he will be cheerful the next day with probability 0.6, so-so with probability 0.2, and glum with probability 0.2. Given that he is so-so on a given day, then he will be cheerful the next day with
Consider the following transition probability matrix:a. What is Pn?b. Obtain φ13(5), the mean occupancy time of state 3 up to five transitions given that the process started from state 1.
The New England fall weather can be classified as sunny, cloudy, or rainy.A student conducted a detailed study of the weather conditions and came up with the following conclusion: Given that it is sunny on any given day, then on the following day it will be sunny again with probability 0.5, cloudy
A taxi driver conducts his business in three different towns 1, 2, and 3. On any given day, when he is in town 1, the probability that the next passenger he picks up is going to a place in town 1 is 0.3, the probability that the next passenger he picks up is going to town 2 is 0.2, and the
Consider the following social mobility problem. Studies indicate that people in a society can be classified as belonging to the upper class (state 1), middle class (state 2), and lower class (state 3). Membership in any class is inherited in the following probabilistic manner. Given that a person
A symmetric random walk {Sn:n = 0, 1, 2,...} starts at the position S0 = k and ends when the walk first reaches either the origin or the position m, where 0
Let X1, X2,... be independent and identically distributed Bernoulli random variables with values ±1 that have equal probability of 1/2. Let K1 and K2 be positive integers, and define N as follows:N = min{n:Sn = K1 or −K2}where Sn =n k=1 Xk n = 1, 2,...is called a symmetric random walk.a. Show
Let X1, X2,... be independent and identically distributed Bernoulli random variables with values ±1 that have equal probability of 1/2. Show that the partial sums Sn =n k=1 Xk k n = 1, 2,...form a martingale with respect to {Xn}.
Let the random variable Sn be defined as follows:Sn =⎧⎨⎩0 n = 0n k=1 Xk n ≥ 1 where Xk is the kth outcome of a Bernoulli trial such that P[Xk = 1] = p and P[Xk = −1] = q = 1 − p, and the Xk are independent and identically distributed. Consider the process {Sn : n = 1, 2,...}.a. For
A one-way street has a fork in it, and cars arriving at the fork can either bear right or left. A car arriving at the fork will bear right with probability 0.6 and will bear left with probability 0.4. Cars arrive at the fork in a Poisson manner with a rate of eight cars per minute.a. What is the
Cars arrive from the northbound section of an intersection in a Poisson manner at the rate of λN cars per minute and from the eastbound section in a Poisson manner at the rate of λE cars per minute.a. Given that there is currently no car at the intersection, what is the probability that a
Suzie has two identical personal computers, which she never uses at the same time. She uses one PC at a time, and the other is a backup. If the one she is currently using fails, she turns it off, calls the PC repairman, and turns on the backup PC. The time until either PC fails when it is in use is
A five-motor machine can operate properly if at least three of the five motors are functioning. If the lifetime X of each motor has the PDF fX(x) = λe−λx, x≥0,λ>0, and if the lifetimes of the motors are independent, what is the mean of the random variable Y, the time until the machine fails?
Joe replaced two lightbulbs, one of which is rated 60 watts with an exponentially distributed lifetime whose mean is 200 hours, and the other is rated 100 watts with an exponentially distributed lifetime whose mean is 100 hours.a. What is the probability that the 60-watt bulb fails before the
Bob has a pet that requires the light in his apartment to always be on. To achieve this, Bob keeps three lightbulbs on with the hope that at least one bulb will be operational when he is not at the apartment. The lightbulbs have independent and identically distributed lifetimes T with PDF fT (t)
Chris is conducting an experiment to test the mean lifetimes of two sets of electric bulbs labeled A and B. The manufacturer claims that the mean lifetime of bulbs in set A is 200 hours, while the mean lifetime of the bulbs in set B is 400 hours. The lifetimes for both sets are exponentially
Customers arrive at a bank according to a Poisson process with an average rate of six customers per hour. Each arriving customer is either a man with probability p or a woman with probability 1−p. It was found that in the first two hours the average number of men who arrived at the bank was
Students arrive at the professor’s office for extra help according to a Poisson process with an average rate of four students per hour. The professor does not start the tutorial until at least three students are available. Students who arrive while the tutorial is going on will have to wait for
Three customers A, B, and C simultaneously arrive at a bank with two tellers on duty. The two tellers were idle when the three customers arrived, and A goes directly to one teller, B goes to the other teller, and C waits until either A or B leaves before she can begin receiving service. If the
Joe is a student who is conducting experiments with a series of lightbulbs.He started with 10 identical lightbulbs, each of which has an exponentially distributed lifetime with a mean of 200 hours. Joe wants to know how long it will take until the last bulb burns out (or fails). At noontime Joe
Customers arrive at the neighborhood bookstore according to a Poisson process with an average rate of 10 customers per hour. Independent of other customers, each arriving customer buys a book with probability 1/8.a. What is the probability that the bookstore sells no book during a particular
An insurance company pays out claims on its life insurance policies in accordance with a Poisson process with an average rate of five claims per week. If the amount of money paid on each policy is uniformly distributed between $2000 and $10,000, what is the mean of the total amount of money that
Suppose X(t) is a Gaussian random process with a mean E[X(t)] = 0 and autocorrelation function RXX(τ) = e−|τ|. Assume that the random variable A is defined as follows:A = B 0X(t)dt where B is a uniformly distributed random variable with values between 1 and 5 and is independent of the random
Suppose X(t) is a Gaussian random process with a mean E[X(t)] = 0 and autocorrelation function RXX(τ) = e−|τ|. Assume that the random variable A is defined as follows:A = 1 0X(t)dt Determine the following:a. E[A]b. σ2 A
Cars arrive at a gas station according to a Poisson process at an average rate of 12 cars per hour. The station has only one attendant. If the attendant decides to take a two-minute coffee break when there were no cars at the station, what is the probability that one or more cars will be waiting
Students arrive for a lab experiment according to a Poisson process with a rate of 12 students per hour. However, the lab attendant opens the door to the lab when at least four students are waiting at the door.What is the probability that the waiting time of the first student to arrive exceeds 20
A Girl Scout troop sells cookies from house to house. One of the parents of the girls figured out that the probability that they sell a set of packs of cookies at any house they visit is 0.4, where it is assumed that they sell exactly one set to each house that buys their cookies.a. What is the
A lady invites 12 people for dinner at her house. Unfortunately the dining table can only seat six people. Her plan is that if six or fewer guests come, then they will be seated at the table (i.e., they will have a sit-down dinner); otherwise she will set up a buffet-style meal. The probability
A sequence of Bernoulli trials consists of choosing components at random from a batch of components. A selected component is classified as either defective or nondefective. A nondefective component is considered to be a success, while a defective component is considered to be a failure. If the
The probability that a patient recovers from a rare blood disease is 0.3.If 15 people are known to have contracted this disease, find the following probabilities:a. At least 10 survive.b. From three to eight survive.c. Exactly six survive.
A sequence of Bernoulli trials consists of choosing seven components at random from a batch of components. A selected component is classified as either defective or nondefective. A nondefective component is considered to be a success, while a defective component is considered to be a failure.If the
15.5 Passengers arrive at a train station according to a Poisson process with a rate of 25 customers per hour. It has been found that 60% of the passengers are females. What is the probability that no male customer arrives at the station over a 15-min period?
15.4 A restaurant has two entrances A and B. Customers arrive at the restaurant through entrance A according to a Poisson process with a rate of five customers per hour, and customers arrive through entrance B according to a Poisson process with a rate of seven customers per hour.a. What is the
15.3 Let fX1; X2; ...g be a sequence of independent and identically distributed random variables with PDF fXðxÞ, and let N be an integer-valued random variable with PMF pNðnÞ, where N and the Xi are independent. Consider a process in which events occur at times X1; X1 1 X2; ...; X1 1 X2 1?1
15.2 A hard-core point process is produced from a Poisson process of rate λ by deleting any point within distance v0 of another point, regardless of whether that point has itself already been deleted. Prove that the rate of the hard-core process is λ expð2 λπv2 0Þ.
15.1 Two classes of customers arrive at Paul’s barber shop: class 1 and class 2. Class 1 customers arrive according to a Poisson process with rate λ1 customers per hour, and class 2 customers arrive according to a Poisson process with rate λ2 customers per hour.a. What is the probability that
14.6 Consider a system that can be modeled by an array of six states labeled 1; 2; ...; 6. Every state makes a transition to itself with probability p and makes a transition to the next higher state with probability 1 2 p. An experiment starts in the Begin state and ends when the End state is
14.5 Consider a system that can be modeled by an array of six states labeled 1; 2; ...; 6. Apart from state 4, which makes a transition to itself with probability p, every other state is visited only once in each experiment that starts in the Begin state and ends when the End state is reached. The
14.4 Consider three coins labeled 1, 2, and 3. When coin 1 is tossed, the probability that it comes up heads is 0.75 and the probability that it comes up tails is 0.25. Similarly, when coin 2 is tossed, the probability that it comes up heads is 0.25 and the probability that it comes up tails is
14.3 Construct the PHMM for the following variable length sequences DOR, DM, DAP, VGBLM. (Hint: Use the following alignment to identify the match, insert, and delete states.) D O 2 2 R D 222 M D A 2 2 P VGB L M
14.2 Consider the HMM shown in Figure 14.19 that has three hidden states 1, 2, and 3, and emits two output symbols: U and V. When it is in state 1, it is equally likely to emit either symbol. When it is in state 2, the probability that it emits the symbol U is 0.1, and the probability that it emits
14.1 Consider an HMM with two states 1 and 2 and emits two symbols: A and B. The statetransition diagram is shown in Figure 14.18.a. Use the Viterbi algorithm to obtain the most likely state sequence that produced the observation sequence {ABBAB}.b. Estimate the probability that the sequence
13.5 Consider a salesman who has offices in two towns called town 1 and town 2. He can be in one of these towns on any particular day but cannot split his time on any day between the two. On any day that he works in town 1, the probability of making a sale is 0.4; similarly, on any day that he is
13.4 Solve Problem 13.3, assuming a discount factor of 0.9.
13.3 A farmer is considering the optimal course of action for his farm each year. The two options are to fertilize the farm and not to fertilize the farm. Optimality is defined such that the farmer will realize the highest expected revenue at the end of 4 years. The conditions (or states) of the
13.2 The price of a certain stock is fluctuating among $10, $20, and $30 from month to month. Market analysis indicates that given that the stock is at $10 in the current month, then in the following month it will be at $10 with probability 0.8 and at $20 with probability 0.2. Similarly, given that
13.1 A recent college graduate is presented with N job offers, one after another. After looking at an offer, she must either accept it and thus terminate the process or reject it. A rejected offer is lost forever. The only information she has at any time is the relative rank of the current offer

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Markov Processes For Stochastic Modeling 2nd Edition Oliver Ibe - Solutions

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