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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

12.6 Consider a system whose environment changes according to a Markov chain. Specifically, Yn is the state of the environment at the beginning of the nth period, where Y 5 fYn; n $ 1g is a Markov
12.5 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 arrive according to a Poisson process with rate
12.4 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
12.3 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
12.1 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 rate μ. Specify the infinitesimal generator,
10.4 Another way to define a diffusion process is as follows. Let μðt; xÞ and σðt; xÞ be continuous functions of t and x, where ðt 0 E½σ2 ðu; BðuÞÞdu ,N Define XðtÞ 5 Xð0Þ 1 ðt 0
10.3 Let fXðtÞ; t $ 0g be a continuous-time continuous-state Markov process whose transition PDF fðx; t; x0; t0Þ satisfies the following forward Kolmogorov equation: @f @t 5 2 @ @x ðaðt;
10.2 Consider a particle whose position, xðtÞ, undergoes the diffusion and damping process dx 5 2 μxdt 1 ð1 2 x2 Þσ dB What is the steady-state PDF of x?
10.1 Show that the FokkerPlanck equation @P @t 5 2 a @P @x 1 D 2 @2 P @x2 has the solution Pðx; tÞ 5 1 ffiffiffiffiffiffiffiffiffiffi 2πDt p e2ðx2aÞ 2=2Dt
9.11 Let Ta be the time until a standard Brownian motion process hits the pointa. Calculate P½T2 # 8.
9.10 The price of a certain stock follows Brownian motion. The price at time t 5 3 is 52. Determine the probability that the price is more than 60 at time t 5 10.
9.9 Let fBðtÞ; t $ 0g be a standard Brownian motion and define the process YðtÞ 5 e2t Bðe2t Þ; t $ 0; that is, fYðtÞ; t $ 0g is the OU process.a. Show that YðtÞ is a Gaussian process.b.
9.8 Let the process fXðtÞ; t $ 0g be defined by XðtÞ 5 B2ðtÞ 2 t, where fBðtÞ; t $ 0g is a standard Brownian motion.a. What is E½XðtÞ?b. Show that fXðtÞ; t $ 0g is a martingale. Hint:
9.7 What is the mean value of the first passage time of the reflected Brownian motion fjBðtÞj; t $ 0g with respect to a positive level x, where BðtÞ is the standard Brownian motion? Determine the
9.6 Consider the Brownian motion with drift YðtÞ 5 μt 1 σBðtÞ 1 x where Yð0Þ 5 x and b , x ,a. Let paðxÞ denote the probability that hits a beforeb. a. Show that 1 2 d2paðxÞ dx2 1 μ
9.5 Let YðtÞ 5 Ðt 0 BðuÞdu, where fBðtÞ; t $ 0g is the standard Brownian motion. Finda. E½YðtÞb. E½Y2ðtÞc. The conditional distribution of YðtÞ, given that BðtÞ 5 x.
9.4 Let T 5 minftjBðtÞ 5 5 2 3tg. Use the martingale stopping theorem to find E½T.
9.3 Let fXðtÞ; t $ 0g be a Brownian motion with drift rate μ and variance parameter σ2. What is the conditional distribution of XðtÞ given that XðuÞ 5 b; u , t?
9.2 Suppose XðtÞ is a standard Brownian motion and YðtÞ 5 tXð1=tÞ. Show that YðtÞ is a standard Brownian motion.
9.1 Assume that X and Y are independent random variables such that XBNð0; σ2Þ and YBNð0; σ2Þ. Consider the random variables U 5 ðX 1 YÞ=2 and V 5 ðX 2 YÞ=2. Show that U and V are
8.10 Consider a CTRW fXðtÞjt $ 0g in which the jump size, Θ, is normally distributed with mean μ and variance σ2, and the waiting time, T, is exponentially distributed with mean 1=λ, where Θ
8.9 Consider a correlated random walk with stay. That is, a walker can move to the right, to the left, or not move at all. Given that the move in the current step is to the right, then in the next
8.8 Consider a cash management scheme in which a company needs to maintain the available cash to be no more than $K. Whenever the cash level reaches K, the company buys treasury bills and reduces
8.7 Consider an asymmetric random walk that takes a step to the right with probability p and a step to the left with probability q 5 1 2 p. Assume that there are two absorbing barriers, a andb, and
8.6 Let N denote the number of times that an asymmetric random walk that takes a step to the right with probability p and a step to the left with probability q 5 1 2 p revisits its starting point.
8.5 Consider the random walk Sn 5 X1 1 X2 1?1 Xn, where the Xi are independent and identically distributed Bernoulli random variables that take on the value 1 with probability p 5 0:6 and the value
8.4 Consider a single-server discrete-time queueing system that operates in the following manner. Let Xn denote the number of customers in the system at time nAf0; 1; 2; ...g. If a customer is
8.3 Chris has $20 and Dana has $30. They decide to play a game in which each pledges $1 and flips a fair coin. If both coins come up on the same side, Chris wins the $2, and if they come up on
currently in state k (that is, Mark has $k), obtain an expression for rk.
8.2 Mark and Kevin play a series of games of cards. During each game each player bets $1, and whoever wins the game gets $2. Sometimes a game ends in a tie in which case neither player loses his
8.1 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
7.25 Consider a closed network with K 5 3 circulating customers, as shown in Figure 7.26. There is a single exponential server at nodes 1, 2, and 3 with service rates μ1; μ2, and μ3, respectively.
7.24 Consider the closed network of queues shown in Figure 7.25. Assume that the number of customers inside the network is K 5 3. Find the joint PMF pN1N2N3 ðn1; n2; n3Þ, if there is a single
7.23 Consider the acyclic Jackson network of queues shown in Figure 7.24, which has the property that a customer cannot visit a node more than once. Specifically, assume that there are four
7.22 Consider the network shown in Figure 7.23, which has three exponential service stations with rates μ1; μ2, and μ3, respectively. External customers arrive at the station labeled Queue 1
7.21 Consider a two-priority queueing system in which priority class 1 (i.e., high-priority) customers arrive according to a Poisson process with rate two customers per hour and priority class 2
7.20 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
7.19 Consider a queueing system in which the interarrival times of customers are the thirdorder Erlang random variable with parameter λ. The time to serve a customer is exponentially distributed
7.18 Consider a finite-capacity G/M/1 queueing that allows at most three customers in the system including the customer receiving service. The time to serve a customer is exponentially distributed
7.17 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,
7.16 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
7.15 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
7.14 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
7.13 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 or she defects
7.12 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
7.11 Consider a birth-and-death process representing a multiserver finite population system with the following birth and death rates: λk 5 ð4 2 kÞλ k 5 0; 1; ...; 4 μk 5 kμ k 5 1; ...; 4a.
7.10 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
7.9 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 h. There is one resident
7.8 A small PBX serving a start-up 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
7.7 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
7.6 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,
7.5 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
7.4 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
7.3 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 h. There are two repairmen who
7.2 Cars arrive at a car wash according to a Poisson process with a mean rate of eight cars per hour. The policy at the car wash is that the next car cannot pass through the wash procedure until the
7.1 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
6.12 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
6.11 In her retirement days, a mother of three grownup 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
6.10 A component is replaced every T time units and upon its failure, whichever comes first. The lifetimes of successive components are independent and identically distributed random variables with
6.9 Customers arrive at a taxi depot according to a Poisson process with rate λ. The dispatcher sends for a taxi where there are N customers waiting at the station. It takes M units of time for a
6.8 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
6.7 Consider a Markov renewal process with the semi-Markov kernel Q given by Q 5 0:6ð1 2 e25t Þ 0:4ð1 2 e22t Þ 0:5 2 0:2 e23t 2 0:3 e25t 0:5 2 0:5 e22t 2 t e22t " #a. Determine the
6.6 Larry is a student who does not 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
6.5 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
6.4 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
6.3 Victor is a student who is conducting experiments with a series of light bulbs. He started with 10 identical light bulbs, each of which has an exponentially distributed lifetime with a mean of
6.2 The Merrimack Airlines company runs a commuter air service between Manchester, NH, and Cape Cod, MA. Because the company is a small one, there is no set schedule for their flights, and no
6.1 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
5.12 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
5.11 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
5.10 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
5.9 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;
5.8 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
5.7 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
5.6 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
5.5 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
5.4 Lazy Chris has three identical light bulbs in his living room that he keeps on all the time. Because of his laziness Chris does not replace a light bulb when it fails. (Maybe Chris does not even
5.3 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
5.2 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
5.1 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
4.10 On a given day Mark is cheerful, so-so, or glum. Given that he is cheerful on a given day, then he will be cheerful again the next day with probability 0.6, so-so with probability 0.2, and glum
4.9 Let fXng be a Markov chain with the state space f1; 2; 3g and transition probability matrix P 5 0 0:4 0:6 0:25 0:75 0 0:4 00:6 2 4 3 5 Let the initial distribution be pð0Þ 5 ½p1ð0Þ; p2ð0Þ;
4.8 Consider the following transition probability matrix: P 5 0:5 0:25 0:25 0:3 0:3 0:4 0:25 0:5 0:25 2 4 3 5a. Calculate f13ð4Þ the probability of first passage from state 1 to state 3 in four
4.7 Consider the following transition probability matrix: P 5 1000 0:75 0 0:25 0 0 0:25 0 0:75 0001 2 6 6 4 3 7 7 5a. Put the matrix in the canonical form P 5 I 0 R Q .b. Calculate the expected
4.6 Consider the following transition probability matrix: P 5 0:5 0:25 0:25 0:3 0:3 0:4 0:25 0:5 0:25 2 4 3 5a. Calculate p13ð3Þ; p22ð2Þ, and p32ð4Þ.b. Calculate p32ðNÞ
4.5 Consider the following transition probability matrix: P 5 0:3 0:2 0:5 0:1 0:8 0:1 0:4 0:4 0:2 2 4 3 5a. What is Pn?b. Obtain φ13ð5Þ, the mean occupancy time of state 3 up to five transitions
4.4 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
4.3 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
4.2 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
4.1 Consider the following transition probability matrix: P 5 0:6 0:2 0:2 0:3 0:4 0:3 0:0 0:3 0:7 2 4 3 5a. Give the state-transition diagram.b. Given that the process is currently in state 1, what
2.15 A symmetric random walk fSnjn 5 0; 1; 2; ...g starts at the position S0 5 k and ends when the walk first reaches either the origin or the position m, where 0 , k , m. Let T be defined by T 5
2.14 Let X1; X2; ... be independent and identically distributed Bernoulli random variables with values 6 1 that have equal probability of 1=2. Let K1 and K2 be positive integers, and define N as
2.13 Let X1; X2; ... be independent and identically distributed Bernoulli random variables with values 6 1 that have equal probability of 1=2. Show that the partial sums Sn 5 Xn k51 Xk k n 5 1; 2;
2.12 Let the random variable Sn be defined as follows: Sn 5 0 X n 5 0 n k51 Xk n $ 1 where Xk is the kth outcome of a Bernoulli trial such that P½Xk 5 1 5 p and P½Xk 521 5 q 5 1 2 p, and the Xk
2.11 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
2.10 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
2.9 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,
2.8 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 fXðxÞ 5 λ e2λx; λ . 0; x $ 0, and if the
2.7 Joe replaced two light bulbs, one of which is rated 60 W with an exponentially distributed lifetime whose mean is 200 h, and the other is rated 100 W with an exponentially distributed lifetime
2.6 Bob has a pet that requires the light in his apartment to always be on. To achieve this, Bob keeps three light bulbs on with the hope that at least one bulb will be operational when he is not at

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