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probability and stochastic modeling

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

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 chain with a state transition probability matrix P. At the beginning of every period, a Bernoulli
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 λ1, and type 2 customers arrive according to a Poisson process with rate λ2. All the m servers
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 down, the time until it is repaired and brought back to service is also exponentially distributed with
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 internal switching probabilities p and q, where q 5 1 2 p, and geometric service times with
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, Q, if the batch size is equally likely to be 1, 2, or 3. 12.2 Consider the superposition of two
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 μðu; BðuÞÞdu 1 ðt 0 σðu; BðuÞÞdBðuÞ t $ 0 Then, fXðtÞ; t $ 0g is a diffusion process
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; xÞfÞ 1 1 2 @2 @x2 ðbðt; xÞfÞ Assume that aðt; xÞ 5 aðtÞ and bðt; xÞ 5 bðtÞ. Show that the
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. Find the covariance function of the process.
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: Start by computing E½XðtÞjBðvÞ; 0 # v # t.
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 CDF of jBðtÞj.
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 dpaðxÞ dx 5 0b. Deduce that paðxÞ 5 e22μb 2 e22μx e22μb 2 e22μac. What is paðxÞ when μ 5 0?
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 independent with UBNð0; σ2=2Þ and VBNð0; σ2=2Þ.
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 and T are independent. Obtain the master equation, Pðx; tÞ, which is the probability that the p
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 step it will move to the right again with probabilitya, to the left with probability b and remain in
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 the cash level to x. Whenever the cash level reaches 0, the company sells enough treasury bills to
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 that the walk starts at the point k, where b , k ,a. a. What is the probability that the walk stops
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. Show that the PMF of N is given by pNðnÞ 5 P½N 5 n 5 βð12βÞ n n 5 0; 1; ... where β 5 jp 2 qj.
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 21 with probability q 5 1 2 p 5 0:4.a. Find the probability P½S8 5 0.b. What value of p maximizes
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 receiving service in time n, then the probability that he finishes receiving service before time n 1 1 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 different sides, Dana wins the $2. The game ends when either of them has all the money. What is the
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 money. Mark is a better player than Kevin and has a probability 0.5 of wining each game, a probability
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 $1. Assume that the balls are drawn with replacement and that Jim starts with $50 with the hope of
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. After receiving service at node 1 or node 2, a customer proceeds to node 3. Similarly, after
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 exponential server at each queue with the service rate indicated.
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 exponential single nodes with service rates μ1; μ2; μ3, and μ4, respectively, such that external
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 according to a Poisson process with rate η. Let N1 denote the steadystate number of customers at Queue
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 (i.e., low-priority) customers arrive according to a Poisson process with rate five customers per hour.
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 the expected waiting time of a customer?
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 with parameter μ. What is the expected waiting time of a customer?
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 with mean 1=μ. As usual, let rn denote the probability that n customers are served during an
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, the service time is exponentially distributed with mean 1=μ. If it comes up tails, the service
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 identical servers, each of which serves at the rate of μ customers per second, but as long as the
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 system returns to the empty state if there are no waiting customers. Customers who arrive while 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 newly arriving customer joins the queue with probability p and balks (i.e., leaves without joining
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 (i.e., leaves the queue without receiving service) is exponentially distributed with a mean of 1=β.
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 clerk takes to serve each customer is exponentially distributed with an unspecified mean. If it is
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. Find, in terms of λ and μ, pk, the probability that there are k customers in the system, k 5 0; 1;
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 PCs are being used are blocked and have to go elsewhere for their Internet access. The time that a
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 repairman at the site so that when a component fails, the repairman immediately swaps it out and
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 blocked and will have to hang up. A blocked call is considered to be lost because the employee will
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 process with a mean rate of 10 customers per hour, and customers that arrive when the facility is full
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, what is the mean service time that the clerk must provide in order that the mean waiting time shall be
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 either attendant is exponentially distributed with a mean of 3 min.a. What is the probability that an
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 4 min.a. What is the probability that a person arriving at the phone booth will have to wait?b. The
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 fix the machines when they fail. The time to fix a machine when it fails is exponentially
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 car in front of it is completely finished. The car wash has a capacity to hold 10 cars, including
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 exponentially distributed with a mean of 3 min. There is only one cashier at the ticket window, and
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 fφijðtÞg of the process.
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 her time next can be modeled by a Markov chain. Thus, if the children are labeled by ages as child
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 PDF fXðxÞ. A cost c1 . 0 is incurred for each planned replacement, and a fixed cost c2 . c1 is
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 taxi to arrive at the depot. When it arrives, the taxi picks up all waiting customers. The taxi
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 going to the fair state with probability 4/5 and to the broken state with probability 1/5. When it is
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 state-transition probability matrix P for the Markov chain fXng.b. Determine the conditional distributions
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 he lives in the suburb, he moves back to the city after one semester. The other half of Larry’s
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 probability that she will buy brand X again is 0.8, and the probability that she will buy brand Y next is
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 components to work, thus when a component fails the machine is shut down until the component is repaired and
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 200 h. Victor wants to know how long it will take until the last bulb burns out (or fails). At
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 reservation is needed for the flights. However, it has been determined that their planes arrive at the
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 down is also exponentially distributed with mean 1=λ. When repaired, the machine is considered to
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 duration of a car is exponentially distributed with mean 1=μ. Let pkðtÞ denote the probability that k
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 system 2 according to a Poisson process with rate λ2. Each system has two identical attendants. The
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 exponentially distributed with mean 1=μ1. When a truck is offloaded, it leaves the warehouse and takes
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; otherwise it remains in station 1 until station 2 is free. Items arrive at station 1 according to a
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 is exponentially distributed with a mean of 1=λ and then either splits into two identical particles
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 exponentially distributed with a mean of 1=μ hours. Customers arrive according to a Poisson process
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 make an attempt to return to the facility. Whenever there are two or fewer customers in the
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 rate λ calls per minute. The call durations are exponentially distributed with a mean of 1=μ
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 notice that the bulb has failed!) However, when all three bulbs have failed, Chris replaces them at
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 mean of 1=λB hours. The PCs also have different repair times. The time to repair PC A when it fails
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 when there is already a customer receiving a haircut. Customers who arrive when Mike is busy and all
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 repairing it immediately. The two PCs fail independently of each other. The time to repair a failed PC
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 with probability 0.2. Given that he is so-so on a given day, then he will be cheerful the next day
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; p3ð0Þ 5 ½0:4; 0:2; 0:4. Calculate the following probabilities:a. P½X1 5 2; X3 5 2; X3 5 1jX0
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 transitions.b. Calculate the mean sojourn time in state 2.
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 absorption times μ2 and μ3.
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 given that the process started from state 1.
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 is sunny on any given day, then on the following day it will be sunny again with probability 0.5,
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 is 0.3, the probability that the next passenger he picks up is going to town 2 is 0.2, and the
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 (state 3). Membership in any class is inherited in the following probabilistic manner. Given that a
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 is the probability that it will be in state 2 at the end of the third transition?c. Given that the
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 minfnjSn 5 0 or mg That is, T is the stopping time.a. Show that E½ST 5 k.b. Define Yn 5 S2 n 2 n and
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 follows: N 5 minfnjSn 5 K1 or Sn 5 2K2g where Sn 5 Xn k51 Xk n 5 1; 2; ... is called a symmetric random
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; ... form a martingale with respect to fXng.
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 are independent and identically distributed. Consider the process fSnjn 5 1; 2; ...g.a. For what
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 probability 0.4. Cars arrive at the fork in a Poisson manner with a rate of 8 cars per minute.a. What is the
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 minute.a. Given that there is currently no car at the intersection, what is the probability that a
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, calls the PC repairman, and turns on the backup PC. The time until either PC fails when it is in
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 lifetimes of the motors are independent, what is the mean of the random variable Y, the time until 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 whose mean is 100 h.a. What is the probability that the 60 W bulb fails before the 100 W bulb?b. What
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 the apartment. The light bulbs have independent and identically distributed lifetimes T with PDF fT

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