Ch6_Exercises

Ch6_Exercises - Exercises 407 raising that matrix to the n...

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Unformatted text preview: Exercises 407 raising that matrix to the n th power (by utilizing k matrix multiplications). It can be shown that the matrix ( I R t/n) 1 will have only nonnegative elements. Remark Both of the preceding computational approaches for approximating P (t) have probabilistic interpretations (see Exercises 41 and 42). Exercises 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 probability h + o(h) . Each mating immedi- ately produces one offspring, equally likely to be male or female. Let N 1 (t) and N 2 (t) denote the number of males and females in the population at t . Derive the parameters of the continuous-time Markov chain { N 1 (t), N 2 (t) } , i.e., the v i , P ij of Section 6.2. *2. Suppose that a one-celled organism can be in one of two stateseither A or B . An individual in state A will change to state B at an exponential rate ; an individual in state B divides into two new individuals of type A at an exponential rate . Define an appropriate continuous-time Markov chain for a population of such organisms and determine the appropriate parameters for this model. 3. Consider two machines that are maintained by a single repairman. Machine i functions for an exponential time with rate i before breaking down, i = 1 , 2. The repair times (for either machine) are exponential with rate . Can we analyze this as a birth and death process? If so, what are the parameters? If not, how can we analyze it? *4. Potential customers arrive at a single-server station in accordance with a Poisson process with rate . However, if the arrival finds n customers already in the station, then he will enter the system with probability n . Assuming an exponential service rate , set this up as a birth and death process and determine the birth and death rates. 5. There are N individuals in a population, some of whom have a certain in- fection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process having rate . When a contact occurs, it is equally likely to involve any of the ( N 2 ) pairs of individuals in the population. If a contact involves an infected and a noninfected individual, then with probabil- ity p the noninfected individual becomes infected. Once infected, an individual remains infected throughout. Let X(t) denote the number of infected members of the population at time t . 408 6 Continuous-Time Markov Chains (a) Is { X(t), t > } a continuous-time Markov chain? (b) Specify its type. (c) Starting with a single infected individual, what is the expected time until all members are infected?...
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Ch6_Exercises - Exercises 407 raising that matrix to the n...

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