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Answer key to Quiz 3.526-F12Problem.Consider a customer service office with two representatives, each one requiresan independent exponentially distributed period of time with mean 10 minutes to set up acustomer. Customers call the office at times of a Poisson process independent of the currentstatus of the representatives with rate 0.2 per minute. If both of the representatives arebusy, the arriving call will be put on hold.If there is already one call on hold, the newcalls will be automatically cancelled. LetXtbe the number of customers connected to theoffice at timet.a) Determine the jump rate matrixQ= (q(i, j)) for the Markov chainXt. (1 pts)b) Use the detailed balance condition to find the stationary distribution. (2 pts)c) Find the limiting fraction of time that no one is on hold. (1 pts).d) Given that exactly one representative is busy, what is the probability that someone isput on hold before both representatives become free? (2 pts)
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Probability theory, Markov chain, exponentially distributed period