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# Poisson Distribution: Birth-Death Process

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The McDougal Sandwich Shop has two windows available for serving customers, who arrive at a Poisson rate of 40/hr. Service time is exponentially distributed with a mean of 2 min. Only one window is open as long as there are three or less customers in the shop. An identical server opens a second window when there are four or more customers in the shop. The manager of the shop helps the two attendants when there are six or more customers in the shop. When the manager is helping, the mean service time for each of the two servers is reduced to 1.5 min. The shop cannot legally hold more than 20 customers. Use the general birth-death process to determine 1) the expected number of customers in the shop 2) the probability that the manager will be helping serve customers and, 3) the probability that the store is full.