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Problem 7 [15 points = 5 + 10] Consider a queuing system M /M / 00 with unlimited number of servers that is functioning under stationary distribution. Customer arrivals form a Poisson process that has rate A = 30 per hour. Service times are independent exponentially distributed with average = [tn1 = 5 minutes. Focus on the departure process, {1705) : t 2 0} assuming that the system is operating under stationary distribution. ConSider a random variable7 T, which is independent of the process, D = {D(t) : t 2 0} and has the density function: 63 fT(t) = a -t2 -exp(6t) for t > 0, where t is measured in hours. Set Y = D(T), which is the count of departed customers by the inspection time. 1. Derive expectation of Y. 2. Evaluate variance of Y