Question: Problem 4 (15 points = 5 + 3 + 3 + 4) Consider a queuing system with a single server, M/M/1. Arrival times are independent

Problem 4 (15 points = 5 + 3 + 3 + 4) Consider a

Problem 4 (15 points = 5 + 3 + 3 + 4) Consider a queuing system with a single server, M/M/1. Arrival times are independent exponentially distributed variables with expectation XT! 10 minutes. Service times are also independent exponentially distributed with expectation = 7.5 minutes. 1. Derive stationary distribution of the number X(t) service requests in a system, m = P(X(t) = m) for any integer m > 0 2. Evaluate expected number E() of customers in the system under stationary distribution 3. Determine the average waiting time, W 4. Find expected idle time, E[and expected busy time, E [B], using formula: El ELT + E[B] - TO Solution Problem 4 (15 points = 5 + 3 + 3 + 4) Consider a queuing system with a single server, M/M/1. Arrival times are independent exponentially distributed variables with expectation XT! 10 minutes. Service times are also independent exponentially distributed with expectation = 7.5 minutes. 1. Derive stationary distribution of the number X(t) service requests in a system, m = P(X(t) = m) for any integer m > 0 2. Evaluate expected number E() of customers in the system under stationary distribution 3. Determine the average waiting time, W 4. Find expected idle time, E[and expected busy time, E [B], using formula: El ELT + E[B] - TO Solution

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