4. Consider an M/M/1/k queueing system where k 1, the arrival rate is , and the...

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4. Consider an M/M/1/k queueing system where k 1, the arrival rate is , and the service rate is . Define p = /. (a) Give the transition matrix of the uniformized process. (b) Is it irreducible? (c) Find the stationary distribution. (d) Pretend that we do know PASTA, and give the stationary distri- bution embedded before arrivals. (e) What fraction of customers are lost, i.e., unable to receive service? (f) What is the expected time that an arriving customer waits for service to begin if the arrival sees i customers already in the system when the customer arrives? (Be careful when i is what value?) Poisson Arrivals See Time Averages The PASTA property refers to the expected state of a queueing system as seen by an arrival from a Poison process. An arrival from a Poisson process observes the system as if it was arriving at a random moment in time. Therefore, the expected value of any parameter of the queue at the instant of a Poisson arrival is simply the long-run average value of that parameter. 4. Consider an M/M/1/k queueing system where k 1, the arrival rate is , and the service rate is . Define p = /. (a) Give the transition matrix of the uniformized process. (b) Is it irreducible? (c) Find the stationary distribution. (d) Pretend that we do know PASTA, and give the stationary distri- bution embedded before arrivals. (e) What fraction of customers are lost, i.e., unable to receive service? (f) What is the expected time that an arriving customer waits for service to begin if the arrival sees i customers already in the system when the customer arrives? (Be careful when i is what value?) Poisson Arrivals See Time Averages The PASTA property refers to the expected state of a queueing system as seen by an arrival from a Poison process. An arrival from a Poisson process observes the system as if it was arriving at a random moment in time. Therefore, the expected value of any parameter of the queue at the instant of a Poisson arrival is simply the long-run average value of that parameter.