### Performance Evaluation

### Final 2009

### Hiroshi Toyoizumi

### 7/?/2009

1. Describe what is PASTA, using examples and figures.

2. Let N(t) be the number of customers in the M/M/1 queue with arrival rate 1 and the service rate 2. Use the state transition diagram (figure!) to derive the steady state probabilityP{N(t) =n}and its meanE[N(t)]. 3. Explain the concept of reversible, and its importance for evaluating queues. 4. Take queues in real life situation and evaluate it. You may use

mathemat-ical arguements and/or intuitions for the evaluation.

5. Suppose there is a tandem queue (two servers in tandem) with Poisson arrival with the rate 1. The service times on each servers are independent and exponentially distributed, but the rate is different. The service rate for the first server is 2 and the second one is 3. On average, how long does it take to leave the second queue from the arrival to the tandem queue? 6. Explain the differences of the ordinary Riemann integral and Ito-stochastic

integral.

7. You can write anything you want.

Remark 1. _{Don’t write lengthy answers. Your answers should be concise and}
focused.

Remark 2. _{Each problem is 10 point worth.}