Handout on the Poisson process and the queuing theory.

1. What is a counting process? What is its typical realization?

2. Give a definition of the Poisson process.

3. Give a definition of a queuing process.

4. What is a M/M/1 process?

5. Explain how you understand the notions of the stationary regime, the stationary distribution, the behavior of the process in the long run.

6. Military vehicles arrive at a service facility according to the Poisson process. Under the condition that the rate at which vehicles arrive is 5 per hour, find

a. the probability that during 6 hours no vehicles will arrive, most three vehicles will arrive;

b. the probability that the waiting time between the third and the fourth vehicle will be greater than 30 min., equal 30 min., greater or equal than 30 min.;

c. the probability that after the first vehicle has arrived, the waiting time for fifth vehicle will be greater than an hour;

d. for the same waiting time - its expected value and the standard deviation.

Answers: \lambda=5. (a) exp{-30}; 39.33* exp{-30}. (b) .082; 0; 0.082.

(c) 104.16( \int_1^{\infty}x^3*exp{-5x}dx). (d) 0.8; 0 .4.

7. Answer the same questions a-d from the previous problem under the condition that the mean inter-arrival time is 30 min. Answers: The same as above, but instead of 5, we should consider \lambda=2.

8. Assume again that the mean inter-arrival time is 30 min, and that the service time for a particular vehicle is exponentially distributed with a mean of 15 min.

a. Will there be a line in the facility? Explain why?

b. For the stationary regime find the probability that there will be no line at a particular moment of time; there will be not more that four vehicles in the facility at a particular moment.

c. In the long run, what share of time the will be no vehicles in the facility; just one vehicle?