随机过程英语讲义-9.pdf

2.2 Properties of Poisson processes Example: Cars arrive at Galveston Beach during spring break. Assume that the interarrival time of cars follows an exponential distribution with parameter λand the sojourn time of a car on the beach follows a probability distribution G. Also, we assume that the sojourn times are independent of each other and the arrival process, the beach can hold an unlimited number of cars, and at time 0 there are no cars on the beach. Let N (t) 1 denote the number of cars that have left the beach at time t and N (t) the number of cars still at the beach at time t. 2 1 2.2 Properties of Poisson processes What can be said about the two random variables N (t) and 1 N (t)? Find means of N (t) and N (t) 2 1 2 X denotes sojourn time N (t): numbers of car left the beach at time t 1 N (t): numbers of car still at the beach at time t 2 {X t-s }⇔ {Car at time t will be type1} P {X t-s} = P {Car at time t will be type 1} P {X t-s}= G (t-s) = P(s) 1 t