Question

4. (Lecture 25, Continuous Time Random Process, 9 pts) Suppose that the total number of requests to a web server received between time 0 and time t, N(t), is given by a Poisson random process with rate 2- 10 requests per minute (a) Find the probability that exactly 8 requests are received in the first 30 seconds. (b) Find the probability that exactly 8 requests are received in the first 30 seconds and that exactly 8 requests are received in the next 30 seconds. (c) Find the probability that the first request occurs before -3 seconds. (d) Find the probability that the first request occurs between 1-1 and 1-3 seconds. (e) Find the probability that the second request occurs before -6 seconds.

Answer 1

a)

rate= 10 request per minute

for 30 second

$\lambda$ = 10/60 * 30 = 5

P(X = 8) = =poisson(8,5,0)

=0.065278

b)

first 30 second and next 30 seconds are independent

hence required probability

= 0.065278^2

= 0.004261

c)

Inter-arrival times follow exponential distribution

P(T < 3)

= 1 -e^(-$\lambda$t)

= 1 - e^(-10/60 * 3)

= 0.393469

d)

P( 1 < T < 3)

= e^(-10/60*1) - e^(-10/60*3)

= 0.23995