## Suppose calls are arriving at a telephone exchange at an average rate of one per second, according to a Poisson arrival process. Find: a) th

Question

Suppose calls are arriving at a telephone exchange at an average rate of one per second, according to a Poisson arrival process. Find: a) the probability that the fourth call after time t = 0 arrives within 2 seconds of the third call; b) the probability that the fourth call arrives by time t = 5 seconds; c) the expected time at which the fourth call arrives.

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2 weeks 2021-09-11T02:44:05+00:00 1 Answer 0

Explanation has been given below

Step-by-step explanation:

a) inter arrival times are exponentially distributed with mean 1/n , where n = rate = 1/sec.

probability distribution function is F(t)=n*exp(-n*t).

reference to any kth packet and the (k-1)th packet

the answer is = integration of F(t).dt with limits 0 to 2 = 1 – exp(-2*n) = 1 – exp(-2)

b)
t=5 , P(q) = exp(-5)*(5)^q/factorial(q)

probability of fourth call within t=5 seconds is =

that is P(4)   P(5)   ……  = 1 – ( P(0)   P(1)   P(2)   P(3) ) ;  put the values and get the answer.

c) number of calls/rate =  4/n = 4 seconds