Question

Customers arrive at a booking office window being manned by a single individual at a rate of 25 per hour. The time required to serve a customer has a mean of 120 seconds. If the space in front of the window can accommodate only 2 customers. Then the probability that a arriving customer will have to wait outside the provided space will be?

Answer 1

Let T0 =  0 be the point first customer arrive and X0 be the time to serve it.

Let the second customer arrive at time T1 and it takes X1 to serve him.

Let T2 be the time of arrival of third customer from the time point T1

Then need to find the probability that X0 > T1 + T2 --- first customer is still being served when third customer arrives and X1 > T2 --- second customer is still being served when third arrives

so need to find P(X0 > T1 + T2 and X1 > T2). Also X0, X1, T1, T2 are independent, as the time of arrival of customer doesnt determine the time it requires to serve them. Also Time require to serve one customer doesn't determine time required to served another one

Now we know that X0, X1 are independent Exp(2min)

and T1, T2 are indepedent exponential (1/25 hr) -- since arrival is poisson process hence interarrival time is exponential

so T1, T2 independent exponential(2.4)

so

Since X0 follows exponential(2 min)

Similarly

So substituting in above and omiting the conditioning part from integrand as they are independent

So since are densities of T1 and T2, respectively hence

----- pdf of exponential(2.4) as it

Substituting values we get

= 0.13 Ans