A gas station is providing the state auto inspection service

A gas station is providing the state auto inspection service for the general public. As soon as the gas station
opens at 7AM, cars arrive for inspection following a Poisson process with rate (arrivals/minute).

(a) Assume that each inspection takes a constant amount time, namely c (minutes). Let W2 denote the
random time that the second vehicle waits between arriving and the time at which its inspection starts.
What is the probability that W2 = 0? What is the expected value of W2?

(b) Answer the same questions above assuming that the service time for the first customer is an exponentially distributed random variable with mean 1/k

Solution

A gas station is providing the state auto inspection service for the general public.

As soon as the gas station opens at 7AM, cars arrive for inspection following a Poisson process with rate (arrivals/minute).

Assume that each inspection takes a constant amount time, namely c (minutes).

Let W2 denote the random time that the second vehicle waits between arriving and the time at which its inspection starts.

We know that waiting time distribution is exponential distribution.

Exponential distribution is continuous distribution.

And we also know that the exact probability in continous distribution is 0.

So Probability that W2 = 0 is 0.

W2 follows exponential distribution with paramneter arrivals per minute.

Expected value of E2 is = 1 /

Answer the same questions above assuming that the service time for the first customer is an exponentially distributed random variable with mean 1/k

the service time for the first customer is exponential distribution with parameter k.