Question

An average of 90 cars per hour arrive at a single-server toll booth. The average service time for each customer is a half minute, and both interarrival times and service times are exponential. For each of the following questions, show your work, including the formula that you are using

What is the probability that the server is idle?

Answer 1

The probability that the server is idle is 0.25.

Explanation:

If N (t) is a Poisson process with rate λ, then the inter-arrival times X₁, X₂, ⋯ are independent and the distribution of X
`_(i)` is Exponential (λ).

The arrival rate of customers at a single-server toll booth is:

λ = 90 cars/hour

The service time for each customer is half a minute.

Then the service rate is:

`\mu=(60)/(0.50)`

μ = 120 cars/hour

Then the probability statement P ( N (t) = n) there are n customers in the system.

`P ( N (t) = n) =((\lambda)/(\mu))^(n)[1-(\lambda)/(\mu)]`

Compute the value of P ( N (t) = 0) as follows:

`P ( N (t) = 0) =((90)/(120))^(0)[1-(90)/(120)]`

`=1* [(120-90)/(120)]`

`=(30)/(120)\\=0.25`

Thus, the probability that the server is idle is 0.25.