Answer:
The probability that the server is idle is 0.25.
Step-by-step 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[tex]_{i}[/tex] 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:
[tex]\mu=\frac{60}{0.50}[/tex]
μ = 120 cars/hour
Then the probability statement P ( N (t) = n) there are n customers in the system.
[tex]P ( N (t) = n) =(\frac{\lambda}{\mu})^{n}[1-\frac{\lambda}{\mu}][/tex]
Compute the value of P ( N (t) = 0) as follows:
[tex]P ( N (t) = 0) =(\frac{90}{120})^{0}[1-\frac{90}{120}][/tex]
[tex]=1\times [\frac{120-90}{120}][/tex]
[tex]=\frac{30}{120}\\=0.25[/tex]
Thus, the probability that the server is idle is 0.25.