During lunch time, customers arrive at a postal office at a rate of lambda equals 36 per hour. The interarrival time of the arrival process can be approximated with an exponential distribution. Customers can be served by the postal office at a rate of mu equals 45 per hour. The service time for the customers can also be approximated with an exponential distribution. For each of the following questions, show your work and use the right notation. Required:
a. Determine the utilization factor.
b. Determine the probability that the system is idle, i. e., no customer is waiting or being served.
c. Determine the probability that exactly one customer is in the system, i. e., no customer is waiting but one is served.

a) utilization factor, P =4/5

b)Probability that the system is idle, P₀=1/5

C) the probability that exactly one customer is in the system,P ₁=4/25

Explanation:

A)

From the question,

Customer arrives at the rate of λ equal 36  per hour

Also,

Customers can be served by the postal office at a rate of μ equals 45 per hour

Therefore, we have that

utilization factor. P = λ / μ

where

λ = 36 / hour

μ = 45 / hour

P= 36 / 45

P= 4/5

The utilization factor is 4/5

b) the probability that the system is idle, i.e., no customer is waiting or being served.

Probability that the system is idle P₀ =1 - P

1 - 4/5

=1/5

C) the probability that exactly one customer is in the system, i.e., no customer is waiting but one is served.

probability that exactly one customer is in the system,P ₁=(λ/μ)¹ x (1-λ/μ)

(36 / 45) x (1-36 / 45)

4/5 x (1-4/5)

4/5 x 1/5

=4/25

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