Question

Consider an espresso stand with a single barista. Customers arrive at the rate of 25 per hour according to a Poisson distribution. Service times are exponentially distributed with a mean service time of 1.5 minutes per customer. Answer: a. What is the service rate per hour for the expresso stand? b. What is the probability that the server is busy? c. What is the probability that the server is idle? d. What is the average number of customers waiting in line? e. What is the average time in minutes a customer spends waiting in line for service?

Answer 1

Answer:

Kindly check explanation

Step-by-step explanation:

Given that:

Arrival rate λ = 25 per hour

Service time = 1.5 minutes per customer

A.) The service rate per hour :

60 / service time = 60 / 1.5 = 40 customers per hour

B.) probability that server is busy

Utilization factor = arrival rate / service rate

= 25 / 40

= 0.625

C.) probability that server is idle :

1 - 0.625 = 0.375

D.) average Number of customers waiting on line ;

λ² / μ(μ - λ)

25² / 40(40 - 25)

625 / 40(15)

= 1.04

Average time spent waiting in line for service :

λ / μ(μ - λ) * 60

25 / 40 ( 40 - 25)

25 / 40(15)

0.0416666 * 60

= 2.5 minutes