Question

1. Arnold’s Muffler Shop has a top mechanic who can install mufflers at an average rate of 3/hour (standard deviation of time is 10 minutes). Customers needing service arrive at the shop on the average of 2/hour (standard deviation of time is 15 minutes). Find the following:
   1. The utilization of the mechanic
   2. The average number of customers in line
   3. The average number of customers in the system
   4. The average time a customer waits before getting a new muffler
   5. The average time a customer spends in the service system

Answer 1

Given information :

Mechanic installs mufflers at the rate of 3 per hour

Standard deviation of time = 10 mins

μ = 3

Customers needing service arrive at the shop on the average of 2/hour

Standard deviation of time = 15 mins

λͅ = 2

a. Utilization rate or mechanic = λͅ / μ

= 2/3

= 66.67%

Mechanic is busy 66.67% of his time

b. Average number of customers in line = square of λͅ / μ (μ - λͅ)

= (2 X 2) / 3 (3-2)

= 4 / 3

= 1.33

              1.33 customers are waiting in line

c. Average number of customers in system = λͅ / (μ - λͅ)

= 2 / (3-2)

= 2

            2 people are waiting in system

d. Average time a customer waits before getting a new muffler = λͅ / μ (μ - λͅ)

= 2 / 3 (3-2)

= 2 / 3

= 4o mins

40 mins are spent waiting for new muffler

e. Average time a customer spends in the service system = 1 / (μ - λͅ)

= 1 / (3-2)

= 1 hour

1 hour is spent waiting in system