Question

Arnold's Muffler Shop in New Orleans, Arnold's mechanic, Reid Blank, is able to install new mufflers at an average rate of 3 per hour, or about 1 every 20 minutes. Customers needing this service arrive at the shop on the average of 2 per hour. Arnold estimates that the cost of customer waiting time, in terms of customer dissatisfaction and lost goodwill, is $50 per hour of time spent waiting in line. Moreover, the salary of Blank is $15/h. A. Calculate the preceding operating characteristic. B. Calculate the total daily cost of the queuing system C. Arnold finds out through the muffler business grapevine that the Rusty Muffler, a cross-town competitor, employs a mechanic named Jimmy Smith who can efficiently install new mufflers at the rate of 4 per hour with a $20 per hour salary. Decide whether or not Arnold switching employee.

Answer 1

To calculate the operating characteristics and costs, the service rate and arrival rate, along with the mechanic salary and cost of customer waiting, are used. The queue model equations determine the average number of customers in the system and their waiting time. Comparing costs between existing and potential new mechanic requires analyzing both labor and customer waiting costs.

Step-by-step explanation:

To calculate the operating characteristics of Arnold's Muffler Shop:

• The mechanic, Reid Blank, installs 3 mufflers per hour, so the service rate (μ) is 3 mufflers per hour.
• Customers arrive at a rate (λ) of 2 per hour.
• The cost of customer waiting is $50 per hour.

1. Reid Blank's salary is $15 per hour.

The average number of customers in the system (L) is given by L = λ / (μ - λ) when μ > λ. Plugging in the values, we get L = 2 / (3 - 2) = 2 customers.

The average time a customer spends in the system (W) is 1 / (μ - λ). W = 1 / (3 - 2) = 1 hour per customer.

For total daily cost (ignoring Reid's working hours), we need to estimate based on the shop's operating hours. Assuming 8-hour workday: The total cost of the queuing system (TC) equals the cost of mechanic's salary plus the cost of customer waiting. TC = (Reid's salary * Operating hours) + (Average number of customers in the system * Cost of waiting * Operating hours). If Reid works 8 hours, TC = ($15 * 8) + (2 * $50 * 8) = $120 + $800 = $920.

To decide whether Arnold should switch to Jimmy Smith, we compare their efficiency and costs:

• Reid Blank: 3 mufflers per hour, costs Arnold $15/hour + customer waiting costs.
• Jimmy Smith: 4 mufflers per hour, costs Arnold $20/hour + potentially lower customer waiting costs due to higher service rate.

Arnold needs to calculate the revised total costs using Jimmy's higher service rate and salary. If the savings from reduced customer waiting times outweigh the increased salary cost, Arnold may consider switching employees.