Question

In the service department of the NAPA (National Auto Parts Association), mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 60 per hour, and a clerk can fill requests at the rate of 30 per hour (exponential). If the cost for a parts clerk is $30 per hour and the cost for a mechanic is $60 per hour, answer the following three questions (Because of the high arrival rate, an infinite source may be assumed.)

• Please analyze the total cost per hour (clerks’ hourly cost and mechanics’ time spent in system) when there are 3 clerks to staff the counter?
• Please analyze the total cost per hour (clerks’ hourly cost and mechanics’ time spent in system) when there are 4 clerks to staff the counter?

Is there a need to have 5 clerk to staff the counter? Please explain why with

Answer 1

Arrival rate Lambda = 60 per hour

service rate mu = 30 per hour

Utilisation rho = 60/30 =2

Lq with rho =2  and number of resources =3 would be 0.8888

avg waiting time per person with 3 clerks = 0.8888 /60

avg waiting time per hour = 60x 0.8888/60 =0.8888 hours

Total cost = 0.8888x60+3x30 = 143.328

In the similar way, Lq with rho =2 and 4 resources would be 0.1739

avg waiting time per hour = 60x0.1739x60 =0.1739

Total cost = 0.1739x60+4x30 = 130.43

There is no need to employ 5 workers as their cost alone would be 30x5 =150, which is higher than both option above.