Machinists who work at a tool-and-die plant must check out tools from a tool center. An average of 10 machinists per hour arrive seeking tools and the interarrival time is exponentially distributed. At present, the tool center is staffed by a clerk who is paid $15 per hour and who takes an average of 4 minutes to handle each request for tools. The handling time is exponentially distributed. Since each machinist produces $10 worth of goods per hour, each hour that a machinist spends at the tool center costs the company $10. The company now needs to decide whether to get one additional clerk or a helper for the current clerk.

1. If an additional clerk is hired, he/she will be paid at $15 per hour. The new clerk will be identical to the current clerk, i.e., he/she can also handle a request for tools in 4 minutes on average. Also, he/she will be helping a machinist by himself/herself. The machinist waiting for a clerk will go to the next available clerk for their requests.

2. If a helper is hired, he/she will be paid at $5 per hour. The helper will help the current clerk in handling machinists’ requests, that is, the helper cannot handle a request by himself/herself; he/she only helps the current clerk. With the helper’s help, the current clerk can handle a machinist’s requests in 3 minutes on average (rather than 4 minutes) and the handling time is still exponentially distributed.

a) Describe the queuing system with the current clerk only (i.e., no additional clerk or helper is hired) and use Kendall’s notation to define the queueing system. After that, calculate the expected hourly cost of the system. In doing so, clearly show your calculations.

b) Describe the queuing system with the current clerk plus the additional clerk and use Kendall’s notation to define the queueing system with the additional clerk. After that, calculate the expected hourly cost of the system. In doing so, clearly show your calculations.

c) Describe the queuing system with the current clerk plus the helper and use Kendall’s notation to define the queueing system with the helper. After that, calculate the expected hourly cost of the system. In doing so, clearly show your calculations.

d) Based on the expected hourly cost, would you hire an additional clerk, or a helper, or none?