From 3:00 P.M. to 8:00 P.M. the local K-Star Supermarket has a steady stream of

Question

From 3:00 P.M. to 8:00 P.M. the local K-Star Supermarket has a steady stream of customers. Customers finish shopping and arrive at the checkout area at a rate of 70 per hour (Poisson distributed). It is assumed that when customers arrive at the cash registers, they will divide themselves relatively evenly so that all the checkout lines contain an equal number. The average checkout time at a register is 7 minutes (exponentially distributed). The store manager’s service goal is for customers to be out of the store within 12 minutes (on average) after they complete their shopping and arrive at a cash register. How many cash registers must the store have open to achieve the manager’s service goal?

Explanation / Answer

= 70 per hour

Wq = average checkout time = 7 minutes = 0.11 hour

W = Average time in the system = 12 minutes = 0.2 hour

As per multiple server model, W = Wq + 1 / Service rate

= 0.2 = 0.11 + 1/ Service rate

Service rate = 11.11 customers / hour or 11 customers per hour

Utilization = / Service rate * Number of cash registers

To achieve the goal, manager would wish for a 100% utilization or slightly less so that people are able to move out of the system in 12 minutes.

100% = 70/(11*Number of cash registers)

Number of cash registers = 70 / 11 = 6.66 cash registerss or ~ 7 cash registers

Average utilization for 7 cash registers = 70 / 7*11 = 70/77 = ~90%

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