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# Waiting Lines (Queuing) / Fast Shop Market

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The Fast Shop Market has a single checkout counter and one employee to serve customers. An average of 24 customers arrives each hour with a Poisson distribution (and therefore, exponentially distributed interarrival times). Customer checkout times are exponentially distributed with a mean of 2 minutes. Customers are checked out according to their order line.

a) Calculate the average length of the checkout line and the average time (minutes) that customers spend waiting in line prior to checkout.

b) The owner of Fast Shop is considering adding a "bagger" to speed up the checkout process. Experiments show that 40 customers can be served per hour with a bagger (exponentially distributed). Recalculate the average length of the checkout line and the average time (minutes) that customers spend waiting in queue for service.

c) The bagger will cost the store employee \$300 per week. The national office has done research that indicates that for each additional minute the average customer waits in line costs a Fast Shop store \$150 per week in lost sales. Is the bagger worth her/his wages?

d) Suppose that Fast Shop finds that in addition to reducing mean processing times, the use of a bagger also reduces the standard deviation of checkout times by 50%. By what percent will customer in-line waiting times be reduced?

#### Solution Preview

The Fast Shop Market has a single checkout counter and one employee to serve customers. An average of 24 customers arrive each hour with a Poisson distribution (and therefore, exponentially distributed interarrival times). Customer checkout times are exponentially distributed with a mean of 2 minutes. Customers are checked out according to their order line.

a) Calculate the average length of the checkout line and the average time (minutes) that customers ...

#### Solution Summary

This solution provides calculations for average time, average length, cost of employees, and percentage reduction.

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