Harriet McNeil, proprietor of McNeil’s Auto Mall, believes that it is good business for her atuomobile dealership to have more customers on the lot than can be served, as she believes this creates an impression that demand for the automobiles on her lot is high. However, she also understands that if there are far more customers on the lot than can be served by her salespeople, her dealership may lose sales to customers who become frustrated and leave without making a purchase.

Ms. McNeil is primarily concerned about the staffing of salespeople on her lot on Saturday mornings (8:00 a.m. to noon), which are the busiest time of the week for McNeil’s Auto Mall. On Saturday mornings, an average of 6.8 customers arrive per hour. The customers arrive randomly at a constant rate throughout the morning, and a salesperson spends an average of one hour with a customer. Ms. McNeil’s experience has led her to conclude that if there are two more customers on her lot than can be served at any time on a Saturday morning, her automobile dealership achieves the optimal balance of creating an impression of high demand without losing too many customers who become frustrated and leave without making a purchase.

Ms. McNeil now wants to determine how many salespeople she should have on her lot on Saturday mornings in order to achieve her goal of having two more customers on her lot than can be served at any time. She understands that occasionally the number of customers on her lot will exceed the number of salespersons by more than two, and she is willing to accept such an occurrence no more than 10% of the time. 

Managerial Report

Ms. McNeil has asked you to determine the number of salespersons she should have on her lot on Saturday mornings in order to satisfy her criteria. In answering Ms. McNeil’s question, consider the following three quesitons: 

1. How is the number of customers who arrive in the lot on a Saturday morning distributed?

2. Suppose Ms. McNeil currently uses five salespeople on her lot on Saturday morning. Using the probability distribution you identified in (1), what is the probability that the number of customers who arrive on her lot will exceed the number of salespersons by more than two? Does her current Saturday morning employment strategy satisfy her stated objective? Why or why not?

3. What is the minimum number of salespeople Ms. McNeil should have on her lot on Saturday mornings to achieve her objective?