# Waiting in Line Queuing theory (also known as waiting-line theory) investigates the problem of…

Waiting in Line Queuing theory (also known as waiting-line theory) investigates the problem of providing adequate service economically to customers waiting in line. Suppose customers arrive at a fast-food service window at the rate of 9 people per hour. With reasonable assumptions, the average time (in hours) that a customer will wait in line before being served is modeled by

where x is the average number of people served per hour. A graph of ƒ(x) for x 7 9 is shown in the figure on the next page.

(a) Why is the function meaningless if the average number of people served per hour is less than 9? Suppose the average time to serve a customer is 5 min.

(b) How many customers can be served in an hour?

(c) How many minutes will a customer have to wait in line (on the average)?

(d) Suppose we want to halve the average waiting time to 7.5 min (1/8 hr). How fast must an employee work to serve a customer (on the average)? (Hint: Let ƒ(x) = 18 and solve the equation for x. Convert the answer to minutes and round  to the nearest hundredth.) How might this reduction in serving time be accomplished?

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