$732$

Chapter $13$ Queuing Theory

CASE $13-2$ Call Center Staffing at Vacations Inc.

Vacations Inc. $($VI$)$ markets time$-$share condominiums throughout North America. One way the company generates sales leads is by offering a chance to win a free mini$-$vacation to anyone who fills out an information card and places it in boxes VI has distributed at various restaurants and shopping malls. All those who fill out the card and indicate an adequate income level subsequently receive a letter from VI indicating they have indeed won the mini$-$vacation. To claim their prize, all the "winner" needs to do is call VI's toll$-$free number. When the "winner" calls the number, they learn that their mini$-$vacation consists of a free dinner, entertainment, and two$-$night stay at one of VI's time$-$share properties; but they must agree to sit through a $2-$hour property tour and sales presentation.

About half the people who call VI's toll$-$free number to claim their prize wind up rejecting the offer after they learn about the $2-$hour property tour. About $40\%$ of those who call accept the mini$-$vacation and do the property tour but don't buy anything. The remaining $10\%$ of those who call the toll$-$free number accept the mini$-$vacation and ultimately purchase a time$-$share. Each mini$-$vacation that VI awards costs the company about $$250.$ Each sale of a time$-$share generates a net profit of $$7,000$ for VI after all commissions and other costs $($including the $$250$ for the buyer's mini$-$vacation$)$ have been paid.

VI's call center operates from $10$ a$.$m$.$ to $10$ p$.$m$.$ daily with $4$ sales representatives and receives calls at a rate of $50$ per hour following a Poisson distribution. It takes an average of $4$ minutes to handle each call with actual times being exponentially distributed. The phone system VI uses can keep up to $10$ callers on hold at any time. Assume those who receive a busy signal don't call back. a$.$ On average, how many customers per hour does each sales person process?

b$.$ What is the expected value of each call to VI's toll$-$free line?

c$.$ Suppose VI pays its phone reps $$12$ per hour. How many phone reps should it employ if it wants to maximize profit?

Please show work utilizing the Q$.$xls queing template for excel with screenshots and explanations of inputs