 From Chapter 9

9.1 L = 4

9.2 See file Ch9.4.xls 1.3% of customers will be denied admittance to the store.

9.6 A 6 minutes service time is a rate of 10 per hour.

To minimize costs, United should have 7 ticket agents. The expected hourly cost would be \$185.26

9.9 a. (i) Since W = .0718 hrs. or 4.308 minutes, the total time a customer spends in the store is 4.308 + 25 = 29.308 minutes

(ii) Lq = .0408

(iii) Pw = 11.228%

b. We assume both checkers work at the same rate and customers spend an average of 25 minutes in the store before checking out.

9.10 a. (i) Since W = .0685 hours or 4.11 minutes, the average time spent in store = 4.111 + 25 = 29.11 minutes, (ii) Lq = .15, (iii) Pw = 40%

b. Since the average customer service time is less and the store saves the space of the additional checkstand it is worthwhile to lease the scanning system.

9.12 9.12 continued

9.12 continued

Ann costs \$42, Bill costs \$16, and Charlie costs \$17.27.

The store should hire Bill with his service rate of 30 per hour. Average hourly costs are lowest if he is hired.

9.17

A service time of 2.5 minutes is a rate of 24 per hour.

The front of the store is an M/M/2 queuing system with  = 20/hr. and  = 24/hr.

9.17 continued

The rear of the store is an M/M/4 queuing system with  = 66/hr. and  = 24/hr

a. Front -- Wq = .0888 hr. = .528 minutes, rear -- Wq = .0136 hr. = .816 minutes

b. Front -- L = 1.008, rear -- L = 3.651

c. P(6 customers in the system at the rear of the store) = .06

d. P(5 or fewer customers in the system at the front of the store) = .99

e. A customer should prefer the front checkstands since the average checkout time is less. W(front) = 0.05 and W(rear) = 0.055.

9.18

We evaluate a system in which  = 86/hr. and  = 24/hr and for which wages are \$16/hr., customer goodwill cost in line is \$20/hr., and customer good will cost while being served is \$10/hr.

To minimize costs it is optimal for Harry’s to have 5 servers. The average system cost per hour = \$136.32.

9.21 Small Store

For the small store the expected profit per day is \$3,289.66

9.21 continued -- Large Store

For the small store the expected profit per day is \$5,526.88. The firm should lease the larger store as the expected daily profit is higher.

9.28

a. Dave and Steve’s should employ 3 clerks.

9.28b.

b. Dave and Steve’s should now employ 4 clerks.

9.29

a. P(X = 4) = 54e-5/4! = .175 (from Poisson distribution with a mean of 5)

b. P(X  3) = (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)) = .0067 + .0337 + .0842 + .1404 = .265 (from Poisson distribution with a mean of 5)

c. P(T  1/20) = 1 - e-15/20 = .5276 (from exponential distribution with a mean of 15)

d. Lq = 1.333

e. W = .2 hrs. = 12 minutes

f. Pw = 66.67%

g. P(W > .10) = e-5(.10) = .6065 (from exponential distribution with a mean of 5)

9.36

a. Proportion of trucks inspected = 1- .59 = .41

b. Po = .0164

c. L = 2.459

9.37

a. We assumed that there was an unlimited customer population, customers will not balk, and there is an unlimited waiting line size.

b. W = .0243 hours or 1.458 minutes