Answers to Modules C7-C8
From Chapter 9
L = 4
9.2 See file Ch9.4.xls
1.3% of customers will be denied admittance to the store.
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
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.
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.
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.
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.
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.
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.
a. Dave and Steve’s should employ 3 clerks.
b. Dave and Steve’s should now employ 4 clerks.
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)
a. Proportion of trucks inspected = 1- .59 = .41
b. Po = .0164
c. L = 2.459
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