CHAPTER 8: MANAGING FLOW VARIABILITY: SAFETY
CAPACITY
8.1 Objective
The objective of this chapter is to give students an understanding of how variability in inflows and
processing can lead to long flow times. In chapter 7 we have focused on make to stock operations and the
use of inventories to dampen the impact of variability on product availability. In this chapter we focus on
8.2 Additional Suggested Readings
The case used is
▪ “Sof-Optics, Inc. (A)”, Harvard Business School 9-681-052 (revised 3/19/91)
8.3 Solutions to the Problem Set
Problem 8.1
Increasing the number of telephone lines from 12 to 13 will
a. Decrease the proportion of customers who get a busy signal. Callers calling when there are 12 people
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Problem 8.2
Merging the two departments will
Problem 8.3
John Doe has promised a total service time of 20 minutes. The processing time at the oven is 15 minutes
Problem 8.4 (M. M. Sprout)
a. We are given:
Average arrival rate Ri = 1/4 per minute,
Average unit capacity 1/Tp = 1/3 per minute,
Number of servers c = 1.
Hence, the total hourly cost = $20 + $5 + $270 = $295/hour.
b. With only four lines and one CSR, we have
Average arrival rate Ri = 1/4 per minute,
Average unit capacity 1/Tp = 1/3 per minute,
Number of servers c = 1,
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c. Upon adding another telephone line, we have
Average arrival rate Ri = 1/4 per minute,
Average unit capacity 1/Tp = 1/3 per minute,
Number of servers c = 1,
Maximum buffer size, K = 4.
d. Upon adding another server (assuming that the fifth line has been added) we have
Average arrival rate Ri = 1/4 per minute,
Average unit capacity 1/Tp = 1/3 per minute,
Number of servers c = 2,
Maximum buffer size K = 3.
Using the Performance.xls spreadsheet we get
Average waiting time Ti = 0.42 minutes,
Average # of customers on hold Ii = 0.105
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Problem 8.5 (Heavenly Mercy Hospital)
a. We have:
Average arrival rate Ri = 18 per hour = 0.3 per minute,
To plan staffing, we know that we should have a utilization of less than 100%, thus:
Utilization = inflow/capacity = 18/hr/(c*2/hr) < 1 so that c > 9
Increasing the number of servers from 10 upward, we have the following results (using the
Performance.xls spreadsheet with infinite queue):
Number of Servers (c)
Avg. Number in System (I)
Avg. Time in System (T)
9.8
Thus hiring 11 servers achieves a turnaround time of 36.45 minutes on average, which is under the
desired target of 40 minutes. The hourly cost of this system is $1,100.
b. Now consider the case where the service time is reduced to 20 minutes but the cost of the
equipment and radiologist is $150 per hour. In this case:
Average arrival rate Ri = 18 per hour = 0.3 per minute,
Number of Servers (c)
Avg. Number in System (I)
Avg. Time in System (T)
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9.7
32.28
Problem 8.6 (First Local Bank)
a. We currently have:
Average arrival rate Ri = 30 per hour = 0.5 per minute,
Average unit capacity 1/Tp = 10 per hour = 1/6 per minute,
Number of servers c = 4,
Average arrival rate Ri = 30 per hour = 0.5 per minute,
Average unit capacity 1/Tp = 15 per hour = 1/4 per minute,
Number of servers c = 4.
Performance of the new system is (using Performance.xls)
Average waiting time Ti = 0.35 mins,
Average time in system T = 4.35 minutes,
Problem 8.7 (Global Airlines)
Average arrival rate, Ri = 52 per hour = 52/60 per minute,
Average unit capacity, 1/Tp = 20 per hour = 1/3 per minute,
Number of servers, c : To be determined,
Increasing the number of servers from 3 upward, we have:
Number
of
Servers c
Server cost
per hour
Average Queue
length Ii
Average
waiting time Ti
Total cost
per hour
3
$60
4.95
5.71
$357
5
$100
0.16
0.19
$109.6
The industry norm of averaging under 3 minutes of waiting can be achieved using only 4 agents.
Problem 8.8 (Henniker Bank)
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claims in the system (for one district) is
( )
07 7 5 5 25. . .=
. Out of those 5.25 – 0.84 = 4.41 will be
waiting to be processed. The total number of claims across all districts is then
3 441 13 23 =. .
.
Problem 8.9.(Burrito King)
a. Reducing variability will reduce the waiting time
Problem 8.10 (V. V. Ranger)
Problem 8.11 (Master Karr)
Arrival
Rate
Service
Time
# of
Servers
Buffer
Capacity
Average
Utilization
Probability of
blocking
Average
queue
length
Part
Ri
Tp
c
K
u
P(block)
II
a
4
1
5
10
79.00%
1.25E-02
1.58
b. Now the number of servers c increases to 6 while the buffer capacity K decreases to 9. Re-calculate
the performance with the spreadsheet (calculations above).
Waiting cost = $60 Ii = $30.77/hr
Problem 8.12 (BizTravel.com)
a. This is the resource pooling idea in the context of one queue vs. multiple queue. With the CRM
software, BizTravel will be able to meet its service guarantee better.
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Problem 8.13 (McBerger)
a. The average waiting time in queue will decrease, because less variability leads to less waiting, by
the queue length formula.