Operations Management & Performance Modeling
1
2
3
4
Operations Strategy
Process Analysis
Lean Operations
Supply Chain Management
5 Capacity Management in Services
– Class 6b: Capacity Analysis and Queuing
» Why do queues build up?
» Performance measures for queuing systems
» The need for safety capacity
» Throughput of queuing system with finite buffer
» Pooling of capacity
6 Total Quality Management
7 Business Process Reengineering
OM&PM/Class 6b
1
Telemarketing at L.L.Bean




During some half hours, 80% of calls dialed received a busy
signal.
Customers getting through had to wait on average 10 minutes
for an available agent. Extra telephone expense per day for
waiting was $25,000.
For calls abandoned because of long delays, L.L.Bean still paid
for the queue time connect charges.
In 1988, L.L.Bean conservatively estimated that it lost $10
million of profit because of sub-optimal allocation of
telemarketing resources.
OM&PM/Class 6b
5
Telemarketing: deterministic analysis
30%
30%
20%
20%
10%
10%
0%
0%
195
40%
180
50%
40%
165
50%
150
60%
135
60%
120
70%
105
80%
70%
90
80%
75
90%
60
90%
45
Flow Time = 8 min
100%
0

100%
30
– one customer every 10
minutes
Flow Time Distribution
15

it takes 8 minutes to
serve a customer
6 customers call per
hour
Probability

Flow Time (minutes)
OM&PM/Class 6b
6
Telemarketing with variability in
arrival times + activity times
80%
15%
60%
10%
40%
5%
20%
0%
0%
– exhibit variability
190
More
180
170
160
150
140
130
120
110
90
100
80
70
60
50
40
30
Flow Time
In reality arrival times
30%
100%
90%
25%
Probability

20
0
– exhibit variability
20%
10
In reality service times
100%
90%
Probability

25%
80%
70%
20%
60%
15%
50%
40%
10%
30%
20%
5%
10%
OM&PM/Class 6b
190
180
170
160
150
More
Flow Time
140
130
120
110
90
100
80
70
60
50
40
30
20
0
0%
10
0%
7
Telemarketing with variability:
The effect of utilization
Average service time =
– 9 minutes
100%
7%
90%
80%
Probability

8%
6%
70%
5%
60%
4%
50%
3%
40%
30%
2%
20%
More
190
180
170
160
150
Flow Time
140
130
120
110
90
100
80
70
60
50
40
30
20
0%
10
10%
0%
0
1%
25%
100%
90%

Average service time =
– 9.5 minutes
Probability
20%
80%
70%
15%
60%
50%
10%
40%
30%
5%
20%
10%
190
180
170
160
150
More
Flow Time
OM&PM/Class 6b
140
130
120
110
100
90
80
70
60
50
40
30
20
0%
10
0
0%
8
Why do queues form?
 utilization:
– throughput/capacity

variability:
– arrival times
– service times
– processor availability
Call #
10
9
8
7
6
5
4
3
2
1
0
0
20
40
60
80
100
TIME
Inventory (# of calls in system)
5
4
3
2
1
0
OM&PM/Class 6b
0
20
40
60
TIME
80
100
9
Cycle Times in White Collar Processes
Industry
Process
Average
Cycle Time
Theoretical
Cycle Time
Process Efficiency
Life Insurance
New Policy
Application
72 hrs.
7 min.
0.16%
Consumer
Packaging
New
Graphic
Design
Consumer
Loan
18 days
2 hrs.
0.14%
24 hrs.
34 min.
2.36%
Hospital
Patient
Billing
10 days
3 hrs.
3.75%
Automobile
Manufacture
Financial
Closing
11 days
5 hrs
5.60%
Commercial
Bank
OM&PM/Class 6b
10
Queuing Systems to model Service
Processes: A Simple Process
Order Queue
“buffer” size K
Sales Reps
processing
calls
Incoming calls
Calls
on Hold
Answered Calls
MBPF Inc. Call Center
Blocked Calls Abandoned Calls
(Busy signal) (Tired of waiting)
OM&PM/Class 6b
11
What to manage in such a process?
 Inputs
– InterArrival times/distribution
– Service times/distribution

System structure
– Number of servers
– Number of queues
– Maximum queue length/buffer size

Operating control policies
– Queue discipline, priorities
OM&PM/Class 6b
12
Performance Measures

Sales
– Throughput R
– Abandonment

Cost
– Server utilization r
– Inventory/WIP : # in queue/system

Customer service
– Waiting/Flow Time: time spent in queue/system
– Probability of blocking
OM&PM/Class 6b
13
Queuing Theory:
Variability + Utilization = Waiting

Throughput-Delay curve:
Actual
Average
Cycle
Time, W
Variability
Theoretical
Cycle Time
m

Pollaczek-Khinchine Form:
100%
Utilization r
– Prob{waiting time in queue < t } = 1 - exp(-t / Ti ) where:
1
r
Ti 
Rp 1  r
OM&PM/Class 6b
Ci2  C p2
2
mean service utilization variability
x
x
time
effect
effect
14
Levers to reduce waiting and increase QoS:
 variability reduction + safety capacity
 How

reduce system variability?
Safety Capacity = capacity carried in excess of
expected demand to cover for system variability
– it provides a safety net against higher than expected arrivals
or services and reduces waiting time
OM&PM/Class 6b
15
Example 1: MBPF Calling Center
one server, unlimited buffer



Consider MBPF Inc. that has a customer service representative
(CSR) taking calls. When the CSR is busy, the caller is put on
hold. The calls are taken in the order received.
Assume that calls arrive exponentially at the rate of one every
3 minutes. The CSR takes on average 2.5 minutes to complete
the reservation. The time for service is also assumed to be
exponentially distributed.
The CSR is paid $20 per hour. It has been estimated that each
minute that a customer spends in queue costs MBPF $2 due to
customer dissatisfaction and loss of future business.
– MBPF’s waiting cost =
OM&PM/Class 6b
16
Example 2: MBPF Calling Center
limited buffer size


In reality only a limited number of people can be put on hold
(this depends on the phone system in place) after which a
caller receives busy signal. Assume that at most 5 people can
be put on hold. Any caller receiving a busy signal simply calls
a competitor resulting in a loss of $100 in revenue.
– # of servers c = 1
– buffer size K = 6
What is the hourly loss because of callers not being able to get
through?
OM&PM/Class 6b
17
Example 3: MBPF Calling Center
Resource Pooling

2 phone numbers
– MBPF hires a second CSR who is
assigned a new telephone number.
Customers are now free to call
either of the two numbers. Once
they are put on hold customers
tend to stay on line since the other
may be worse ($111.52)

50%
Queue Server
50%
Queue Server
1 phone number: pooling
– both CSRs share the same
telephone number and the
customers on hold are in a single
queue ($61.2)
Queue
Servers
OM&PM/Class 6b
18
Example 4: MBPF Calling Center
Staffing
 Assume
that the MBPF call center has a total of 6
lines. With all other data as in Example 2, what is the
optimal number of CSRs that MBPF should staff the
call center with?
– c=3
OM&PM/Class 6b
19
Class 6b Learning objectives

Queues build up due to variability.

Reducing variability improves performance.

If service cannot be provided from stock, safety capacity must
be provided to cover for variability.

Tradeoff is between cost of waiting, lost sales, and cost of
capacity.

Pooling servers improves performance.
OM&PM/Class 6b
20
National Cranberry Cooperative
Hourly Berry Arrivals
2500
2298
2000
1792
1713
1680
1477
1335
1395
1500
1269
1341
1317
Bbls
1032
1016
1000
539
500
0
0
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Time
OM&PM/Class 6b
21
Real Processes exhibit variability in
order placement time and type
National Cranberry on Sept 23, 1970
Histogram of Truck Weights
40
40
35
35
Frequency (# of trucks)
Frequency (# of trucks)
Histogram of Truck inter-delivery times
30
25
20
15
10
5
30
25
20
15
10
5
0
0
0
2
4
6
8
10
12
14
16
Truck interarrival time (min)
OM&PM/Class 6b
18
20
0
4
8
12
16
20
24
28
32
36
40
Truck Weight (Kpounds)
22