V. Capacity Planning in Services
OM

Matching Supply and Demand

The Service Process

Performance Measures

Causes of Waiting

Economics of Waiting

Management of Waiting Time

The Sof-Optics Case
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S. D. Deshmukh
Matching Supply and Demand

Goods vs. Services
– Make to Stock vs. Make to Order
– Produce in advance vs. on demand
– Safety Inventory vs. Safety Capacity

Examples
–
–
–
–
–
–
–
OM
Banks (tellers, ATMs, drive-ins)
Fast food restaurants (counters, drive-ins)
Retail (checkout counters)
Airline (reservation, check-in, takeoff, landing, baggage claim)
Hospitals (ER, OR, HMO)
Call centers (telemarketing, help desks, 911 emergency)
Service facilities (repair, job shop, ships/trucks load/unload)
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S. D. Deshmukh
The DesiTalk Call Center
The Call Center Process
Incoming Calls
(Customer Arrivals)
Calls
on Hold
(Service Inventory)
Blocked Calls
Abandoned Calls
(Due to busy signal) (Due to long waits)
OM
Sales Reps
Processing
Calls
Answered Calls
(Customer Departures)
(Service Process)
Calls In Process
(Due to long waits)
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S. D. Deshmukh
The Service Process

Customer Inflow (Arrival) Rate (Ri)
– Inter-arrival Time = 1 / Ri

Processing Time Tp
– Processing Rate per Server = 1/ Tp

Number of Servers (c)
– Number of customers that can be processed simultaneously


OM
Total Processing Rate (Capacity) = Rp= c / Tp
Buffer Capacity (K)
– Maximum Queue Length
S. D. Deshmukh
Operational Performance Measures








OM

Flow time T =
Ti
+
Tp
Inventory I
=
Ii
+
Ip
Flow Rate R =
Min (Ri, Rp)
Stable Process =
Ri < Rp,, so that R = Ri
Little’s Law: I = Ri  T, Ii = Ri  Ti, Ip = Ri  Tp
Capacity Utilization  = Ip / c = Ri  Tp / c = Ri / Rp < 1
Safety Capacity Rs = Rp - Ri
Number of Busy Servers = Ip= c  = Ri  Tp
Fraction Lost Pb = P(Blocking) = P(Queue = K)
S. D. Deshmukh
Financial Performance Measures

Sales
– Throughput Rate
– Abandonment Rate
– Blocking Rate

Cost
– Capacity utilization
– Number in queue / in system

Customer service
– Waiting Time in queue /in system
OM
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S. D. Deshmukh
Flow Times with Arrival Every 4 Secs
Customer
Number
Arrival
Time
Departure
Time
Time in
Process
1
0
5
5
2
4
10
6
3
8
15
7
4
12
20
8
5
16
25
9
6
20
30
10
3
7
24
35
11
2
8
28
40
12
9
32
45
13
10
36
50
14
10
9
Customer Number
8
7
6
5
4
1
0
10
20
30
40
50
Time
What is the queue size?
What is the capacity utilization?
OM
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S. D. Deshmukh
Flow Times with Arrival Every 6 Secs
Arrival
Time
Departure
Time
Time in
Process
10
1
0
5
5
9
2
6
11
5
8
3
12
17
5
4
18
23
5
5
24
29
5
6
30
35
5
7
36
41
5
2
8
42
47
5
1
9
48
53
5
10
54
59
5
What is the queue size?
What is the capacity utilization?
OM
Customer Number
Customer
Number
7
6
5
4
3
0
10
20
30
40
50
60
Time
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S. D. Deshmukh
Effect of Variability
Customer
Number
Arrival
Time
Processing
Time
Time in
Process
1
0
7
7
2
10
1
1
3
20
7
7
4
22
2
7
5
32
8
8
6
33
7
14
7
36
4
15
8
43
8
16
9
52
5
12
10
54
1
11
10
9
8
Customer
7
6
5
4
3
2
1
0
10
20
30
40
50
60
70
Time
Queue Fluctuation
4
What is the queue size?
What is the capacity utilization?
Number
3
2
1
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
OM
Time
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S. D. Deshmukh
Effect of Synchronization
Customer
Number
Arrival
Time
Processing
Time
Time in
Process
1
0
8
8
2
10
8
8
8
3
20
2
2
7
4
22
7
7
6
5
32
1
1
5
6
33
1
1
4
7
36
7
7
3
8
43
7
7
2
9
52
4
4
1
10
54
5
7
10
9
0
10
20
30
40
50
60
70
What is the queue size?
What is the capacity utilization?
OM
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S. D. Deshmukh
Conclusion



OM
If inter-arrival and processing times are
constant, queues will build up if and only if the
arrival rate is greater than the processing rate
If there is (unsynchronized) variability in interarrival and/or processing times, queues will
build up even if the average arrival rate is less
than the average processing rate
If variability in interarrival and processing
times can be synchronized (correlated), queues
and waiting times will be reduced
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S. D. Deshmukh
Summary: Causes of Delays and Queues

High Unsynchronized Variability in
– Interarrival Times
– Processing Times

High Capacity Utilization  = Ri / Rp, or
Low Safety Capacity Rs = Rp – Ri, due to
– High Inflow Rate Ri
– Low Processing Rate Rp = c/ Tp
OM
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S. D. Deshmukh
The Queue Length Formula
Ii
ρ

Utilization
effect
2(c 1)
Ci2  C p2
1 ρ
2
x
Variability
effect
where   Ri / Rp, where Rp = c / Tp, and
Ci and Cp are the Coefficients of Variation
(Standard Deviation/Mean) of the inter-arrival
and processing times (assumed independent)
OM
S. D. Deshmukh
Throughput- Delay Curve
Average
Flow
Time T
Variability
Increases
Tp
Utilization (ρ)
OM
100%

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S. D. Deshmukh
Computing Performance Measures

Given
– Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2
– Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1
– c=1

Compute
– Capacity Utilization  = Ri / Rp = 0.833
– Ci = 3.937/6 = 0.6562
– Cp = 2.8284/5 = 0.5657

Queue Length Formula
– Ii = 1.5633

Hence
– Ti = Ii / R = 9.38 seconds, and Tp = 5 seconds, so
– T = 14.38 seconds, so
– I = RT = 14.38/6 = 2.3966
OM
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S. D. Deshmukh
Effect of Increasing Capacity

Given
– Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2
– Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1
– c=2

Compute
– Capacity Utilization  = Ri / Rp = 0.4167
– Ci = 3.937/6 = 0.6562
– Cp = 2.8284/5 = 0.5657

Queue Length Formula
– Ii = 0.07536

Hence
– Ti = Ii / R = 0.45216 seconds, and Tp = 5 seconds, so
– T = 5.45216 seconds, so
– I = RT = 5.45216/6 = 0.9087
OM
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S. D. Deshmukh
The Exponential Model

Poisson Arrivals
– Infinite pool of potential arrivals, who arrive
completely randomly, and independently of one
another, at an average rate Ri  constant over time

Exponential Processing Time
– Completely random, unpredictable, i.e., during
processing, the time remaining does not depend on
the time elapsed, and has mean Tp

OM
Computations
– Ci = Cp = 1
– If c = 1, T = 1/(Rp  Ri), then I = Ri  T,...
– If c ≥ 2, and K < ∞ , use Performance.xls
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S. D. Deshmukh
Example


OM
Interarrival time = 6 secs, so Ri = 10/min
Tp = 5 secs = 0.833 mins
c
ρ
1
0.833
2
0.417
Rs
Ii
Ti
T
I
0.0333 4.162
0.416
0.499
4.995
0.2333 0.175
0.018
0.101
1.0087
S. D. Deshmukh
Synchronization


Matching Capacity with Demand
Capacity
– Short term Control
– Long term Planning

Demand
– Pricing
– Scheduling
OM
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S. D. Deshmukh
Performance Improvement Levers

Capacity Utilization / Safety Capacity
– Demand Management (arrival rate)
» Peak load pricing
– Increase Capacity (processing rate)
» Number of Servers (scale)
» Processing Rate (speed)

Variability Reduction
– Arrival times
» Scheduling, Reservations, Appointments
– Processing times
» Standardization, Specialization, Training

Synchroniztion
– Matching capacity with demand
OM
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S. D. Deshmukh
Effect of Pooling
Ri/2
Server 1
Queue 1
Ri
Ri/2
Server 2
Queue 2
Server 1
Ri
Queue
Server 2
OM
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S. D. Deshmukh
Effect of Buffer Capacity

Process Data
– Ri = 20/hour, Tp = 2.5 mins, c = 1, K = # Lines – c

OM
Performance Measures
K
4
5
6
Ii
1.23
1.52
1.79
Ti
4.10
4.94
5.72
Pb
0.1004
0.0771
0.0603
R
17.99
18.46
18.79

0.749
0.768
0.782
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S. D. Deshmukh
Economics of Capacity Decisions

Cost of Lost Business Cb
– $ / customer
– Increases with competition

Cost of Buffer Capacity Ck
– $/unit/unit time

Cost of Waiting Cw
– $ /customer/unit time
– Increases with competition

Cost of Processing Cs
– $ /server/unit time
– Increases with 1/ Tp

OM
Tradeoff: Choose c, Tp, K
– Minimize Total Cost/unit time
= Cb Ri Pb + Ck K + Cw I (or Ii) + c Cs
S. D. Deshmukh
Optimal Buffer Capacity

Cost Data
– Cost of telephone line = $5/hour, Cost of server = $20/hour,
Margin lost = $100/call, Waiting cost = $2/customer/minute

OM
Effect of Buffer Capacity on Total Cost
K
$5(K + c)
$20 c
$100 Ri Pb
$120 Ii
4
25
20
200.8
147.6
TC
($/hr)
393.4
5
30
20
154.2
182.6
386.4
6
35
20
120.6
214.8
390.4
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S. D. Deshmukh
Optimal Processing Capacity
OM
c
K=6–c
Pb
Ii
1
5
0.0771
1.542
TC ($/hr) =
$20c + $5(K+c) +
$100Ri Pb+ $120 Ii
$386.6
2
4
0.0043
0.158
$97.8
3
3
0.0009
0.021
$94.2
4
2
0.0004
0.003
$110.8
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S. D. Deshmukh
Performance Variability

Effect of Variability
– Average versus Actual Flow time

Time Guarantee
– Promise

Service Level
– P(Actual Time  Time Guarantee)

Safety Time
– Time Guarantee – Average Time

Probability Distribution of Actual Flow Time
– P(Actual Time  t) = 1 – EXP(- t / T)
OM
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S. D. Deshmukh
Flow Time Management: Review

Waiting occurs due to
– low processing capacity in relation to the inflow rate
– variability in inter-arrival and processing times

Waiting can be reduced by
–
–
–
–

managing demand
pooling arrival streams
increasing capacity (number of servers  service rate)
reducing the variability in arrivals and processing
Optimal level of service involves a tradeoff
– cost of waiting, lost business and cost of service
OM
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S. D. Deshmukh
Flow Time Management Levers

Manage Arrivals
– Demand Management: Price incentives
– Pool arrivals

Increase Capacity
– Scale: Servers, Part-timers, customer participation
– Speed: Simplify, Automation, Information, Training

Decrease Variability
– Arrivals: Forecast, Reservations, Pooling
– Processing: Standardize

OM
Reduce Impact of Waiting
– Comfortable, Distract, Entertain, Perception
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S. D. Deshmukh