Chapter 8 Managing Flow Variability 1-16
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Transcript Chapter 8 Managing Flow Variability 1-16
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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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
–
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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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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)
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Sales Reps
Processing
Calls
Answered Calls
(Customer Departures)
(Service Process)
Calls In Process
(Due to long waits)
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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
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Operational Performance Measures
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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)
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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
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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?
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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?
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Customer Number
Customer
Number
7
6
5
4
3
0
10
20
30
40
50
60
Time
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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
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Time
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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?
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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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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
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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)
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Throughput- Delay Curve
Average
Flow
Time T
Variability
Increases
Tp
Utilization (ρ)
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100%
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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
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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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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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Example
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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
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Synchronization
Matching Capacity with Demand
Capacity
– Short term Control
– Long term Planning
Demand
– Pricing
– Scheduling
OM
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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
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Effect of Pooling
Ri/2
Server 1
Queue 1
Ri
Ri/2
Server 2
Queue 2
Server 1
Ri
Queue
Server 2
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Effect of Buffer Capacity
Process Data
– Ri = 20/hour, Tp = 2.5 mins, c = 1, K = # Lines – c
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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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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
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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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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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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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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
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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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