Transcript OPSM 901: Operations Management

Koç University
OPSM 301: Operations Management
Session 19:
Flow variability
Zeynep Aksin
[email protected]
Announcements
 Midterm 2-December 14 at 18:30 CAS Z48, CAS
Z08
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Does not include Midterm 1 topics
LP, Inventory, Variability (Congestion+Quality)
LP: from course pack
Inventory Ch6 excluding 6.7, Ch 7.1, 7.2, 7.3
Chapter 8 excluding 8.6 and 8.8 (this week)
Chapter 9 (next week)
Components of the Queuing System
Visually
Customers
come in
Customers are
served
Customers
leave
Flow Times with Arrival Every 4 Secs
(Service time=5 seconds)
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
What is the queue size? Can we apply Little’s Law?
What is the capacity utilization?
20
30
Time
40
50
Flow Times with Arrival Every 6 Secs
(Service time=5 seconds)
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?
Customer Number
Customer
Number
7
6
5
4
3
0
10
20
30
Time
40
50
60
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
Time
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
What is the queue size?
What is the capacity utilization?
10
9
0
10
20
30
40
50
60
70
Conclusion
 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 inter-arrival 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
To address the “how much does variability
hurt” question: Consider service processes
 This could be a call center or a restaurant or a ticket
counter
 Customers or customer jobs arrive to the process; their
arrival times are not known in advance
 Customers are processed. Processing rates have some
variability.
 The combined variability results in queues and waiting.
 We need to build some safety capacity in order to reduce
waiting due to variability
Why is there waiting?
 the perpetual queue: insufficient capacity-add
capacity
 the predictable queue: peaks and rush-hourssynchronize/schedule if possible
 the stochastic queue: whenever customers
come faster than they are served-reduce
variability
A measure of variability
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Needs to be unitless
Only variance is not enough
Use the coefficient of variation
C or CV= s/m
Interpreting the variability measures
Ci = coefficient of variation of interarrival times
i) constant or deterministic arrivals
Ci = 0
ii) completely random or independent arrivals Ci =1
iii) scheduled or negatively correlated arrivals Ci < 1
iv) bursty or positively correlated arrivals
Ci > 1
Specifications of a Service Provider
Reneges or abandonments
Arriving
Customers
Waiting
Pattern
Demand
Pattern
Service
Provider
Waiting
Customers
Served
Customers
Service Time
Resources
• Human resources
• Information system
• other...
Leaving
Customers
Satisfaction
Measures
Distribution of Arrivals
 Arrival rate: the number of units arriving per
period
– Constant arrival distribution: periodic, with exactly
the same time between successive arrivals
– Variable (random) arrival distributions: arrival
probabilities described statistically
• Exponential distribution for interarrivals
• Poisson distribution for number arriving
• CV=1
Service Time Distribution
 Constant
– Service is provided by automation
 Variable
– Service provided by humans
– Can be described using exponential distribution CV=1
or other statistical distributions
The Service Process
 Customer Inflow (Arrival) Rate (Ri) ()
– Inter-arrival Time = 1 / Ri
 Processing Time Tp (unit load)
– Processing Rate per Server = 1/ Tp (µ)
 Number of Servers (c)
– Number of customers that can be processed simultaneously
 Total Processing Rate (Capacity) = Rp= c / Tp (cµ)
Operational Performance Measures
() Ri
waiting
processing
R ()
e.g10 /hr
10 /hr
Tw?








10 min, Rp=12/hr
Flow time T
=
Tw
+
Tp (waiting+process)
Inventory I
=
Iw
+
Ip
Flow Rate R
=
Min (Ri, Rp)
Stable Process =
Ri < Rp,, so that R = Ri
Little’s Law: I = R  T,
Iw = R  Tw, Ip = R  Tp
Capacity Utilization  = Ri / Rp < 1
Safety Capacity = Rp – Ri
Number of Busy Servers = Ip= c  = Ri  Tp
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 (i.e. long service
time, or few servers)
The psychology of waiting
 waiting as psychological punishment
 keep the customer busy
 keep them entertained
 keep them informed
 break the wait up into stages
 in multi-stages, its the end that matters
The psychology of waiting
 waiting as a ritual insult
 sensitivity training
 make initial contact
 waiting as a social interaction
 prevent injustice
 improve surroundings
 design to minimize crowding
 get rid of the line whenever possible
Reducing perceived wait
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Understand psychological thresholds
Distract customers (mirrors, music, information)
Get customers out of line (numbers, call-back)
Inform customers of wait (over-estimate)
Keep idle servers out of sight
Maintain fairness (FCFS)
Keep customers comfortable
Is a queue always bad?
 queues as a signal for quality
 doctors
 business schools
 restaurants
 other people demand similar things
 the advantage of being in
A solution:
Add capacity to remove a persistent line?
 You want others to be there to signal quality
 Risks of being in versus out: its an unstable
proposition!
 Don’t want to relate everything to price
The challenge: matching demand and
supply
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changing number of servers
changing queue configuration
changing demand
managing perceptions