OPSM 301: Operations Management
Session 19:
Flow variability
Koç University
Zeynep [email protected]
Announcements
Midterm 2-December 14 at 18:30 CAS Z48, CAS Z08– 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 Customers come income in
Customers are Customers are servedserved
Customers Customers leaveleave
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
7 24 35 11
8 28 40 12
9 32 45 13
10 36 50 14
0 10 20 30 40 50
Time
1
2
3
4
5
6
7
8
9
10
Cust
omer
Num
ber
What is the queue size? Can we apply Little’s Law?What is the capacity utilization?
Customer Number
Arrival Time
Departure Time
Time in Process
1 0 5 5
2 6 11 5
3 12 17 5
4 18 23 5
5 24 29 5
6 30 35 5
7 36 41 5
8 42 47 5
9 48 53 5
10 54 59 5
0 10 20 30 40 50 60
Time
1
2
3
4
5
6
7
8
9
10
Cust
omer
Num
ber
Flow Times with Arrival Every 6 Secs (Service time=5 seconds)
What is the queue size?What is the capacity utilization?
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
0 10 20 30 40 50 60 70
Time
1
2
3
4
5
6
7
8
9
10
Cu
sto
mer
Queue Fluctuation
0
1
2
3
4
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
Time
Nu
mb
er
Effect of Variability
What is the queue size?What is the capacity utilization?
Customer Number
Arrival Time
Processing Time
Time in Process
1 0 8 8
2 10 8 8
3 20 2 2
4 22 7 7
5 32 1 1
6 33 1 1
7 36 7 7
8 43 7 7
9 52 4 4
10 54 5 7 0 10 20 30 40 50 60 70
1
2
3
4
5
6
7
8
9
10
Effect of Synchronization
What is the queue size?What is the capacity utilization?
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-hours-synchronize/schedule if possible
the stochastic queue: whenever customers come faster than they are served-reduce variability
A measure of variability
Needs to be unitless Only variance is not enough Use the coefficient of variation C or CV= /
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
ServiceProvider
Leaving Customers
Waiting Customers
Demand Pattern
Resources
• Human resources
• Information system
• other...
Arriving Customers
Satisfaction Measures
Reneges or abandonments
Waiting Pattern
Served Customers
Service Time
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
Flow time T= Tw + Tp (waiting+process)
Inventory I = Iw + Ip
Flow Rate R = Min (Ri, RpStable 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
waiting processing() Ri
e.g10 /hr
R ()
10 /hr
10 min, Rp=12/hrTw?
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
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
changing number of servers changing queue configuration changing demand managing perceptions