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Ko ç Un iversity. OPSM 301: Operations Management. Session 19: Flow variability. Zeynep Aksin zaksin @ku.edu.tr. 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

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opsm 301 operations management

Koç University

OPSM 301: Operations Management

Session 19:

Flow variability

Zeynep Aksin


  • 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)
flow times with arrival every 4 secs service time 5 seconds
Flow Times with Arrival Every 4 Secs(Service time=5 seconds)

What is the queue size? Can we apply Little’s Law?

What is the capacity utilization?

flow times with arrival every 6 secs service time 5 seconds
Flow Times with Arrival Every 6 Secs (Service time=5 seconds)

What is the queue size?

What is the capacity utilization?

effect of variability
Effect of Variability

What is the queue size?

What is the capacity utilization?

effect of synchronization
Effect of Synchronization

What is the queue size?

What is the capacity utilization?

  • 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
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
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
A measure of variability
  • Needs to be unitless
  • Only variance is not enough
  • Use the coefficient of variation
  • C or CV= s/m
interpreting the variability measures
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
Specifications of a Service Provider

Reneges or abandonments

Arriving Customers

Waiting Pattern

Demand Pattern

Leaving Customers



Satisfaction Measures

Waiting Customers

Served Customers

Service Time

  • Resources
  • Human resources
  • Information system
  • other...
distribution of arrivals
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
Service Time Distribution
  • Constant
    • Service is provided by automation
  • Variable
    • Service provided by humans
    • Can be described using exponential distributionCV=1 or other statistical distributions
the service process
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
Operational Performance Measures



() Ri

e.g10 /hr

R ()

10 /hr


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
Summary: Causes of Delays and Queues
  • High Unsynchronized Variability in
    • Interarrival Times
    • Processing Times
  • High Capacity Utilization r = 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
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
The challenge: matching demand and supply
  • changing number of servers
  • changing queue configuration
  • changing demand
  • managing perceptions