PRODUCTIONSOPERATIONS MANAGEMENT

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PRODUCTION/OPERATIONS MANAGEMENT
Waiting Line
Chapter 19
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Youve Been There Before!

• The other line always
• moves faster.
• If you change lines,
• the one you left will start
• to move faster than the
• one youre in.

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n535732.shtml
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Why is there waiting?

• Waiting lines occur naturally because of two
  reasons

1. Customers arrive ________, not at evenly
placed times nor at predetermined times 2.
Service requirements of the customers are
________. (Think of a bank for example)
randomly
variable

• Because of these two reasons, waiting lines form
  even in __________ systems.

underloaded
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Is there a cost of waiting?

• Quantifiable costs
• When the customers are internal (e.g employees
  waiting for making copies), salaries paid to the
  employees
• Cost of the space of waiting (e.g. patient
  waiting room)
• Loss of business (lost profits)
• Hard to quantify costs
• Loss of customer goodwill
• Loss of social welfare (e.g. patients waiting for
  hospital beds)

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Capacity -Waiting Trade-off

• Waiting lines can be reduced by increasing
  capacity
• More service counters
• Adding workers to increase speed

? COST!
Queuing Analysis

• Mathematical analysis of waiting lines
• Goal minimize sum of
• Service capacity costs
• Customer waiting costs
• Economic analysis can be done using either BEA or
  NPV

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Queuing Analysis
Total cost
Customer waiting cost
Capacity cost


Total cost
Cost
Cost of service capacity
Cost of customers waiting
Optimum
Service capacity
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Waiting Line System

• A waiting line system consists of two components
• The ________________ (people or objects to be
  processed)
• The _____________________
• Whenever demand _______ available capacity, a
  waiting line or queue forms

customer population
process or service system
exceeds
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Waiting Line System
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Waiting Line Examples
Situation Arrivals Servers Service Process
Bank Customers Teller Deposit etc. Doctors Patie
nt Doctor Treatmentoffice Traffic
Cars Light Controlledintersection passage
Assembly line Parts Workers Assembly Tool
crib Workers Clerks Check out/in tools
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Waiting Line Terminology

• ______ versus ______ populations
• Is the number of potential new customers affected
  by the number of customers already in queue?
• _______
• When an arriving customer chooses not to enter a
  queue because its already too long
• ________
• When a customer already in queue gives up and
  exits without being serviced
• ________
• When a customer switches back and forth between
  alternate queues in an effort to reduce waiting
  time

Finite
infinite
Balking
Reneging
Jockeying
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Characteristics of Waiting Line

• The service system is defined by
• The number of __________
• The number of _______
• The ___________ of servers
• The arrival and service _______
• The service ______ rules
• Population Source
• - Infinite source customer arrivals are
  unrestricted
• - Finite source number of potential customers
  is limited

waiting lines
servers
arrangement
patterns
priority
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Number of Lines

• Waiting lines systems can have single or multiple
  queues.
• Single queues avoid ________ behavior all
  customers are served on a __________________
  fashion (perceived fairness is high)
• Multiple queues are often used when arriving
  customers have ________ characteristics (e.g.
  paying with cash, less than 10 items, etc.) and
• can be readily __________

jockeying
first-come, first-served
differing
segmented
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Arrangement of Servers

• Single servers or multiple, parallel servers
• Arrangement of servers
• _______ phase systems require customers to visit
  more than one server
• Example of a multi-phase, multi-server system
  with a single waiting line

Multiple
1
4
Depart
Arrivals
C
C
C
C
C
2
5
3
6
Phase 1
Phase 2
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Arrival and Service Pattern

• _____ rate
• The average number of customers arriving per time
  period
• Modeled with _______ distribution
• _______ rate
• The average number of customers that can be
  served during the same period of time
• Modeled using the _________ distribution

Arrival
Poisson
Service
Poisson
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Modeling the Arrival Process
Inter-arrival Time Time between two
consecutive arrivals (a random
quantity) Number Number of arrivals in a unit
time period (also a random quantity) How to
represent this randomness ? ? use common
probability distributions
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Modeling the Arrival Rates

• Number of events that
• occur in an interval of
• time
• Example Number of
• customers that arrive
• in 15 min.
• Mean? (e.g., 4/hr)
• Probability

• 
  ? 0.5

? 6.0
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Modeling Service Time

• Service time, time
• between arrivals
• Example Service
• time is 20 min.
• Mean service rate?
• e.g., customers/hr.
• Mean service time1/ ?
• Probability

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Arrival Rate and Inter-Arrival Time

• Represent Inter-arrival time with an Exponential
  probability distribution with mean (1/?).

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Service Rate and Service Time

• Determine the service rate (?) number of
  customers that can be served per period
• Represent Service time with an Exponential
  probability distribution with mean (1/ ?).

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Important Remarks

• The use of exponential distributions is a
  _________. (it makes mathematical analysis
  simpler)
• You have to ______ these distributions used in
  the waiting line models. (by collecting data and
  then checking for a distribution that fits the
  data, via ____________ such as goodness of fit
  test)
• Poisson arrival is realistic but not exponential
• service

convention
justify
statistical tests
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Queuing Discipline

• First Come First Served (FCFS) is the most
  common, and perhaps the most fair one.
• There are other rules that prioritize customers
• Emergency customers first
• Highest Profit Customers are first
• etc.

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Common Performance Measure

• The average number of customers waiting in the
  line or in the system)
• The average waiting time of customers in the line
  or in the system)
• The system ____________ ( of capacity use)
• Probability that an arrival will have to wait

utilization rate
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Queuing Models-Infinite Source

• ? arrival rate, ? service rate (be careful of
  the units)
• c number of channels (it is also sometimes
  represented by M)
• ? utilization
• Lq Average number of customers in queue
• Ls Average number of customers in system
• Wq Average Waiting time in queue
• Ws Average Waiting time in system
• Pn Prob. of n customers in system

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Basic Relationships (Steady State)

• Utilization (should be lt1,M of servers)
• Average Number in Service
• Average Number in Line (Lq ) model depen.
• Average Number in System
• Average Time Customers Wait in Line
  Wait in System

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Basic (Steady State) Relationships

• One of the most important relationship in queuing
  theory is called Littles Law
• Ls ?Ws
• Lq ?Wq
• Intuitive explanation?

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Two Popular Waiting Line Models

• 1. Model 1 Single Channel
• Model 3 Multiple Channel
• Single phase
• Poisson arrivals
• Exponential service times
• FCFS queue discipline
• No limit on the waiting line length

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Model 1-Single Channel
? Other service measures can be obtained from the
basic relationship formulas.
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Model 1-Single Channel
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Example

• A help desk in the computer lab serves students
  on a first-come, first served basis. On average,
  15 students need help every hour. The help desk
  can serve an average of 20 students per hour.
• Based on this description, we know
• ? __
• ? __

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15
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Average Utilization
15
_____
0.75 or 75
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where M is number of servers
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Average Number of Students in the System
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Average Number of Students Waiting in Line
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Average Time a student Spends in the System
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Average Time a Student Spends Waiting in the Line
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Probability of n Students in Line
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Model 3 Multi-Servers withSingle Line
? Lq formula is complicated. Use Table at the
end of this chapter to read the Lq value.

• ? Use the basic relationship formulas to
  calculate other service measures
• Increasing the number of servers will bring down
  the waiting cost but increase the capacity cost

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Model 3 Formula
Average waiting time for an arrival not
immediately served
Probability that an arrival will have to wait
for service
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Extending Model 1
? Other measures from the basic formulas by
replacing ? with ?eff ?(1- PK) (Why?)
Note that K includes the person in service
also in this case not necessary to have ? lt
1.
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Limitations of Waiting Line Models

• Mathematical analysis becomes very complicated or
  intractable for more _______ waiting lines. Some
  examples
• Non-Poisson arrivals
• Non-exponential service times
• Complex customer behavior (e.g. customers
  switching between lines, or leaving after some
  time etc)
• multiple phase systems
• ? _________ can be useful analysis tool

complex
Simulation
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Remember ? ? Are Rates

• ? Mean number of
• ________ per time period
• e.g., 3 units/hour
• ? Mean number of
• people or items ______
• per time period
• e.g., 4 units/hour
• 1/ ?15 minutes/unit

If average service time is 15 minutes, then µ is
_____________
arrivals
4 customers/hour
served
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Take Away

• When you complete this chapter, you should
• be able to
• Identify or Define
• The assumptions of the three basic
  waiting-line
• models
• Explain or be able to use
• How to apply waiting-line models
• How to conduct an economic analysis of
• queues