Operations Management MD021

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Operations Management(MD021)

• Waiting Line Models

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Agenda

• Waiting Lines and Service Design
• Queueing Theory
• Queueing Models

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Waiting Lines (Queues) and Service Design
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When do waiting lines NOT occur?

• System variability is minimal/nonexistent
• Customer arrivals can be scheduled
• They are not random
• Customers arrive when they are scheduled to
  arrive
• Service times are known and constant
• Customers do not make any undue demands or
  exhibit random behaviors that affect system
  performance

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Why do waiting lines occur?

• Waiting lines occur due to
• Larger arrival rate than servicing (departure)
  rate
• Randomness/Variability
• Customers usually arrive at random intervals
• Variability in order lengths some orders take
  longer than others

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Waiting Time vs. Utilization
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Managerial implications of high service system
utilization and long waiting lines

• Costly to provide waiting space
• Possible loss of business
• Customers refusing to wait
• Customers leaving
• Loss of customer goodwill
• Reduction in customer satisfaction
• Congestion may disrupt other business operations

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Service design involves choosing appropriate
lines for a service environment

• Hot dog stand on street corner single hot dog
  vendor (server) with a single line and a
  potentially huge number of customers
• Copier repair single server with finite set of
  customers (copiers breaking down)
• Bank single line with multiple tellers
  (servers)
• Grocery stores multiple lines with multiple
  checkers (one line per server)
• Amusement park system of queues as customers
  move from ride to ride

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Companies employ many tactics to decrease
negative impact of waiting

• Waiting in lines does not add enjoyment
• Waiting in lines does not generate revenue
• Potential tactics
• Try to disguise waiting/distract customers minds
  while waiting
• Tell customers how long their wait will be
• Get customers into processing as soon as
  possible, so customers feel they are being
  processed

Waiting lines are non-value added occurrences
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Companies employ many tactics to decrease
negative impact of waiting

• Reduce perceived waiting time
• Magazines in waiting rooms
• Radio/television
• In-flight movies
• Filling out forms
• Derive benefits from waiting
• Place impulse items near checkout
• Advertise other goods/services

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Managers want to optimize certain performance
measures

• System Utilization
• Average Number of Customers Waiting
• Average Customer Time in System
• Waiting time processing time
• Typically want this to be small
• Average Customer Waiting Time
• Typically, you dont want to keep the customer
  waiting for an unreasonable amount of time
• Probability of Excessive Waiting
• Would like customer to wait less than T time
  units
• Customer Waiting Costs
• Service Costs
• Probability of Lost Sales
• Would like to minimize

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Queueing Theory
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Waiting Lines

• Queuing theory Mathematical approach to the
  analysis of waiting lines.
• Goal of queuing analysis is to minimize the sum
  of two costs
• Customer waiting costs
• Service capacity costs

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Objective of Queuing Analysis is to balance
waiting costs vs. capacity costs
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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Features of Queuing Systems
Calling Population
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Features of Queueing Systems

• Calling Population
• Composition
• Homogeneous customers
• Heterogeneous customers (multiple classes of
  customers)
• How many groups? Characteristics of groups?
• Size
• Infinite source customer arrivals are
  unrestricted
• Finite source number of potential customers is
  limited

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Features of Queueing Systems

• Arrival process
• The demand for a service has temporal and spatial
  components that determine how demand arrives at
  the service system
• Nature
• Discrete/Fixed
• Scheduled
• Stochastic/Random
• Inter-arrival times (period between arrivals) are
  commonly assumed to be distributed exponentially
• Customer Behaviors
• Balking seeing length of waiting line, and
  leaving before entering line
• Reneging leaving the waiting line after being
  in it for a while

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Exponential Distribution
Relative Frequency
0.15 0.10 0.05 0
Time
0 1 2 3 4 5 6 7 8 9 10 11
Useful for Representing Inter-Arrival
Times Service Times
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Poisson Distribution
Relative Frequency
Customers Per Time Unit
Useful for Representing Arrival Rates Service
Rates
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Features of Queueing Systems

• Servicing process
• The tasks that must be accomplished to complete
  the service
• The number of servers
• The location of servers where the tasks are
  completed
• The length of time for completing each task

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Features of Queueing Systems

• Queue configuration
• The number of queues, their locations, their
  spatial requirements, and their effects on
  customer behavior
• Single queue vs. multiple queues
• Queue traffic controller? (bouncer, lady in bank
  at lunchtime yelling out which line is empty)
• Allowing jockeying behavior customers leave one
  line for another shorter one

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A queue system with multiple servers and multiple
stages/phases
Multiple channel
Multiple phase
Channel A server in a service system
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Features of Queueing Systems

• Potential customer behaviors
• Patient
• Customers enter the waiting line and remain until
  served
• Reneging
• Waiting customers grow impatient and leave the
  line
• Jockeying
• Customers may switch to another line
• Balking
• Upon arriving, decide the line is too long and
  decide not to enter the line

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Features of Queueing Systems

• Queue discipline
• A policy established by management to select the
  next customer from the queue for service
• First-Come, First-Served (FCFS)
• Person who arrived first goes next
• Shortest Processing Time
• Job that will take the shortest time to process
  goes next
• Priority Based Queueing Disciplines
• Bouncer at bar lets in the beautiful people
  first rest of us have to wait (or tip heavily
  to get past bouncer) until space is available
  late at night
• Emergency room helps people with really bad
  injuries first

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Queueing Models
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Analytical Modeling of Simple Queues

• Identify what type of queue you are modeling
  (number of queues, number of servers, etc.)
• Collect data about behaviors (arrival process,
  service times), and identify distributions of
  data (e.g., Poisson, etc.)
• Look in a queueing book, and find equations for
  calculating the expected performance of those
  queueing systems
• Plug and chug
• See if 2 servers provides enough capacity. If
  not, recalculate for 3 servers, then for 4, until
  you find a number of servers that provides
  sufficient capacity.

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Analytical Modeling of Complex Queues

• Analytical approach for complex queueing systems
• Hire a statistician who knows how to model a
  queue as a Stochastic Process
• Hire a statistician who can build a simulation
  model of the system

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Analytical ModelsWe will study steady state
systems

• States of Interest
• Transient State Performance
• System is NOT in equilibrium
• Parameters of system change over time
• Steady State Performance
• System is in equilibrium
• Parameters are stable over time
• Allows us to calculate long-run performance of a
  queueing system

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A/B/C Queue Notation used to represent a variety
of queues

• A Interarrival Time Distribution
• M exponential distribution
• D deterministic (constant)
• Ek Erlang distribution
• G general distribution
• B Service Time Distribution
• M, D, Ek, or G
• C Number of Servers
• 1, 2, 3, 8

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There are many types of queues of interest to
operations managers

• Analytical Models for Estimating Capacity
• Single Server
• M/M/1, M/G/1, M/D/1, G/G/1
• Multiple Server
• M/M/c, M/G/8 etc.
• Multiple Stage
• Markov Chain models

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Infinite-Source Queuing Models

• Single channel, exponential service time (M/M/1)
• Single channel, constant service time (M/D/1)
• Multiple channel, exponential service time
  (M/M/S)
• Multiple priority service, exponential service
  time

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Queue Notation

• ? customer arrival rate
• µ service rate per server
• Lq average number of customers waiting for
  service
• Ls average number of customers being served
• r average number of customers being served
• ? system utilization
• Wq average time that customers wait in line
• Ws average time customers spend in the system
• 1/µ service time
• P0 probability of zero units in the system
• Pn probability of n units in the system
• M number of servers
• Lmax maximum expected number waiting in line

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Basic Queue Performance

• System utilization
• ? ?/Mµ
• Average number of customers being served
• r ?/µ
• Average number of customers
• Waiting in line for service Lq
• In the system Ls Lq r
• Average times
• Waiting in line Wq Lq/?
• In the system Ws Wq 1/µ Ls/?

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The Classic Queue M/M/1
Identical Customers
Queue
Server

• M/M/1 queueing model
• Assumptions
• Infinite or very large population of customers
• Poisson arrival rate
• Single waiting line w/ infinite length, but no
  balking or reneging
• First-Come, First Served
• Poisson service times

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M/M/1 Queuing Formulas

• Average number of customers waiting in line
• Lq ?2/µ(µ - ?)
• Probability of zero units in the system
• P0 1 (?/µ)
• Probability of n units in the system
• Pn P0(?/µ)n
• Probability of less than n units in the system
• Pltn 1 - (?/µ)n

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M/D/1 Queue
Identical Customers
Queue
Server

• M/D/1 queueing model
• Assumptions
• Constant Service Time

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M/D/1 Queueing Formulas

• Average number of customers waiting in line
• Lq ?2/2µ(µ - ?)

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M/M/S Queue
Identical Customers
Queue
Pool of Identical Servers

• M/M/3 queueing model
• Assumptions
• Infinite or very large population of customers
• Poisson arrival rate
• Single waiting line w/ infinite length, but no
  balking or reneging
• First-Come, First Served
• Poisson service times

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Priority Queues
Gunshot Wound
Priority Queue
Pool of Servers
Broken Arm
Non-Priority Queue
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Priority Queueing

• Littlefield Technologies
• Stage 2 Machines
• Priority to Step 2
• Highest priority Jobs requiring stage 2 next
• Low priority Jobs requiring stage 4 next
• Priority to Step 4
• Highest priority Jobs requiring stage 4 next
• Low priority Jobs requiring stage 2 next

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Priority Queueing Model
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Finite-Source Queueing Model

• Appropriate for situations when there is a
  calling population that is finite (countable and
  small)
• Major difference Arrival rate of customers in a
  finite situation is affected by the length of the
  waiting line
• The more customers are already waiting, the fewer
  that can arrive
• If everyone is already in waiting line, then no
  additional customers can arrive everyone is
  already there

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Finite-source queuing modelcustomer groups
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Finite-Source Queueing Formulas