Waiting Line Models For Waiting Line Models For Service ImprovementService Improvement
by:by:Shannon DuffyShannon DuffyKatie McPartlinKatie McPartlinJason JacklowJason JacklowAmanda HoltzAmanda Holtz
B.J. KoB.J. Ko
Economic Optimization ModelEconomic Optimization Model
EOMWhich has developed
using queuing analysis
EOMEOM
Used at L.L. Bean for telemarketing operations
To determine the optimal number of telephone trunks for incoming calls
The number of agents scheduledThe queue capacity
– The maximum number of customers who are put on hold to wait for an agent
Queuing models are usedQueuing models are used
To determine the economic impact of busy signals
Customer waiting timeLost orders
Decision about waiting linesDecision about waiting lines
Are based on averages for customer arrivals and service times
They are used to computer operation characteristics– Which are average of values for characteristics
that describe the performance of a waiting line system
Elements of a waiting lineElements of a waiting line
Basic element is queue– Which is a single waiting line– Which consists of arrivals, servers, and waiting
line structure– Single-channel queuing system
The Calling PopulationThe Calling Population
Is the source of the customers to the queuing system, and it can be either infinite or finite
Infinite– Calling population assumes such a large
number of potential customers that is always possible for one more customer to arrive to be served
– Ex. – grocery store, bank
Finite calling populationFinite calling population
Has a specific, countable number of potential customers– Ex. – repair facility in a
shop
Arrival RateArrival Rate
Is the rate at which customers arrive at the service facility during a specified period of time
This rate can be estimated from empirical data derived from studying the system or a similar system, or it can be average of these empirical data
Service timesService times
The time required to serve a customer, is more frequently described by the negative exponential distribution
Service must be expressed as a rate to be compatible with the arrival rate
Customers must be served faster than they arrive or an infinitely large queue will build up
Queue Discipline and LengthQueue Discipline and Length
Queue discipline is the order in which waiting customers are served
Most common type is first come, first served
Infinite QueueInfinite Queue
Can be of any size with no upper limit and is the most common queue structure– Ex. Movie theater
line
Finite QueueFinite Queue
Is limited to size– Ex. Driveway at
bank
Basic Waiting Line StructuresBasic Waiting Line Structures
There are four basic structures according to the nature of the service facilities– Single-channel, single-phase– Single-channel, multiple-phase– Multiple-channel, single-phase– Multiple-channel, multiple-phase
Channels and PhasesChannels and Phases
Channel– Is the number of parallel servers for servicing
arriving customers
Phases– Denotes the number of sequential servers each
customer must go through to complete service
Poisson DistributionPoisson Distribution
The Poisson Distribution is a discrete distribution which takes on the values X=0,1,2,3… – It is often used as a model for the number of
events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period.
Single-Channel, Single-Phase Single-Channel, Single-Phase ModelsModels
Most basic of the waiting line structuresFrequently used variation
– Poisson arrival rate, exponential service times– Poisson arrival rate, general distribution of
service times– Poisson arrival rate, constant service items– Poisson arrival rate, exponential service times
with a finite queue and a finite calling population
Basic Single-Server ModelBasic Single-Server Model
Assume the following– Poisson arrival rate– Exponential service times– First-come, first-served queue discipline– Infinite queue length– Infinite calling population
Constant Service timesConstant Service times
The single-server model with Poisson arrivals and constant service times is a queuing variation that is of particular interest in operations management, since the most frequent occurrence of constant service times is with automated equipment and machinery.– This model has direct applications for many
manufacturing operations
Finite Queue LengthFinite Queue Length
Since some waiting lines systems the length of the queue may be limited by the physical area in which the queue forms;– Space may permit only a limited number of
customers to enter the queue– Variation of the single-phase, single-channel
queuing model
Finite Calling PopulationsFinite Calling Populations
The population of customers from which arrivals originate is limited, such as the number of police cars at a station to answer calls
Multi-Server ModelsMulti-Server Models
Two or more independent servers in parallel serve a single waiting line
The number of servers must be able to serve customers faster than they arrive
Definition of VariablesDefinition of Variables
Pn = Probability of n Units in System
= Mean Number of Arrivals per Time Period = Mean Number of People or Items Served per
Time Period Ls = Average Number of Units in the System
Ws = Average Time a Unit Spends in the System (Wait Time + Service Time)
Definition of VariablesDefinition of VariablesContinuedContinued
Lq = Average Number of Units in the Waiting Line
Wq = Average Time a Unit Spends Waiting in the Line
= Utilization Factor for the System (Percent of Time the Servers are Busy)
P0 = Probability of 0 Units in the System
Single Channel, Single PhaseSingle Channel, Single Phase
Ls = /( Ws = 1/( Lq = Wq = P0 = Pn =n
– In all Cases,
Single Channel, Single Phase Single Channel, Single Phase ExampleExample
For cars arriving/hourcars serviced/hourLs = /(= 2/(3-2) = 2 cars in System
Ws = 1/(= 1/(3-2) = 1 hour average time spent in system
Lq == 22/3(3-2) = 1.33 cars waiting
Single Channel, Single Phase Single Channel, Single Phase Example Cont.Example Cont.
Wq = = 2/(3(3-2)) = 2/3 hour or 40 minute average time waiting
= 2/3 = 66.7% Utilization of Mechanic P0 = = 1-(2/3) = 1/3 = 0.33 probability
there are no cars in system = 0.33 P1 =1 = (2/3)12/3)) = 1/9
probability there is 1 car in system = 0.11
The EndThe End
Any Questions?