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?