Waiting Line Theory

8/3/2019 Waiting Line Theory

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Stay in Queue: Short Video

Something we can all relate to

http://www.youtube.com/watch?v=IPxBKx

U8GIQ&feature=related


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Queuing Theory Introduction

Definition and Structure

Characteristics

Importance Models

Assumptions

Examples

Measurements

Apply it to SCM


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What is the Queuing Theory?

Queue- a line of people or vehicles waiting

for something

Queuing Theory- Mathematical study ofwaiting lines, using models to show

results, and show opportunities, within

arrival, service, and departure processes


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Structure

Balking CustomersReneging Customers


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Customer Behaviors

Balking of Queue

Some customers decide not to join the queue due totheir observation related to the long length of queue,insufficient waiting space or improper care while

customers are in queue. This is balking, and, thus,pertains to the discouragement of customer for not

joining an improper or inconvenient queue.

Reneging of Queue Reneging pertains to impatient customers. After being in

queue for some time, few customers become impatientand may leave the queue. This phenomenon is called asreneging of queue.


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Characteristics Arrival Process

The probability density distribution that determines thecustomer arrivals in the system.

Service Process The probability density distribution that determines the customer

service times in the system.

Numberof Servers Number of servers available to service the customers.

Numberof Channels Single channel

N independent channels

Multi channels

Numberof Phases/Stages Single Queue

Series or Tandem

Cyclic -Network

Queue Discipline -Selection forService First com first served (FCFS or FIFO)

Last in First out (LIFO)

-Random-Priority


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Importance of the Queuing

Theory-Improve Customer Service, continuously.

-When a system gets congested, the service

delay in the system increases. A good understanding of the relationship

between congestion and delay is essential

for designing effective congestion control

for any system.

Queuing Theory provides all the tools

needed for this analysis.


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Queuing Models

Calculates the best number of servers tominimize costs.

Different models for different situations(Like SimQuick, we noticed differentmeasures for arrival and service times)

Exponential

Normal Constant

Etc.


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Queuing Models Calculate:

Average number of customers in the systemwaiting and being served

Average number of customers waiting in the

line Average time a customer spends in the

system waiting and being served

Average time a customer spends waiting in

the waiting line or queue. Probability no customers in the system

Probability n customers in the system

Utilization rate: The proportion of time the

system is in use.


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Assumptions

Different for every system.

Variable service times and arrival times are

used to decide what model to use.

Not a complex problem:

Queuing Theory is not intended for complex

problems.We have seen this in class, where this

are many decision points and paths to take. Thiscan become tedious, confusing, time consuming,

and ultimately useless.


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Examples of Queuing Theory

Outside customers (Commercial Service Systems)-Barber shop, bank teller, cafeteria line

Transportation Systems-Airports, traffic lights

Social Service Systems-Judicial System, healthcare

Business or IndustrialProduction lines


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How the Queuing Theory is used in

Supply Chain Management Supply Chain Management use simulations andmathematics to solve many problems.

The Queuing Theory is an important tool used tomodel many supply chain problems. It is used tostudy situations in which customers (or ordersplaced by customers) form a line and wait to beserved by a service or manufacturing facility.Clearly, long lines result in high response times and

dissatisfied customers. The Queuing Theory may beused to determine the appropriate level of capacityrequired at manufacturing facilities and the staffinglevels required at service facilities, over the nominalaverage capacity required to service expected

demand without these surges.


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When is the Queuing Theory

used? Research problems

Logistics

Product scheduling Ect


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Terminology

Customers: independent entities that arrive

at random times to a server and wait for

some kind of service, then leave.

Server: can only service one customer at a

time; length of time depends on type of

service. Customers are served based on

first in first out (FIFO)

Time: real, continuous, time.


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Queue: customers that have arrived at

server and are waiting for their service tostart

Queue Length at time t: number of

customers in the queue at that time

Waiting Time: how long a customer has to

wait between arriving at the server and

when the server actually starts the service


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Littles Law

The mean queue length or the average

number of customers (N) can be

determined from the following equation:

N= T

lambda is the average customer arrival

rate and T is the average service time for a

customer.

* Finding ways to reduce flow time can lead

to reduced costs and higher earnings


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Poisson DistributionPoisson role in the arrival and service process:

Poisson (orrandom) processes: means that the

distribution of both the arrival times and the

service times follow the exponential distribution.

Because of the mathematical nature of this

exponential distribution, we can find many

relationships based on performance which help

us when looking at the arrival rate and service

rate.Poisson process.An arrival process where

customers arrive one at a time and where the

interval s between arrivals is described by

independent random variables


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Factors of a Queuing System

When do customers arrive? Are customer arrivals increased during a certain time

(restaurant- Dennys: breakfast, lunch, dinner) Or is

the customer traffic more randomly distributed (a caf-

starbucks)

Depending on what type of Queue line, How

much time will customers spend

Do customers typically leave in a fixedamount of time?

Does the customer service time vary with the

type of customer?


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Important characteristics

Arrival Process: The probability

distribution that determines the customer

arrivals in the system.

Service Process: determines the

customer service times in the system.

Numberof Servers:Amount of servers

available to provide service to the

customers


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Queuing systems can then be classified as

A/S/n

A (Arrival Process) and S (Service Process)

can be any of the following:

Markov (M): exponential probability density

(Poisson Distribution)

Deterministic (D): Customers arrival is

processed consistently

N: Number of servers

G: General, the system has n number of

servers


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Notation

A/B/x/y/z

A = letter for arrival distribution

B = letter for service distribution

x = number of service channels y = number allowed in queue

z = queue discipline


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Examples of Different Queuing

Systems

M/M/1 (A/S/n)

Arrival Distribution: Poisson rate (M) tells

you to use exponential probability Service Distribution: again the M signifies

an exponential probability

1 represents the number of servers


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M/D/n

-Arrival process is Poisson, but service isdeterministic.

The system has n servers.

ex: a ticket booking counter with n cashiers.G/G/n

- A general system in which the arrival and

service time processes are both random


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Poisson Arrivals M/M/1 queuing systems assume a Poisson arrival

process.This Assumptions is a good approximation for the arrival

process in real systems:

The number of customers in the system is very large.

Impact of a single customer on the performance of the

system is very small, (single customer consumes a verysmall percentage of the system resources)

All customers are independent (their decision to use thesystem are independent of other users)

Cars on a Highway

Total number of cars driving on the highway is verylarge.

A single car uses a very small percentage of the highwayresources.

Decision to enter the highway is independently made byeach car driver.


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Summary

M/M/1: The system consists of only oneserver. This queuing system can be appliedto a wide variety of problems as any systemwith a very large number customers.

M/D/n: Here the arrival process is poison andthe service time distribution is deterministic.The system has n servers. Since allcustomers are treated the same, the servicetime can be assumed to be same for allcustomers

G/G/n: This is the most general queuingsystem where the arrival and service timeprocesses are both arbitrary. The system has

n servers.


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Pros and Cons of Queuing Theory

(END)

Helps the user to easilyinterpret data by lookingat different scenariosquickly, accurately, andeasily

Can visually depictwhere problems may

occur, providing time tofix a future error

Applicable to a widerange of topics

Based on

assumptions ex.Poisson Distributionand service time

Curse of variability-congestion and waittime increases asvariability increases

Oversimplificationof model

Positives Negatives


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L

I

M

I

T

A

T

I

ON

S

Mathematical models put a

restriction on finding realworld solutions

Ex: Often assume infinite

customers, queue capacity,

service time, In reality there aresuch limitations.

Relies too heavily on behavior

and characteristics of peopleto work smoothly with the

model


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Types of Queuing Systems

A population consists of either an infinite

or a finite source.

The number of servers can be measuredby channels (capacity of each server) or

the number of servers.

Channels are essentially lines.

Workstations are classified as phases in a

queuing system.


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Single Channel Single Phase: Trucks

unloading shipments into a dock.


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Single Line Multiple Phase:Wendys Drive

Thru -> Order + Pay/Pickup


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Multiple Line Single Phase:Walgreens

Drive-Thru Pharmacy


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Multiple Line Multiple Phase: Hospital

Outpatient Clinic, Multi-specialty


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Measuring Queuing System

Performance

Average number of customers waiting (in

the queue or in the system)

Average time waiting Capacity utilization

Cost of capacity

The probability that an arriving customerwill have to wait and if so for how long.


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Queuing Model Analysis

Two simple single-server models help

answer meaningful questions and also

address the curse of utilization and the

curse of variability.

One model assumes variable service time

while the other assumes constant service

time.


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Three Important Assumptions

1: The system is in a steady state. The meanarrival rate is the same as the meandeparture rate.

2: The mean arrival rate is constant. This rateis independent in the sense that customerswont leave when the line is long.

3: The mean service rate is constant. Thisrate is independent in the sense that servers

wont speed up when the line is longer.


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Parameters For Queuing

Models = mean arrival rate = average number ofunits arriving at the system per period.

1/ = mean inter arrival time, time betweenarrivals.

= mean service rate per server =

average number of units that a server can

process per period.

1/ = mean service time

m = number of servers


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Parameter Examples

(mean arrival rate) = 200 cars per hour through a tollbooth

If it takes an average of 30 seconds to exchangemoney at a toll booth, then:

(mean inter arrival time) = 1/30 cars per second 60 seconds/minute * 1/30 cars per second = 2 cars per

minute

2 cars per minute * 60 minutes/hour = 120 cars perhour

Thus, with 200 cars per hour coming through () andonly 120 cars being served per hour (), the ratio of/ is 1.67, meaning that the toll booth needs 2 serversto accommodate the passing cars.


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Performance Measures

System Utilization = Proportion of the timethat the server is busy.

Mean time that a person or unit spends in the

system (In Queue or in Service) Mean time that a person or unit spendswaiting for service (In Queue)

Mean number of people or units in the system(In Queue or in Service)

Mean number of people or units in line forservice (In Queue)

Probability ofn units in the system (In Queueor in Service)


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Formulas For Performance

Measures m = Total Service Rate = Number of Servers *

Service Rate of Each Server

System Utilization = Arrival Rate/Total ServiceRate = /m

Average Time in System = Average time inqueue + average service time

Average number in system = average number inqueue + average number in service

Average number in system = arrival rate *average time in system

Average number in queue = arrival rate *average time in queue


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Performance Formulas (contd.)

Though these seem to be common sense,

the values of these formulas can easily be

determined but depend on the nature of

the variation of the timing of arrivals and

service times in the following queuing

models:


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System Measurements

Drive-Thru Example:

If one car is ordering, then there is one unit in

service.

If two cars are waiting behind the car in service,then there are two units in queue.

Thus, the entire system consists of 3 customers.


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The Curse of Utilization

One hundred percent utilization may sound goodfrom the standpoint of resources being used tothe maximum potential, but this could lead topoor service or performance.

Average flow time will skyrocket as resourceutilization gets close to 100%.

For example, if one person is only taking 3

classes next semester, they will probably havean easier time completing assignments thansomeone who is taking 5, even though theperson taking 5 classes is utilizing their timemore in terms of academics.


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The Curse of Variability

When you remove variance from service

time, lines decrease and waiting time does

as well. Thus, as variability increases, then

line congestion and wait times increase as

well.


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The Curse of Variability (contd.)

The sensitivity of system performance to

changes in variability increases with

utilization.

Thus, when you try to lower variance, it is

more likely to pay off when the system has a

higher resource utilization.

To provide better service, systems with highvariability should operate at lower levels of

resource utilization than systems with lower

variability.


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The Curse of Variability (contd.)

Exponential distribution shows a high degree

of variability; the standard deviation of service

time is equal to the mean service time.

Constant service times shows no variation at

all.

Therefore, actual performance is better than

what the M/M/1 (Exp.) model predicts andworse than what the M/D/1 (Const.) model

predicts.


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Questions?


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T

hank You!

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Waiting Line Theory