Basic -- Introduction to Queuing Theory

7/31/2019 Basic -- Introduction to Queuing Theory

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Stay in Queue: Short Video

Something we can all relate to…

You must, must, must see this video as it is both short and also illustrates comically

the problems of queuing (before you review the rest of this document, go see this short video).

• http://www.youtube.com/watch?v=IPxBKxU8GIQ&feature=related

http://www.youtube.com/watch?v=IPxBKxU8GIQ&feature=related






Queuing Theory Introduction

• Definition and Structure

• Characteristics

• Importance• Models

• Assumptions

• Examples• Measurements

• Apply it to SCM



What is the Queuing Theory?

• Queue- a line of people or vehicles waitingfor something

• Queuing Theory- Mathematical study ofwaiting lines, using models to showresults, and show opportunities, withinarrival, service, and departure processes



Structure

Inputsource

QueueDiscipline

ServiceFacility

ServedCustomers

Balking CustomersReneging Customers



Customer Behaviors

• Balking of Queue • Some customers decide not to join the queue due to

their 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.



Characteristics• Arrival Process

– The probability density distribution that determines the customerarrivals in the system.

• Service Process – The probability density distribution that determines the customer service

times in the system.• Number of Servers

– Number of servers available to service the customers.• Number of Channels

– Single channel – N independent channels – Multi channels

• Number of Phases/Stages – Single Queue – Series or Tandem – Cyclic -Network

• Queue Discipline -Selection for Service – First com first served (FCFS or FIFO) – Last in First out (LIFO)

-Random-Priority



Importance of the QueuingTheory

-Improve Customer Service, continuously.

-When a system gets congested, the servicedelay in the system increases.

• A good understanding of the relationshipbetween congestion and delay is essentialfor designing effective congestion controlfor any system.

• Queuing Theory provides all the toolsneeded for this analysis.



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.



Queuing Models Calculate:

• Average number of customers in the systemwaiting and being served

• Average number of customers waiting in theline

• Average time a customer spends in thesystem 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.



Assumptions

• Different for every system.

• Variable service times and arrival times areused to decide what model to use.

• Not a complex problem:

– Queuing Theory is not intended for complexproblems. 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.



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 Industrial –Production lines



How the Queuing Theory is used inSupply 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 anddissatisfied customers. The Queuing Theory may be

used to determine the appropriate level of capacityrequired at manufacturing facilities and the staffinglevels required at service facilities, over the nominalaverage capacity required to service expecteddemand without these surges.



When is the Queuing Theoryused?

• Research problems

• Logistics

• Product scheduling• Ect…



Terminology

Customers: independent entities that arriveat random times to a server and wait forsome kind of service, then leave.

Server: can only service one customer at atime; length of time depends on type ofservice. Customers are served based on

first in first out (FIFO)

Time: real, continuous, time.



• Queue: customers that have arrived at serverand are waiting for their service to start

• Queue Length at time t: number of customersin the queue at that time

• Waiting Time: how long a customer has to wait

between arriving at the server and when theserver actually starts the service



Little’s Law

• The mean queue length or the average numberof customers (N) can be determined from thefollowing 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 toreduced costs and higher earnings



Poisson Distribution Poisson role in the arrival and service process:

Poisson (or random) processes: means that thedistribution of both the arrival times and the servicetimes follow the exponential distribution. Because ofthe mathematical nature of this exponentialdistribution, we can find many relationships basedon performance which help us when looking at thearrival rate and service rate.

Poisson process. An arrival process wherecustomers arrive one at a time and where the

interval s between arrivals is described byindependent random variables



Factors of a Queuing System

• When do customers arrive? – Are customer arrivals increased during a certain time(restaurant- Denny’s: breakfast, lunch, dinner) Or is thecustomer traffic more randomly distributed (a café-starbucks)

• Depending on what type of Queue line, Howmuch time will customers spend

• Do customers typically leave in a fixed amountof time?

• Does the customer service time vary with thetype of customer?



Important characteristics

• Arrival Process: The probabilitydistribution that determines the customerarrivals in the system.

• Service Process: determines thecustomer service times in the system.

• Number of Servers: Amount of serversavailable to provide service to thecustomers



• Queuing systems can then be classified asA/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



NotationA/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



Examples of Different QueuingSystems

• M/M/1 (A/S/n)

• Arrival Distribution: Poisson rate (M) tells youto use exponential probability

• Service Distribution: again the M signifies anexponential probability

• 1 represents the number of servers



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



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

process.

This Assumptions is a good approximation for the arrivalprocess 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 very

large.

• A single car uses a very small percentage of the highwayresources.• Decision to enter the highway is independently made by

each car driver.



Summary

• M/M/1: The system consists of only one server.This queuing system can be applied to a widevariety of problems as any system with a verylarge number customers.

• M/D/n: Here the arrival process is poison andthe service time distribution is deterministic. Thesystem has n servers. Since all customers aretreated the same, the service time can beassumed to be same for all customers

• G/G/n: This is the most general queuing systemwhere the arrival and service time processes areboth arbitrary. The system has n servers.



Pros and Cons of Queuing Theory

(END)

• Helps the user to easilyinterpret data by looking

at different scenariosquickly, accurately, andeasily

• Can visually depictwhere problems mayoccur, providing time to

fix a future error• Applicable to a widerange of topics

• Based onassumptions ex.

Poisson Distributionand service time• Curse of variability-

congestion and waittime increases asvariability increases

• Oversimplificationof model

Positives Negatives



LI

MIT

ATI

ONS

• Mathematical models put arestriction on finding real

world solutions – Ex: Often assume infinite

customers, queue capacity,

service time, In reality there aresuch limitations.

• Relies too heavily on behaviorand characteristics of peopleto work smoothly with themodel



Types of Queuing Systems

• A population consists of either an infiniteor a finite source.

• The number of servers can be measuredby channels (capacity of each server) orthe number of servers.

• Channels are essentially lines.

• Workstations are classified as phases in aqueuing system.




• Single Channel Single Phase: Trucksunloading shipments into a dock.




• Single Line Multiple Phase: Wendy’s Drive

Thru -> Order + Pay/Pickup




• Multiple Line Single Phase: WalgreensDrive-Thru Pharmacy




• Multiple Line Multiple Phase: HospitalOutpatient Clinic, Multi-specialty



Measuring Queuing SystemPerformance

• Average number of customers waiting (inthe 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.



Queuing Model Analysis

• Two simple single-server models helpanswer meaningful questions and alsoaddress the curse of utilization and the

curse of variability .

• One model assumes variable service timewhile the other assumes constant service

time.



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 customerswon’t leave when the line is long.

• 3: The mean service rate is constant. Thisrate is independent in the sense that serverswon’t speed up when the line is longer.



Parameters For QueuingModels

• λ = mean arrival rate = average number of

units arriving at the system per period.

• 1/λ = mean inter arrival time, time between

arrivals.

• μ = mean service rate per server =

average number of units that a server can

process per period.

• 1/μ = mean service time

• m = number of servers



Parameter Examples

• λ (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:

• μ (mean inter arrival time) = 1/30 cars per second

• 60 seconds/minute * 1/30 cars per second = 2 cars perminute

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

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



Performance Measures

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

• Mean time that a person or unit spends in thesystem (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 of n units in the system (In Queue

or in Service)



Formulas For Performance Measures

• m μ = Total Service Rate = Number of Servers *Service Rate of Each Server

• System Utilization = Arrival Rate/Total ServiceRate = λ/m μ

• Average Time in System = Average time inqueue + average service time• Average number in system = average number in

queue + average number in service• Average number in system = arrival rate *

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



Performance Formulas (contd.)

• Though these seem to be common sense,the values of these formulas can easily bedetermined but depend on the nature of

the variation of the timing of arrivals andservice times in the following queuingmodels:



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.



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.



The Curse of Variability

• When you remove variance from servicetime, lines decrease and waiting time doesas well. Thus, as variability increases, then

line congestion and wait times increase aswell.



The Curse of Variability (contd.)

• The sensitivity of system performance tochanges in variability increases withutilization.

• Thus, when you try to lower variance, it ismore likely to pay off when the system has ahigher resource utilization.

• To provide better service, systems with highvariability should operate at lower levels ofresource utilization than systems with lowervariability.



The Curse of Variability (contd.)

• Exponential distribution shows a high degreeof variability; the standard deviation of servicetime is equal to the mean service time.

• Constant service times shows no variation atall.

• Therefore, actual performance is better than

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

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Basic -- Introduction to Queuing Theory

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