Waiting Line TheoryWaiting Line Theory

Akhid Yulianto, SE, MSc (log)Akhid Yulianto, SE, MSc (log)

BankBank CustomersCustomers TellerTeller Deposit etc.Deposit etc.

Doctor’sDoctor’s PatientPatient DoctorDoctor TreatmentTreatmentofficeoffice

Traffic Traffic CarsCars LightLight ControlledControlledintersection intersection passage passage

Assembly lineAssembly line PartsParts WorkersWorkers AssemblyAssembly

Tool cribTool crib WorkersWorkers ClerksClerks Check out/in toolsCheck out/in tools

Situation Arrivals Servers Service Process

Waiting Line ExamplesWaiting Line Examples

Structure of a Waiting Line Structure of a Waiting Line SystemSystem

Queuing theoryQueuing theory is the study of waiting is the study of waiting lines. lines.

Four characteristics of a queuing system Four characteristics of a queuing system are: are: the manner in which customers arrivethe manner in which customers arrive the time required for servicethe time required for service the priority determining the order of the priority determining the order of

serviceservice the number and configuration of the number and configuration of

servers in the systemservers in the system..

Level of serviceLevel of service

CostCost

Service costService costTotal waiting line cost

Total waiting line cost

Waiting time costWaiting time cost

Optimal

Waiting Line CostsWaiting Line Costs

Structure of a Waiting Line Structure of a Waiting Line SystemSystem Distribution of ArrivalsDistribution of Arrivals

Generally, the arrival of customers Generally, the arrival of customers into the system is a into the system is a random eventrandom event. .

Frequently the arrival pattern is Frequently the arrival pattern is modeled as a modeled as a Poisson processPoisson process..

Distribution of Service TimesDistribution of Service Times Service time is also usually a random Service time is also usually a random

variable. variable. A distribution commonly used to A distribution commonly used to

describe service time is the describe service time is the exponential distributionexponential distribution..

Poisson ProbabilityPoisson Probability

x = Tingkat kedatanganx = Tingkat kedatangan λλ = rata rata kedatangan per = rata rata kedatangan per

periodeperiode e = 2.71828e = 2.71828

!)(

xexPx

Eksponential ProbabilityEksponential Probability

µ =jumlah unit yang di layani per µ =jumlah unit yang di layani per periodeperiode

e = 2.71828e = 2.71828

tettimeserviceP 1)(

Structure of a Waiting Line Structure of a Waiting Line SystemSystem Queue DisciplineQueue Discipline

Most common queue discipline is Most common queue discipline is first come, first served (FCFS)first come, first served (FCFS). .

An elevator is an example of last An elevator is an example of last come, first served (LCFS) queue come, first served (LCFS) queue discipline.discipline.

Other disciplines assign priorities to Other disciplines assign priorities to the waiting units and then serve the the waiting units and then serve the unit with the highest priority first.unit with the highest priority first.

Structure of a Waiting Line Structure of a Waiting Line SystemSystem

SS11

SS11

SS22

SS33

CustomerCustomerleavesleaves

CustomerCustomerleavesleaves

CustomerCustomerarrivesarrives

CustomerCustomerarrivesarrives

Waiting lineWaiting line

Waiting lineWaiting line

SystemSystem

SystemSystem

Queuing SystemsQueuing Systems

A A three part codethree part code of the form of the form AA//BB//kk is used to is used to describe various queuing systems. describe various queuing systems.

AA identifies the arrival distribution, identifies the arrival distribution, BB the service the service (departure) distribution and (departure) distribution and kk the number of the number of channels for the system. channels for the system.

Symbols used for the arrival and service Symbols used for the arrival and service processes are: processes are: MM - Markov distributions - Markov distributions (Poisson/exponential), (Poisson/exponential), DD - Deterministic - Deterministic (constant) and (constant) and GG - General distribution (with a - General distribution (with a known mean and variance). known mean and variance).

For example, For example, MM//MM//kk refers to a system in which refers to a system in which arrivals occur according to a Poisson distribution, arrivals occur according to a Poisson distribution, service times follow an exponential distribution service times follow an exponential distribution and there are and there are kk (sometimes others say s) servers (sometimes others say s) servers working at identical service rates. working at identical service rates.

Queuing System Input Queuing System Input CharacteristicsCharacteristics

= the average arrival = the average arrival raterate 1/1/ = the average = the average timetime between between

arrivalsarrivals µ µ = the average service = the average service raterate

for each serverfor each server 1/1/µ µ = the average service = the average service timetime = the standard deviation of = the standard deviation of

the service the service timetime

Analytical FormulasAnalytical Formulas For nearly all queuing For nearly all queuing

systems, there is a systems, there is a relationship between the relationship between the average time a unit spends in average time a unit spends in the system or queue and the the system or queue and the average number of units in the average number of units in the system or queue. system or queue.

These relationships, known as These relationships, known as Little's flow equationsLittle's flow equations are: are:

LL = = WW and and LLqq = = WWqq

Analytical FormulasAnalytical Formulas When the queue discipline is FCFS, When the queue discipline is FCFS,

analytical formulas have been derived for analytical formulas have been derived for several different queuing models including several different queuing models including the following: the following: MM//MM/1/1 MM//MM//kk MM//GG/1/1 MM//GG//kk with blocked customers cleared with blocked customers cleared MM//MM/1 with a finite calling population/1 with a finite calling population

Analytical formulas are not available for all Analytical formulas are not available for all possible queuing systems. In this event, possible queuing systems. In this event, insights may be gained through a insights may be gained through a simulation of the system. simulation of the system.

M/M/1M/M/1 Ls = average number of units in Ls = average number of units in

the system (waiting and being the system (waiting and being served)served)

Ws = average time a unit spends Ws = average time a unit spends in the systemin the system

Lq = average number of units Lq = average number of units waiting in the queuewaiting in the queue

Wq = Average time a unit Wq = Average time a unit spends waiting in the queuespends waiting in the queue

Utilization factor for the systemUtilization factor for the system Probability of 0 units in the Probability of 0 units in the

systemsystem Probability of more than k units Probability of more than k units

in the system, where n is the in the system, where n is the number of units in the systemnumber of units in the system

1

0

2

1

1

k

kn

q

q

s

s

P

P

W

L

W

L

MM//MM//kk Queuing System Queuing System

Multiple channels (with one central waiting Multiple channels (with one central waiting line)line)

Poisson arrival-rate distributionPoisson arrival-rate distribution Exponential service-time distributionExponential service-time distribution Unlimited maximum queue lengthUnlimited maximum queue length Infinite calling populationInfinite calling population Examples:Examples:

Four-teller transaction counter in bankFour-teller transaction counter in bank Two-clerk returns counter in retail storeTwo-clerk returns counter in retail store

M/M/SM/M/S Ls = average number of units Ls = average number of units

in the system (waiting and in the system (waiting and being served)being served)

Ws = average time a unit Ws = average time a unit spends in the systemspends in the system

Lq = average number of units Lq = average number of units waiting in the queuewaiting in the queue

Wq = Average time a unit Wq = Average time a unit spends waiting in the queuespends waiting in the queue

Probability of 0 units in the Probability of 0 units in the systemsystem

qsq

sq

s

M

s

mM

n

n

M

s

LWW

LL

LPMM

W

forM

MM

Mn

P

pMM

L

1

1!1

!1

!1

1!1

02

1

0

0

02