QUEUING THEORY

QUEUING THEORY[M/M/C MODEL]

Student Adviser:-Assist.Prof. Sanjay Kumar

Student:-Ram Niwas Meena

Semester:-Fourth

“Delay is the enemy of efficiency” and “Waiting is the enemy of utilization”

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OVERVIEW

What is queuing theory? Examples of Real World Queuing Systems? Components of a Basic Queuing Process A Commonly Seen Queuing Model Terminology and Notation Little’s Formula The M/M/1 – model Example M/M/c Model

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Mathematical analysis of queues and waiting times in stochastic systems. Used extensively to analyze production and service processes

exhibiting random variability in market demand (arrival times) and service times.

Queues arise when the short term demand for service exceeds the capacity Most often caused by random variation in service times and

the times between customer arrivals. If long term demand for service > capacity the queue will

explode!

Queuing theory is the mathematical study of waiting lines (or

queues) that enables mathematical analysis of several related

processes, including arriving at the (back of the) queue, waiting in the queue, and being served by the Service Channels at the front of the queue.

WHAT IS QUEUING THEORY?

What is Transient & Steady State of the system?Queuing analysis involves the system’s behavior over time. If the operating characteristics vary with time then it is said to be transient state of the system.If the behavior becomes independent of its initial conditions (no. of customers in the system) and of the elapsed time is called Steady State condition of the system

What do you mean by Balking, Reneging, Jockeying?BalkingIf a customer decides not to enter the queue since it is too long is called BalkingRenegingIf a customer enters the queue but after sometimes loses patience and leaves it is called RenegingJockeyingWhen there are 2 or more parallel queues and the customers move from one queue to another is called Jockeying

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QUEUING MODELS CALCULATE: Average number of customers in the system waiting 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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Commercial Queuing Systems Commercial organizations serving external customers Ex. , bank, ATM, gas stations…

Transportation service systems Vehicles are customers or servers Ex. Vehicles waiting at toll stations and traffic lights, trucks

or ships waiting to be loaded, taxi cabs, fire engines, elevators, buses …

Business-internal service systems Customers receiving service are internal to the organization

providing the service Ex. Inspection stations, conveyor belts, computer support …

Social service systems Ex. Judicial process, hospital, waiting lists for organ

transplants or student dorm rooms …

Examples of Real World Queuing Systems?

Prabhakar Car Wash

enter exit

Population ofdirty cars

Arrivalsfrom thegeneral

population …

Queue(waiting line)

Servicefacility Exit the system

Exit the systemArrivals to the system In the system

Arrival Characteristics•Size of the population•Behavior of arrivals•Statistical distribution of arrivals

Waiting Line Characteristics•Limited vs. unlimited•Queue discipline

Service Characteristics•Service design•Statistical distribution of service

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The calling population The population from which customers/jobs

originate The size can be finite or infinite (the latter is

most common) Can be homogeneous (only one type of

customers/ jobs) or heterogeneous (several different kinds of customers/jobs)

The Arrival Process Determines how, when and where customer/jobs

arrive to the system Important characteristic is the customers’/jobs’

inter-arrival times To correctly specify the arrival process requires

data collection of inter arrival times and statistical analysis.

Components of a Basic Queuing Process (II)

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The queue configuration Specifies the number of queues

Single or multiple lines to a number of service stations

Their location Their effect on customer behavior

Balking and reneging Their maximum size (# of jobs the queue can

hold) Distinction between infinite and finite

capacity

Components of a Basic Queuing Process (III)

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The Service Mechanism Can involve one or several service facilities with one or several

parallel service channels (servers) - Specification is required The service provided by a server is characterized by its service

time Specification is required and typically involves data

gathering and statistical analysis. Most analytical queuing models are based on the assumption

of exponentially distributed service times, with some generalizations.

The queue discipline Specifies the order by which jobs in the queue are being

served. Most commonly used principle is FIFO. Other rules are, for example, LIFO, SPT, EDD… Can entail prioritization based on customer type.

Components of a Basic Queuing Process (IV)

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A Commonly Seen Queuing Model (I)

C C C … CCustomers (C)

C S = Server

C S

•

•

•

C S

Customer =C

The Queuing System

The Queue

The Service Facility

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Service times as well as inter arrival times are assumed independent and identically distributed If not otherwise specified

Commonly used notation principle: (a/b/c):(d/e/f)

a = The inter arrival time distribution b = The service time distribution c = The number of parallel servers d= Queue discipline e = maximum number (finite/infinite) allowed in the system f = size of the calling source(finite/infinite)

Commonly used distributions M = Markovian (exponential/possion) –arrivals or departurs distribution

Memoryless D = Deterministic distribution G = General distribution

Example: M/M/c Queuing system with exponentially distributed service and inter-arrival

times and c servers

A Commonly Seen Queuing Model (II)

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Example – Service Utilization Factor

• Consider an M/M/1 queue with arrival rate = and service intensity =

• = Expected capacity demand per time unit• = Expected capacity per time unit

μλ

CapacityAvailableDemandCapacity

ρ

*cCapacityAvailable

DemandCapacity

• Similarly if there are c servers in parallel, i.e., an M/M/c system but the expected capacity per time unit is then c*

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The state of the system = the number of customers in the system

Queue length = (The state of the system) – (number of customers being served)

n =Number of customers/jobs in the system at time t

Pn(t) =The probability that at time t, there are n customers/jobs

in the system.

n =Average arrival intensity (= # arrivals per time unit) at n

customers/jobs in the system

n =Average service intensity for the system when there are n customers/jobs in it.

=The utilization factor for the service facility. (= The expected

fraction of the time that the service facility is being used)

Terminology and Notation

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Pn = The probability that there are exactly n customers/jobs in the system (in steady state, i.e., when t)

L = Expected number of customers in the system (in steady state)

Lq = Expected number of customers in the queue (in steady state)

W = Expected time a customer spends in the system

Wq= Expected time a customer spends in the queue

Notation For Steady State Analysis

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Assume that n = and n = for all n

Assume that n is dependent on n

Little’s Formula

WL qq WL

WL qq WL

0nnnPLet

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Assumptions - the Basic Queuing Process Infinite Calling Populations

Independence between arrivals The arrival process is Poisson with an expected arrival

rate Independent of the number of customers currently in the

system The queue configuration is a single queue with

possibly infinite length No reneging or balking

The queue discipline is FIFO The service mechanism consists of a single server

with exponentially distributed service times = expected service rate when the server is busy

The M/M/1 - model

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n= and n = for all values of n=0, 1, 2, …

The M/M/1 Model

0

1 nn-1

2 n+1

L=/(1- ) Lq= 2/(1- ) = L-

W=L/=1/(- ) Wq=Lq/= /( (- ))

Steady State condition: = (/) < 1

Pn = n(1- )P0 = 1- P(nk) = k

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Situation Patients arrive according to a Poisson process with

intensity ( the time between arrivals is exp() distributed.

The service time (the doctor’s examination and treatment time of a patient) follows an exponential distribution with mean 1/ (=exp() distributed)

Þ The SMS can be modeled as an M/M/c system where c=the number of doctors

Example – SMS Hospital

Data gatheringÞ = 2 patients per hourÞ = 3 patients per hour

Questions– Should the capacity be increased from 1 to 2

doctors?– How are the characteristics of the system (, Wq, W,

Lq and L) affected by an increase in service capacity?

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Interpretation To be in the queue = to be in the waiting room To be in the system = to be in the ER (waiting or under treatment)

Is it warranted to hire a second doctor ?

Summary of Results – SMS Hospital

Characteristic One doctor (c=1) Two Doctors (c=2) 2/3 1/3

P0 1/3 1/2

(1-P0) 2/3 1/2

P1 2/9 1/3

Lq 4/3 patients 1/12 patients

L 2 patients 3/4 patients

Wq 2/3 h = 40 minutes 1/24 h = 2.5 minutes

W 1 h 3/8 h = 22.5 minutes

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In steady state the following balance equation must hold for every state n (proved via differential equations)

Generalized Poisson queuing model

The Rate In = Rate Out Principle:

Mean entrance rate = Mean departure rate

0ii 1P

• In addition the probability of being in one of the states must equal 1

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0011 PP

State Balance Equation

0

1

n

11112200 PPPP

nnn1n1n1n1n P)(PP

Generalized Poisson queuing model

01

01 PP

1

2

12 PP

1nn

1nn PP

11PP:ionNormalizat0i 321

210

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10

1

00i

C0 C2

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Steady State Probabilities

Expected Number of customers in the System and in the Queue Assuming c parallel servers

Steady State Measures of Performance

10P 0PP nn

0n

nPnL

cn

iq PcnL )(

COMPONENTS OF A QUEUING SYSTEM

Arrival Process

ServersQueue or Waiting Line

Service Process

Exit

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The M/M/c Model (I)

1c1c

0n

n

0 )c/((11

!c)/(

!n)/(

P

,2c,1cnforPc!c

)/(

c,,2,1nforP!n)/(

P

0cn

n

0

n

n

0

2 (c-1)

c

1 cc-2

2 c+1

c

c-1

(c-2)

• Generalization of the M/M/1 model– Allows for c identical servers working independently from

each other

Steady State Condition:

=(/c)<1

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W=Wq+(1/)

Little’s Formula Wq=Lq/

The M/M/c Model (II)

02

c

cnnq P

)1(!c

)/(...P)cn(L

• A Condition for existence of a steady state solution is that = /(c) <1

Little’s Formula L=W= (Wq+1/ ) = Lq+ /

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An M/M/c model with a maximum of K customers/jobs allowed in the system If the system is full when a job arrives it is denied

entrance to the system and the queue. Interpretations

A waiting room with limited capacity (for example, the ER at County Hospital), a telephone queue or switchboard of restricted size

Customers that arrive when there is more than K clients/jobs in the system choose another alternative because the queue is too long (Balking)

The M/M/c/K – Model (I)

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The state diagram has exactly K states provided that c<K

The general expressions for the steady state probabilities, waiting times, queue lengths etc. are obtained through the balance equations as before (Rate In = Rate Out; for every state)

The M/M/c/K – Model (II)

0

2 (c-1)

c

1 K-1c-1

2 Kc cc

3

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An M/M/c model with limited calling population, i.e., N clients

A common application: Machine maintenance c service technicians is responsible for keeping N service

stations (machines) running, that is, to repair them as soon as they break

Customer/job arrivals = machine breakdowns Note, the maximum number of clients in the system = N

Assume that (N-n) machines are operating and the time until breakdown for each machine i, Ti, is exponentially distributed (Tiexp()). If U = the time until the next breakdown

Þ U = Min{T1, T2, …, TN-n} Uexp((N-n))).

The M/M/c//N – Model (I)

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• The State Diagram (c service technicians and N machines)– = Arrival intensity per operating machine– = The service intensity for a service technician

• General expressions for this queuing model can be obtained from the balance equations as before

The M/M/c//N – Model (II)

0 N (N-1) (N-(c-1))

2 (c-1)

c

1 N-1c-1

2 Nc c3

Reference:-

1. Operations research: an introduction

By:- Hamdy A. Taha2 . Operations research By:-P.Sankar Iyar