Queuing Theory

AMIT SINGH DAHALG5638545

Outline:

IntroductionCharacteristics of Queuing SystemConfiguration of Service SystemSoftware for SimulationLimitation of Queuing Theory

Introduction: a waiting line of "customers" requiring service from one or more

servers

Example: -Waiting for service in bank and at reservation counter -Waiting for a train or bus

-Waiting at doctor’s clinic

when the short term demand for service exceeds the capacity of facilities

QT, mathematical study of waiting lines (or queues)

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

Characteristics of Queuing System:

1. The Arrival Pattern

2. The Service Mechanism

3. The Queue Discipline

4. Number of Service Channels

1. The Arrival Pattern: Arrivals can be measured as the arrival rate or the inter-arrival

time(time between arrivals)

 Inter-arrival time =1/ arrival rate

(i) Balking: The customer may decide not to enter the queue upon arrival, because it is too long.

(ii) Reneging: The customer may decide to leave the queue after waiting a certain time in it.

(iii) Jockeying: If there are multiple queues in parallel the customers may switch between them

2. Service Mechanism: Service pattern can be deterministic or stochastic Can also be batched or bulked services service rate may be state-dependent Note: difference in Arrival pattern and Service

pattern because services cannot be initiated if the Queue is empty

3. Queue Discipline Vital, the way of selecting customers for services

Can be:i)First in First Out (FIFO)

Ii)Last in First Out (LIFO)

iii)Service in Random Order (SIRO)

iv)Priority Schemes. Priority schemes are :

-Preemptive:customer with higher priority displaces the

customer with low priority and get services-Non-preemptive:

customer with the higher priority wait for the current service to be completed before getting the service

4. Number of Service Channels: Single Queue:

-have only one queue to get different services

-customer get services based on the availability of the server

4. Number of Service Channels(Contd…) Parallel Queue

-have multiple queues to get different services

-the customer can change the queue, if the number of customer deceases in the other queue(Jockeying)

Configuration of Service System Single Server-Single Queue:

-Example: student standing in the library queue

Parallel Server-Single Queue:-Example: Bank transaction, many token numbers are served one-by-one according to available Server

Several Server-Several Queue-Example: different cash counters in electricity office

Services facilities in a series:-Example: may be product manufacturing that requires steps like cutting, drilling, grinding, operations, packaging

Software for Simulation: Model represented using Kendall- Lee Notation

(a/b/c):( d/e/f)where,

a = arrival rate distributionb = service rate distributionc = no. of parallel service channels

(identical service)d = service disciplinee = maximum no. allowed in the systemf = calling source

Common notation:M =Poissonl/Exponential ratesG = General Distribution of Service TimeEk-=Erlangian Distribution

Information Flows:

Queuing Model

Lq

λ

n

μ

L

Wq

W

Pn

ρ

Notation:

λc = effective mean arrival rate λ = λc if queue is infinite λe = λ - [expected number who balk if the queue is finite] W = expected waiting time of a customer in the system Wq = expected waiting time of a customer in the queue L = expected no. of customers in the system Lq = expected number of customers in the queue Po = probability of no customers in the system Pn = probability of n customers in the system ρ = traffic intensity= λ/nμρc= effective traffic intensity= λe/μ

MULTIPLE CHANNEL-POISSON/EXPONENTIAL MODEL [(M/M/C):(GD/ α /α)] 

Characteristics:

1. Input population is infinite.

2. Arrival rate has a Poisson Distribution

3. There is more than one server.

4. Service time is exponentially distributed with mean1/μ. [λ<μ]

5. System capacity is infinite.

6. Balking and reneging are not allowed.

Queuing Theory Limitation

Queuing models are quite complex and cannot be easily understood

Form of theoretical distribution applicable to given queuing situation is not easily known

If the queuing discipline is not FIFO based, then the study of queuing problem become more difficult

References: DEMO: https://www.stat.auckland.ac.nz/~

stats255/qsim/qsim.html

http://homepages.inf.ed.ac.uk/jeh/Simjava/queueing/mm1_q/mm1_q.html

http://en.wikipedia.org/wiki/Queueing_theory

Zukerman, Moshe: Introduction to Queueing Theory and Stochastic Teletraffic Models.

http://homes.cs.washington.edu/~lazowska/qsp/

http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html

http://www.eventhelix.com/realtimemantra/congestioncontrol/queueing_theory.htm

THE ENDAny Questions???