QUEUING THEORY SOORYA PRAVEEN NAIR ROLL NO. 11 EC 2009 ME-EXTC RAIT

ContentsIntroductionDefinitionsApplication Delay analysisLittles theoremBasic queuing modelsQueuing model classificationsQueuing system variableBasic multiplexer model- M/M/1/KEffect of scale on performanceAverage packet delay in a networkGeneral multiplexer model- M/G/1M/M/C/C model and Erlangs B formulaReferences .

1.IntroductionQueue Customers that have arrived at server but are waiting for their service to start are in the queue.Queuing theory- -a collection of mathematical models of various queuing system. - make an analytical model of customers needing service, and use that model to predict queue length and waiting timesWhy study Queuing Theory? Basic tool for network performance evaluation. Help to manage the network traffic towards performance

2. Definitions (Bose) the basic phenomenon of queuing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.

(Wolff) The primary tool for studying these problems [of congestions] is known as queuing theory.

(Kleinrock) We study the phenomena of standing, waiting, and serving, and we call this study Queuing Theory." "Any system in which arrivals place demands upon a finite capacity resource may be termed a queuing system.

3.Applications of Queuing TheoryTelecommunicationsTraffic control Determining the sequence of computer operationsPredicting computer performanceHealth services (e.g.. control of hospital bed assignments)Airport traffic, airline ticket salesLayout of manufacturing systems.

4. Delay analysis A basic model for a delay/loss system:

Time spent in system = T No. customers in system = N(t) Fraction of arriving customers that are lost or blocked = Pb Long term arrival rate = Average no of messages/second that pass through = throughput

Key system variables

Arrival rates and traffic loads

5.Littles theorem

The average number of customers in a stable system (over some time interval) is equal to their average arrival rate multiplied by their average time in the system

E[N] = E[T] (without blocking)

E[N] = (1 Pb) E[T] (with blocking)

Application of Littles theorem

Consider an entire network of queuing systems

By littles formula we are finding the average delay experienced by a packet in traversing a packet -switching network.

6. Basic Queuing Models

Arrival process

Service times

Resources are denoted by servers because their function is to serve customer requests

The time required to service a customer is called the service time, X

Processing capacity, m, is given by m= 1/E[X] customers/s

7.Queuing model classification

8.Queuing system variables

By Littles theorem : average number in the system and average delay in the system are related by E[N] =(1-pb) E[T] average number of customers in queue and average waiting time are related by E[Nq ] =(1-pb) E[W] average number of customers in service and average service time are related by E[Ns] =(1-pb) E[X]

Offered(traffic) load a= l E[X] = L/M Erlangs - rate at hich work arrives at the system

Carried load = E[X]L(1-Pb)=l/m(1-Pb) = a(1-Pb) -average rate at which system does work

Utilisation r=E[Ns] /c= l/cm (1-Pb) - Average fraction of servers that are in use

9.M/M/1/KBasic Multiplexer Model

Average packet length is E[L] bits/packet,Transmission line speed is R bits/second,Mean packet processing rate is m= R/E[L] packets/secondMaximum K packets are allowed in the system before the buffer overflows (K1in buffer and 1in processor) If > m the system will loose packetsIf < m the system may loose packets due to occasional surges in arrivals or long consecutive service times (long packets)

. M/M/1/K

M/M/1/K

.M/M/1/K

.M/M/1/K

.M/M/1/K

m/m/1/

10. Effect of scale on performance

11.Average Packet Delay in a Network

12. M/G/1General Model

Pollaczek-khinchin formula

..M/G/1General Model

We can use the above to model various multiplexing schemes in networks.

13.M/M/C/C model and Erlangs B formula

14. References

1.Kleinrock, L., Communication Nets: Stochastic Message Flow and Delay, McGraw-Hill, New York, 19642.Cohen, J.W., The Single Server Queue, North-Holland, Amsterdam, 19693.Kleinrock, L., Queueing Systems, Vol. I: Theory, Wiley-Interscience, New York, 19754.Kleinrock, L., Queueing Systems, Vol. II: Computer Applications, Wiley-Interscience, New York, 19765.Betserkas D. and Gallager R., Data Networks, Prentice-Hall, Englewood Cliffs, NJ, 19876.Leon-Garcia A. and Widjaja I., Communication Networks: Fundamental Concepts and Key Architectures, McGraw-Hill, New York, 2004