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On the program of the spectral method for On the program of the spectral method for computing the stationary probability vector for a computing the stationary probability vector for a

BMAP/G/1 queueBMAP/G/1 queue

Shoichi NishimuraShoichi NishimuraNaohiko YatomiNaohiko Yatomi

Department of Mathematical Information ScienceDepartment of Mathematical Information ScienceTokyo University of ScienceTokyo University of Science

JapanJapan

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BMAP/G/1 by the spectral methodBMAP/G/1 by the spectral methodPurposePurpose To release the program of the spectral method for computing the To release the program of the spectral method for computing the

stationary probability vector for a BMAP/G/1 queuestationary probability vector for a BMAP/G/1 queue

The spectral methodThe spectral method One of analytical methods introduced in [5]One of analytical methods introduced in [5]

Application of a BMAPApplication of a BMAP A BMAP captures characteristics of real IP traffic in [4]A BMAP captures characteristics of real IP traffic in [4]

Websites [6]Websites [6] http://www.rs.kagu.tus.ac.jp/bmapq/http://www.rs.kagu.tus.ac.jp/bmapq/ http://www.astre.jp/bmapq/http://www.astre.jp/bmapq/

In figures...In figures...

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BMAP/G/1 by the spectral methodBMAP/G/1 by the spectral method

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DefinitionsDefinitions MM the size of the underlying Markov process the size of the underlying Markov process the transition rate matrix with an arrival of batch size the transition rate matrix with an arrival of batch size kk the z-transform ofthe z-transform of

the traffic intensitythe traffic intensity a distribution function of the service time with meana distribution function of the service time with mean

the boundary vectorthe boundary vector the stationary probability vectorthe stationary probability vector

inverse Fast Fourier Transforminverse Fast Fourier Transform

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Spectral method for the vector Spectral method for the vector gg Theorem 1Theorem 1 ([5]) ([5]) 　　 There are M zeros There are M zeros of of

in , wherein , where

Theorem 2Theorem 2 ([5])([5])

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Double for-loop iterationDouble for-loop iteration an increasing sequencean increasing sequence the zeros of inthe zeros of in

is directly obtained !

The modified Durand-Kerner (D-K) methodThe modified Durand-Kerner (D-K) method

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stationary probability vectorstationary probability vector a sufficiently large integer such that is negligiblea sufficiently large integer such that is negligible the the NNth root of the unityth root of the unity

Proposition 4Proposition 4 ([5]) ([5])

(inverse Fast Fourier Transform)(inverse Fast Fourier Transform)

(spectral resolution)(spectral resolution)

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ProgramProgramSome functions to realize various purposes of researchersSome functions to realize various purposes of researchers a constant service a constant service oror a gamma distribution a gamma distribution just after service completion epochs just after service completion epochs oror at arbitrary time at arbitrary time the stationary probability vector the stationary probability vector oror only the stationary probability only the stationary probability

Programming LanguageProgramming Language Decimal BASICDecimal BASIC

double precisiondouble precision

graphical observationsgraphical observations

easy treatment of complex numberseasy treatment of complex numbers

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Main ideasMain ideas Idea 1Idea 1.. (Reduction of computational time and amount of memory) (Reduction of computational time and amount of memory)

Dx( , ,0) Dx( , ,1) Dx( , ,2) Dx( , ,3) Dx( , ,4)Dx( , ,0) Dx( , ,1) Dx( , ,2) Dx( , ,3) Dx( , ,4)

batch(0)=0 batch(1)=1 batch(2)=10 batch(3)=100 batch(4)=1000batch(0)=0 batch(1)=1 batch(2)=10 batch(3)=100 batch(4)=1000

Idea 2.Idea 2. (Increasing the stability of the iteration) (Increasing the stability of the iteration)

cf. [1] O. Aberthcf. [1] O. Aberth

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Idea 3Idea 3.. (Reduction of computational time) (Reduction of computational time)

In most loops, we escape from the loop if all intermediate values hardly In most loops, we escape from the loop if all intermediate values hardly move from the previous values.move from the previous values.

Idea 4Idea 4.. (Keeping stability of the iteration) (Keeping stability of the iteration) some some ss : computational error / iteration error : computational error / iteration error

the same the same ss : Set and compute again. : Set and compute again.

Ignore all the computation at that Ignore all the computation at that ss and go to the next and go to the next s.s.

Main ideasMain ideas

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Traffic data available on WIDE projectTraffic data available on WIDE project(http://www.wide.ad.jp/wg/mawi/) ; the record of Feb, 28th ,2004(http://www.wide.ad.jp/wg/mawi/) ; the record of Feb, 28th ,2004

Comparison of a BMAP and raw IP traffic:Comparison of a BMAP and raw IP traffic: Arrivals per unit time, the stationary probability of a queueing Arrivals per unit time, the stationary probability of a queueing

model.model.

Numerical exampleNumerical example

For For MM=9, rate matrices are =9, rate matrices are estimated by the estimated by the EM algorithmEM algorithm..

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IP traffic （ unit time 0.001sec.） BMAP（ unit time 0.001sec.）

IP traffic （ unit time 0.01sec.） BMAP（ unit time 0.01sec.）

Arrivals per unit timeArrivals per unit time

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IP traffic （ unit time 0.1sec.） BMAP（ unit time 0.1sec.）

IP traffic （ unit time 1sec.） BMAP（ unit time 1sec.）

Arrivals per unit timeArrivals per unit time

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IP trafficIP traffic BMAP/BMAP/

D/1D/1IP trafficIP traffic

BMAP/BMAP/D/1D/1

meanmean 1539.81539.8 1554.51554.5 c.vc.v 1.0741.074 0.9710.971

s.d.s.d. 1654.51654.5 1510.81510.8 P(idle)P(idle) 8.3E-48.3E-4 9.2E-49.2E-4

IP trafficIP trafficBMAP/BMAP/

D/1D/1IP trafficIP traffic

BMAP/BMAP/D/1D/1

meanmean 3144.23144.2 2711.52711.5 c.vc.v 1.1331.133 1.1471.147

s.d.s.d. 3605.83605.8 2821.82821.8 P(idle)P(idle) 5.6E-45.6E-4 6.1E-46.1E-4

IP trafficIP trafficBMAP/BMAP/

D/1D/1IP trafficIP traffic

BMAP/BMAP/D/1D/1

meanmean 8605.48605.4 8131.88131.8 c.vc.v 0.9210.921 1.0231.023

s.d.s.d. 7929.17929.1 8314.48314.4 P(idle)P(idle) 2.2E-42.2E-4 2.3E-42.3E-4

Stationary probability & StatisticsStationary probability & StatisticsIP traffic

BMAP/D/1

IP traffic

BMAP/D/1

IP traffic

BMAP/D/1

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Conclusions & next problemsConclusions & next problems

Large batch sizesLarge batch sizesEstimation by the EM algorithmEstimation by the EM algorithm

The program for general-purposesThe program for general-purposesGenerality, stability, preciseness Generality, stability, preciseness and computational speedand computational speed

Characteristics of IP trafficCharacteristics of IP traffic - Arrivals per unit time- Arrivals per unit time - Queue length distribution- Queue length distribution

http://www.rs.kagu.tus.ac.jp/bmapq/http://www.rs.kagu.tus.ac.jp/bmapq/http://www.astre.jp/bmapq/http://www.astre.jp/bmapq/

Realize the computation Realize the computation in high precision.in high precision.(ex. Rewriting in C++)(ex. Rewriting in C++)

Next problemNext problem

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We will perform our program later We will perform our program later if there is a request .if there is a request .

Thank you .Thank you .