Shoichi Nishimura Naohiko Yatomi Department of Mathematical Information Science

1. On the program of the spectral method for computing the stationary probability vector for a BMAP/G/1 queue Shoichi Nishimura Naohiko Yatomi Department of Mathematical Information Science Tokyo University of Science Japan

2. BMAP/G/1 by the spectral method Purpose • To release the program of the spectral method for computing the stationary probability vector for a BMAP/G/1 queue The spectral method • One of analytical methods introduced in [5] Application of a BMAP • A BMAP captures characteristics of real IP traffic in [4] Websites [6] • http://www.rs.kagu.tus.ac.jp/bmapq/ • http://www.astre.jp/bmapq/ In figures...

3. [5] Method [4] Application [6] Program BMAP/G/1 by the spectral method

4. Definitions • M the size of the underlying Markov process • the transition rate matrix with an arrival of batch size k • the z-transform of • the traffic intensity • a distribution function of the service time with mean • the boundary vector • the stationary probability vector inverse Fast Fourier Transform

5. Spectral method for the vector g • Theorem 1([5]) There are M zeros of in , where • Theorem 2([5])

6. Double for-loop iteration • an increasing sequence • the zeros of in The modified Durand-Kerner (D-K) method is directly obtained !

7. stationary probability vector • a sufficiently large integer such that is negligible • the Nth root of the unity • Proposition 4 ([5]) (inverse Fast Fourier Transform) (spectral resolution)

8. Program Some functions to realize various purposes of researchers • a constant service or a gamma distribution • just after service completion epochs or at arbitrary time • the stationary probability vector or only the stationary probability Programming Language • Decimal BASIC double precision graphical observations easy treatment of complex numbers

9. cf. [1] O. Aberth Main ideas • Idea 1. (Reduction of computational time and amount of memory) Dx( , ,0) Dx( , ,1) Dx( , ,2) Dx( , ,3) Dx( , ,4) batch(0)=0 batch(1)=1 batch(2)=10 batch(3)=100 batch(4)=1000 • Idea 2. (Increasing the stability of the iteration)

10. Main ideas Idea 3. (Reduction of computational time) In most loops, we escape from the loop if all intermediate values hardly move from the previous values. Idea 4. (Keeping stability of the iteration) • some s : computational error / iteration error • the same s : Set and compute again. • Ignore all the computation at that s and go to the next s.

11. Numerical example Traffic data available on WIDE project (http://www.wide.ad.jp/wg/mawi/) ; the record of Feb, 28th ,2004 Comparison of a BMAP and raw IP traffic: • Arrivals per unit time, the stationary probability of a queueing model. For M=9, rate matrices are estimated by the EM algorithm.

12. Arrivals per unit time IP traffic （unit time 0.001sec.） BMAP（unit time 0.001sec.） IP traffic （unit time 0.01sec.） BMAP（unit time 0.01sec.）

13. Arrivals per unit time IP traffic （unit time 0.1sec.） BMAP（unit time 0.1sec.） IP traffic （unit time 1sec.） BMAP（unit time 1sec.）

14. IP traffic BMAP/D/1 IP traffic BMAP/D/1 IP traffic BMAP/D/1 Stationary probability & Statistics

15. [5] Method [4] Application • Large batch sizes • Estimation by the EM algorithm • Characteristics of IP traffic - Arrivals per unit time - Queue length distribution • The program for general-purposes • Generality, stability, preciseness and computational speed [6] Program Conclusions & next problems Next problem Realize the computation in high precision. (ex. Rewriting in C++) • http://www.rs.kagu.tus.ac.jp/bmapq/ • http://www.astre.jp/bmapq/

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