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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
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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...
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BMAP/G/1 by the spectral method
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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
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Spectral method for the vector g

Theorem 1 ([5]) There are M zeros
in

of
, where
Theorem 2 ([5])
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Double for-loop iteration


an increasing sequence
the zeros of
in
The modified Durand-Kerner (D-K) method
is directly obtained !
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stationary probability vector




a sufficiently large integer such that
the Nth root of the unity
is negligible
Proposition 4 ([5])
(inverse Fast Fourier Transform)
(spectral resolution)
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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
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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)
cf. [1] O. Aberth
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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.
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Numerical example
Traffic data available on WIDE project
(http://www.wide.ad.jp/wg/mawi/) ; the record of Feb, 28th ,2004
For M=9, rate matrices
are
estimated by the EM algorithm.
Comparison of a BMAP and raw IP traffic:
 Arrivals per unit time, the stationary probability of a queueing
model.
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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.）
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Arrivals per unit time
IP traffic （unit time 0.1sec.）
IP traffic （unit time 1sec.）
BMAP（unit time 0.1sec.）
BMAP（unit time 1sec.）
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Stationary probability & Statistics
IP traffic
BMAP/
D/1
mean
1539.8
1554.5
s.d.
1654.5
1510.8
IP traffic
BMAP/
D/1
mean
3144.2
2711.5
s.d.
3605.8
2821.8
IP traffic
BMAP/
D/1
mean
8605.4
8131.8
s.d.
7929.1
8314.4
IP traffic
IP traffic
BMAP/
D/1
c.v
1.074
0.971
P(idle)
8.3E-4
9.2E-4
IP traffic
BMAP/
D/1
c.v
1.133
1.147
P(idle)
5.6E-4
6.1E-4
IP traffic
BMAP/
D/1
c.v
0.921
1.023
P(idle)
2.2E-4
2.3E-4
BMAP/D/1
IP traffic
BMAP/D/1
IP traffic
BMAP/D/1
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Conclusions & next problems
Large batch sizes
Estimation by the EM algorithm
Characteristics of IP traffic
- Arrivals per unit time
- Queue length distribution
Next problem
The program for general-purposes Realize the computation
Generality, stability, preciseness in high precision.
and computational speed
(ex. Rewriting in C++)
http://www.rs.kagu.tus.ac.jp/bmapq/
http://www.astre.jp/bmapq/
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Thank you .
We will perform our program later
if there is a request .
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