Macrosystem models of flows in communication-computing networks(GRID-technology)

Yuri S. Popkov

Institute for Systems Analysisof the Russian Academy of Sciences

popkov@isa.ru

GRID — distributed computer

Real-time operation mode

• network as a computer• response time is a random value

which depends on the flows in network

• random delay• random delay depends on flows in

network

A ),( 2111 xxfx B),( 2122 xxfx

])[],[(][])1[(

])[],[(][])1[(

221222

121111

nhxnhxhfnhxhnx

nhxnhxhfnhxhnx

Transportation flows in Moscow traffic system (middle of the day)

T = 25 min

Change of transportation flows in Moscow traffic system (morning)

T = 32 min

Change of transportation flows in Moscow traffic system (evening)

T = 29 min

GRID — Stochastic network — Dynamic system

History

Transportation networks (passanger, cargo)

Pipe-line networks (oil, gas)

Computer networks (Internet, Intranet)

Energy networks

GRID

State

Distribution ofInformation

flows

Stochastic factors

Inertia

Dynamic stochastic network

Macrosystem theory

GRID states

• Spatial distribution of information and computing resources

relaxation time

• Distribution of correspondence flows

relaxation time

)(tX

)(tYr

f

fr

Problems for study

A. Formation of quasi-stationary states of corresponding flows

B. Spatial-temporary evolution of network: interaction between “slow” and “fast” processes in network

GRID phenomenology

Network Correspondences

Flows )(tI

AssignmentMacrostate

)]([)( tytY ij - correspondence flows

Model of quasi-stationary states

Probabilistic characteristics

Time interval

t tt t

Information and computing resources Number of information portions )(tX

Correspondence flows Number of information portions per time unit )(tY

Prior probabilities

ttXAtXB ),(),(

Flows )(tY

Volumes ttYtG )()(

Generalized Boltzmann information entropy

),(ln)(),,(

tXbe

gtgttGH

ij

ijijB

Model of quasi-stationary states

Probabilistic characteristics

Generalized Fermi-Dirac information entropy

)),()(ln())()((),(

~ln),,( tgtCtgtCtXbe

ggttGH ijijijij

ij

ijijF

),(1

),(),(

~

tXb

tXbtXb

ij

ijij

Throughputssijijij EEE ,,, 21

Feasible correspondence flows

mij

sm

trij

ij E

EtC

1

)(

max)(

ijij Ctg )(0Volume of correspondences

Model of quasi-stationary states

Feasible sets

General model

),,(max)](,,,[

ttXDGtXttGH

WгдеtWtg kkij

Mjiij

k

,)(,

j

iiijij njttXtg ,1,)()(

— transmission cost of an information portion for ( i j ) – correspondenceijCost constraints

— transmission cost of an information portion per time unit for i–th resourcei

- demands

Balance constraints

i

jjijij mjtQttqtg ,1),()()(

- throughput constraints

otherwise

karctobelongsencecorrespondjiofroutekij ,0

)(,1

– throughput of k-th arc

I. MQSS for constant capacity of correspondences

II. MQSS for variable capacity of correspondences

III. MQSS for small network loading

Classification of the model of quasi-stationary states (MQSS)

),(max,)),(,( tXDYCtXYHF

rkWy

mjqy

nitXy

ij

kkijij

ijij

iij

ijij

,1

,1

,1)(

:D

),(max,))(),(,,( tXDYtCtXtYHF

ijij

iij

ijij

mjqy

nitXy

,1

,1)(:D

),(max,))(,( tXDYtXYHB

ijij

iij

ijij

mjqy

nitXy

,1

,1)(:D

Illustration of adequacy of the MQSS(transport network)

Dynamic models of stochastic network

Regional structure of network

— volume of computing resources in i-th region (slow variables)

)(tX i

— information flows between regions i and j (fast variables)

)(tyij

)()(0)()(0

tCtYtMtX

)()(0 tCtY or

Change factors of information and computing resources

• ageing (depends on X(t))• renewal (external influence U(t))• information flows (Y(t))

Change factors of information flows

• information and computing resources (X(t))• demand (Q(t))• information flows (Y(t))

Dynamic model

)](),(),([)];(),(),([~

tQtXtYФdt

dYtYtUtXF

dt

dX

А. Resource dynamic

- positiveness

),,(

)(,0)(),,,()(),|(~

,10),|,,,0,,,(~

111

YUXFXdt

dX

FXwhereYUXFXYUXF

niYUXXXXF

iii

niii

- boundedness

};,1),()();()(0:{

0),|(

ijnjtMtXtMtXX

XдляYUXF

iijji

ii

Example:

niYUsXYUbYUXF iiii ,1),(),(),|(

Model types

1. Ageing with constant rate

constbYPXbYXF

i

iiiii

),(

2. Ageing and renewal with constant rate

constbb

YPXUbbYXF

ii

iiiiii

~,

~),(

3. Renewal with constant rate

constb

YPXUbUXF

i

iiiii

~

~),(

P – (m x n) matrix; Pi – i –th row of matrix P; Yi – i –th column of matrix Y;

B. Quasi-stationary states of the information flows distribution

)(),,(max

xDYtXYH

General dynamic model of stochastic network

))(|),,(max(arg),(*

))),(*,()),(*,((

xDYtXYHtXY

tXYUsXtXYUbXdt

dX

Positive dynamic system with entropy operator

Conclusion

GRID-technologyHardware, software, technical tools and etc.

GRID as a systemInformation and computing resources, information

flows, distributed on-line computing

Why it is necessary to studySystem properties of GRID?

Interestingly: new class of dynamic systems

Usefully: active and strategic control, prediction

Tools

Macrosystem modelling

Quasi-stationary states Resources evolution

Entropy maximization modelsModels of dynamic systems with

entropy operator

Numerical methods, sensitivity, smothness

Existing, boundedness, stability