1. Macrosystem models of flows in communication-computing networks (GRID-technology) Yuri S. Popkov Institute for Systems Analysis of the Russian Academy of Sciences popkov@isa.ru

2. GRID — distributed computer

3. 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 AB

4. Transportation flows in Moscow traffic system (middle of the day) T = 25 min

5. Change of transportation flows in Moscow traffic system (morning) T = 32 min

6. Change of transportation flows in Moscow traffic system (evening) T = 29 min

7. GRID — Stochastic network — Dynamic system History Transportation networks (passanger, cargo) Pipe-line networks (oil, gas) Computer networks (Internet, Intranet) Energy networks GRID State Distribution of Information flows Stochastic factors Inertia Dynamic stochastic network Macrosystem theory

8. GRID states Spatial distribution of information and computing resources relaxation time Distribution of correspondence flows relaxation time 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

9. GRID phenomenology NetworkCorrespondences Flows Assignment Macrostate - correspondence flows

10. Model of quasi-stationary states Probabilistic characteristics Time interval Information and computing resources  Number of information portions Correspondence flows  Number of information portions per time unit Prior probabilities Flows Volumes Generalized Boltzmann information entropy

11. Model of quasi-stationary states Probabilistic characteristics Generalized Fermi-Dirac information entropy Throughputs Feasible correspondence flows Volume of correspondences

12. Model of quasi-stationary states Feasible sets General model — transmission cost of an information portion for ( i j ) – correspondence Cost constraints — transmission cost of an information portion per time unit for i –th resource - demands Balance constraints - throughput constraints – throughput of k -th arc

13. 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)

14. Illustration of adequacy of the MQSS (transport network)

15. Dynamic models of stochastic network Regional structure of network — volume of computing resources in i -th region (slow variables) — information flows between regions i and j (fast variables) 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) )

16. Dynamic model А. Resource dynamic - positiveness - boundedness Example:

17. Model types 1. Ageing with constant rate 2. Ageing and renewal with constant rate 3. Renewal with constant rate P – (m x n) matrix; P i – i –th row of matrix P ; Y i – i –th column of matrix Y ; B. Quasi-stationary states of the information flows distribution

18. General dynamic model of stochastic network Positive dynamic system with entropy operator

19. Conclusion GRID-technology Hardware, software, technical tools and etc. GRID as a system Information and computing resources, information flows, distributed on-line computing Why it is necessary to study System properties of GRID? Interestingly: new class of dynamic systems Usefully: active and strategic control, prediction Tools Macrosystem modelling Quasi-stationary statesResources evolution Entropy maximization models Models of dynamic systems with entropy operator Numerical methods, sensitivity, smothness Existing, boundedness, stability