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ODE and Discrete Simulation or Mean Field Methods for Computer and Communication Systems

ODE and Discrete Simulation or Mean Field Methods for Computer and Communication Systems. Jean-Yves Le Boudec EPFL MLQA , Aachen, September 2011. Contents. Mean Field Interaction Model Convergence to ODE Finite Horizon: Fast Simulation and Decoupling assumption

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ODE and Discrete Simulation or Mean Field Methods for Computer and Communication Systems

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  1. ODE and Discrete SimulationorMean Field Methods for Computer and Communication Systems Jean-Yves Le Boudec EPFL MLQA, Aachen, September 2011 1

  2. Contents • Mean Field Interaction Model • Convergence to ODE • Finite Horizon: Fast Simulation and Decouplingassumption • Infinite Horizon: Fixed Point Method and Decouplingassumption 2

  3. 1 Mean Field Interaction Model 3

  4. Mean Field • A model introduced in Physics • interaction between particles is via distribution of states of all particle • An approximation method for a large collection of particles • assumes independence in the master equation • Why do we care in I&C? • Model interaction of manyobjects: • Distributedsystems, communication protocols, gametheory, self-organizedsystems 4

  5. Mean Field Interaction Model • Time is discrete • N objects, N large • Object n has state Xn(t) • (XN1(t), …, XNN(t)) is Markov • Objects are observable onlythroughtheir state • “Occupancy measure”MN(t) = distribution of object states at time t 5

  6. Mean Field Interaction Model • Time is discrete • N objects, N large • Object n has state Xn(t) • (XN1(t), …, XNN(t)) is Markov • Objects are observable onlythroughtheir state • “Occupancy measure”MN(t) = distribution of object states at time t • Theorem[Gast (2011)] MN(t) is Markov • Called “Mean Field Interaction Models” in the Performance Evaluation community[McDonald(2007), Benaïm and Le Boudec(2008)] 6

  7. A Few Examples Where Applied Never again ! E.L. 7

  8. Example: 2-Step Malware Mobile nodes are either `S’ Susceptible `D’ Dormant `A’ Active Time is discrete Nodes meet pairwise (bluetooth) One interaction per time slot, I(N) = 1/N; mean field limit is an ODE State space is finite = {`S’ , `A’ ,`D’} Occupancy measure isM(t) = (S(t), D(t), A(t)) with S(t)+ D(t) + A(t) =1 S(t) = proportion of nodes in state `S’[Benaïm and Le Boudec(2008)] Possible interactions: Recovery D -> S Mutual upgrade D + D -> A + A Infection by active D + A -> A + A Recovery A -> S Recruitment by Dormant S + D -> D + D Direct infection S -> D Direct infection S -> A 8

  9. Simulation Runs, N=1000 nodes Node 1 State = D State = A State = S Node 2 Node 3 D(t) Proportion of nodes In state i=1 A(t) Proportion of nodes In state i=2 9

  10. Sample Runs with N = 1000 10

  11. Example: TCP and ECN ECN Feedback q(R(t)) 1 ECN router n queue length R(t) N N connections • [Tinnakornsrisuphap and Makowski(2003)]At, every time step, all connections update their state: I(N)=1 • Time isdiscrete, meanfieldlimitisalso in discrete time (iteratedmap) • Similarexamples: HTTP Metastability[Baccelli et al.(2004)Baccelli, Lelarge, and McDonald]Reputation System [Le Boudec et al.(2007)Le Boudec, McDonald, and Mundinger] 11

  12. The Importance of Being Spatial • Mobile node state = (c, t)c = 1 … 16 (position) t ∊ R+ (age of gossip) • Time iscontinuous, I(N) = 1 • OccupancymeasureisFc(z,t) = proportion of nodesthatat location c and have age ≤ z[Age of Gossip, Chaintreau et al.(2009)] no class 16 classes Qqplots simulation vs meanfield 12

  13. What can we do with a Mean Field Interaction Model ? Large Nasymptotics, Finite Horizon fluid limit of occupancy measure (ODE) decouplingassumption (fast simulation) Issues When valid How to formulate the fluid limit Large t asymptotic Stationary approximation of occupancy measure Decoupling assumption Issues When valid 13

  14. E. L. 2. Convergence to ode 14

  15. IntensityI(N) • I(N) = expectednumber of transitions per object per time unit • A meanfieldlimitoccurswhenwere-scale time by I(N)i.e. weconsiderXN(t/I(N)) • I(N) = O(1): meanfieldlimitis in discrete time [Le Boudec et al (2007)]I(N) = O(1/N): meanfieldlimitis in continuous time [Benaïm and Le Boudec (2008)] 15

  16. The Mean Field Limit • Under verygeneral conditions (givenlater) the occupancymeasure converges, in law, to a deterministicprocess, m(t),called the meanfieldlimit • Finite State Space => ODE 16

  17. Mean Field Limit N = +∞ Stochastic system N = 1000 17

  18. Sufficient Conditions for Convergence • [Kurtz 1970], seealso [Bordenav et al 2008], [Graham 2000] • Sufficientconditonverifiable by inspection:Example: I(N) = 1/NSecond moment of number of objectsaffected in one timeslot = o(N) • Similarresultwhenmeanfieldlimitis in discrete time [Le Boudec et al 2007] 18

  19. Example: Convergence to Mean Field • Rescale time suchthat one time step = 1/N • Number of transitions per time stepisbounded by 2, thereforethereis convergence to meanfield = = = 19

  20. Formulating the Mean Field Limit • Drift = sum over all transitions ofproba of transition xDelta to system state MN(t) • Re-scale drift by intensity • Equation for meanfieldlimitis dm/dt = limit of rescaled drift • Can beautomatedhttp://icawww1.epfl.ch/IS/tsed drift = = = = 20

  21. Convergence to Mean Field E. L. E.L. • For the finite state space case, there are many simple results, oftenverifiable by inspectionFor example [Kurtz 1970] or [Benaim, Le Boudec 2008] • For the general state space, thingsmaybe more complex(fluidlimitzis not an ODE, e.g. [Chaintreau et al, 2009]) 21

  22. 3. FINITE HORIZON :FastSimulation and Decouplingassumption 22

  23. Convergence to Mean Field Limitis Equivalent to Propagation of Chaos 23

  24. Propagation of Chaos = DecouplingAssumption • (Propagation of Chaos)kobjects are asymptoticallyindependentwithcommonlawequal to the meanfieldlimit, for anyfixedk • (DecouplingAssumption) (alsocalledMean Field Approximation, or Fast Simulation)The law of one objectisasymptotically as if all otherobjectsweredrawnrandomlywith replacement fromm(t) 24

  25. The TwoInterpretations of the Mean Field Limit At any time t Thus for larget : Prob (node n is dormant) ≈ 0.3 Prob (node n is active) ≈ 0.6 Prob (node n is susceptible) ≈ 0.1 m(t) approximatesboth the occupancymeasure MN(t) the state probability for one objectat time t, drawnatrandomamongN 25

  26. « Fast Simulation » • pNj(t|i) is the probabilitythat a nodethatstarts in state i is in state j at time t: • Then wherep(t|i)is a continuous time, non homogeneousprocess • Same ODE as meanfieldlimit, but withdifferent initial condition pdf of node 2 occupancy measure (t) pdf of node 1 pdf of node 3 26

  27. The DecouplingAssumption • The evolution for one object as if the otherobjectshad a state drawnrandomly and independentlyfrom the distribution m(t) • Is validover finite horizon whenevermeanfield convergence occurs • Can beused to analyze or simulateevolution of k objects 27

  28. 4. Infinite Horizon: Fixed Point Method and Decouplingassumption 28

  29. The Fixed Point Method Decouplingassumptionsaysdistribution of prob for state of one objectisapprox. m(t)with We are interested in stationaryregime, i.ewe do This is the « Fixed Point Method » Example: in stationaryregime: Prob (node n is dormant) ≈ 0.3 Prob (node n is active) ≈ 0.6 Prob (node n is susceptible) ≈ 0.1 Nodes m and n are independent 29

  30. Example: 802.11 Analysis, Bianchi’s Formula 802.11 single cell mi = proba one node is in backoff stage I = attempt rate  = collision proba See [Benaim and Le Boudec , 2008] for this analysis ODE for meanfieldlimit Solve for Fixed Point: Bianchi’s Fixed Point Equation [Bianchi 1998] 30

  31. 2-Step Malware, Again Same as before except for one parameter value : h = 0.1 instead of 0.3 The ODE does not converge to a unique attractor (limit cycle) The equationF(m) = 0has a unique solution (red cross) – but itisnot the stationaryregime ! 31

  32. ExampleWhereFixed Point MethodFails In stationaryregime, m(t) = (D(t), A(t), S(t)) follows the limit cycle Assume you are in stationaryregime (simulation has run for a long time) and you observe that one node, say n=1, is in state ‘A’ It is more likelythat m(t) is in region R Therefore, itis more likelythatsomeothernode, say n=2, isalso in state ‘A’ This issynchronization R h=0.1 32

  33. Joint PDFs of TwoNodes in StationaryRegime Stationary point of ODE Mean of Limit of N = pdf of one node in stationary regime pdf of node 2 in stationary regime, given node 1 is A pdf of node 2 in stationary regime, given node 1 is D pdf of node 2 in stationary regime, given node 1 is S 33

  34. Whereis the Catch ? Decouplingassumptionsaysthatnodesm and n are asymptoticallyindependent There ismeanfield convergence for thisexample But wesawthatnodesmay not beasymptoticallyindependent … isthere a contradiction ? 34

  35. Markov chainisergodic • The decouplingassumptionmay not hold in stationaryregime, even for perfectlyregularmodels Mean Field Convergence ≠ mi(t) mj(t) mi(t) mj(t) 35

  36. Result 1: Fixed Point MethodHoldsunder (H) Assume that (H) ODE has a unique global stable point to which all trajectories converge Theorem [e.g. Benaim et al 2008] : The limit of stationary distribution of MNisconcentrated on thisfixed point The decouplingassumptionholds in stationaryregime 36

  37. Here:Birkhoff center = limit cycle  fixed point Theorem in [Benaim] says that the stochastic system for large N is close to the Birkhoff center, i.e. the stationary regime of ODE is a good approximation of the stationary regime of stochastic system Result 2: Birkhoff Center 37

  38. StationaryBehaviour of Mean Field Limitis not predicted by Structure of Markov Chain • MN(t)is a Markov chain on SN={(a, b, c) ≥ 0, a + b + c =1, a, b, c multiples of 1/N} • MN(t)isergodic and aperiodic • Depending on parameter, thereis or is not a limit cycle for m(t) SN(for N = 200) h = 0.3 h = 0.1

  39. Example: 802.11 withHeterogeneousNodes • [Cho et al, 2010]Two classes of nodeswithheterogeneousparameters (restransmissionprobability)Fixed point equation has a unique solution, but thisis not the stationaryprobaThere is a limit cycle 39

  40. Result 3: In the Reversible Case, the Fixed Point MethodAlways Works • DefinitionMarkov Process X(t) on enumerable state E space, with transition rates q(i,j)isreversibleiff 1. itisergodic 2. p(i) q(i,j) = p(j) q(j,i) for somep • Stationary points = fixed points • If processwithfiniteNisreversible, the stationarybehaviourisdeterminedonly by fixed points. 40

  41. A Correct Method • 1. Writedynamical system equationsin transientregime • 2. Study the stationaryregime of dynamical system • if converges to unique stationary point m*thenmakefixed point assumption • elseobjects are coupled in stationaryregime by meanfieldlimitm(t) • Hard to predictoutcome of 2 (except for reversible case) 41

  42. Conclusion • Meanfieldmodels are frequent in large scalesystems • Validity of approachisoften simple by inspection • Meanfieldisboth • ODE for fluidlimit • Fast simulation usingdecouplingassumption • Decouplingassumptionholdsatfinite horizon; may not hold in stationaryregime. • Stationaryregimeis more thanstationary points, in general (except for reversible case) 42

  43. Thank You … 43

  44. References 44

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