Upload
lavinia-combs
View
22
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Mean field models of interacting objects: fluid equations, independence assumptions and pitfalls. Jean-Yves Le Boudec EPFL October 2009. Abstract. We consider a generic model of N interacting objects, where each object has a state and interaction between objects is Markovian, - PowerPoint PPT Presentation
Citation preview
1
Mean field models of interacting objects:
fluid equations,independence assumptions and pitfalls
Jean-Yves Le Boudec
EPFL
October 2009
2
AbstractWe consider a generic model of N interacting objects, where eachobject has a state and interaction between objects is Markovian,i.e. the evolution of the system depends only on the collection ofstates at any point in time. This is quite a general modelingframework, which was successfully applied to model many forms ofcommunication protocols. When the number of objects N is large, oneoften uses simplifying assumptions called "mean fieldapproximation", "fluid approximation", "fixed point method" or"decoupling assumption". In this tutorial we explain the meaning ofthese four concepts and show that the first two, namely mean fieldapproximation and fluid approximation, are generally valid. However,we also show that the last two, namely fixed point method anddecoupling assumption, require more care, as they may not be valid,even in simple cases. We give sufficient conditions under which theyare valid. We illustrate the concepts with the analysis of the802.11 WiFi protocol.
This slide show is available on my home page under “talks” or directly at http://ica1www.epfl.ch/PS_files/lebSlides.htm
3
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
4
A Simple Model of Interacting ObjectsTime is discrete
N objects
Object n has state Xn(t) 2 {1,…,I}
(X1(t), …, XN(t)) is Markov
Objects can be observed only through their state
N is large, I is small
Called “Mean Field Interaction Models” in the Performance Evaluation community (But mean field has other meanings in physics, see later)
Example 1: N wireless nodes, state = retransmission stage k
Example 2: N wireless nodes, state = k,c (c= node class)
Example 3: N wireless nodes, state = k,c,x (x= node location)
5
Example: 2-step malware propagation
Mobile nodes are either`S’ Susceptible`D’ Dormant`A’ Active
3 states
N nodes
Nodes meet pairwise (bluetooth)
Possible interactions:
1. RecoveryD -> S
2. Mutual upgrade D + D -> A + A
3. Infection by activeD + A -> A + A
4. RecoveryA -> S
5. Recruitment by DormantS + D -> D + D
6. Direct infectionS -> A
6
Details of the ExampleTo specify the previous model entirely, ( = to be able to simulate it), we need to specify the transition matrixIn a compact form, we define probas of each type of transition.
1. RecoveryD -> S
2. Mutual upgrade D + D -> A + A
3. Infection by activeD + A -> A + A
4. RecoveryA -> S
5. Recruitment by Dormant
S + D -> D + D6. Direct infection
S -> A
Simulation algorithm:At every time step
Pick one case with prob as given in table
sum of probs is less than 1, possible to do nothing at one step(case 1) Pick one node uniformly at random among all nodes that are in state ‘D’(case 2) Pick one pair of nodes uniformly at random among all pairs nodes that are in state ‘D’(case 3) Pick one node among ‘A’ nodes, and one among ‘D’ nodes, each uniformly at randometcS, D, A are the numbers of nodes in state `S’, `D’, `A’
A(t)Proportion of nodes In state i=2
7
Simulation Runs, N=1000 nodesNode 1
Node 2
Node 3
D(t)Proportion of nodes In state i=1
State = DState = AState = S
8
Sample Runs with N = 1000
Simplified Analysis 1 :Decoupling Assumption (Transient Regime)
9
1. RecoveryD -> S
2. Mutual upgrade D + D -> A + A
3. Infection by activeD + A -> A + A
4. RecoveryA -> S
5. Recruitment by Dormant
S + D -> D + D6. Direct infection
S -> A
Simplified Analysis 2Decoupling Assumption (Stationary
Regime)
Solve for (D,A,S)
Has a unique solution
10
1. RecoveryD -> S
2. Mutual upgrade D + D -> A + A
3. Infection by activeD + A -> A + A
4. RecoveryA -> S
5. Recruitment by Dormant
S + D -> D + D6. Direct infection
S -> A
11
Issues
When is decoupling assumption valid ?
How to formulate the ODE ?
Is stationary regime of ODE an approximation of stationary regime of original system ?
12
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
13
Scaling Assumptions
We want to simplify the model for large N, we need scaling assumptions
Let WN(t) be the (random) number of objects that do a transition at time slot t when there are N objects
Informally, the main scaling assumptions are:The expectation of WN(t) tends to a constant as N growsThe second moment of WN(t) remains bounded as N grows
i.e., for large N, the probability that this object makes a transition is O(1/N)
This is equivalent to a time scale assumption: the time slot duration is O(1/N)
14
Formal Statements
Definition: Occupancy MeasureMN
i(t) = fraction of objects in state i at time tExample: MN(t) = (D(t), A(t), S(t))
Definition: drift = expected change to MN(t) in one time slot
The scaling assumptions are:
15
Writing the Drift Without Error
Drift = sum over all transitions of
proba of transition£Delta to the system state MN(t)
Can be automated
http://icawww1.epfl.ch/IS/tsed
16
Example
17
Fluid Approximation Theorem
Under the scaling assumption:
stochastic system MN(t) can be approximated by fluid limit (t), solution of the ODE:
Rescaled drift of MN(t)
18
Example
19
Fluid limitN = +1
Stochastic system
N = 1000
20
Fluid Approximation Theorem
Definition: Re-Scaled Occupancy measure
[Benaïm, L] :
21
Computing the Mean Field Limit
Compute the drift of MN and its limit over intensity
22
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
23
Propagation of Chaos
Convergence to an ODE implies “propagation of chaos” [Sznitman, 1991]
This says that, for large N, any k objects are ≈ independentJustifies Decoupling Assumption (transient)
mean field limit
24
Mean Field Independence
At any time tk nodes are asymptotically independent
Thus for large t :Prob (node n is dormant) ≈ 0.3Prob (node n is active) ≈ 0.6 Prob (node n is susceptible) ≈ 0.1
25
Fast Simulation Result
A stronger result than propagation of chaos – does not require exchangeability
[Tembine, L et al], 2009
Assume we know the state of object n at time 0; we can approximate its evolution by
Replacing all other objects collectively by the ODE
The state of object n is a jump process, with transition matrix driven by the ODE
PNi,j (m) is the transition probability for one object, given that the state of the system
is m
Note: Knowing the transition matrix PN (m) is not enough to be able to simulate (or
analyze) the system with N objectsBecause there may be simultaneous transitions of several objects (on the example, up to 2)
However, the fast simulation says that, in the large N limit, we can consider one (or k) objects as if they were independent of the other N-k
(XN1(t/N), MN(t/N)) can be approximated by the process (X1(t), m(t)) where m(t)
follows the ODE and X1(t) is a jump process with time-dependent transition matrix A(m(t)) where
Example
26
The state of one object is a jump process with transition matrix:
where m = (D, A, S) depends on time (is solution of the ODE)
27
AN
Example
28
pdf of node 1
pdf of node 2
pdf of node 3
occupancy measure(t)
ODEs
Let pNj(t|i) be the probability that a node that starts in state i is
in state j at time t:
The fast simulation result says that
With the ODEs:
Computing the Transition Probability
PNi,j (m) is the transition probability for one object, given that
the state if m
30
31
The Mean Field Approximation
Common in Physics
Consists in pretending that XNm(t), XN
n(t) are independent in the time evolution equation
It is asymptotically true for large N, at fixed time t, for our model of interacting objects
Also called “decoupling assumption” (in computer science)
32
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
33
Stationary Regime
Original system (stochastic):(XN(t)) and (MN(t)) are Markov, finite state space, discrete timeAssume either one is irreducible, thus has a unique stationary proba N
For large N, how does N relate to the stationary regime of the ODE ?
Law of MN(t) N
(t) ???
t -> +1
N -> +1
Assume (H) the ODE has a unique stable point m* to which all trajectories converge
Theorem Under (H)
i.e.
(1) m* is the limit of N for large N [N = stat. prob. of (X1N(t),
…, XNN(t) ]
and
(2) decoupling assumption holds in stationary regime
34
Under (H), Decoupling Assumption Holds In Stationary Regime
35
Example
In stationary regime:
Prob (node n is dormant) ≈ 0.3Prob (node n is active) ≈ 0.6 Prob (node n is susceptible) ≈ 0.1
Nodes m and n are independent
We are in the good case: the diagram commutes
Law of MN(t) N
(t) m*
t -> +1
N -> +1
t -> +1
N -> +1
36
Counter-Example
The ODE does not converge to a unique attractor (limit cycle)
Assumption H does not hold; does the decoupling assumption still hold ?
Same as beforeExcept for one
parameter value
h = 0.1 instead of 0.3
37
Decoupling Assumption Does Not Hold HereIn Stationary Regime
In stationary regime, m(t) = (D(t), A(t), S(t)) follows the limit cycle
Assume you are in stationary regime (simulation has run for a long time) and you observe that one node, say n=1, is in state ‘A’
It is more likely that m(t) is in region R
Therefore, it is more likely that some other node, say n=2, is also in state ‘A’
This is synchronization
R
38
Numerical Example
Mean of Limit of N = pdf of one node in stationary regime
Stationary point of ODE
pdf of node 2 in stationary regime, given node 1 is D
pdf of node 2 in stationary regime, given node 1 is S
pdf of node 2 in stationary regime, given node 1 is A
Simplified Analysis 2Decoupling Assumption (Stationary
Regime)
Solve for (D,A,S)Has a unique solution
10
1. RecoveryD -> S
2. Mutual upgrade D + D -> A + A
3. Infection by activeD + A -> A + A
4. RecoveryA -> S
5. Recruitment by Dormant
S + D -> D + D6. Direct infection
S -> A
40
Where is the Catch ?
Fluid approximation and fast simulation result say that nodes m and n are asymptotically independent
But we saw that nodes may not be asymptotically independent
… is there a contradiction ?
The Diagram Does Not Commute
For large t and N:
where T is the period of the limit cycle
42
Generic Result for Stationary Regime
Original system (stochastic):(XN(t)) is Markov, finite, discrete timeAssume it is irreducible, thus has a unique stationary proba N
Let N be the corresponding stationary distribution for MN(t), i.e.
P(MN(t)=(x1,…,xI)) = N(x1,…,xI) for xi of the form k/n, k integer
Theorem
Birkhoff Center: closure of set of points s.t. m2 (m)Omega limit: (m) = set of limit points of orbit starting at m
43
Here: Birkhoff center = limit cycle fixed point
The theorem 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
ExampleAt fixed t, the large N limit is deterministic (Dirac)
In stationary regime it is not
Stationary regime is periodicSampling algorithm for the stationary regime
pick t uniformly at random in a periodSet m = (t)
QuizMN(t) is a Markov chain on E={(a, b, c) ¸ 0, a + b + c =1, a, b, c multiples of 1/N}
E (for N = 200)
A. MN(t) is periodic, this is why there is a limit cycle for large N.
B. For large N, the stationary proba of MN tends to be concentrated on the blue cycle.
C. For large N, the stationary proba of MN tends to a Dirac.
D. MN(t) is not ergodic, this is why there is a limit cycle for large N.
Take Home Message
For large N the decoupling assumption holds at any fixed time t
It holds in stationary regime under assumption (H)ODE has a unique global stable point to which all trajectories converge
Otherwise the decoupling assumption may not hold in stationary regime
It has nothing to do with the properties at finite NIn our example, for h=0.3 the decoupling assumption holds in stationary regimeFor h=0.1 it does not
Study the ODE !
46
47
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
48
The Fixed Point Method
Commonly used to model protocol performance, finite state machines etc
When valid, works as followsNodes 1…N each have a state in {1,2,…,I}Assume N is large and therefore nodes are independent (decoupling assumption)Let i be the proba that any given node n is in state I. Write the equilibrium equations using the independenceCan often be cast as a fixed point equation for , solved numerically by iteration
The Fixed Point Method Consists in Finding the Stationary Points of the ODE
The transition matrix for one object depends on the occupancy measure, assumed equal to
This is the same as
or
For large N it is the same as the drift
Thus is a fixed point of the ODE
This is justified if assumption (H) holds, otherwise not
50
Existence and Unicity of a Fixed Point are not Sufficient for Validity of Fixed Point Method
Essential assumption is
(H) () converges to a unique m*
It is not sufficient to find that there is a unique stationary point, i.e. a unique solution to F(m*)=0
Counter Example on figure(XN(t)) is irreducible and thus has a unique stationary probability
There is a unique stationary point ( = fixed point ) (red cross)
F(m*)=0 has a unique solutionbut it is not a stable equilibrium
The fixed point method would say hereProb (node n is dormant) ≈ 0.1Nodes are independent
… but in realityWe have seen that nodes are not independent, but are correlated and synchronized
51
Mean of limit of N = pdf of one node in stationary regime
Stationary point of ODE
52
Correct Use of Fixed Point Method
Verify scaling assumption
Write ODE
Study stationary regime of ODE, not just fixed point
Verify assumption (H), i.e. there is a unique attractor to which all solutions converge
53
Example: Bianchi’s Formula
Example: 802.11 single cell
mi = proba one node is in backoff stage I= attempt rate = collision proba
Solve for Fixed Point:
54
Bianchi’s Formula requires analysis of ODEThe fixed point solution satisfies “Bianchi’s Formula” [Bianchi]
Another interpretation of Bianchi’s formula [Kumar, Altman, Moriandi, Goyal]
= nb transmission attempts per packet/ nb time slots per packet
assumes collision proba remains constant from one attempt to next
(H) true in single cell system [Bordenave,McDonald,Proutière] for q0< ln 2 and K= 1 [Sharma, Ganesh, Key] and for K=1
Method to porve (H) uses majorization by linear systems of ODE
Unclear what happens for heterogeneous systems
55
Contents
A Simple Model of Interacting Objects
Fluid Approximation
Mean Field Approximation, Fast Simulation
Stationary Regime and the Decoupling Assumption
The Fixed Point Method
Useful Extensions
56
Useful Extensions
Systems with a global resource that is updated at every transition [Bordenave,McDonald,Proutière], [Benaïm, L]
Multiclass systems [Banerjee et al ] [Chaintreau, L., Ristanovic]
Spatial model : XNn(t) = ( i, c) where c
= location on grid
Continuous state space (instead of discrete) [Chaintreau, L., Ristanovic]
XNn(t) = ( a, c) where a 2 [0, 1)
57
Conclusion
Convergence to Mean Field:
We have found a simple framework, easy to verify, as general as can beMultiple synchronized transitions are possible in the model
Under very large assumptions, first order properties can be predicted by the limit ODE
But for stationary regime one needs to study the ODE for itself
original system being ergodic does not imply ODE converges to a fixed point
Decoupling assumption holds (in stationary regime) if ODE has a unique attractor to which all trajectories converge
Otherwise, it may not hold
58
[Benaïm, L] “A Class Of Mean Field Interaction Models for Computer and Communication Systems”, Performance Evaluation, April 2008
[Tembine, L et al] Hamidou Tembine , Jean Yves Le Boudec, Rachid ElAzouzi, Eitan Altman, "Mean Field Asymptotic of Markov Decision Evolutionary Games and Teams", May 2009 " in proceedings of GameNets
[L,Mundinger,McDonald]
[Benaïm,Weibull]
[Sharma, Ganesh, Key]
[Bordenave,McDonald,Proutière]
References
59
[Chaintreau, L., Ristanovic] A. Chaintreau, J.-Y. Le Boudec and N. Ristanovic, “The age of Gossip: Spatial Mean Field Regime”, ACM Sigmetrics 2009
[Banerjee et al ] N Banerjee, MD Corner, D Towsley, BN Levine, “Relays, base stations, and meshes: enhancing mobile networks with infrastructure”, Mobicom 2008
Sznitman]
[Bianchi]
[Kumar, Altman, Moriandi, Goyal]
Solution to Quiz
The only correct answer is B.
MN is aperiodic (there are cycles of length 3 and 4)