1. Mixing TIME OF PARALLEL Glauber Dynamics and QuEUE LENGTH BOUNDS FOR CSMA Scheduling
R. Srikant
University of Illinois at Urbana-Champaign
Joint work with Libin Jiang, Mathieu Leconte, Jian Ni and Jean Walrand (Berkeley)

2. Introduction
Glauber dynamics is a powerful tool to generate randomized, approximate solutions to combinatorially difficult problems (statistical physics, approximate counting, graph coloring)
Glauber Dynamics Inspired CSMA Scheduling Algorithms
Low complexity, fully distributed
Can achieve maximum throughput
However, queueing performance is not well understood
In this work we analyze the mixing time of a generalized version of the Glauber dynamics with parallel updates. We use the result to derive queue-length bounds for the CSMA algorithm based on this dynamics.

3. Scheduling in Wireless Networks
Wireless links may not be able to transmit simultaneously due to interference
A scheduling algorithm determines which links can access the medium in each time instant
Interference relationships among wireless links is often represented by the conflict graph
active(scheduled)
disabled
inactive

4. Conflict (Interference) Graph
Each vertex in the conflict graph represents a wireless link.
An edge connects two vertices if the corresponding wireless links interfere with each other.
Feasible schedule: a set of vertices (links) which are not neighbors in the conflict graph (an independent set). 5 2 7 1 4 6 3
Example of feasible schedule: {1, 4, 7}
Represented by a binary vector x = (1, 0, 0, 1, 0, 0, 1)
xi=1 if link i is included in the schedule and 0 otherwise

5. Throughput Optimality
Associate each link i with a weight wi . The weight of a schedule x is w(x) = i2x wi.
Want the following probability of picking schedule x:
In wireless networks, if weights are chosen as appropriate functions of queue lengths, an algorithm which chooses schedules from this distribution is throughput-optimal.

6. Traditional Glauber Dynamics
Randomly choose a vertex v.
If all neighbors of v are inactive, v decides to become:
active w.p. pv=/(1+)
inactive w.p. 1-pv
Else, inactive.
: fugacity. 5 2 7 1 4 6 3
vertex 3 is selected to update: stays inactive (no choice)
vertex 6 is selected to update: becomes active w.p. p6
vertex 5 is selected to update: becomes inactive w.p. 1-p5

7. Parallel Glauber Dynamics (PGD)
Randomly select an update set m with probability qm: a set of vertices that may decide to change their states; other vertices keep their states unchanged.
For each vertex v 2 m do
If no vertices in its neighborhood N(v) were active in the previous slot, v will decide to become
active with probability pv=v/(1+v) xv=1
inactive with probability1-pv: xv=0
Else, v will be inactive: xv=0

8. Illustration of PGD-CSMA
Current schedule: x(t)={1, 5}
Select an update set: m={3, 5, 6}
Allowed decisions for links in m:
link 3: x3=0 (no choice)
Link 5: x5=0 (w.p. 1-p5)
link 6: x6=1 (w.p. p6)
Other links’ states unchanged
New schedule: x(t+1)={1, 6} 5 2 7 1 4 6 3

9. Dynamics of Schedules
x(t) evolves as a Discrete-Time Markov Chain (DTMC)
Proposition. If the probability of updating every link is positive, the steady-state probability of using schedule x has the following product-form: By letting i =exp(wi), we have

10. Throughput and Fugacities
Let  be the capacity region of the network: set of arrival rate vectors that can be stabilized by some scheduling algorithm.
Lemma 1: Given any arrival rate vector  2 , there exist fugacities  such that PGD-CSMA can serve  (i.e., mean service rate is greater than or equal to mean arrive rate for all links).
Lemma 2: If 2 (1/) , where  is the maximum vertex degree in the conflict graph, then the fugacities required to support  satisfy
For Lemma 2, if v strictly lies in 1/Delta Lambda, then the equation becomes strictly inequality.

11. Main Result
Theorem 1: Suppose the network has n links and  is independent of n. If the arrival rate vector  2  for some  < 1/ which is also independent of n, then there exist fugacities  such that in the steady-state, the expected queue length per link is O(log n) under PGD- CSMA.
By Lemma 2, for such , we can find  such that the mean service rate si is great than the mean arrival rate i for all links, we use this  in PGD-CSMA
For comment 2, the trick is to select rho’such that rho < rho’<1/Delta, (rho’/rho)v still strictly lies in (1/Delta) Lambda.

12. Queue Length Analysis (1)
ai(t)
xi(t)
ai(t+1)
xi(t+1)
Qi(t)
Qi(t+1)
slot t
slot t+1
slot t+2
Time slotted system
ai(t): # of packets arriving at link i in slot t
xi(t): scheduling variable 2 {0,1} (determined by PGD-CSMA)
Qi(t): queue length of link i at the end of slot t
Queue dynamics:
Direct analysis of queue lengths is hard because scheduling variables are correlated both spatially and temporally.
So we need to analyze how fast the PGD Markov chain converges to steady-state (mixing time).

13. Mixing Time of a Markov Chain
Roughly speaking, the time required to reach steady-state
The variation distance between two distributions ,  is defined as:
The mixing time Tmix of the MC is the time required for the MC to get close to the stationary distribution:

14. Queue Length Analysis (2)
Tmix slots
sample schedules from steady-state dist.
T slots
Consider T=Tmix/ slots (the Markov chain of schedules will converge to steady-state in Tmix slots, and after that, schedules seem to be sampled independently from steady-state distribution)
Lyapunov function
Compute drift:
Negative drift if  < mini (si-i) and queue lengths are sufficiently large, then (Q(t), x(t)) is positive recurrent.

15. Queue Length Analysis (3)
In steady-state, further we can prove:
Existing approaches use conductance method which yields exponential bounds on Tmix
We prove a logarithmic bound on Tmix for graphs with bounded degree using the coupling method

16. Coupling
(X(t), Y(t)) is a coupling of the Markov chain if both {X(t)} and {Y(t)} are copies of the MC, and once X(t)=Y(t), then X(t+1)=Y(t+1) henceforth.
X(t) 1 1 1 1 1 1 2 2 2 2 2 2 … 3 3 3 3 3 3
Y(t) 4 4 4 4 4 4
t=0
t=1
t=2
t=3
t=4
t=5

17. Coupling Theorem
Let d(x,y) be some distance metric between two states.
Theorem: Suppose there exist a constant  < 1 and a coupling (X(t),Y(t)) of the MC such that Then the mixing time is bounded by
Tmix · log(De)/(1-) where D is the ratio between max and min distances.

18. Path Coupling Theorem
Under the coupling theorem, we have to check the condition for all pairs of schedules to determine (estimate) .
Bubley&Dyer’97 introduced the path coupling theorem, under which, in our context, we only need to check those x and y which are different at only one link, for example x = (1, 0, 0, 0, 1, 1, 0) y = (1, 0, 0, 0, 1, 0, 0)

19. Coupling on Conflict Graph
Distance metric: weighted Hamming distance with weights f(v) for all vertex v.
Coupling: both chains select the same update set and use the same coin toss when a vertex in the update can be added to both schedules
X(t) = (1, 0, 0, 0, 1, 1, 0)
X(t+1)= (1, 0, 0, 0, 0, 1, 0)
Y(t)= (1, 0, 0, 0, 1, 0, 0)
Y(t+1)=(1, 0, 0, 0, 0, 1, 0) 5 5 2 2 7 7 1 1 4 4 6 6 3 3
d(X(t+1), Y(t+1)) = 0
d(X(t), Y(t)) = f(6)

20. Useful Lemma
Lemma: Consider a pair of adjacent schedules x and y that differ only at v, we have
If v is selected to update (with prob. qv), distance will be decreased by f(v)
If a neighbor w 2 N(v) is selected to update (with prob. qw) and w decides to become active (with prob. w/(1+w), distance will be increased by f(w)

21. Main Theorem for Fast Mixing
Theorem: For any weight function f(v)>0 of v2V, let m = min f(v), M = max f(v), D=M/m, if then the mixing time of PGD is bounded by

22. Condition for Fast Mixing
Choose f(v)=dv/qv where dv is the degree of vertex v in the conflict graph, then if v < 1/(dv-1) for all v
where
M = max f(v), m = min f(v), D = M/m.

23. Queue Length Analysis (4)
For bounded-degree conflict graphs (dv · ), using a simple distributed randomized scheme, qv can be lower bounded by some constant (“constant” means “independent of n”), e.g., 0.5+1, so both M and D can be upper bounded by some constants.
When arrival rate vector  2  for some  < 1/ where  is independent of n, then v · 1/( -1)- for some constant , so  can also be lowered bounded by some constant.
Therefore, Tmix=O(log n), and by our previous queue length analysis, E[Qi(t)] = O(log n) for every link i.

24. Summary
CSMA can achieve 100% throughput (throughput-optimal) in ad hoc wireless networks with a fully distributed implementation
Use Glauber Dynamics as a distributed, randomized algorithm to solve the max- weight independent set problem
The average queue length (delay) grows logarithmically with the size of the network when the arrival rate lies in a fraction of the capacity region
Fast mixing of Parallel Glauber Dynamics
The fraction is lower bounded by 1/, where  is the maximum vertex degree in the conflict graph
CSMA can stabilize the network queues for all arrival rates in the capacity region.
capacity region
Low-delay region
When the arrival rate lies in this region, the delay grows logarithmically with the size of the network under CSMA.