Mixing TIMES and QuEUE LENGTH BOUNDS FOR GLAUBER DYNAMICS-BASED CSMA Scheduling

1. Mixing TIMES and QuEUE LENGTH BOUNDS FOR GLAUBER DYNAMICS-BASED 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 Inspired CSMA Scheduling Algorithms • Low complexity, fully distributed • Can achieve maximum throughput • However, queueing performance is not well understood • The mean queue length can be related to the amount of time (mixing time) that the Glauber dynamics takes to converge to equilibrium. • We study the mixing time as a function of the nodes in the network.

3. Conflict Graph of a Wireless Network • 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 iis included in the schedule and 0 otherwise

4. Throughput Optimality • Associate each link i with a weightwi. The weight of a schedule x is w(x) = i2xwi. • Want the followingprobability 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.

5. Parallel Glauber Dynamics (PGD) • Randomly select an update set mwith probability qm: a set of vertices that are allowed to change their states; other vertices do not change their states. • For each vertex v 2m do • If no vertices in its neighborhood N(v) were active in the previous slot, v will decide to become • activewith probability pv=v/(1+v) xv=1 • inactive with probability1-pv: xv=0 • Else, v will be inactive: xv=0

6. 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

7. Dynamics of Schedules • x(t) evolves as a Discrete-Time Markov Chain (DTMC) • Proposition(Ni & S.) If the probability of updating every link is positive, the steady-state probability of using schedule xhas the following product-form:By letting i =exp(wi), we have • Discrete-time version of the algorithm studied by Jiang and Walrand

8. Main Result • Theorem 1: Consider a network with n links and maximum interference degree . If the arrival rate vector  2 for some  < 1/ which is also independent of n, then • there exist fugacities such that the queue lengths are stable, and • the expected queue length per link is O(log n) under PGD-CSMA.

9. Queue Length Evolution • Time slotted system; fugacities chosen to ensure stability • 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: • Exact computation of mean queue lengths appears to be hard because the schedules are correlated both spatially and temporally. ai(t) xi(t) ai(t+1) xi(t+1) Qi(t+1) Qi(t) slot t slot t+1 slot t+2

10. Drift Analysis • Sample the system once every T=Tmix/ slots (Tmixis the mixing time ; T is sufficiently large for the Markov chain to reach steady-state; thus, the empirical average service given to a link over T time slots is “close” to the steady-state mean) • Test function • Compute drift: • Setting E(drift)=0 yields a bound on E(Qi). Tslots

11. Queue Length Bound • Expected queue length is upper bounded by a linear function of the mixing time: • Often the conductance method is used to estimate Tmix. Leads to bounds which grow exponentially in n, the number of links in the network. • We prove a logarithmic bound on Tmixfor graphs with bounded degree using the coupling method

12. 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 timeTmix of the MC is the time required for the MC to get close to the stationary distribution:

13. Coupling • (X(t), Y(t)) is a coupling if both {X(t)} and {Y(t)} are two “copies” of a Markov chain, 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

14. Coupling Theorem • Let d(x,y)be some distance metric between two states of the Markov chain (NOT the variation distance). • Theorem: Suppose there exist a constant  < 1 and a coupling (X(t),Y(t)) of the MC such thatThen the mixing time is bounded by Tmix· log(De)/(1-)where D is the ratio between max and min distances between two distinct states.

15. Path Coupling Theorem • Under the coupling theorem, we have to check the condition for all pairs of schedules to 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 examplex = (1, 0, 0, 0, 1, 1, 0) y = (1, 0, 0, 0, 1, 0, 0)

16. Coupling on Conflict Graph X(t+1)= (1, 0, 0, 0, 0, 1, 0) X(t) = (1, 0, 0, 0, 1, 1, 0) Y(t)= (1, 0, 0, 0, 1, 0, 0) Y(t+1)=(1, 0, 0, 0, 0, 1, 0) • 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 5 5 2 2 7 7 1 1 4 4 6 6 3 3 d(X(t), Y(t)) = f(6) d(X(t+1), Y(t+1)) = 0

17. 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 decrease 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 increase by f(w)

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

19. 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

20. Proof of Main Result • For bounded-degree conflict graphs (dv·), using a simple distributed randomized scheme, qv can be lower bounded by some constant (i.e., independent of n), 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] = O(log n) for every link i.

21. Summary • 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 • Polynomial growth in delay can also be shown for the case where the weights are chosen adaptively as a function of queue lengths 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.