MIXING TIMES AND QUEUE LENGTH
BOUNDS FOR GLAUBER DYNAMICSBASED CSMA SCHEDULING
R. Srikant
University of Illinois at Urbana-Champaign
Joint work with Libin Jiang, Mathieu Leconte, Jian Ni
and Jean Walrand (Berkeley)
Introduction
2


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.
Conflict Graph of a Wireless Network
4

5
2
1
7
4
3

6
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
Example of feasible schedule: {1, 4, 7}
vertices (links) which are not
Represented by a binary vector x = (1, 0, 0, 1, 0, 0, 1)
neighbors in the conflict
xi=1 if link i is included in the schedule and 0 otherwise
graph (an independent set).

Throughput Optimality
5


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.
Parallel Glauber Dynamics (PGD)
7


Randomly select an update set m with probability qm: a
set of vertices that are allowed to change their states;
other vertices do not change their states.
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
Illustration of PGD-CSMA
8

5
2


1
7
4
3
6


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}
Dynamics of Schedules
9
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 x has the following product-form:

 ( x) 
1
Z

i
ix
By letting i =exp(wi), we have
 Discrete-time version of the algorithm studied by Jiang
and Walrand
Main Result
11

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
a) there exist fugacities  such that the queue lengths
are stable, and
b) the expected queue length per link is O(log n) under
PGD-CSMA.
Queue Length Evolution
12
ai(t)
xi(t) ai(t+1)
slot t

Qi(t)
xi(t+1)
slot t+1
Qi(t+1)
slot t+2
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.
Drift Analysis
13
T slots

Sample the system once every T=Tmix/ slots (Tmix is 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).
Queue Length Bound
14



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 Tmix for graphs with
bounded degree using the coupling method
Mixing Time of a Markov Chain
15



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:
Coupling
16

X(t)
Y(t)
(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.
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
t=0
t=1
t=2
t=3
t=4
t=5
…
Coupling Theorem
17


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 that
Then the mixing time is bounded by
Tmix · log(De)/(1-)
where D is the ratio between max and min distances
between two distinct states.
Path Coupling Theorem
18

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 example
x = (1, 0, 0, 0, 1, 1, 0)
y = (1, 0, 0, 0, 1, 0, 0)
Coupling on Conflict Graph
19
X(t) = (1,(1,
X(t+1)=
0,0,
0,0,
0, 0,
1, 0,
1,1,
0)0)
5
2
1
5
2
7
4
3
Y(t)= (1, 0, 0,
Y(t+1)=(1,
0,0,
0,0,
1,0,
0,1,
0)0)
1
6
7
4
3
6
d(X(t), Y(t))
d(X(t+1),
Y(t+1))
= f(6)= 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
Useful Lemma
20

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.
If a neighbor
wq
2v),N(v) is selected to
distance will decrease
by f(v)(with prob. q ) and w decides to
update
w
become active (with prob. w/(1+w),
distance will increase by f(w)
Fast Mixing
21

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
Condition for Fast Mixing
22

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
Proof of Main Result
23



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.
Summary
24

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.
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