R. SrikantCoordinated Science Laboratory and

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Joint work with Jian Ni and Bo Tan

Hybrid Q-CSMA: A Distributed Scheduling Algorithm for Wireless

Networks

Wireless Networks

Links may not be able to transmit simultaneously due to interference.

Scheduling algorithm determines which links transmit at each time instant.

Performance metrics: throughput and delay.

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Throughput-Optimal Scheduling

A schedule is a collection of links that can be activated simultaneously.

MaxWeight Scheduling (centralized, high complexity) [Tassiulas-Ephremides ‘92] Associate a weight with each link, equal to its queue lengthFind schedule x which maximizes w(x); w(x): weight of a

schedule x is the sum of the weights of the links in the schedule

Observation [Eryilmaz-Srikant-Perkins’05]: Throughput-optimal even under the following modification: pick the max-weight schedule with high probability, going to one as the weight of the MWS goes to infinity

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Distributed AlgorithmsJiang-Walrand (‘08): Distributed algorithms which

pick schedule x with probability

Distribution realized using a continuous-time model.Also see Boorstyn et al (‘87), Rajagopalan-Shah-Shin

(’08). Related work: Marbach, Eryilmaz, Ozdaglar (‘07)

Goal: Discrete-time model which explicitly models contentions and allows the algorithm to be combined with heuristics leading to dramatic delay reduction

Z

ex

xw )(

)(

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Modeling Assumption

Divide each time slot into a control slot and a data transmission slot:

Links contend in control mini-slots to determine a collision-free schedule in the data slot.

Collisions are allowed in the control mini-slotsA Key Result: Two control mini-slots are

sufficient to achieve the product-form distribution. (Even one mini-slot is sufficient, thanks to Libin Jiang.)

time slot t time slot t+1

control mini-slots data slot control mini-slots data slot

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Interference Graph

Each vertex in the interference graph represents a link in the network.

If two links interfere with each other, they are neighbors in the interference graph.

A feasible schedule: a set of nodes such that no two nodes have an edge between them

We consider one-hop traffic only.

a

b

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d

e

g

f

schedule x = {a, d, g}

a dg

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Basic Scheduling Algorithm

Step 1. In control slot t, select a “decision schedule” m(t): a set of links that may decide to change their state from the previous slot; other links cannot change their state

Step 2. For any link i in m(t) doIf no links in its conflict set N(i) were active in the previous

data slot, link i will decide to becomeactive with probability pi: xi(t)=1inactive with probability 1-pi: xi(t)=0

Else, link i will be inactive: xi(t)=0

Step 3. In the data slot, use x(t) as the transmission schedule.

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Illustration of Scheduling Algorithm

Current schedule: {a, e}Decision schedule, m(t):

{c, f}Allowed decisions for

links in m(t):Link c, xc(t)=0 (no

choice)Link f, xf(t)=1 (w.p. pi)

Other links’ states are unchanged.

New schedule x(t)={a, e, f}

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fc

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Product-Form DistributionSchedule Evolution is a Markov chainProposition 1. If the set of possible decision schedules includes all the links, then the DTMC

is reversible and the steady-state probability of using schedule x is

Proof:

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xi i

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p

pZ

p

p

Zx

1

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1)(

(x) p(x,y) = (y) p(y,x)10

Throughput Optimality

Choose pi for link i (whose weight is wi) as

pi/(1-pi)=exp(wi),

then the probability of choosing a schedule x with weight w(x) is given by

Thus, a schedule with a large weight is picked with high probability.

Question: How to pick the decision schedule?

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ee

Zp

p

Zx

xw

xi

w

xi i

i i

)(1

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1)(

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Queue-Length Based CSMA (Q-CSMA)

Each time slot is divided into a data slot and control mini-slots

The control mini-slots are used to determine the decision schedule in a distributed manner; each link i selects a random control mini-slot Ti in [1,W].

Roughly, the idea is that a link will send a message announcing its intent to make a decision during its chosen control mini-slot if it does not hear such a message from its neighbors.

data slotcontrol mini-slots

link i : Ti = 3 (W = 4)

INTENT Message

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Case 1If link i hears an INTENT message from a link in its

neighborhood N(i) before its chosen mini-slot, it will keep its state unchanged from the previous time-slot.

If it was active in the previous time slot, it will continue to be active; will be inactive otherwise.

data slotcontrol mini-slots

link i : Ti = 3

data slotcontrol mini-slots

link j : Tj = 2

INTENT Message

Case 2Otherwise, link i will broadcast an INTENT

message to links in N(i) in the Ti-th control mini-slot.

Case 2: If there is a collision, link i will not change its state.

data slotcontrol mini-slots

link i : Ti = 3

data slotcontrol mini-slots

link j : Tj = 3

INTENT Message

INTENT Message

Case 3If there is no collision, link i will make its decision:

If no links in N(i) were active in the previous data slot, then link i’s state is chosen as follows:

active with probability pi

inactive with probability1-pi Otherwise: inactive

data slotcontrol mini-slots

link i : Ti = 3

data slotcontrol mini-slots

link j : Tj = 4

INTENT Message

Key Property of Q-CSMA

Proposition 2. The Q-CSMA algorithm achieves the product-form distribution if the window size W¸ 2.Any maximal schedule will be selected as the

decision schedule with positive probability.The set of maximal schedules includes all the links.

Modification: Works even if W=1. A link chooses to participate in the decision schedule with probability ½. Again, one can show that the above result is still valid.

PerformanceQ-CSMA is a randomized algorithm, the delay

performance can be badWhat are the alternatives?

MaxWeight algorithm: Performance is very good; but high complexity,

centralized implementationMaximal matching:

Add links to the schedule till no more links can be added

Very low complexity; decentralized implementation?; throughput can be small in certain networks

Longest Queue First (LQF) or Greedy Maximal Matching (GMS)

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LQF/GMSAlgorithm:

add link with the longest queue to the scheduleRemove the added link and its “neighbors” from

the graph and repeatvery low complexity; distributed implementation?

Networks that are unstable under maximal scheduling can be stable under LQFDimakis-Walrand, 2006; Brzezinski-Zussman-

Modiano, 2006; Joo-Lin-Shroff, 2008; Leconte-Ni-Srikant, 2009

Performance is very good in simulations; but not always provably throughput-optimal

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Hybrid Q-CSMAChoose a weight threshold w0; choose a

schedule with probability p(x) (defined previously) among those links whose weights exceed the threshold

Add additional links with weight smaller than the threshold, if possible, using a distributed approximation of the longest-queue-first policy

Key Result: the hybrid algorithm is still throughput optimal; Question: does it improve performance over Q-CSMA?

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Simulation Evaluation (1)24-Link Grid Network

(one-hop interference model)

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(pk

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(a) All the three algorithms

LQFQ-CSMAHybrid Q-CSMA

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Q (

pkt)

(b) The two algorithms with good delay performance

LQFHybrid Q-CSMA

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Simulation Evaluation (2)9-Link Ring Network

(two-hop interference model)

0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9

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4x 10

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Q (

pkt)

LQFQ-CSMAHybrid Q-CSMA

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Ongoing workPerformance of Hybrid Q-CSMA

Relationship between mixing time of the Markov chain and expected delays

Mixing time estimates are reasonable at light loads but not at heavy loads

w/ Jiang and Walrand

Paradigm shift: Finite-sized flows Instability with fading (van de Ven-Borst-Schneer ‘09)Very different algorithms are needed, somewhat

surprisingly being greedy is good (Liu-Ying-Srikant ‘09)

Ad hoc networks are very different, w/ Shroff and Tan

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Ongoing WorkParadigm shift: packets with deadlines

MaxWeight works here too!: Hou-Borkar-Kumar (‘09), Hou-Kumar (‘09), Hou-Kumar (‘09)

Derivation using purely optimization considerations: Jaramillo-Srikant ; allows extensions to ad hoc networks, fits into the dual decomposition view of network architecture (parallels the interpretation of the Tassiulas/Ephremides result in Lin/Shroff, Neely/Modiano/Li, Eryilmaz/Srikant and Stolyar)

GMS/LQF type ideas seem to work here tooTCP timeout and heavy-tailed file-sizes

Impact of wireless link losses on files with heavy-tailed distributed file sizes (w/ Towsley)

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SummaryQ-CSMA can achieve max throughput in

wireless networks with a fully distributed implementation.

Performance can be improved dramatically by using a hybrid algorithm, combining Q-CSMA with approximations of longest queue first algorithm.

Ongoing work addresses extensions, and several other network control problems in complex wireless networks

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