r. srikant coordinated science laboratory and department of electrical and computer engineering...
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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
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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
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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.
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schedule x = {a, d, g}
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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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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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(x) p(x,y) = (y) p(y,x)10
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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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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
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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
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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
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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.
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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)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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(a) All the three algorithms
LQFQ-CSMAHybrid Q-CSMA
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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(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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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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