R. Srikant

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Hybrid Q-CSMA A Distributed Scheduling
Algorithm for Wireless Networks

• R. Srikant
• Coordinated Science Laboratory and
• Department of Electrical and Computer Engineering
• University of Illinois at Urbana-Champaign
• Joint work with Jian Ni and Bo Tan

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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 length
• Find 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-Perkins05
  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 Algorithms

• Jiang-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-Sh
  in (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-slots
• A 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 t1
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.

schedule x a, d, g
e
b
g
g
a
a
d
d
f
c
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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) do
• If no links in its conflict set N(i) were active
  in the previous data slot, link i will decide to
  become
• active with probability pi xi(t)1
• inactive 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

e
e
b
g
a
a
d
f
f
f
c
c
c
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Schedule Evolution Markov Chain

• If both x(t-1) and m(t) are feasible, then x(t)
  is also feasible.
• x(t) evolves as a discrete-time Markov chain
  (DTMC) (if m(t) is picked at random in each time
  slot).
• x can make a transition to y if and only if xy
  is feasible and there exists a decision schedule
  m such that x? y µ m.

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Product-Form Distribution

• Schedule Evolution is a Markov chain
• Proposition 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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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.

INTENT Message
link i Ti 3 (W 4)
data slot
control mini-slots
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Case 1

• If 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.

INTENT Message
link j Tj 2
data slot
control mini-slots
link i Ti 3
data slot
control mini-slots
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Case 2

• Otherwise, 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.

INTENT Message
link j Tj 3
data slot
control mini-slots
INTENT Message
link i Ti 3
data slot
control mini-slots
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Case 3

• If 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 is state is chosen as
  follows active with probability pi inactive
  with probability1-pi
• Otherwise inactive

link j Tj 4
data slot
control mini-slots
INTENT Message
link i Ti 3
data slot
control mini-slots
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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 W1. 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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Performance

• Q-CSMA is a randomized algorithm, the delay
  performance can be bad
• What are the alternatives?
• MaxWeight algorithm
• Performance is very good but high complexity,
  centralized implementation
• Maximal 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/GMS

• Algorithm
• add link with the longest queue to the schedule
• Remove the added link and its neighbors from
  the graph and repeat
• very low complexity distributed implementation?
• Networks that are unstable under maximal
  scheduling can be stable under LQF
• Dimakis-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-CSMA

• Choose 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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Simulation Evaluation (2)
9-Link Ring Network (two-hop interference model)
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Ongoing work

• Performance 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 Work

• Paradigm 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 too
• TCP 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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Summary

• Q-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