Scheduling%20in%20Wireless%20Networks:%20Interference%20Model%20and%20Scheduling

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Scheduling in Wireless Networks Interference
Model and Scheduling

• Hyang-Won Lee

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Lecture Outline

• Wired vs. wireless
• Interference and time-varying channel
• Impact of scheduling
• Interference model
• Node-exclusive interference model
• 2-hop interference model
• Throughput-optimal scheduling
• Algorithms and complexity
• Reducing complexity

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Wireless Link
Wired links no interference
- Independent of each other - Fixed link rate
Wireless links broadcast nature
interference
Interference
Link rate depends on scheduling
Scheduling is important
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Wireless Link
Wired link time-invariant
Fixed link capacity (rate)
Wireless link time-varying
Varying link rate
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Importance of Scheduling

• Two-user case example
• Scheduling policy 1
• User A in slot 1, user B in slot 2, user A in
  slot 3,
• Scheduling policy 2
• User B in slot 1, user A in slot 2, user B in
  slot 3,
• Optimal scheduling must take into account channel
  quality

rate
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User A
1
User B
Slot 1
Slot 2
time
User A thruput User B thruput Total thruput
Scheduling 1 0.5 0.5 1
Scheduling 2 1 1 2
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Network Model

• Network model
• Link exists between two nodes if they can
  communicate with each other
• Time-slotted system
• Fixed scheduling interval
• Any link ls rate is 1 if interference-free
• Time-varying rate will be briefly discussed later
• Single-hop traffic
• No routing issue

Leave the network
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Interference Model

• Node-exclusive model
• For each node, at most one of its incident links
  can be activated
• example
• Any feasible link activation is a matching
• Also called primary or 1-hop interference model
• Applications
• Bluetooth, etc.

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Link Activation Vector

• Link activation vector
• Binary vector indicating active/inactive links
• Feasible (link activation) region F
• Set of all feasible link activation vectors
• Finite set

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1
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10
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Stability Region

• Network stability region
• Admissible rate vector
• The arrival rate vector ? is admissible if there
  exist ? such that

Admissible region
Stability region
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Max-Weight Scheduling

• Max-weight scheduling (MWS)
• example
• In 1-hop interference model, MWS finds
• Maximum weight matching (link weight backlog)
• Polynomial time solvable

MWS
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Throughput Optimality

• Theorem
• The max-weight scheduling stabilizes the network
  whenever ? is an admissible rate vector
• For proof, recall strong stability condition
• Lyapunov function
• Strong stability criterion

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Proof

• Conditional Lyapunov drift
• i.i.d. assumption on arrival gives
• The rest of the proof follows the same procedure
  in the previous lecture
• Use the conditions

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Proof (contd.)

• We have
• Sum up over all i
• Use this to write
• Quite enough done!

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2-hop Interference Model

• If a link is active, any link within its 2-hop
  neighbors cannot be active
• Example
• Applications
• IEEE 802.11-based MAC

Remove 1-hop neighbors
Remove 2-hop neighbors
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Max-Weight Scheduling

• Max-weight scheduling
• Note F is different from the one in
  node-exclusive model
• The max-weight scheduling is also throughput
  optimal in 2-hop interference model
• True for general K-hop interference model

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Distributedness and Complexity

• Multi-hop network
• Algorithm should be implemented in distributed
  manner
• Complexity of max-weight scheduling
• 1-hop model
• Maximum weight matching O(n3)
• No known distributed algorithm
• K-hop model, K2
• Maximum weight independent set NP-hard
• No efficient algorithm
• Need to consider suboptimal algorithms enabling
  distributed implementation

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Reducing Complexity
Max-Weight Scheduling
Complexity reduction Distributedness
Randomized approach (optimal)
Suboptimal approach
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Impact of Suboptimal Scheduling

• Suboptimal scheduling
• ?-approximation algorithm
• Theorem impact of suboptimal scheduling
• The ?-approximation scheduling algorithm
  stabilizes the network whenever ? is an
  admissible rate vector in ??, i.e., there exists
  ? such that

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Interpretation

• ?-approximation scheduling algorithm can stably
  support the arrival in ?-reduced stability region

Max-weight scheduling
?-approx. scheduling
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Proof

• Main part of proof

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Example of Suboptimal Scheduling

• Maximal scheduling
• A scheduling to which no more links can be added
  without violating interference constraint
• Example (1-hop interference model)
• Greedy maximal scheduling
• Pick a link with heaviest weight
• Remove all links interfering with the selected
  link
• Repeat until no more links can be added
• Example (1-hop interference model)

graph
Not maximal
maximal
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Greedy Maximal Scheduling

• Performance bound in 1-hop interference model
• See Preis99
• Consequence throughput performance of GMS
• Stabilize half of stability region
• In fact,
• Maximal scheduling (may not be greedy) achieves
  the same result
• Maximal for positive backlog, i.e., ql(t)gt0
• See Chaporkar05

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15
x
Link 2 can be selected by GMS as long as
xgt15 1/2(1515)
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2
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Preis99 R. Preis, Linear time
1/2-approximation algorithm for maximum weighted
matching in general graphs,. In 16th STACS,
Trier, Germany, 1999.
Chaporkar05 P. Chaporkar, K. Kar, and S.
Sarkar, Throughput Guarantees Through Maximal
Scheduling in Wireless Networks, 43rd Annual
Allerton, 2005
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Maximal Scheduling in K-hop Model

• Interference degree d(G) of graph G
• Maximum number of non-interfering links in every
  interference region
• Interference region of a link set of all links
  interfering with that link
• For example, in 1-hop, d(G)2
• Maximal scheduling stabilizes 1/d(G) of stability
  region
• See Chaporkar05
• Distributed implementation
• See Hoepman04 and references therein

Hoepman04 J.-H. Hoepman, Simple Distributed
Weighted Matchings, CoRR cs.DC/0410047, 2004
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Randomized Approach

• Framework (for each time slot t)
• Generate a new scheduling rnew
• Compare r(t-1) and rnew
• Pick the better of the two
• r(t) betterr(t-1), rnew having larger weight
• Requirements
• For example, randomly generated scheduling
  satisfies this constraint
• This approach achieves throughput-optimality

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Randomized Approach

• NxN input-queued switch
• Optimal matching (MWM) O(N3)
• Randomized algorithm O(N)
• Significant complexity reduction
• Originally proposed in Tassiulas98
• Multi-hop networks
• Perfect comparison may not be possible
• Generalized in Modiano06
• Studied impact of imperfect comparison

Tassiulas98 L. Tassiulas, Linear complexity
algorithms for maximum throughput in radio
networks and input queued switches, IEEE
INFOCOM98. Modiano06 E. Modiano, D. Shas, G.
Zussman, Maximizing Throughput in Wireless
Networks via Gossiping, ACM SIGMetric/Performance
06.
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Some Generalizations

• Tighter bound for greedy maximal scheduling
  Joo07
• Time-varying link rate
• Max-weight scheduling with link weights
  ql(t)cl(t) for every link l
• Multi-hop traffic
• Routing issue has to be addressed
• Tassiulas92, Neely05

Joo08 C. Joo, X. Lin, and N. B. Shroff,
Performance Limits of Greedy Maximal Matching
in Multi-hop Wireless Networks,'' in IEEE
CDC07. Tassiulas92 L. Tassiulas and A.
Ephremides, Stability properties of onstrained
queueing systems and scheduling policies for
maximum throughput in multihop radio networks,
IEEE Trans. Automat. Contr., 1992. Neely05 M.
Neely, E. Modiano and C. Rohrs, "Dynamic Power
Allocation and Routing for Time-Varying Wireless
Networks," IEEE JSAC 2005.