Energy-Efficient Rate Scheduling in Wireless Links A Geometric Approach

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Energy-Efficient Rate Scheduling in Wireless
Links A Geometric Approach

• Yashar Ganjali
• High Performance Networking Group
• Stanford University

Joint work with Mingjie Lin February 9,
2005 Networking, Communications, and DSP
Seminar University of California Berkeley
yganjali_at_stanford.edu http//www.stanford.edu/yga
njali
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Introduction and MotivationRate Scheduling
Problem

• Setting. A transmitter sending packets to a
  receiver over a wireless link.
• Observation. If we reduce the transmission rate,
  we can save energy.
• Constraint. Low transmission rate means higher
  delays for packets.

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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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Rate Scheduling Problem

• Given.
• A sequence of N packets
• ti Instantaneous arrival time of packet i
• Li Length of packet i
• di Departure deadline for packet i
• A wireless channel with power function w(r)
• Find. A feasible rate schedule, which minimizes
  the energy.

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Wireless Channel Transmission Power Function

• Represents energy/bit as a function of the
  transmission rate r.
• w(r) gt 0
• w(r) is monotonically increasing in r and
• w(r) is strictly convex in r.
• The energy required to transmit a packet of
  length L is w(r)L.

Energy/Bit
Transmission Rate
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Feasible Schedule

• Transmission Schedule. For packet i
• Start transmitting at time si and
• finish transmission by time fi.
• R(t) for any time between si and fi.
• Feasible Transmission Schedule.
• For all i in 0,N, 0 ti si fi di T
  and
• 0 s1 lt f1 s2 lt f2 sN lt fN lt T.
• Data transmitted during si,fi equals Li.

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Packet Reordering

• In a setting with no constraints on the packet
  arrivals and departure deadlines, reordering can
  reduce the transmission energy.
• Theorem. When reordering is allowed, optimal rate
  scheduling problem is NP-hard.

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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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Previous Results

• A lot of research on transmission power control
  schemes.
• Mostly try to mitigate the effect of
  interference.
• Results range
• Distributed power control algorithms
• Determining information theoretic capacity
  achievable under interference limitations
• Most power control schemes maximize the amount of
  information sent for a given average power
  constraint.

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Previous Results (Contd)

• Uysal, Prabhakar, El Gamal 2002
• Minimizing energy subject to time constraints
• Arbitrary arrivals
• Single departure deadline
• Assumes instantaneous arrivals and departures
• Algebraic Approach
• Runs in O(N2) time

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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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RT Diagrams
Accumulative Amount of Data
Time
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Feasible Schedules

• Feasible schedule ?
• Curve C on the RT-diagram
• simple, and continuous
• lies inside RT polygon
• connects the two endpoints of the polygon and
• Is monotonically increasing in time.

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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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Optimal Rate Schedules on RT Diagrams

• Claim. To find the optimal rate schedules, we
  just need to find the shortest path inside the RT
  polygon, which connects its two endpoints.
• We need to consider piece-wise linear schedules.
• Among those, the shortest path corresponds to the
  optimal rate schedule.

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Piece-wise Linearity

• Lemma. During any time interval with no
  arrivals/departures transmission rate must remain
  fixed.

• Proof. A simple application of Jensens
  inequality to w(r)xr and the random variable
  YR(t)

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RT Diagrams
Accumulative Amount of Data
Time
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Main Theorem

• Theorem. The shortest path connecting the two
  endpoints of the RT Polygon corresponds to the
  schedule with minimum amount of energy
  consumption.

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Proof of the Main Theorem

• Only need to consider piece-wise linear
  schedules.
• Mathematical induction on M the number of
  segments.
• If M1
• We have a single arrival, and departure.
• Based on the lemma that we just showed, rate must
  remain fixed.
• This corresponds to the straight line connecting
  the two endpoints (i.e. the shortest path).

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Proof of the Main Theorem (Contd)

• Let us assume for Mltk, the claim is true.
• Want to show that for Mk, the shortest path
  corresponds to the optimal schedule.
• We prove this step by contradiction.
• Let us assume the shortest path between the
  endpoints represents schedule ?.
• There is another schedule ? which consumes less
  energy.

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Proof of the Main Theorem (Contd)

• Case 1. ? and ? intersect at some point.

?
?
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Proof of the Main Theorem (Contd)

• Case 2. ? and ? do not intersect.

?
?
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Proof of the Main Theorem (Contd)

• Case 2. ? and ? do not intersect.

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Main Theorem

• None of the two cases is possible.
• Therefore ? and ? must be the same.
• In other words the shortest path inside the RT
  polygon corresponds to the schedule with minimum
  energy consumption.
• This result does not depend on the wireless
  channel power function.

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Shortest Path Problem

• This is a classic problem in computational
  geometry.
• If we have a triangulation of the polygon, we can
  find the shortest path in O(N) time Lee,
  Preparata 85
• Triangulation can be found in linear time
  Tarjan, Van Wyk 86.
• Our problem is simpler due to its special
  structure.

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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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Special Case
Accumulative Amount of Data
t1
t2
t3
t4
d
Time
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Extensions
Accumulative Amount of Data
t1
t2
t3
t4
d1
d2
d3
d4
Time
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Outline

• Rate scheduling problem
• Previous results
• RT diagrams
• Shortest path optimal rate schedule
• Special cases and extensions
• Online algorithms
• Summary and conclusion

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Online Scheduling Problem

• Given (at each point t in time)
• packet arrivals ti up to the present
• departure time di
• Length Li of packets and
• a wireless channel with power function w(r).
• Find the transmission rate, i.e. R(t), such that
• departure deadlines are met and
• the total amount of energy used to transmit
  packets is minimized.

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Competitive Ratio

• An online rate scheduling algorithm ALG is
  c-competitive if there is a constant a such that
  for any finite input sequence I,
• ALG(I) c.OPT(I) a
• ALG(I) and OPT(I) denote the cost of the
  schedule produced by ALG, and optimal offline
  algorithm, respectively.
• We call c the competitive ratio.

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No Constant Competitive Ratio

• Theorem. For any constant c, no online rate
  scheduling algorithm is c-competitive, unless it
  misses some departure deadlines.

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No Constant Competitive Ratio
Accumulative Amount of Data
RU
Time
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Optimistic Online Scheduling Algorithm (OOSA)

• Idea. Use the best decision based on the arrivals
  up to the present.
• Algorithm. Construct the RT diagram, and apply
  the optimal offline scheduling algorithm.
• Properties
• It is a greedy algorithm.
• It is always feasible.
• Works even if the arrivals are not instantaneous.

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OOSA
Accumulative Amount of Data
Time
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Pessimistic Online Scheduling Algorithm (POSA)

• Idea. Assume the worst possible arrivals in the
  future.
• Assumptions.
• All packets are of the same length L.
• Each packet departs exactly D units after its
  arrival.
• Algorithm. If there are k packets in the system,
  send with rate kL/D.

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POSA
Accumulative Amount of Data
t1
t2
t3
t4
Time
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Properties of POSA

• It is always feasible
• Compare to M/D/? queue.
• Theorem. For a fixed packet length L, and a fixed
  departure deadline D, there is a constant c such
  that POSA is c-competitive.
• This constant can be huge, as L/D grows.
• When L/D is small, the constant is small.
• We can show that for fixed L and D OOSA always
  outperforms POSA.
• In other words, OOSA is also c-competitive in
  this setting.

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Performance of Online Algorithms
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Summary and Conclusion

• Introduced RT diagrams
• Shortest path optimal schedule
• Works in special cases/extensions
• More profound implications
• No constant competitive ratio for online
  algorithms
• For fixed L and D, we have c-competitive online
  algorithms.

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Thank You!

• Questions?

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Extra Slides

• EXTRA SLIDES

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Packet Reordering

• Theorem. When reordering is allowed, optimal rate
  scheduling problem is NP-hard.
• Sketch of the proof.
• 2k1 packets of length L1, , L2k1
• For all i, 1 i 2k we have si 0, di T
• s2k1 T/3, and d2k3 2T/3
• L2k1 gtgt Li

2k1
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Piece-wise Linearity

• w(r).r is a convex function of r.
• t uniformly distributed in tA, tB.
• Y R(t)
• w(EY).EY
• Ew(Y).Y)
• Jensens inequality

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Algebraic Approach
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Comparison
Best Previous Result New Approach
Works in a special setting Runs in O(N2) Algebraic Works in a general setting Runs in O(N) Geometric Leads to fast online algorithms Can be applied to other problems