minimal energy routing with latency quality-of-service ...9/17 yannis paschalidis, boston university...

Minimal Energy Routing with LatencyQuality-of-Service Guarantees

Yannis [email protected], http://ionia.bu.edu/

Center for Information and Systems Engineering (CISE),Department of Manufacturing Engineering, and

Department of Electrical and Computer EngineeringBoston University

November 17, 2006Sensor Network Consortium Meeting

Boston University

Intro Model Performance Routing algorithms Conclusions

Minimal energy routing with QoS guarantees

Focus on low data rate SNETs servingtime-critical applications (e.g., surveillance,environmental monitoring).

A common strategy to preserve energy is to haveSNET nodes shut down their radios (go to“sleep”).

Hence, SNETs can operate in an event-basedsensing mode.

This creates challenges for (timely) routing datato the gateway.

Trade-off: Long hops (lots of energy) vs. shorthops (large delay).

2/17 Yannis Paschalidis, Boston University Sensor Network Routing:


Outline

System Model.

Performance metrics and analysis.

Routing algorithms.

Numerical results.

Conclusions.

3/17 Yannis Paschalidis, Boston University Sensor Network Routing: Intro


System Model

Continuous time t.

N nodes, 1 gateway (node N + 1): (V , E ).

s(k) d(k)k

X tk: channel state between nodes s(k) and d(k) at time t

Stochastic process, independent of all the other channels.Two-state Markov process: 1 (“good”) and 0 (“bad”).

Y ti: state of node i at time t; 1 (ON) or 0 (OFF)

Two-state Markov process.Y t

N+1 = 1, for all t.

Energy consumption model for each link k ∈ E

ck : power consumption before connection is established(Watt).gk : energy consumption after connection is established(Joule).

4/17 Yannis Paschalidis, Boston University Sensor Network Routing: Model


Comments on the System Model

Channel model.

Finite state Markov model for Rayleigh fading channels.Gilbert-Elliot model.

Sleeping schedule model.

No time synchronization required.Randomized transition between ON and OFF (good forsecurity).

Few data at a low rate.

Data is generated at a slow time scale.Transmission, propagation and queueing delays negligible.Only causes of latency

Bad channel conditions.

Sleeping nodes.



Objective

Let p the path from node 1 to the gateway.

Necessary and sufficient condition for successful delivery onlink k at time t: channel is good (X t

k= 1) and downstream

node is ON (Y td(k) = 1).

Lk : (random) time needed for node s(k) to successfullydeliver a packet to d(k).

Total time for packet delivery on path p:∑

k∈p Lk .

Energy consumption on path p: Tp =∑

k∈p(ckLk + gk).

Latency probability P(Lp ≥ d) for a constant d on path p.

Objective

Characterize E(Tp) and P(Lp ≥ d), and optimize by selectingrouting strategies.



Performance Metrics

Expected energy consumption

E(Tp) =∑

k∈p E(Tk)

(We can obtain a closed-form expression)

Latency probability P(Lp ≥ d) = P(∑

k∈p Lk ≥ d)

Requires convolution of latencies on all links in path p ...Computationally expensive for long paths.Significant communication overhead.We have developed tight approximations of P(Lp ≥ d) withmuch lower complexity.

7/17 Yannis Paschalidis, Boston University Sensor Network Routing: Performance


Bound on latency probability P(Lp ≥ d)

Latency probability is hard to compute exactly. Consider boundsand approximations ...

Chernoff bound: For any θ ≥ 0

P(Lp ≥ d) ≤ exp

∑

k∈p

Λk(θ) − θd

,

where Λk(θ) is the logarithmic moment generating function ofLk (it depends on the transition probability matrices of channeland sleeping schedule Markov chains).



Large Deviations-type asymptotic for P(Lp ≥ d)

Let λk the maximum eigenvalue of Hk (a matrix that depends onthe transition probabilities of the Markov chains).

Theorem

For any p ∈ P, we have

limd→∞

1

dlog P(Lp ≥ d) = max

k∈pλk .

where λk is a quantity that characterizes link k.

Interpretation: P(Lp ≥ d) ≈ edmaxk∈p λk , i.e., it decreases

exponentially w.r.t. d .

“Bottleneck link”.

Convenient to characterize and update.



Energy vs. delay trade-off

Generally, E(Tp) and P(Lp ≥ d) cannot be minimized on thesame path p ∈ P.

A possible formulation to capture the trade-off

minp∈P∑

k∈p E(Tk)

s.t. P(Lp ≥ d) ≤ �.

Resource constrained shortest path problem. NP-hard.

Alternative objective: weighted sum of E(Tp) and (Chernoffbound) exponent of P(Lp ≥ d).

minp∈P

(

E(Tp) + β minθ≥0

(Λp(θ) − θd)

)

where β is a positive constant, Λp(θ) =∑

k∈p Λk(θ).

Nontrivial global optimization problem.

10/17 Yannis Paschalidis, Boston University Sensor Network Routing: Routing algorithms


Energy vs. delay trade-off (cont.)

Exchange the order of minimization

minθ≥0

(

−βθd + minp∈P

(E(Tp) + βΛp(θ))

)

The inner minimization is a shortest path problem.Objective function: continuous, piecewise convex w.r.t. θ.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8

θ

min

p E(T

p)+Λ

p(θ)

−θ d

Typical shape of the objective function



Solution approaches

Approach I: Convex polynomial underestimation approach(centralized).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8

θ

min

p E

(Tp)

+Λp(

θ)−θ

d

Typical shape of the objective function



Solution Approaches (cont.)

Approach II: Simulated annealing based approach.

Scoring function:

v(p) = E(Tp) + β minθ≥0

(Λp(θ) − θd)

The second term is a surrogate for β log P(Lp ≥ d).Large deviations-based scoring function:

log P(Lp ≥ d) ≈ log(−λpE(Lp)) + dλp

v(p) = E(Tp) + β log(−λpE(Lp)) + βdλp

where λp , max{λk | k ∈ p}.Arbitrarily choose an initial path.Change the path configuration locally.Accept/reject the new path according to the Metropoliscriterion.Distributed algorithm.



Simulated annealing algorithm

(Simulated annealing algorithm)


annealing.wmvMedia File (video/x-ms-wmv)


Optimization over duty cycles

Nodes have power consumption qi (i = 1, . . . , N) when theyare ON, but NOT transmitting any packets.

What is the fraction of time nodes should stay ON?

This adds a new dimension to the problem over which one canfurther optimize.

Solution approach: simulated annealing algorithm combinedwith local minimization methods.



Numerical results: solution quality comparison

Setting:

300 networks with random topologies.Each network has 50 nodes and 1 gateway.

fmin: benchmark optimal value obtained by exhaustive search.

Empirically, centralized method always finds the optimalsolution.Simulated annealing based algorithm: objective value f ∗.

f ∗−fmin|fmin|

5% 10% 15% 20%

% of cases 87.0% 88.0% 92.7% 95.0%

Mean = 2.2%, Std = 8.8%.

Using our large deviations approximation: 1 order ofmagnitude faster with less communication overhead andsimplified computation.Further optimizing over duty cycles: In 92/100 instances hadimprovement, on average by 16.5%.



Summary

Considered the expected energy consumption and the latencyprobability in sensor networks in the presence of varyingchannel conditions and sleeping schedules.

Energy vs. latency trade-off.

Large deviations asymptotic for the latency probability.

Centralized and distributed solution approaches:

There are significant gains from optimizing the duty cycle andthe sleeping schedule.We considered a randomized sleeping schedule (no timesynchronization, security advantages).

17/17 Yannis Paschalidis, Boston University Sensor Network Routing: Conclusions

IntroModelPerformanceRouting algorithmsConclusions

minimal energy routing with latency quality-of-service ...9/17 yannis paschalidis, boston university...

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