CENS, April 20 2007 1
Modeling Wireless Sensor Networks
Bhaskar KrishnamachariMing Hsieh Department of Electrical Engineering
USC Viterbi School of Engineering
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Overview
• Mathematical modeling provides fundamental insights into:
1. Link layer behavior
2. Protocol design
3. Scaling and architecture
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1. Modeling Link Layer Behaviorin Low Power Wireless Networks
Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007.
Dongjin Son, Bhaskar Krishnamachari, John Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ’06 + Ongoing work
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• Two simplified models form the basis of >95% of the literature on wireless networks:
X
Circular radio range with perfect reception within &zero reception outside
Collision with simultaneous transmissions within range
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Link Quality Variation with Distance
From Wooet al. ‘03
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An Explanatory Model• Basic idea: compose the following two functions (a) SNR
versus distance with (b) PRR versus SNR
Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007.
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Bimodal PRR Distribution• A majority of the links are either good (above 90%) or bad (below 10%), matching empirical findings (e.g., Cerpa et al. ’05)
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Expectation and Varianceof Packet Reception Rate
Justifies the presence of “long links”
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• Models Incorporated into simulators:– TOSSIM (Berkeley)– Castalia (NICTA, Australia)
• Standalone code at http://ceng.usc.edu/~anrg/downloads.html
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X
Conservative protocol assumption: always a collision
Concurrent Transmissions
Reality: SINR makes the difference
Son, Krishnamachari, Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ‘06
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SINR-view of Interference
)2(8)exp5.01( 10 lfSINRPRR
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Feasibility of Concurrent Transmissions
P1g11/(P2g21+N) ≥
P2g22/(P1g12+N) ≥
S1 R1
R2S2
g11g12
g21
g22
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Linear Topology Case
Counter-intuitive“embedding” of simultaneousconversations
S1 R1 R2S2
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2. MAC Design for ScalableData Collection
Kiran Yedavalli, Bhaskar Krishnamachari, "Enhancement of the IEEE 802.15.4 MAC Protocol for Scalable Data Collection in Dense Sensor Networks", USC Computer Engineering Technical Report CENG-2006-14, November 2006.
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State of the Art: IEEE 802.15.4
• Specifies both PHY and MAC layers for low-power, low-rate embedded wireless networks.
• The MAC protocol is a slotted CSMA with binary exponential back-off
• 256 nodes allowed by standard
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p-persistent CSMA Model
…
: Idle Slot
: Collision Slot
: Successful Slot
epoch
…… … ……
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Delay and Energy Expressions
1. Average expected epoch delay
2. Average expected epoch energy consumption
ξR: Energy Consumption per node per time slot in the Receive State
ξT: Energy Consumption per node per time slot in the Transmit State
1
( 1)(1 )[ ](1 )
n
n n
L L pE Tnp p
1
2 1
( 1)(1 )[ ](1 ) (1 )
n
n R Tn n
L L p LE Ep p p
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Optimality• Delay
• Energy2
1 , 1( , )
2 ( 1)( 1), 1
( 1)( 1)
Topt
Ln
p n Ln n n L n
Ln n L
2 2 ( 1)( 1) 4 ( 1)( 1)( , ) ,
( 1)( 1) 2 ( 1)( 1)E T
optR
n n n L L n np n L
n n L L n
If ξR = ξT, the same transmission probability optimizes both delay and energy simultaneously.
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A Useful Optimality Criterion
0 10 20 30 40 50 60 70 80 90 1000.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of nodes in an epoch (n)
E[T
Idle
,n]/E
[Tn]
L = 5
poptT (n,L)
poptT (n,L) - 0.003
poptT (n,L) + 0.003
,[ ] 2 1, , ( , ) , ,[ ] ( 1)( 2 1 1)Idle nT
optn
E T L Lp p n LE T L L
When the number of contending nodes is high, this provides sensitive feedback that can be used to adapt the access rate
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Receiver Feedback Enhancement
• Receiver performs measurement and broadcast
• Window update rule:
• All contending nodes change the window size simultaneously
,
2 1( 1)( 1)( 2 1 1)
,Current
Idle Currentnext
Current
L LWTL L
WT
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Results
50 55 60 65 70 75 80 85 90 95 1000
20
40
60
80
Number of Contending Nodes
Thro
ughp
ut (K
bps)
Packet Length = 50 Bytes
IEEE 802.15.4Enhanced IEEE 802.15.4
50 55 60 65 70 75 80 85 90 95 1000
2
4
6
8
Number of Contending Nodes
Ene
rgy
(mJo
ules
) IEEE 802.15.4Enhanced IEEE 802.15.4
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3. Fair and Efficient Rate Control for Data Gathering
Avinash Sridharan and Bhaskar Krishnamachari, "Maximizing Network Utilization with Max-Min Fairness in Wireless Sensor Networks," to be presented at 5th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), April 2007.
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Problem Formulation• Allocate rates to each
source to (a) ensure fairness, and (b) efficient use of available bandwidth.
• Closely related prior work by Rangwala et al. SIGCOMM ’06 – focuses primarily on fairness and proposes a TCP-like AIMD mechanism
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• Receiver capacity interference model – source rates from node’s sub-tree and its interfering neighbors’ sub-trees must not exceed available bandwidth
Problem Formulation
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Validating Capacity Model
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• Bottleneck rate turns out to be the minimum supply/demand ratio:
• This can be calculated easily given the tree, interference graph, and receiver bandwidths
P1: Solving for Fairness
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P2: Solving for Efficiency
• Duality-based approach based on the classic work on optimization flow control by Low & Lapsley ’99
• Introduce new dual variables (shadow prices) that weigh resource constraints
• Yields distributed algorithms with market auction interpretation
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Structure of P2’s Lagrange Dual
• Each router sets a price for its bandwidth
• The rate for each source depends on sum-price of routers affected by its flow
sum-price
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• Increment the shadow prices in the direction of the negative sub-gradient (determined by source rates)
• Choose source-rates to maximize component function (determined by shadow prices)
• In general, this could be a very slow iterative process…
Subgradient Optimization
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Good News
• Numerical evidence: setting all shadow prices to 1 provides near-optimal solutions in one iteration!
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Resulting Heuristic
1. First determine and allocate min rate to all sources
2. Give rank to each source that is inversely proportional to the number of downstream receivers whose bandwidth it consumes;
3. Allocate saturating rates to flows, in rank order
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Simulation Results
CDF of difference from optimal solution
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Ongoing Work
• Test-bed Implementation
• Cross-layer extensions
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4. Fundamental Scaling Lawsfor Store and Query Sensor Networks
Joon Ahn and Bhaskar Krishnamachari, "Fundamental Scaling Laws for Energy-Efficient Storage and Querying in Wireless Sensor Networks", ACM MobiHoc, May 2006.
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• Race between increasing supply and demand:- Energy and storage- Application-specific event and query traffic
• The winner of this race determines scalability.
In a Nutshell
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• N nodes deployed in a 2D area with constant density for time T
• m atomic events and qi queries for the ith event, all uniformly distributed
• Can create ri replicas for event i to reduce search cost (at the expense of increased replication cost)
Preliminaries
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Data-Centric Querying Approaches
• Unstructured: expanding ring searches, random walks.
• Structured: Geographic Hash Table, DIFS, DIM
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Energy Cost Scaling
• Creplication = c1
r : # of copies of an event
N : # of nodes
• Csearch(unstructured) = c2 • Csearch(structured) = c3
EVENTEVENT REPLICATIONUNSTRUCTURED QUERYSTRUCTURED QUERY
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Energy Optimization Formulation
S : total storage sizem : the total number of eventsqi : the query rate for ith eventri : the number of copies of ith event
Cs(ri) : the expected minimum search cost of ith event
Cr(ri) : the expected replication cost of ith event
Cr(r) = c1 Cs(r) = c2
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Optimization Solution
Minimizer
The Optimized Total Cost
(inactive constraint)
(active constraint)
qi : # of queries for event i
N : # of nodesS : total storage
sizem : # of events
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Optimal Total Cost
Simplified, assuming : q : # of queries per event
N : # of nodesS : total storage
sizem : # of eventsif
if
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Illustration of Energy Scaling
m : # of eventsq : # of queries
per event
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I - Storage and Energy Scalability Results
Energy ConditionThe energy requirement per node is boundedif and only if mq1/2 = O(N1/4)
Energy constraint is stricter than storage constraint
m : # of eventsq : # of queries per eventN : # of nodes
Storage ConditionA network scales efficiently with bounded storage per node
if mq1/2 = o(N3/4)
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II - Fixed Energy Budget Results
S – successful operation region
N : # of nodese: per-node energy budget
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III - Network Lifetime Scaling Results
Network Lifetime as a function of Network Size
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Summary• Only certain classes of applications can be sustained in arbitrarily
large sensor networks.
• Specifically, if mq1/2 = O(N1/4) for unstructured networks, and mq2/3 = O(N1/2) for structured networks:
a. The network can operate with bounded energy and storage per node.
b. The network lifetime does not decrease with network size for a given energy budget.
• The results can be reinterpreted to understand how to tier sensor networks into zones with localized queries
• These results generalize in a straightforward manner to 1D and 3D deployments. 3D deployments are inherently more scalable.
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Final Thoughts
“In theory, theory and practice are the same; in practice, they’re different.”
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Thanks
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