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Maximizing the Functional Lifetime of Sensor Networks Arvind Giridhar, P.R. Kumar Coordinated Science Laboratory 4/28/2009 CSL, University of Illinois 1 Data Gathering Sensor Networks Network consisting of multiple sensors, one sink Data collection: – Each sensor has data to be sent to sink (bi bits for sensor i) – Periodically repeated Limited energy: – Sensor i has energy level Ei How to route data to maximize network lifetime? 4/28/2009 CSL, University of Illinois 2 Related Work Tassiulas et al (2001) – Maximum lifetime routing – Derived equivalent linear programming formulation – Developed distributed algorithms to converge to optimal solution Sankar and Liu (2004), Madan and Lall (2004) consider similar problems Bharadwaj, Chandrakasan (2001) – Upper bounds on lifetime based on total energy consumption Our contribution: Consider simple regular networks – Analytically solve/provide sharp bounds, using properties of cost function 4/28/2009 CSL, University of Illinois 3 Network Model Model wireless network as complete graph Any node can transmit to any other node…. ….but at a cost Cost function for communication Energy to transmit = bf (d(i, j)) b bits from node i to node j Energy to receive = bfR 4/28/2009 CSL, University of Illinois 4 Model of Information bits Information bits modeled as fluid – Incompressible and infinitely divisible – Node can transmit arbitrary non-negative quantity of information to any other node 2 2 1 2 b1 b1 1 2 A routing flow is a set of non-negative reals {ij :1 i n, 0 j n} denoting quantities of information transmitted between every ordered pair of nodes 4/28/2009 CSL, University of Illinois 5 Flow Routing Network Energy constraints Given {(x1 , y1 ),...,(xn , yn )} {E1 , E2 ,..., En } Data Collection Task + {b1 ,b2 ,...,bn } We seek flows which route all the bits to the sink while respecting the energy constraints n n That is, ij f (dij ) ki fR Ei , for each i Energy constraints j0 k 1 n n ij ki nbi , for each i Flow conservation j0 k 1 ij 0, for each i, j. 4/28/2009 CSL, University of Illinois 6 Functional Lifetime Let be a flow for task {b1,b2 ,...,bn } The functional lifetime of the network is the maximum number of repetitions L of the flow which is feasible given limited node energies {E1, E2 ,..., En } L repetitions of task {b1,b2 ,...,bn } task {Lb1, Lb2 ,..., Lbn } L repetitions of flow flow L L can be non-integral We seek the routing flow which maximizes the functional lifetime 4/28/2009 CSL, University of Illinois 7 No Direct Dependence on Time Functional lifetime is measured in number of “rounds,” not number of time units – Serves as metric of cost-to-lifetime for data collection task Actual lifetime (in time units) will thus depend on the rate of data generation or sampling of the environment Functional lifetime does not depend on time to transmit Thus, interference does not appear in formulation – Transmissions can be staggered in time so that no two transmitters ever interfere 4/28/2009 CSL, University of Illinois 8 Linear Program Formulation Min z (1) subject to: n n ij f (dij ) ki f R Ei z, for each i j0 k 1 z is the maximum n n fraction of node energy ij ki nbi , for each i consumed in routing j0 k 1 ij 0, for each i, j. LEMMA: The functional lifetime is the inverse of the optimal value z* in the linear program (1). The corresponding flow {ij:1 i,j n} scaled by 1/ z* achieves the optimum. 4/28/2009 CSL, University of Illinois 9 Cost Function Consider following cost function: f (x) Et x e x 2 path loss exponent 0 dispersion coefficient Inverse of distance-attenuation function of signal power over wireless medium Assumption: fR f (dmin )(1 0.5 1 ) For =3, require fR 0.75 f1 4/28/2009 CSL, University of Illinois 10 Network Topology Linear regular topology – Regularly spaced sensors on E1 E2 E3 En a line 0 1 2 3 n – Sink node at one corner b1 b2 b3 bn – Inter-node distance d – Define fi : f (id) 4/28/2009 CSL, University of Illinois 11 “Near-Far Flow” Near-far flow is a flow for which: 1. Each node transmits only to nearest node towards sink, and to the sink • For each node i, ij 0 for 0<j i * 2. Fraction of energy consumed same for all nodes 1 * 1 * 1 * (10 f1 21 fR ) * (21 f1 20 f2 32 fR ) L * * (nn1 f1 n0 fn ) * E1 E2 En Tx En. Rx. En. 4/28/2009 CSL, University of Illinois 12 Uniqueness of Near-far Flow The near-far flow is solution to following linear equations 10 f1 21 fR E1z* * * 20 f2 21 f1 32 fR E2 z* * * * L n0 fn nn 1 f1 En z* * * Equations have unique solution…. 10 21 b1 * * 20 21 32 b2 * * * …but solutions may not be non- L negative n0 nn 1 bn * * Thus, near-far flow exists only if ii 1, i*0 0,1 i n * 4/28/2009 CSL, University of Illinois 13 Main Result THEOREM: Consider the linear regular network. 1. The functional lifetime is upper bounded as follows n Ei n n f1 f gi 1 f j i 1 i 1 i j Tf gin & & n1 n fR b g n i i 1 f i 1 j i j 2. If a near-far flow exists, then the above upper bound is achievable LEMMA: If bifi/Ei and Ei are non-decreasing in i, then a near-far flow exists – Lemma provides sufficient conditions 4/28/2009 CSL, University of Illinois 14 Proof Outline Construct feasible solution to dual program with same objective function value Weak duality proves upper bound Proof of feasibility of dual solution depends on particular structure of the cost function f(.) – Through the following inequality LEMMA: Given a 0,b c 1, 2, 0,fR f (1)(1 0.5 1 ) ( f (a b) f (a))( f (b c) f (b)) ( f (a b c) f (b c))( f (b) fR ) where f (x) : Et e x x 4/28/2009 CSL, University of Illinois 15 Crossing Paths in a Flow are Suboptimal Equivalently… 1 2 1 2 1 2 1 4/28/2009 CSL, University of Illinois 16 Example 1 b1=b2=…=bn-1=bn=b, E1/f1=E2/f2=…=En/fn=E Optimal strategy consists of each node transmitting directly to sink – Lifetime is E/b – Does not depend on fR or n n Ei n n n f gi g E i 1 i E i 1 i Tf n n = b b b g n i i gin i 1 i 1 4/28/2009 CSL, University of Illinois 17 Example 2 b1=b2=…=bn-1=0, bn=b, E1=E2=…=En=E Under optimal strategy: – Node n transmits to sink and node n-1 – Every other node i transmits only to node i-1 – Amounts transmitted are determined by equalizing energy consumed nn 1 f1 (b nn 1 ) fn ( fR f1 )nn 1 * * * f E 1 R fn Tf ( fR f1 )b 4/28/2009 CSL, University of Illinois 18 Regular Planar Network Nodes located on concentric rings N rings Ring i has Mi nodes and radius iR Total of N(N+1)M/2 nodes Sink node at center of circle Other regular networks, such as square lattice, can also be considered – Similar results hold, but more analysis is more cumbersome 4/28/2009 CSL, University of Illinois 19 Mapping to Linear Network 3M 2M M b 3j b 2j b 1j j1 j1 j1 3M 2M R M E 3j E 2j E 1j j1 j1 j1 Map each ring to a “super-node” Super-nodes located on a line at intervals of length R Energy and information bits of each super-node are sums of respective quantities over corresponding ring Lifetime of linear network upper bounds the lifetime of original network 4/28/2009 CSL, University of Illinois 20 Upper bound Therefore… n 1 Mi n n f1 f E j gi 1 f j i1 i j1 j i1 Tf n Mi gi n n Ý Ý f R b j gin 1 f j i1 j1 j i f i f (iR) Ý Ý THEOREM: Under “circular symmetry” of node energies and bits, and existence of a near-far flow for the linear network of super-nodes, a lifetime of K times the above upper bound is achievable, where f (R) K f (R 1 8 / M ) 4/28/2009 CSL, University of Illinois 21 Scaling of Lifetime Take 2, 0, fR 0, d 1 E1 E2 ... En E b1 b2 ... bn b Choice of parameters gives best case lifetime For these parameters, we compare optimal flow to simple nearest neighbor scheme 4/28/2009 CSL, University of Illinois 22 Linear Network Case Simple nearest neighbor scheme – Transmit all to nearest neighbor towards sink E E 1 Tf1 n b nf1 ( f1 f R ) bi f R b1 i1 Optimal flow Summation in denominator can be computed E E 1 1 Tf* n i bf1 (n 1) 1 log n Ratio goes to 1 as bf1 i1 i 1 n network size grows large 4/28/2009 CSL, University of Illinois 23 Planar Network Case Consider circular planar network Similar calculation gives 2E log N Tf* bf (R) N(N 1) While the nearest neighbor scheme has a lifetime lower bounded as follows 2E Optimal is better by Tf1 Kbf (R)N(N 1) factor of log N 4/28/2009 CSL, University of Illinois 24 Implications Thus, simple nearest neighbor communication scheme is nearly optimal Such a scheme is preferable from an operational point of view – Lower interference and medium access contention – Need not use multiple power levels Broader implication: Optimal lifetime scales poorly with network size 4/28/2009 CSL, University of Illinois 25 Future Work Problem needs to be solved for all values of per-node energies and information quantities – Lifetime may be determined by the energy/number of bits of a single node - degenerate case – Need to determine conditions when this happens, and what happens otherwise May need to consider more general cost functions Tradeoff between lifetime and delay? 4/28/2009 CSL, University of Illinois 26

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posted: | 4/29/2009 |

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