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Maximizing the Functional Lifetime of Sensor Networks Arvind Giridhar P.R. Kumar Coordinated Science Laboratory Coordinated Science Laboratory University of Illinois, Urbana. University of Illinois, Urbana. Email: giridhar@uiuc.edu Email: prkumar@uiuc.com Abstract— The functional lifetime of a sensor network is deﬁned as the the total energy consumed may not be optimal for network lifetime, maximum number of times a certain data collection function or task can due to the distributed nature of the communication burden as well as be carried out without any node running out of energy. The speciﬁc task energy supplies. considered in this paper is that of communicating a speciﬁed quantity of information from each sensor to a collector node. The problem of Slight variants of this problem, with essentially the same modeling ﬁnding the communication scheme which maximizes functional lifetime assumptions and notion of lifetime (i.e., time until the ﬁrst node can be formulated as a linear program, under “ﬂuid-like” assumptions failure) have been considered in a number of papers in the literature, on information bits. This paper focuses on analytically solving the linear in the context of energy aware or maximum lifetime routing. Chang program for some simple regular network topologies. The two topologies considered are a regular linear array, and a and Tassiulas [2] consider the problem of maximizing lifetime regular two-dimensional network. In the linear case, an upper bound given a certain set of source-destination information rates that must on functional lifetime is derived, as a function of the initial energies and be supported. Similar approaches, leading to linear and/or integer quantities of data held by the sensors. Under some assumptions on the programming formulations which are basically the same as in this relative amounts of the energies and data, this upper bound is shown to be achievable, and the exact form of the optimal communication strategy paper, have been employed in [3–5]. Most of these papers have is derived. For the regular planar network, upper and lower bounds on focused on ﬁnding distributed algorithms to ﬁnd the optimal routing functional lifetime, differing only by a constant factor, are obtained. strategy to maximize lifetime, without speciﬁcally analyzing what Finally, it is shown that the simple collection scheme of transmitting this solution is. only to nearest neighbors, yields a nearly optimal lifetime in a scaling In this work, we consider a restricted version of the problem sense. considered in the above papers. Two regular spatial topologies are I. I NTRODUCTION studied, a linear array and a planar circularly symmetric network. Each of these consist of many sensors and one collector. Under these Consider a network of sensors, each with a certain quantity of restrictions, we are able to provide sharp analytical bounds, and in data to be communicated to a designated collector node. The sensor some cases exact solutions, to the functional lifetime problem. These nodes are low power devices, while the collector node might be analytical results cannot directly be translated to practical distributed an information processing center, or a more energy-rich device algorithms, since constructing the optimal schemes requires non-local functioning as a gateway to a higher bandwidth wireless network information. The advantage of an analytical approach however lies in (such as in [1]). Information packets can be relayed - there is no the structural insight that it provides; while the linear programs could compression or processing allowed - through an arbitrary sequence of equally well be solved numerically, the numbers may not provide nodes before reaching the collector. There is an energy cost associated such insight. Speciﬁcally, this paper attempts to shed light on natural with each transmission, which consists of a cost to transmit, that is questions of interest such as: a function of the transmitter-receiver distance, as well as a cost to • How does lifetime depend on the relative quantities and distri- receive. bution among sensors of data that is to be collected? The above is a somewhat idealized model of a sensor network, • What is the structure of routing strategies that are optimal with with a very simple deﬁnition of a data collection task. In this respect to lifetime? paper, we examine the notion of functional lifetime, which is deﬁned • How does lifetime scale with the size of the network, and/or as follows: Given a quantity of data and an energy budget (for quantity of data to be collected? transmitting and receiving) at each node, the functional lifetime is • How well do simple routing strategies perform in comparison the maximum number of times the task of delivering all the data to the optimal strategy? to the collector node can be repeated before some node runs out of energy. Alternatively, it is the maximum common scale factor by We take the approach of addressing these questions for simple which all the quantities of data at all the sensors can be scaled, while network topologies, which nevertheless are fairly good representatives ensuring that it can be delivered to the collector node without some of multi-hop networks. Indeed, our results are indicative of what can node running out of energy. be expected in the broader class of “more or less” regular networks In general, a sensor network may be simultaneously performing a as well. Our main contributions can be summarized as follows: variety of tasks, including sensing, processing and communication. 1) We provide an upper bound on the functional lifetime for In such a context a natural question to ask is the “cost” to longevity regular linear networks, which is valid for arbitrary energy associated with carrying out a speciﬁc task of interest, and how proﬁles and data distribution. such a cost is to be minimized. The per task functional lifetime 2) Under some restrictions on the energy proﬁle and data distribu- deﬁned above addresses such a question for communication tasks; tion, this upper bound is shown to be achievable, and the exact its inverse represents the maximum fraction of any node’s energy, form of the optimal solution is given. This optimal solution i.e., or “fraction of lifetime” of the network, that is consumed in causes all nodes to die simultaneously. performing the task. The important point to note is that minimizing 3) Similar results are obtained for a class of regular planar networks. Collector 4) The optimal solutions are compared to a simple suboptimal 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 routing strategy, which consists of each node forwarding all 1 2 3 n data to its nearest neighbor in the direction of the collector. Fig. 1. A regular linear sensor network. The results indicate that the simple strategy is nearly optimal in a scaling sense. All the above results depend on a speciﬁc property of the energy cost Consider the following linear program: function, which is derived in Lemmas 3 and 4 in the appendix. Min z (1) Analytical upper bounds on lifetime are also derived in [6]. These subject to: bounds are not very sharp for the “many-to-one” information ﬂows that we consider here, since they deal with total energy consumption 1 λij f (d(i, j)) + λji fR ≤ z, 1 ≤ i ≤ n (2) rather than per-node energy consumption. Ei 0≤j≤n 1≤j≤n One minor point needs to be noted. While most of the afore- mentioned papers have considered lifetime as being in actual time λij − λji = bi , 1 ≤ i ≤ n (3) units, as a function of sets of desired information rates, we have 0≤j≤n 1≤j≤n deﬁned functional lifetime as being the number of times a task can λij ≥ 0, 1 ≤ i, j ≤ n. be repeated. At one level, this is merely an issue of semantics; one Here λij denotes the amount of information transmitted from node i deﬁnition can be mapped to the other by mapping a time unit to to node j, and z is essentially the maximum of the normalized costs, a “round,” which is the time taken for one repetition of the data or energy cost divided by initial energy, incurred by each node. The collection task. However, in the former approach one also needs to set of λij ’s satisfying the constraints speciﬁes a communication ﬂow address the feasibility of a particular communication ﬂow, which is from the set of sensors to the collector node. speciﬁed in bits per time unit, given the bandwidth and interference constraints of a wireless network. This formulation is in a different form than the linear program described in [2], but the two can be mapped to each other via simple In our formulation, however, there is no limitation on the time change of variables. The following lemma establishes the connection taken to perform the required communication. For instance, we could between the functional lifetime problem and the linear program (1). allow for the extremely inefﬁcient strategy of only one transmitter Lemma 1: The solution to the optimization problem (1) yields in the entire network being active at a time. Such a strategy might the strategy that maximizes the number of times the information have an extremely large associated delay, but the energy consumption collection process described above can be repeated before some node would be as speciﬁed by our modeling assumptions. Thus, our dies. The functional lifetime is given by z1 , where z ∗ is the optimal ∗ formulation clearly separates the issue of lifetime optimization from value of the objective function. scheduling, delay, and achievable throughputs. In practice, however, Proof. Assuming the ﬂuidity of information, any feasible solution to sensor network routing algorithms must jointly consider all these. (1) can be translated to a communication scheme with appropriate Characterizing the tradeoffs between lifetime, throughput and delay scheduling, since the ﬂow conservation constraints (3) are satisﬁed. remains an open problem. If λ∗ is the solution to the optimization problem, with optimal value The outline of the rest of the paper is as follows. Section II bi λ∗ describes the model and linear programming formulation. Sections z ∗ , one can replace all the bi ’s by z ∗ and the λ∗ ’s by zij . Then, ij ∗ III and IV deal with the linear and planar networks, respectively. from the constraints (2), we have that for each i, Section V obtains scaling laws for the optimal lifetime, followed by 1 a discussion of conclusions and future work. λij f (d(i, j)) + λji fR ≤ Ei , (4) z∗ 0≤j≤n 1≤j≤n II. M ODEL , A SSUMPTIONS AND F ORMULATION OF L INEAR meaning that no sensor runs out of energy. Thus, the optimal P ROGRAM communication scheme can repeat the operation z1 times without ∗ any sensor running out of energy. The general problem setting we consider is very similar to those On the other hand, any communication scheme to repeat the considered in [2–5]. Suppose there are sensors 1, 2, . . . , n located on operation K times, i.e., transmit Kbi bits from each node to the a plane, along with a collector node labeled 0. Sensor i located at collector, must involve sending bits between nodes in such a way so (xi , yi ), has bi bits to send to the collector node, and has initial as to satisfy the ﬂow conservation constraints (3). Furthermore, the energy level Ei . The energy consumed by node i in sending m total energy consumed by each node i must be no more than Ei , units of information to node j, which is a distance d(i, j) away, so a constraint of the form (4) must be satisﬁed as well. Thus, the is mf (d(i, j)), while the energy consumed by j (for j = 0; we do constraints (3) and (2) are necessary as well as sufﬁcient, and z1 is ∗ not count the energy consumed by the collector node) in receiving the maximum number of times the operation can be repeated. m units of information, is fR m, for a given constant fR ≥ 0. We The linear program given above can be numerically solved for any seek the communication scheme that maximizes the number of times number of nodes, any spatial placement, and energy function. The the information collection process, i.e., communicating bi bits from rest of the paper, however, will be devoted to explicitly solving this each sensor to the collector node, can be repeated before one of the problem in some restricted cases. sensors dies, i.e., has no remaining energy. III. F UNCTIONAL L IFETIME OF R EGULAR L INEAR NETWORKS We make the simplifying assumption that information is inﬁnitely divisible and incompressible. As a consequence of this “ﬂuid-like” Consider the uniform linear conﬁguration (Figure 1), where n assumption, ﬂow conservation is preserved. sensors spaced evenly at a distance d from each other on a line, with a single collector node located at the leftmost point. The collector z ∗ is lower bounded as follows: node is located at the origin, and sensor i at the point (id, 0). n f 1− f1 n The energy cost function f (·) is given by b j=i+1 j i=1 i n f 1+ fR z∗ ≥ j=i j n f . (7) f (x) = Et xα eγx , (5) n Ei j=i+1 1− f1 j i=1 fi n f 1+ fR j=i j where α ≥ 2 is the path loss exponent, γ ≥ 0 the absorption coefﬁcient, and Et is some positive constant. We make the following Proof. We ﬁnd a feasible solution to the dual program of (6) with assumption on the constant fR , which represents the energy to receive objective function equal to the RHS of (7). The result then follows a unit of information: by weak duality. Assumption: fR ≤ Et dα (1 − (1/2)α−1 ). Associating the ﬁrst n constraint equations in (6) with vari- This is a fairly reasonable condition; for α = 3, it states that ables y1 , y2 , . . . , yn , and the last n constraint equations with 3 the power to receive must be no more than 4 times the power to w1 , w2 , . . . , wn , the dual linear program has the following form: transmit to the nearest neighbor. As stated, it is a sufﬁcient condition for Lemma 4 to hold; we expect that a somewhat weaker condition Max b1 y1 + b2 y2 + . . . + bn yn (8) would also sufﬁce. Nonetheless, the structure of the energy cost subject to: function, together with this assumption, are critical to all the results E1 w1 + E2 w2 + . . . + En wn ≤ 1 in this paper. The only speciﬁc property we require of the energy cost function is stated in Lemma 4, and this property is essential −f1 w1 + y1 ≤ 0 for any of our results to hold. Essentially, what it describes is the −f1 w1 − fR w2 + y1 − y2 ≤ 0 relative costs of short and long hops; the disadvantage of a long hop −f2 w1 − fR w3 + y1 − y3 ≤ 0 is that due to the sharp degradation in signal power with distance, it is ··· more inefﬁcient in terms of transmit power than multiple short hops (see [7]). However, multiple short hops involve more receptions and −fn−1 w1 − fR wn + y1 − yn ≤ 0 thus increase the power spent in receiving, and in addition increase −f2 w2 + y2 ≤ 0 the energy consumption of the relaying nodes. The optimal strategy −f1 w2 − fR w1 + y2 − y1 ≤ 0 described in Subsection II B optimally balances these two effects. −f1 w2 − fR w3 + y2 − y3 ≤ 0 However, if power to receive is too high (i.e., the assumption on fR is violated), then relaying becomes too expensive, and such a strategy ··· is no longer optimal. −fn−2 w2 − fR wn + y2 − yn ≤ 0 The form of the cost function f (·) given by (5), and the assumption ··· on fR will be implicitly assumed for the rest of the paper. −fn wn + yn ≤ 0 Denote fi := f (id) for brevity. The linear program (1) has the −fn−1 wn − fR w1 + yn − y1 ≤ 0 following form: ··· Min z (6) −f1 wn − fR wn−1 + yn − yn−1 ≤ 0 subject to: w1 , w2 , . . . , wn ≥ 0. E1 z − λ1j f|j−1| + λj1 fR ≥ 0 Consider the following choices of yi ’s and wi ’s: For each 1 ≤ i ≤ 0≤j≤n 1≤j≤n n, set n f 1− f1 j=i+1 j n f E2 z − λ2j f|j−2| + λj2 fR ≥ 0 j=i 1+ fR j yi 0≤j≤n 1≤j≤n yi = n f and wi = . (9) 1− f1 fi ··· n Ei j=i+1 n f j i=1 fi 1+ fR j=i j En z − λnj f|j−n| + λjn fR ≥ 0 0≤j≤n 1≤j≤n Then, b y = z ∗ in equation (7), so the result is proved if i i i λ10 + λ12 + . . . + λ1n − λ21 − λ31 − . . . − λn1 = b1 this set of choices actually constitutes a feasible solution to the dual. λ20 + λ21 + . . . + λ2n − λ12 − λ32 − . . . − λn2 = b2 Clearly, the wi ’s are all non-negative. Now, ··· n 1− f1 f Ei j=i+1 j λn0 + λn1 + . . . + λnn−1 − λ1n − . . . − λn−1n = bn fi n f 1+ fR j=i j λij ≥ 0, for each 1 ≤ i, j ≤ n. Ei wi = n f = 1. 1− f1 i i n Ei j=i+1 j i=1 fi n f 1+ fR j=i j A. Upper Bound on Lifetime Also, −fi wi + yi = 0 for each 1 ≤ i ≤ n, by deﬁnition of wi . Next, We now give the ﬁrst main result, which provides an upper bound it must be proved that for each 1 ≤ j ≤ n, 1 ≤ k ≤ n, to functional lifetime via linear programming duality. Theorem 1: Let z ∗ be the optimal value of the linear program (6). −f|k−j| wj − fR wk + yj − yk ≤ 0. For k > j, we have k 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 fk−j i=j+1 1 − f1 fi k n −fk−j wj + yj = 1− k−1 yk Collector fj 1 + fR i=j fi Fig. 2. The optimal communication “ﬂow” for node k. fR < 1+ yk . fk Thus, the only remaining case is in which k < j, for which we Suppose that the equations are indeed solvable and that the need to prove that appropriate solutions are all nonnegative. We can then explicitly calculate the objective function z ∗ . Solving (11) and (14), we get fj−k fR 1− yj ≤ 1+ fk E1 ∗ z + b1 fR fj fk f1 f1 λ∗ 10 = fR , (17) j 1+ fR i=k+1 1 − f1 fi f1 = 1+ yj . E1 ∗ fk j−1 1 + fR f1 z − b1 i=k fi λ∗ 21 = fR . (18) In other words, 1+ f1 fj−k j 1− f1 Substituting these in (12) and (15) and solving for λ∗ , we have 32 i=k+1 fi 1− ≤ j−1 . fj 1+ fR E1 (1 − f1 ) b1 (1 − f1 ) i=k+1 fi fR E2 f1 f2 f2 λ∗ (1 + 32 ) = z∗ + fR − fR − b2 . This is proved in Lemmas 3 and 4 in the Appendix. f2 f2 1+ f1 1+ f1 The crucial step in this proof is the validity of Lemma 3, which guarantees the feasibility of the conjectured dual solution. The Repeating this calculation, we obtain the following expression for validity of this Lemma depends on the exact structure of the cost λ∗ kk−1 , function f (·), as will be evident in its proof. k−1 k−1 f1 k−1 k−1 f1 Ei j=i+1 1− fj j=i+1 1− fj B. Attaining the Dual Upper Bound λ∗ kk−1 =z ∗ k−1 − bi k−1 . (19) fi 1+ fR 1+ fR i=1 j=i fj i=1 j=i fj It turns out that the RHS of (7) is the objective function value corresponding to a particular assignment of variables for the primal By setting λ∗ n+1n = 0 (since there is no node n + 1), we then obtain problem, which is however not always feasible. But when it is, it is the following expression for z ∗ : automatically optimal due to strong duality. We now describe how to n f 1− f1 obtain this assignment of variables. n b j=i+1 j i=1 i n f 1+ fR Consider the following 2n equations: z∗ = j=i j n f , (20) 1− f1 Feasible ﬂow constraints (10) n Ei j=i+1 j i=1 fi n fR λ∗ f1 + λ∗ fR 10 21 = z ∗ E1 (11) j=i 1+ f j λ∗ f2 20 + λ∗ f1 21 + λ∗ fR 32 = z ∗ E2 (12) which is the same as the upper bound derived in Theorem 1. The ··· above calculations show that the equations (10) do indeed admit λ∗ fk + λ∗ ∗ z ∗ Ek solutions. However, these solutions correspond to a valid ﬂow only k0 kk−1 f1 + λk+1k fR = (13) if the λ∗ ’s and λ∗ k0 kk−1 ’s are non-negative. Duality thus yields the ··· following result. λ∗ fn + λ∗ n0 nn−1 f1 = z ∗ En , Theorem 2: If the set of equations (10) have non-negative solu- λ∗ 10 − b1 = λ∗ 21 (14) tions, the corresponding communication ﬂow, in which each node i λ∗ + λ∗ − b2 = λ∗ (15) communicates only to the two nodes, node i − 1 and the collector 21 20 32 node 0, is optimal with respect to the primal linear program (6). The ··· normalized cost incurred by each node under this ﬂow is the same, λ∗ ∗ k−1k−2 + λk−10 − bk−1 = λ∗ kk−1 (16) and consequently all nodes in the network die at the same time. The ··· optimal functional lifetime is given by the inverse of the expression λ∗ ∗ λ∗ in (20). n−1n−2 + λn−10 − bn−1 = nn−1 Figure 2 shows the form of the optimal strategy. The proof of λ∗ nn−1 + λ∗ n0 − bn = 0, Theorem 2, depending as it does on the construction of the dual with unknowns λ∗ ii−1 and λ∗ i0for each 1 ≤ i ≤ n, and z ∗ . If solution and the solubility of the set of equations (10), seemingly these equations are solvable, and if the λ∗ ’s and λ∗ ’s are all ii−1 i0 gives no intuitive reason why such a particular combination of long nonnegative, then the solution to these equations clearly yields a and short hops is optimal. In fact, it is possible to construct a feasible solution to the original linear program (6) (after setting all direct proof that such a strategy is optimal, by employing variational the other ﬂows to zero). The communication ﬂow deﬁned by this arguments to show that any other strategy is suboptimal. Such a proof solution consists of each node sending a part of its information to is much more lengthy than the one provided here. It turns out that the collector node, and the rest to its nearest neighbor in the direction the inequality proved in Lemma 4 is one of the key steps in this of the collector. The exact quantities are chosen in such a way as to alternate proof. equalize the normalized costs incurred by each node (if possible, Theorem 2 provides sufﬁcient conditions which are purely in terms otherwise the solution is not valid). This corresponds to all nodes of the bi ’s and Ei ’s. However, the form of the conditions is somewhat depleting their energy supply at the same time. involved. The following lemma gives simpler sufﬁcient conditions. 1 1 1 0 0 0 Lemma 2: If bEfi and Ei are non-decreasing in i, then the set of i 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 i equations (10) have non-negative solutions. 0 1 Proof. We ﬁrst prove that λ∗ 00 0 0 0 0 0 11 1 1 1 1 1 k+1k ≥ 0 for all k. For convenience, 11 1 1 1 1 1 00 0 0 0 0 0 j 1− f1 f 0 1 000000000 111111111 denote g(i, j) := l=i+1 l . Substituting (19), this is the same 000000000 111111111 j fR 1 0 11 1 1 11 1 00 0 0 00 0 1+ f 00 0 0 00 0 11 1 1 11 1 l=i l as proving that 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 k b g(i, k) z∗ ≥ i=1 i . 0 0 0 1 1 1 k Ei 1 1 1 0 0 0 0 1 i=1 fi g(i, k) Fig. 3. The regular planar network Cross multiplying and canceling the common terms, we need to prove that n k Ei +fR bk bi g(i, n) g(i, k) f1 fi 1 − fk−1 i=k+1 i=1 = fR (Ek−1 z ∗ − (fR + f1 )λ∗ k−1k−2 + fR bk−1 ) n Ei k 1+ fk−1 ≥ g(i, n) bi g(i, k) . fR f1 fi fk−1 + fR − fR + fk−1 i=k+1 i=1 +bk−1 fR + bk fR Now, for any k + 1 ≤ l ≤ n, and 1 ≤ m ≤ k, by assumption we 1+ fk−1 have bl Em ≥ bm El . This implies that the lmth cross term in the f m fl ≥ fR bk + f1 bk−1 (23) LHS is greater than or equal to the lmth cross term in the RHS, for ≥ 0, each l and m. This proves that λ∗ k+1k ≥ 0 for all k ≥ 0. What is left to prove is that λ∗ ≥ 0 for all k. From (17), it is k0 where (22) follows from the assumption that Ek ≥ Ek−1 , and (23) clear that λ∗ > 0. Solving for λ∗ in (13) and (16), we obtain that 10 k0 from the induction hypothesis. for k ≥ 2, fR 1 λ∗ (1 + k0 )= (Ek z ∗ − (fR + f1 )λ∗ kk−1 + fR bk ). IV. F UNCTIONAL L IFETIME OF A R EGULAR P LANAR N ETWORK fk fk Thus, we need to prove that for 2 ≤ k ≤ n, Consider the planar network shown in Figure 3. The center node ∗ Ek z − (fR + f1 )λ∗ + fR bk ≥ 0. is the collector, and there are N concentric circles, each containing kk−1 nodes along its circumference. The ith ring, of radius iR, contains We prove this using induction on k. For the base case k = 2, we M i nodes, equally spaced along the circumference. There are thus have a total of N(N+1) M nodes. This particular conﬁguration for the 2 network is chosen simply as a convenient example of a regular planar E2 z ∗ − (fR + f1 )λ∗ + fR b2 21 network, due to its circular symmetry. Any other regular arrangement E1 ∗ f1 z − b1 of nodes would admit similar results, though the analysis might be ≥ E1 z ∗ − (fR + f1 ) fR + fR b2 (21) more cumbersome. 1+ f1 The direct approach of the last section is considerably harder in = b1 f1 + b2 fR ≥ 0, the planar case. However, we can use the following simple idea to where (21) follows by substituting (18) and the assumption that E2 ≥ obtain an upper bound. Suppose that all intra-ring communication, E1 . Suppose now that the induction hypothesis is true for k − 1. i.e., communications between nodes belonging to the same ring, have Substituting for λ∗ k−1k−2 in (19), we get that zero cost. Further, let the cost of a unit of transmission from any f1 Ek−1 ∗ node in ring i to any node in ring j = i be f ((i − j)R), i.e. as if the 1− fk−1 fk−1 z − bk−1 distance between the two nodes were equal to the distance between λ∗ kk−1 = fR λ∗ k−1k−2 + fR . 1+ 1+ the two rings (when in fact the former is larger than or equal to the fk−1 fk−1 latter). Since all link or hop communication costs are thereby either Then, reduced or remain the same, it is clear that such a modiﬁed cost network would have functional lifetime greater than or equal to the Ek z ∗ − (fR + f1 )λ∗ kk−1 + fR bk original planar network. Therefore, an upper bound on the lifetime f1 of this modiﬁed cost network is also an upper bound on the lifetime 1− ≥ Ek−1 z ∗ − (fR + f1 ) fk−1 λ∗ of the original network. fR k−1k−2 1+ fk−1 The treatment of this modiﬁed cost network is simpliﬁed by the Ek−1 ∗ fact that it is equivalent to the following linear network: Replace z − bk−1 −(fR + f1 ) fk−1 + fR bk (22) the ith ring with a super-node si , located at a distance iR from the fR 1+ fk−1 collector node. The initial energy of this super-node is j Eij , and fR fR f1 the number of bits initially held by it is equal to ij bij , where Eij Ek−1 + Ek−1 fk−1 − E fk−1 k−1 − E fk−1 k−1 = z∗ and bij are the amount of energy and number of bits respectively fR 1+ fk−1 of the j th node in ring i of the original planar network. The set of 1− f1 super-nodes thus forms a linear array with inter-node distance R. The fk−1 (fR + f1 )bk−1 +(fR + f1 ) fR λ∗ k−1k−2 + fR upper bound of Theorem 1 can therefore be directly applied (with 1+ fk−1 1 + fk−1 fi now denoting f (iR)) to obtain the following upper bound on the a dividing j by i, i.e., j = pi + q. This node communicates a fraction Bi Ai +Bi of its total load to the collector node, and the remaining to ring (i−1). Of these remaining bits, fractions i−1 and i−1−q respectively q i−1 (p(i−1)+q−1)2π are transmitted to nodes M (i−1) and (p(i−1)+q)2π of the i − M (i−1) θ 1th ring. This ensures that for each i ≥ 2, each node in the (i − 1)th ring e b d i receives a total of i−1 Bi bits from ring i. To see this, suppose that each node in the ith ring sends the same number of bits to the (i−1)th ′ ′ ring. In the (i − 1)th ring, node (p (i−1)+q )2π receives a fraction M (i−1) f q ′ +1 from node (p′ i+q ′ +1)2π in ring i, and a fraction i−1−q ′ from i−1 Mi i−1 ′ ′ (p i+q )2π i node in ring i, for a total of times the number of bits c Mi i−1 sent by each node in ring i to ring i − 1. Backwards induction from Fig. 4. Bounding the constant K. the outermost node, together with the ﬂow conservation constraints, complete the proof. What is the normalized cost incurred by each node? The distance 1 functional lifetime ∗ zp of the regular planar network: between each node in the ith ring and the collector node is iR. Mi n f However, the distance between each such node and the node(s) it 1− f1 N j=1 Eij j=i+1 j transmits to in the (i − 1)th ring is greater than R. Let the largest n f 1 i=1 fi j=i 1+ fR j such distance for any ring be KR. In that case, each node in the ith ∗ ≤ n f . (24) ring incurs a normalized cost that is less than K times the normalized zp N Mi 1− f1 i=1 b j=1 ij j=i+1 n f j cost incurred by the ith super-node under the optimal scheme derived 1+ fR j=i j above (because total load and energy are scaled by the same factor This upper bound is valid for any set of Eij ’s and bij ’s. M i). To provide lower bounds on functional lifetime, however, we will The factor K can be bounded as follows. Consider Figure 4. Note 2π have to impose further restrictions on the energies and information that the angle θ = M i , |af | = iR, and |f c| = R. The two nodes in th quantities of the sensor nodes. Suppose that it is further true that for the i ring nearest to node j must both be within the arc ef d. Thus, each ring i, the maximum distance between j and any of these nodes is no more than |cd|. We then have, Eij = Ei and bij = ¯i , for all 1 ≤ j ≤ M i, ¯ b (25) and that ¯i fi |cd|2 = |bc|2 + |bd|2 b ¯ and iEi are non-decreasing in i. (26) ¯ Ei = (iR(1 − cos θ) + R)2 + (iR sin θ)2 , Then, Lemma 2 guarantees the achievability of the following func- = R2 (1 + 2i(i + 1)(1 − cos θ)), tional lifetime for the linear network of super-nodes: = R2 (1 + 2i(i + 1)2 sin2 (θ/2), n f π 2 N ¯ Ei M i j=i+1 1− f1 j ≤ R2 (1 + 2i(i + 1)2( ) , i=1 fi n fR Mi 1 1+ f π ≥ j=i j . (27) ≤ R2 (1 + 8( )2 ). zp∗ n 1− f1 f M N ¯i M i b j=i+1 n f j Thus, i=1 1+ fR π j=i j f (R 1 + 8( M )2 ) Now we construct a routing scheme for the actual planar network K≤ . (28) f (R) which achieves this solution to within a constant factor. This scheme The results of this section are summarized in the following attempts to mimic the optimal scheme for the linear super-node theorem. network, with each ring equally dividing the ﬂow that it transmits Theorem 3: Consider the regular planar network of Figure 3. Its to the next inner ring among all the nodes in that inner ring. The optimal functional lifetime is upper-bounded by the right hand side constant factor gap arises because the distance between two nodes in of (24). If the initial energies and information quantities in the planar successive rings is larger than the inter-ring distance. network further satisfy conditions (25) and (26), then there exists a Let λ∗ ∗ ii−1 and λi0 be the respective amounts of information 1 routing strategy achieving a functional lifetime of K times the right communicated from super-node si to super-node si−1 and to the hand side of (27), where K is given by (28). collector node respectively, under the optimal ﬂow. Deﬁne Ai := λ∗ii−1 λ∗ It should be noted that as m → ∞, the factor K approaches 1, Mi , and Bi := M i . i0 yielding asymptotic optimality. This indicates that the routing strategy The nodes in the ﬁrst ring communicate all their bits to the described above is nearly optimal in large dense networks. collector. This scheme will ensure that each node in the ith ring has a total load of Ai + Bi bits, i.e., each node receives an equal V. S CALING I MPLICATIONS amount of incoming ﬂow, and further that each such node will incur We now evaluate the simple routing strategy consisting of commu- the same cost. nicating only with the nearest neighbor in the direction of the collec- Consider the ith ring, for i ≥ 2. Index each node by the angle of tor. Because it involves only short hop communication, this scheme is the radial line connecting the node to the origin. These angles are an appealing one for operational reasons, such as minimizing medium multiples of M i . Consider the node located at the angle 2πj . Let 2π iM access contention. However, from the point of view of lifetime, it is p and q respectively be the quotient and remainder obtained upon not clear whether it is a good routing strategy, because it places a higher load on nodes close to the sink. Thus, it is worthwhile to more powerful communication infrastructure, are more expensive to compare it with the optimal strategy derived in this paper. Henceforth, maintain, whereas sensor nodes are cheap. On the other hand, the it will be referred to as simply the nearest neighbor strategy. number of sensor nodes per collector node is limited by such lifetime Consider ﬁrst the linear case. Let α = 2, γ = 0, and fR = 0. and throughput considerations. One of the motivations for performing These are the smallest values within the range we consider, so the in-network processing is precisely to limit the communication burden lifetime obtained will be the largest possible. Let Ei = E and bi = b on nodes which are close to the collector. for all i, and let the inter-node distance be d = 1. Under the nearest neighbor strategy, the node next to the collector is the the bottleneck, VI. C ONCLUSIONS AND F UTURE W ORK and it will transmit nb bits. Hence, the functional lifetime under this E The main contribution of this paper is to provide sharp bounds, scheme is bf1 n . Now, consider the lifetime corresponding to the optimal strategy. and in some cases exact solutions, to the functional lifetime problem We have for spatially regular networks. Even here, some gaps remain. The n n characterization of the optimal solution holds only under certain f1 (j − 1)(j + 1) i(n + 1) conditions on the energies and trafﬁc requirements of the network. 1− = = . fj j2 (i + 1)n Nevertheless, we believe that these conclusions hold broadly for large j=i+1 j=i+1 sensor networks. Substituting in (24), we get Another interesting extension involves considering more general n 1 1 E i=1 i(i+1) cost functions. In practice, power levels belong to a discrete set, = n i and all pairs of nodes may not be connected. Such effects could be z∗ bf1 i=1 i+1 n 1 1 modeled by suitably changing the cost function. The key property E i=1 i − i+1 of the cost function that underlies all our results is the inequality in = n 1 bf1 i=1 1 − i+1 Lemma 4. It would be worth investigating if a similar result holds E 1 1 for more general functions. ≈ . One major limitation of our model is that we treat information as bf1 n + 1 1 − log n n incompressible, whereas in-network processing [8] and compression Thus, not only is the optimal lifetime of the same order as the of correlated sources might substantially reduce the relaying burden. lifetime under the nearest neighbor strategy, but the ratio of the two Indeed, our results suggest the need for such techniques to increase converges to 1 as the size of the network goes to ∞. Thus, in the lifetime. linear case at least, the nearest neighbor strategy is nearly optimal. A similar calculation can be performed for the regular planar ACKNOWLEDGMENTS network of Figure 3. The nearest neighbor strategy in this case involves each node in the ith ring transmitting to its two nearest This material is based upon work partially supported by neighbors in the (i − 1)th ring just as in the scheme described in the AFOSR under Contract No. F49620-02-1-0217, NSF under Con- last section. The nodes in the ﬁrst ring would then incur the greatest tract Nos. NSF ANI 02-21357 and CCR-0325716, USARO un- N Kf (R) ib normalized cost, which would be i=1 = N(N+1)bf (R) . der Contract Nos. DAAD19-00-1-0466 and DAAD19-01010-465, E 2E The functional lifetime associated with this scheme is thus upper DARPA/AFOSR under Contract No. F49620-02-1-0325, and DARPA 2E bounded by KN(N+1)bf (R) . On the other hand, the optimal functional under Contact Nos. N00014-0-1-1-0576. lifetime can be bounded by substituting γ = 0 and α = 2 in (24), to get R EFERENCES N [1] A. Mainwaring, D. Culler, J. Polastre, R. Szewczyk, and J. Anderson, 1 1 E i=1 i+1 “Wireless sensor networks for habitat monitoring,” in WSNA ’02: Pro- = N ceedings of the 1st ACM international workshop on Wireless sensor z∗ bf (R) i2 i=1 i+1 networks and applications. ACM Press, 2002, pp. 88–97. N 1 [2] J.-H. Chang and L. Tassiulas, “Maximum lifetime routing in wireless E i=1 i+1 sensor networks,” IEEE Trans. Netw., vol. 12, no. 4, pp. 609–619, 2004. = N bf (R) 1 i − 1 + i+1 [3] A. Sankar and Z. Liu, “Maximum lifetime routing in wireless ad-hoc i=1 networks,” in Proceedings of IEEE Infocom ’04, Hong Kong, 2004, pp. E 2 logN ≈ N−1 log N 1089–1097. bf (R) N (N + 1) N+1 − N(N+1) [4] K. Kalpakis, K. Dasgupta, and P. Namjoshi, “Efﬁcient algorithms for 2E maximum lifetime data gathering and aggregation in wireless sensor ≈ logN. networks,” Comput. Networks, vol. 42, no. 6, pp. 697–716, 2003. bf (R)N (N + 1) [5] R. Madan and S. Lall, “Distributed algorithms for maximum lifetime There is thus a log N factor improvement in this case. Note that for routing in wireless sensor networks,” in Proceedings of IEEE Globecom ’04, Dallas, 2004, pp. 748–753. larger values of α, the ratio between the gap in performance between [6] M. Bhardwaj, A. Chandrakasan, and T. Garnett, “Upper bounds on the the optimal scheme and the nearest neighbor scheme will further lifetime of sensor networks,” in Proceedings of IEEE ICC 2001, Helsinki, reduce. Finland, 2001, pp. 785–790. Some conclusions follow from these scaling results. First, the [7] S. Narayanaswamy, V. Kawadia, R. Sreenivas, and P. Kumar, “Power lifetime scales down sharply with the size of the network. This is control in ad-hoc networks: Theory, architecture, algorithm and im- plementation of the COMPOW protocol,” in Proceedings of European not very surprising in itself, but underscores one of the problems Wireless Conference 2002, Florence, Italy, 2002, pp. 156–162. with the “many sensors one collector” network conﬁguration. This [8] A. Giridhar and P. Kumar, “Computing and communicating functions over seems to be one of the major tradeoffs to be made in sensor networks, sensor networks,” IEEE Journal on Selected Areas of Communication, to since collector nodes, which are essentially gateways to a larger and appear. A PPENDIX We can cancel out the common factor eγa . Letting L(a) := Lemma 3: For any 1 ≤ k < j, e−γa dRHS(a) , R(a) := e−γa dLHS(a) , we have a a fj−k j 1− f1 fi L(a) = (α + γ(a + b))(a + b)α−1 b(b + c)α−1 eγb i=k+1 1− ≤ j−1 , +γbc(a + b)α−1 (b + c)α−1 eγb fj 1+ fR i=k+1 fi +(γa + α)c(a + b)α−1 (b + c)α−1 eγb where fi := (id)α eγ(id) , with α ≥ 2 and γ ≥ 0. −(α + γ(a + b))(a + b)α−1 cα ). (30) Proof. We prove this for ﬁxed k, by induction on j. For j = k + 1, the LHS and the RHS are the same, so the inequality is true with Let the four summands in (30) be L1 , L2 , L3 , L4 . Now, equality. Suppose it is true for some j > k. From the induction hypothesis, R(a) = (γ(a + b) + α)(a + b + c)α−1 eγa bα eγb j+1 f1 f1 +γc(a + b + c)α−1 eγa bα eγb 1− fj−k 1− fj+1 i=k+1 j fi ≥ 1− fR . +K(α + γ(a + b + c))(a + b + c)α−1 . (31) 1+ fR fj 1+ fj i=k+1 fi Let the three summands in (31) be R1 , R2 , R3 . Since (a+b)(b+c) > It is thus enough to prove that (a + b + c)b, we have L1 > R1 and L2 > R2 . It remains to prove 1− f1 that L3 + L4 ≥ R3 . Now, fj−k fj+1 fj+1−k 1− ≥ 1− . fj 1+ fR fj+1 L3 ≥ (α + γa)(1 + γb)c(a + b)(b + c)α−1 fj = (α + γa + γαb + γ 2 ab)c(a + b)(b + c)α−1 Cross multiplying and rearranging terms, this is equivalent to proving that ≥ (α + γ(a + b + c))c(a + b)(b + c)α−1 , (32) fj−k f1 −f1 fj +fj fj+1−k ≥ fj+1 fj−k +fR (fj+1 −fj−k+1 ). (29) since α ≥ 2 and b ≥ c. Thus, Let a := k, b := j − k, c := 1. Adding and subtracting fj+1−k fj−k , L3 + L4 > (α + γ(a + b + c))c(a + b)α−1 ((b + c)α−1 − cα−1 ). and canceling out the common factor d2α (recall that fi = f (id)), Since c ≥ 1, it remains to prove that this is equivalent to proving that (a + b)α−1 ((b + c)α−1 − cα−1 ) ≥ K(a + b + c)α−1 . (33) ((a + b)α eγ(a+b) − bα eγb )((b + c)α eγ(b+c) − cα eγc ) ≥ α γ(a+b+c) α γ(b+c) α γb ′ (a+b)α−1 ((b+c)α−1 −cα−1 ) ((a + b + c) e − (b + c) e )(b e + fR ), Consider the expression P (a) := (a+b+c)α−1 . Differ- ′ fR entiating with respect to a, we get where fR := E t dα . This in turn is proved in Lemma 4, which follows. dP (a) c(a + b)α−2 ((b + c)α−1 − cα−1 ) = (α − 1) da (a + b + c)α Lemma 4: Given any real numbers a ≥ 0, b ≥ c ≥ 1, α ≥ 2, ≥ 0. (34) γ ≥ 0, K ≤ 1 − (1/2)α−1 , Therefore, the ratio is minimum at a = 0. Thus, ((a + b)α eγ(a+b) − bα eγb )((b + c)α eγ(b+c) − (c)α eγ(c) ) ≥ P (a) ≥ P (0) ((a + b + c)α eγ(a+b+c) − (b + c)α eγ(b+c) )(bα eγb + K). cα−1 = bα−1 (1 − ) (b + c)α−1 Proof. The common factors eγ(b+c) can be cancelled out from both 1 ≥ 1 − ( )α−1 ≥ K. sides. Denote the left hand side and right hand side of the resulting 2 inequality, as functions of the variable a, respectively by LHS(a), Note: The conditions on a, b, c and K for the above inequality to hold and RHS(a). may seem somewhat arbitrary. This is mainly due to the presence of F (a) := LHS(a) − RHS(a) is a differentiable function of a, the constant K. If K = 0, the inequality has a more symmetric and F (0) = 0. It is enough to prove that dF (a) > 0 for all a > 0. da structure, and is true for any a, b, c ≥ 0, and α > 1. If, further, We have the constant γ = 0, the inequality follows in straightforward fashion dLHS(a) from Jensen’s inequality, due to the convexity of the function xα = (γ(a + b) + α)(a + b)α−1 eγa ((b + c)α eγb − cα ). da (but not only from its convexity; the inequality does not hold for all Also, convex functions). We are not aware if any form of this inequality is dRHS(a) already known. = (γ(a + b + c) + α)(a + b + c)α−1 eγa (bα eγb + K). a