A Simple Cooperative Transmission Protocol for Energy-Efficient Broadcasting Over Multi-Hop Wireless Networks

1 A Simple Cooperati v e T ransmission Protocol for Ener gy-Ef ficient Broadcasting Over Multi-Hop W ireless Networks Aravind Kailas, Lakshmi Thanayankizil, and Mary Ann Ingram Abstract —This paper analyzes a broadcasting technique for wireless multi-hop sensor netw orks that uses a form of coop- erative diversity called opportunistic lar ge arrays (OLAs). W e propose a method for autonomous scheduling of the nodes, which limits the nodes that relay and sa ves as much as 32% of the transmit energy compar ed to other broadcast appr oaches, without requiring Global Positioning System (GPS), individual node addressing, or inter-node interaction. This energy-saving is a result of cross-layer interaction, in the sense that the Medium Access Control (MA C) and routing functions are partially ex- ecuted in the Physical (PHY) layer . Our proposed method is called OLA with a transmission threshold (OLA-T), where a node compares its receiv ed power to a threshold to decide if it should forward. W e also inv estigate OLA with variable threshold (OLA-VT), which optimizes the thresholds as a function of level. OLA-T and OLA-VT are compared with OLA broadcasting without a transmission threshold, each in their minimum energy configuration, using an analytical method under the orthogonal and continuum assumptions. The trade-off between the number of OLA levels (or hops) r equir ed to achieve successful network broadcast and transmission energy saved is inv estigated. The results based on the analytical assumptions are confirmed with Monte Carlo simulations. Index T erms —Cooperative Diversity , Network Broadcast, Wir eless Communications, Wireless Sensor Networks I . I N T RO D U C T I O N C OOPERA TIVE transmission for W ireless Sensor Net- works (WSNs) is a relay technique where multiple, spatially separated radios cooperate to transmit the same message so that the recei ver can deriv e diversity gain from the multiple transmissions [1], [2]. Because of the di versity gain, the transmitters can dramatically lower their transmit po wers and sav e energy without sacrificing reliability . This paper pro- poses a “transmission threshold” method for autonomous node scheduling as an extension to the existing Opportunistic Large Array (OLA) cooperativ e transmission broadcast protocol [3]. W e will refer to the e xisting method as ‘Basic OLA ’ in this paper . In a multi-hop ad hoc network, broadcasting is a significant operation to support numerous applications. For example, broadcasting is used in “Hello” messages and route discovery in ad hoc networks and queries in WSNs. Flooding, one of the earliest broadcast protocols for multi-hop transmissions, where Mary Ann Ingram, Aravind Kailas and Lakshmi Thanayankizil are with the School of Electrical and Computer Engineering, Georgia Institute of T ech- nology , Altanta, GA 30332-0250, USA. The authors gratefully acknowledge support for this research from the National Science Foundation under CNS- 0721269 and CNS-0721296. all nodes relay the receiv ed message, is energy inefficient and unreliable, as it leads to sev ere contention, collision and redundancy , a situation referred to as br oadcast storm [4]. Many broadcast strategies hav e been proposed to av oid the broadcast storm. One energy ef ficient type of broadcast strategy is the broadcast tree. A popular example is the Broadcast Incremental Power (BIP) algorithm, which was proposed and analyzed in [5]. In a broadcast tree, the relay nodes are carefully chosen so that, ideally , each node receives the message the first time from exactly one relay . While this approach reduces considerably the number of collisions in a broadcast, it is fundamentally inconsistent with cooperati ve transmission. Furthermore, the identification of relay nodes in a broadcast tree requires significant network ov erhead, especially for mobile networks. Our proposed protocol does not require that a node kno ws its geographical location, so we mention a fe w other broad- casting protocols that also do not require this information. The Border Node Retransmission Based Probabilistic Protocol privile ges the retransmission by nodes located at the radio border of the transmitter [6]. Border nodes are identified through single hop exchanges of “Hello” messages and hence this scheme requires no location or signal strength information. In another work by Cartigny et al. [7], the authors considered a Relati ve Neighbor Graph (RNG) relay subset protocol [7], where only a subset of nodes relay the message from the source. Pairs of nodes are assumed to be able to ev aluate their relative distance with integration of a positioning system or a signal strength measure. Since both [6] and [7] require neighbor information, these protocols will not scale well with node density . An OLA is a collection of nodes that transmit the same message at approximately the same time. They do this without coordination between each other , but they naturally fire at approximately the same time in response to energy recei ved from a single source or another OLA [8]. Carrier sensing must be disabled to permit OLA transmission to take place. Because all the transmissions within an OLA are repeats of the same message, the signal received from an OLA has the same model as a multipath channel. Small time of fsets (because of different distances and computation times) [9], and small frequency offsets (because each node has a different oscillator frequency) appear as excess delays and Doppler shifts, respectiv ely . As long as the receiv er, such as a RAKE receiv er, can tolerate the ef fecti ve delay and Doppler spreads of the receiv ed signal, decoding should proceed normally . 2 The primary benefits of an OLA transmission are its spatial div ersity gain and its lack of network ov erhead. Even though many nodes may participate in an OLA transmission, ener gy can still be saved because all nodes can reduce their transmit powers dramatically and lar ge fade mar gins are not needed. Even in non-fading channels, the array gain in an OLA transmission may be desirable for applications where there is a lo w maximum po wer constraint, resulting, for example from sev ere cost or heat restrictions. Further , in [8], the Basic OLA algorithm was shown to yield an energy savings of about 5 dB compared to the BIP algorithm. Other OLA works include the following. In the centralized broadcasting scheme Accumulative Br oadcast [10], the order and po wer le vels of node transmissions are chosen to minimize the total po wer consumption in general multi-hop networks. While the second problem can be solv ed by means of linear programming tools, the first problem was found to be NP complete [11], [12]. In [13], an optimal trivial schedule was found for a dense OLA network that allocated power and order of transmission according to node distance from the source. The Dual Threshold Cooperative Broadcast (DTBC) was in- troduced in [14] as a way to save even more energy compared to the Basic OLA broadcast, by allo wing a node to join an OLA only if its receiv ed signal power is less than a gi ven threshold. Ho wever the DTBC concept was not analyzed in [14]. Our paper analyzes OLA-Threshold (OLA-T), which is the same, but independently deriv ed, concept as DTBC. Our paper also extends the concept to allow the thresholds to vary from OLA to OLA. OLA with variable threshold (OLA-VT) can be optimized to minimize total energy in a broadcast. OLA-VT can also be used to control OLA sizes, thereby enabling certain other protocols, such as the OLA Concentric Routing Algorithm (OLA CRA), which does upstream routing in WSNs [16]. OLA-T and OLA-VT can both be shown to be suboptimal trivial schedules [13], with the virtues of simple implementation and good performance. Finally , tw o important features that all the proposed schemes inherit from Basic OLA is that individual nodes are not addressed and they do not need location information for routing. Lack of addressing makes the protocols scalable with node density . Not needing location knowledge for routing makes the proposed protocol suitable for applications where location information is either not av ailable or too expensi ve or energy consuming to obtain or exploit. I I . S Y S T E M M O D E L Half-duplex nodes are assumed. For the purpose of anal- ysis, the nodes are assumed to be distributed uniformly and randomly over a continuous area with average density ρ . The originating node is assumed to be a point source at the center of the giv en network area. W e assume a node can decode and forward a message without error when its receiv ed Signal-to- Noise Ratio (SNR) is greater than or equal to a modulation- dependent threshold [15]. Assumption of unit noise v ariance transforms the SNR threshold to a recei ved power criterion, which is denoted as the decoding threshold τ d . W e note that the decoding threshold τ d is not explicitly used in real receiv er operations. A real receiv er alw ays just tries to decode a message. If no errors are detected, then it is assumed that the receiver po wer must ha ve exceeded τ d . In contrast, the proposed transmission threshold of OLA-T w ould be explicitly compared to an estimate of the recei ved SNR. Let the source power , relay transmit po wer , and the relay transmit po wer per unit area be denoted by P s , P r , and P r = ρP r , respecti vely . Follo wing [15], we assume the deterministic model , which implies that the power receiv ed at a node is the sum of the po wers from each of the node transmissions. This model implies node transmissions are orthogonal. Howe ver , because non-orthogonal transmissions also produce similarly shaped OLAs [15], the basic OLA-T concept should work for them as well although the theoretical results would ha ve to be modified. Furthermore, we sho w in Section VI that the deterministic result can approximated in a Rayleigh f aded channel by using distrib uted 4-th order transmit div ersity in the OLA transmissions. Again following [15], we assume a continuum of nodes in the network, which means that we let the node density ρ become very large ( ρ → ∞ ) while P r is kept fixed. Section VI will also sho w that results of the continuum assumption can be approximated with a “reasonable” node density for a network. The path loss function in Cartesian coordinates is given by l ( x, y ) = ( x 2 + y 2 ) − 1 , where ( x, y ) are the normalized coordinates at the receiv er . As in [15], distance d is normalized by a reference distance, d 0 . Let power P 0 be the receiv ed power at d 0 . Recei ved power from a node distance d away is P rec = min( P 0 d 2 , P 0 ) [15]. As in [15], the aggregate path- loss from a circular disc of radius r 0 at an arbitrary distance p > d 0 from the source is given by f ( r 0 , p ) = Z r 0 0 Z 2 π 0 l ( p − r cos θ , r sin θ ) r drdθ , = π ln p 2 | p 2 − r 2 0 | . (1) Then the receiv ed po wer at a distance p from the source, P p is giv en by P p = P r π ln p 2 | p 2 − r 2 0 | . W e note that the normalized relay transmit power , P r , is actually the SNR received by a node at the reference distance aw ay from a single relay node. I I I . B A S I C O L A In a Basic OLA broadcast [8], a node relays immediately if it can decode and if it has not relayed before. The aim is to succeed in broadcasting the message over the whole network. The source node transmits a message and the group of neighboring nodes that receiv e and decode the message form Decoding Lev el 1 ( D L 1 ), which is the disk enclosed by the smallest circle in Fig. 1(a). Next, each node in D L 1 transmits the message. These transmitting nodes in D L 1 constitute the first OLA. Ne xt, nodes outside of DL 1 receiv e the superposition of relayed copies of the message. Nodes in this group that can decode the message constitute D L 2 , which are represented as the ring between D L 1 and the next bigger concentric circle in Fig. 1(a). All the nodes in a decoding le vel form an OLA, which in turn generates the next decoding le vel. From [15], for a fixed P r , the maximum value of τ d such that 3 the relayed signal will be propagated in a sustained manner by concentric OLAs satisfies τ d ≤ π (ln 2) P r . (2) Fig. 1(a) illustrates this phenomenon for a giv en network area (defined in Fig. 1 by the dashed line). Fig. 1. (a) Broadcast using Basic OLA, (b) Broadcast using OLA with transmission threshold (OLA-T). Only nodes in the grey areas relay . I V . O L A W I T H T R A N S M I S S I O N T H R E S H O L D ( O L A - T ) The energy ef ficiency of OLAs can be improved preventing the nodes whose transmissions have a negligible effect on the formation of the next OLA from participating in the relaying. By definition, a node is near the forward boundary if it can only barely decode the message. The state of bar ely decoding can be determined in practice by measuring the av erage length of the error vector (the distance between the received and detected points in signal space), conditioned on a successful CRC check. On the other hand, a node that receiv es much more power than is necessary for decoding is more likely to be near the source of the message. The OLA-T method is simply Basic OLA with the additional transmission criterion that the node’ s recei ved SNR must be less than a specified transmission threshold , τ b . The difference between the two thresholds is giv en by τ b − τ d =  . W e will also refer to the Relativ e T ransmission Threshold (R TT), defined as τ b τ d . In other words, τ d and τ b define the range of received po wers that correspond to the “significant” boundary nodes. The concept and analysis of OLA-T are the original contributions of this paper . Fig. 1(b) illustrates this concept. The grey strips in Fig. 1(b) represent OLAs within each decoding level. Unlik e the approach depicted in Fig. 1(a), the nodes that compose an OLA are only a subset of the nodes in a decoding lev el. Before the analysis of OLA-T , it is important to point out that the transmission threshold, τ b , is only one of the ways to achiev e energy savings. For example, it is also possible to sav e on energy by varying the relay transmission power , P r , of the sensors (depending on their le vel) across the network. OLA- T can be thought of as an extreme quantization of v ariable power allocation, and therefore will not be as power efficient as an optimal continuous power allocation. Ho wever OLA-T has the advantage of essentially no network overhead, making it potentially applicable to highly mobile networks. W e note that Basic OLA transmission has been proposed for unicast transmission because of its lack of overhead [17]. Fig. 2. Outer radii, r d,k , and the inner radii, r b,k versus k For radios that consume substantial energy when receiving and decoding, Basic OLA might not be advantageous for unicast, since every node recei ves and decodes. Howe ver , in OLA-T , a node doesn’t need to decode the data if it is not a relay and not the destination. If the ener gy spent determining that a node should relay can be made extremely small, then OLA-T might be an attractiv e unicast scheme. Ho we ver , in this paper we consider only broadcasting and transmit energy . W e acknowledge that although OLA-T saves energy com- pared to Basic OLA in a single broadcast, the nodes selected by OLA-T for relaying will drain their batteries quickly , because the same nodes are always selected for a fixed source in a static netw ork. In this case, OLA-T would cause a netw ork partition ev en earlier than Basic OLA because the relays use a slightly higher transmit power . Ho we ver , the opposite will be true if the source location v aries randomly , or if the nodes mov e about randomly . Even for a fixed source and a static network, network lifetime can be extended relativ e to Basic OLA by modifying OLA-T to use mutually exclusi ve sets of nodes on consecutiv e broadcasts. This new technique, which we call Alternating OLA-T (A-OLA-T) [18] builds on the results reported in this paper . Finally , we note that to decode, a node in an OLA-T netw ork receiv es energy from just one decoding lev el. Multiple le vels are not ganged to form a very thick OLA as in [14] nor are OLA transmissions at dif ferent times from dif ferent decoding lev els combined as in [10]. Instead, the emphasis of OLA-T is on forming thin, widely separated OLAs. A. Analysis of OLA-T Br oadcast for Constant  In this section, the OLA boundaries are determined as functions of the decoding le vel k , for the case when the transmission threshold is constant over the network. The case of v ariable τ b is treated in Section V. For the constant  , we are able to deriv e the closed-form expression by slightly modifying the continuum approach in [15]. Let the outer and inner boundary radii for the k -th OLA ring be denoted as r d,k and r b,k , respecti vely . The boundaries 4 can be found recursi vely using P r [ f ( r d,k , r j,k +1 ) − f ( r b,k , r j,k +1 )] = τ j , j ∈ { b, d } . (3) Applying (1) yields τ j P r = π ln | r 2 j,k +1 − r 2 b,k | | r 2 j,k +1 − r 2 d,k | , which yields r 2 d,k = β ( τ d ) r 2 d,k − 1 − r 2 b,k − 1 β ( τ d ) − 1 , r 2 b,k = β ( τ b ) r 2 d,k − 1 − r 2 b,k − 1 β ( τ b ) − 1 , (4) where β ( τ ) = exp  τ / ( π P r )  . Next, the recursi ve problem is cast as a matrix difference equation as follo ws:  r 2 d,k +1 r 2 b,k +1  =  α ( τ d ) + 1 − α ( τ d ) α ( τ b ) + 1 − α ( τ b )   r 2 d,k r 2 b,k  , where α ( τ ) = [ β ( τ ) − 1] − 1 . Using the initial conditions, r d, 1 = q P s τ d , and r b, 1 = q P s τ b , the following solution is obtained by solving this first-order difference equation using Z-transforms: r 2 d,k = η k 1 − η k 2 A 1 − A 2 , r 2 b,k = ζ k 1 − ζ k 2 A 1 − A 2 , (5) where A 1 = α ( τ d ) − α ( τ b ) , A 2 = 1 , A 1 6 = 1 , for i ∈ { 1 , 2 } , η k i =  [ A i + α ( τ b )] P s τ d − α ( τ d ) P s τ b  ( A i ) k − 1 , and ζ k i =  [1 + α ( τ b )] P s τ d + [ A i − α ( τ d ) − 1] P s τ b  ( A i ) k − 1 . The radii given by (5) hav e been plotted in Fig.2 on a logarithmic scale, as functions of the OLA index. The low , moderate, and high values of  are 0.2, 0.43, and 1.2 (in terms of R TT in dB, 0.79, 1.55, and 3.42), respectiv ely . Where network broadcast is achie ved, the radii gro w in an unbounded fashion, with a rate that increases with level index, k . W e have observed that for some values of  , such as  = 0 . 43 , the radii increase at a sub-linear rate with respect to k , up to a certain point and then the increases are faster than linear for all higher k (that we test). From observ ation of the abo ve de velopment and of the Basic OLA condition for broadcast in (2), we find that the variables τ d , τ b ,  , and P s always appear divided by P r ( P r cancels in the ratio P s /τ d ). In particular , we giv e τ d /P r the name Decoding Ratio (DR), because it can be sho wn to be the ratio of the receiv er sensiti vity (i.e. minimum power for decoding at a giv en data rate) to the po wer receiv ed from a single relay at the ‘distance to the nearest neighbor , ’ d nn = 1 / √ ρ . If ρ is a perfect square, then the d nn would be the minimum distance between the nearest neighbors if the nodes were arranged in a uniform square grid. For a fixed P r and τ d , energy is reduced for OLA-T by minimizing  (and hence, τ b ) . W e next deriv e an expression for  min for a fixed DR. First, we re-write (5) as shown below . z k + 1 = Az k , (6) where z k =  r 2 d,k , r 2 b,k  T , and A =  α ( τ d ) + 1 , − α ( τ d ); α ( τ b ) + 1 , − α ( τ b )  T . It can be seen that A 1 and A 2 are the eigenv alues of A . For infinite network broadcast, the Fig. 3. MR TT , in dB versus DR OLA rings must continue to grow implying that the system described by (5) must be “unstable, ” i.e., | A 1 | > 1 [19]. Since, radii are always positi ve and α ( τ d ) > α ( τ b ) > 0 by design, A 1 > 1 becomes a necessary and suf ficient condition for infinite network broadcast. Setting A 1 = 1 would gi ve us an expression for the minimum  . A 1 = 1 , ⇒ α ( τ d ) − α ( τ b ) = 1 , ⇒ 1  exp  τ d P r π  − 1  − 1 = 1  exp  τ d +  P r π  − 1  . Collecting the τ d terms and solving for  results in  min = ( − 1) ( P r π ln  2 − exp  τ d P r π   + τ d ) , (7) and the following necessary and sufficient condition: 2 ≥ exp  τ d P r π  + exp  − τ d −  P r π  . (8) W e remark that when  → ∞ , OLA-T becomes Basic OLA, and (8) becomes the same condition (2) that was deriv ed in [15]. So for infinite network broadcast using OLA-T ,  >  min , or equi v alently , τ b > τ b min . Fig. 3 sho ws Minimum Relati ve T ransmission Threshold (MR TT), τ b min /τ d , in dB, versus the DR. For example, for DR = 1 , the minimum transmission threshold is about 1.8 dB higher than the decoding threshold. It can also be inferred that theoretically , it is possible for OLA- T to achiev e infinite network broadcast with an infinitesimally small τ b min and DR. Howe ver , a very small MR TT may not be very ef fective if the precision in the estimate of the SNR is not good enough. B. Energy Consumption for a Given Broadcast In this section, we compare the total radiated energy during a successful OLA-T broadcast to that of a successful Basic OLA broadcast. As τ b → ∞ (or  → ∞ ), the OLA-T rings 5 grow in thickness until they become the same as the OLA decoding le vels as in [15]. On the other hand, as τ b → τ d , one would expect the transmitting strips to start thinning out . In other words, the inner and outer radii for each OLA become close and the OLA areas decrease. If the DR is allowed to diminish (e.g. as P r increases for a fixed τ d ) as τ b → τ d , successful broadcast can be maintained in a continuum network, ev en though OLAs become very thin. If the transmit energy consumptions for Basic OLA and OLA-T are compared for the same P r , then it can be shown that OLA-T saves over 50% of the energy consumed by Basic OLA [16]. Ho we ver , Basic OLA can achie ve successful broadcast at a lo wer P r according to (2). Hence, we need to compare these two protocols for a fixed v alue of τ d (i.e. data rate) such that each is in its minimum energy configuration. The energy consumed by OLA-T in the first L lev- els is mathematically expressed, in energy units, as ξ L = P r T s L X k =1 π ( r 2 d,k − r 2 b,k ) , where T s is the length of the message in time units. The Fraction of transmission Energy Saved (FES) for OLA-T relativ e to Basic OLA can be expressed as FES = 1 − P r ( O T ) L X k =1  r 2 d,k − r 2 b,k  P r ( O ) r 2 d,L , (9) where P r ( O T ) and P r ( O ) are the lowest values of P r that would guarantee successful broadcast using OLA-T and Basic OLA, respectiv ely . If we multiply the numerator and denominator of the ratio by 1 /τ d , and substitute P r ( O ) /τ d by its smallest value of π ln 2 , we can re-write (9) as FES = 1 − π ln 2 L X k =1  r 2 d,k − r 2 b,k   τ d /P r ( O T )  r 2 d,L . (10) Fig.4 shows FES versus DR for various network sizes (i.e. numbers of decoding le vels or hops). W e learn that there is a small dependence of FES on the number of lev els, but it quickly diminishes after 50 levels. W e observe that the FES is positiv e over the range of DR v alues we consider . For example, at DR = 0 . 5 , FES is about 0.25. This means that at their respectiv e lo west energy le vels (OLA-T at τ b min , and Basic OLA at P r ( O T ) ), OLA-T sa ves about 25% of the energy used by Basic OLA at this DR. FES increases as DR decreases and attains a maximum of about 32% for infinitesimally small values of DR. The results gi ven so far hav e been in terms of normalized units. W e would no w like to consider some examples of un- normalized v alues for these v ariables to giv e an idea of what power lev els and node densities can achiev e the v arious values of DR and FES. The DR ratio was previously defined as τ d / P r , where τ d is the required SNR for decoding, P r is the normalized relay transmit power , and ρ is the node density in number of nodes per area, where area is normalized by the square of the reference distance, d 2 0 . Expanded in terms of Fig. 4. V ariation of FES with DR and the numbers of le vels (network size) T ABLE I E X AM P L E S O F U N - N O R MA L I Z ED V A R I AB L E S Example P t Node Density RX sens. d nn DR (dBm) (nodes/area) (dBm) (m) 1 -56.00 2.65 nodes/m 2 -90.00 0.61 1.5 2 -56.00 2.65 nodes/m 2 -94.77 0.61 0.5 3 -34.95 1 node/16 m 2 -90.00 4.00 0.5 4 -43.98 1 node/4 m 2 -90.00 2.00 1.0 5 -20.97 9 nodes/3.60 km 2 -90.00 20.00 0.5 un-normalized variables, we can write DR as DR =  Receiv er Sensitivity in mW σ 2 n   P t G t G r σ 2 n  λ 4 π d 0  2  (# nodes ) d 2 0 Area in m 2  , (11) where P t is the relay transmit power in mW , G t and G r are the transmit and receive antenna gains, σ 2 n is the thermal noise power , λ is the wavelength in meters, and d 0 is the reference distance in meters. Suppose that the radio frequency is 2.4 GHz ( λ = 0 . 125 m), and the antennas are isotropic ( G t = G r = 1 ). Then (11) can be simplified to DR =  Receiv er Sensitivity in mW  10 4  P t in mW   (# nodes ) Area in m 2  . (12) T able I sho ws fiv e different examples of un-normalized variables and their resulting d nn and DR v alues. W e observe that DR = 0 . 5 can be obtained in Examples 2, 3, and 5, ranging from high density (2.65 nodes/m 2 ) to low density (9 nodes/3.60 km 2 ). W e also observe that the high density cases, Examples 1 and 2, correspond to very low transmit powers. V . O L A - T B RO A D C A S T W I T H V A R I A B L E  The OLA-based cooperati ve transmission techniques pre- sented so far in the paper in volve just a single fixed  for the whole wireless system. A shortcoming of this technique is that the radii gro wth is polynomial and the OLA rings keep 6 growing bigger, expending more energy than is needed, to cov er a gi ven network area. W e are moti v ated, therefore, to in vestigate how much more energy can be saved by letting each level ha ve a different  . W e call the resulting broadcast protocol OLA-V ariable Threshold (OLA-VT). The Genetic Algorithm (GA) is adopted to determine the sequence of {  k } that yields the minimum OLA-T ener gy per broadcast, for a given τ d , P r , and fixed number of decoding lev els. T wo different constraints are considered. For each constraint, the radii are computed for the optimized {  k } , and the FES is computed, assuming Basic OLA is in its minimum energy configuration. Figure 2 suggests that a criterion for successful broadcast is e ventual upw ard conca vity of the curve. T o capture this, we define the k -th Double Difference (DD) as D D k = ( r d,k +2 − r d,k +1 ) − ( r d,k +1 − r d,k ) . Constraint T ype 1 is that D D k > 0 for k ≥ 4 ; the total number of le vels or hops is fixed, but no constraint is made on the physical size of the network. Constraint T ype 2, on the other hand, fixes the number of le vels and the physical size of the network. The key difference is that Constraint T ype 2 requires that the outer radius of the last decoding level be greater than the specified network radius. Fig.5 plots the FES as a function of network radius. Con- straint T ype 1 is evaluated for a maximum of 20 levels (dashed line), and Constraint T ype 2 is e v aluated for 10 lev els (dotted line) and 20 lev els (dash-dot). Both Constraint T ype 2 cases fix the network radius to be 25 distance units. As an example, for the 20-le vel case, the Constraint T ype 2 algorithm minimizes broadcast energy with the constraint that r d, 20 > 25 . The fixed  case (solid line) is included for reference and requires 150 lev els to reach a radius of 25. All OLA-T and -VT examples share the same DR of 0.9, and P s /P r of 4.31 dB. The fix ed  case uses the MR TT of 1.56 dB. The points on each curve are the FES values calculated for each radius in the sequence { r d, 1 , r b, 2 , r d, 2 , r b, 3 , . . . } . Since the FES is a function of whole lev els and not partial le vels, we just define the FES for r b,k to be equal to the FES for r d,k − 1 ; this enables us to identify OLA widths as the widths of the flat parts of the curve. The first non-zero point represents the FES at r d, 1 , since the FES at r b, 1 is zero. Even though the constraints inv olve a fixed number of levels or physical network size, the FES v alue at a particular radius, r , indicates the FES as though the network were truncated to have radius r . For e xample, after 2 OLAs (i.e. at the right edge of the second plateau), the constant  curve indicates an FES of about 0.25 at a radius of about 3. This means that a network of radius 3 that uses the fixed  of 0.4113 to form two OLAs will achieve 25% ener gy savings ov er the minimum ener gy Basic OLA for network of radius 3. W e note that the network radius in Fig.5 is normalized by the reference distance. This means that if d 0 = 1 m, then a network with normalized radius of 5 has an un-normalized radius of 5 m. On the other hand, if d 0 = 100 m, then the same normalized network radius represents an un-normalized radius of 500 m. When d 0 increases in (11), to maintain the same normalized relay transmit power , the un-normalized transmit po wer must increase by a factor of d 2 0 , and to maintain the same normalized density , the un-normalized density must Fig. 5. FES comparisons for variable  k versus fixed  decrease by d 2 0 . In other words, if d 0 increases by a factor of 10, then the DR and hence the FES can be conserved by having the nodes spread out so that inter-node un-normalized distances increase by a factor of 10, and having the un- normalized relay transmit po wer increase by a factor of 100. Let us first compare the Constraint T ype 1 curve to the fixed  curve. W e notice that the fixed  curve starts high and then decays down to about 0.2. The Constraint T ype 1 curve, on the other hand drops to ne gati ve FES values and then climbs up to a final v alue of about 0.3. That the final value of 0.3 is higher than the FES of the fixed  curv e for the same network radius of approximately 5 is e vidence that variable  can be more energy efficient than fixed  . The FES is negativ e because the P r for OLA-T is larger than the P r for Basic OLA, while the first few OLAs of OLA-T are allowed to be lar ge and comparable to the first few OLAs of Basic OLA in size. The step sizes or hop distances for the fix ed  curve decrease smoothly with network radius, while the step sizes for the Constraint T ype 1 curve are on the same order for the first 4 lev els, until the FES reaches 0.2, and then the step sizes decrease significantly . Relativ ely small step sizes should be OK as long as the density is high enough so that the OLA ring is sev eral d nn thick. Constraint T ype 2 curv es drop down to much lower FES values and eventually climb back up to about 0.2. At first glance, it may seem that the v ariable  case does no better than the constant  case, until one considers that the variable  case reaches the same FES in only 10 or 20 steps, while constant  requires 150 steps. A d 0 of 10 m, for example, would result in a Constraint T ype 2 network of radius 250 m, with OLA sizes that would be reasonable for ρ on the order of 1 node/4 m 2 , as in Example 4 in T able I. V I . S I M U L A T I O N S W I T H F A D I N G A N D F I N I T E N O D E D E N S I T I E S Throughout our analysis we have assumed the deterministic model (nodes in the network relay in orthogonal channels) and a continuum of nodes (node density , ρ, → ∞ ) [15]. In this 7 section, we use Monte Carlo simulations to show that results based on the continuum and deterministic assumptions can be approximated by networks of finite density with Rayleigh fading channels. First we will address the continuum assump- tion and second we will address the orthogonality assumption. W e will ev aluate performance in terms of the Probability of Successful Broadcast (PSB), where a successful broadcast is when 99% of the nodes in the network can decode the message. Normalized values are used in each case. Fig. 6 is a plot of the PSB as a function of R TT in dB for various choices of network densities. The step function that represents the continuum assumption is also plotted. The Monte Carlo results hav e been obtained from a simulation of 400 random networks. The parameters used for the Monte Carlo trials are as follo ws. The nodes were uniformly and randomly distributed over a circular area of radius 17 distance units with the sink node located at the center of the network area. The source po wer , P s , was chosen to be 3 and decoding threshold, τ d was 1. Nodes in the first lev el used a fixed  of 2.5 (in terms of R TT in dB, 5.44) for all the trials. R TT for all the other lev els follo w the horizontal axis. From Fig. 6, we observe that at high v alues of network densities (10 in this case), the results from the trials approach the continuum plot, which represents an upper bound on the PSB for an R TT . At d 0 = 1 m, ρ = 10 corresponds to an un-normalized node density of 10 nodes/m 2 . At this high node density , the PSB drops sharply at a value of R TT , below which broadcast fails. Next, we consider fading channels. This time, we assume 1500 nodes to be uniformly and randomly distributed over the same circular network area. All the other parameters are kept unchanged throughout the 100 network realizations. Rayleigh fading is assumed. The transmitted signals are assumed to be Direct Sequence Spread Spectrum (DSSS), and the receiv ers are assumed to have a 4-th order RAKE processor . Let m denote the desired div ersity order . W e assume that the chip width is larger than the delay spread of the real multipath channel. T o ensure di versity gain in the RAKE receiv er , each relay chooses a transmit delay randomly from a windo w of 4 chip delay choices. m = 1 means there is just one transmission path and is no di versity [9], [20]. Likewise, m = 2 means 2 fingers are excited in the RAKE receiver . W e assume the chip pulse width to be 500 ns which is consistent with MICA sensor mote specifications [21]. From Fig. 7, we observe that with m = 3 , the performance in a fading channel is very similar to the deterministic channel. V I I . C O N C L U S I O N In this paper , we proposed and analyzed a novel energy- efficient strate gy that lev erages the cooperativ e advantage in multi-hop wireless networks. By self-scheduling the transmis- sions in a network, significant energy savings are realized. W e hav e shown how OLA-T can sav e a maximum of about 32% of the energy of a Basic OLA broadcast, and OLA-VT sav es additional energy with no overhead and no central control. For fixed-size networks, OLA-VT simplifies boundary-matching. The trade-offs between the FES and the number of hops ov er a gi ven network size were discussed using the OLA-VT . Fig. 6. PSB as a function of R TT and node density , ρ Fig. 7. PSB as a function of R TT and div ersity order , m Further work is needed in terms of performance for practical estimators of SNR and practical synchronization. A C K N O W L E D G M E N T The authors are thankful for the revie wers’ comments. The authors are also thankful to Prof. Shui-Nee Chow , from the School of Mathematics at the Georgia Institute of T echnology for helpful discussions. R E F E R E N C E S [1] A. Sendonaris, E. Erkip, and B. Aazhang,“User Cooperation – part i: Sys- tem Description, part ii: Implmentation Aspects and Performance Anal- ysis, ” IEEE T rans. Commun. , vol. 51, no. 11, pp. 1927–48, Nov . 2003. [2] J. N. Laneman, D. Tse, and G. W . 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