Investigating Multiple Alternating Cooperative Broadcasts to Enhance Network Longevity

Investigating Multiple Alternating Cooperative Broadcasts to Enhance Network Longevity Aravind Kailas School of Electrical and Coputer Engineering Georgia Institute of Technology Atlanta, Georgia 3033-050, USA Eail: aravindk@ieeeorg Mary Ann Ingra School of Electrical and Coputer Engineering Georgia Institute of Technology Atlanta, Georgia 3033-050, USA Eail: ai@ecegatechedu Abstract We propose a broadcast protocol that is based on a for of cooperative transission called the Opportunistic Large Array (OLA) Multiple SNR (or transission) thresholds are used to define utually exclusive sets of OLAs, such that the union of the sets includes all the nodes in the network The new protocol, tered Alternating OLA with Transission Threshold (A-OLA-T), exercises a different set of OLAs on each consecutive broadcast fro the sae sink until all sets have transitted once Then the sequence repeats Thus, broadcasts consue energy efficiently and uniforly over the network, and A-OLA-T is especially well suited for static networks The transission thresholds are optiized to axiize the network life if broadcasts were the only transissions In this paper, we first optiize triples of broadcasts, and then extend the optiization for a higher nuber of broadcasts I INTROUCTION In this paper, we propose and analyze a Mediu Access Control (MAC)-free alternating sets broadcast strategy to increase the network longevity for ultihop wireless networks The Alternating OLA with a Transission Threshold (A- OLA-T) algorith introduced in this paper uses a siple for of Cooperative Transission (CT) called the Opportunistic Large Array (OLA) [1], and is an extension of a previous non-alternating OLA with a Transission Threshold (OLA-T) algorith [] By having two or ore nodes cooperate to transit the sae essage, CT-based strategies offer the spatial diversity benefits of an array transitter enabling a draatic signal-to-noise ratio (SNR) advantage in a ultipath fading environent [3], [4] This advantage can be used to save transit energy [3], [4] The Opportunistic Large Array (OLA), a siple for of CT, is a group of nodes that behave without coordination between each other, but naturally fire at approxiately the sae tie in response to energy received fro a single source or another OLA [1] All the transissions within an OLA are repeats of the sae wavefor; therefore the signal received fro an OLA has the sae odel as a ultipath channel Sall tie offsets (because of different distances and coputation ties) and sall frequency offsets (because each node has a different oscillator frequency) are like excess delays and oppler shifts, respectively As long as the receiver, such as a RAKE receiver, can tolerate the effective delay and The authors gratefully acknowledge support for this research fro the National Science Foundation under grant CNS-07196 oppler spreads of the received signal and extract the diversity, decoding can proceed norally Even though any nodes ay participate in an OLA transission, total transission energy can still be saved because all nodes can reduce their transit powers draatically and large fade argins are not needed When used for broadcasting, nodes repeat if they haven t repeated the essage before, and the resulting OLAs will propagate, foring concentric ring shaped OLAs that will eventually include all nodes, under a condition on relay power and receiver sensitivity [1]; we refer to this broadcast schee as Basic OLA OLA with Transission Threshold (OLA- T) applies an SNR threshold to liit the relaying nodes to those at the edge of the decoding range [], [5] Even though these border nodes ust transit at a higher power than for Basic OLA to sustain propagation [], total transit energy is still saved because only a fraction of the nodes relay Basic OLA and OLA-T share the iportant feature that no individual nodes are addressed Given that the node density is sufficient to sustain OLA transission, the coplexity of these broadcast protocols is absolutely independent of node density, aking OLA-based broadcasting very attractive for extreely high density wireless networks However, OLA-T has a proble for a fixed source in a static network, because the sae nodes relay in every broadcast Therefore, the OLA nodes die (drain their batteries) first and the network loses sensitivity in the OLA areas Alternating OLA-T (A-OLA-T) was proposed in [7] to reedy this proble In A-OLA-T, the transission threshold is used to divide all the nodes in the network into two utually exclusive sets of OLAs The first set is exercised in the initial OLA- T broadcast, while the second set is exercised in the second broadcast Each succeeding broadcast alternates between the two sets, thereby saving transit energy via OLA-T, but also draining the batteries uniforly across the network Conditions were derived in [7] to ensure sustained propagation of both sets of OLAs The contribution of the present paper is to extend -set A- OLA-T to -set A-OLA-T where > The condition for sustained propagation for -set A-OLA-T is derived and the energy savings for -set A-OLA-T relative to Basic OLA is shown

Fig 1 A-OLA-T with 3 alternating utually exclusive sets of OLAs II SYSTEM MOEL For our analysis, we adopt the notation and assuptions of [], ost of which were used earlier in [8] Half-duplex nodes are assued to be distributed uniforly and randoly over a continuous area with average node density ρ The originating node is assued to be a point source at the center of the given network area We assue a node can ecode and Forward (F) a essage without error when its received SNR is greater than or equal to a odulation-dependent threshold [8] Assuption of unit noise variance transfors the SNR threshold to a received power criterion, which is denoted as the decoding threshold τ d The Transission Threshold, τ b is used explicitly in the receiver to copare against the received SNR The thresholds, τ d and τ b, define a range of received powers that correspond to the significant boundary nodes, which for the OLA Further, we define the Relative Transission Threshold (RTT) as R = τ b τ d Continuing to follow [8], we assue a non-fading environent The noralized path loss function in Cartesian coordinates is given by l(x, y) = (x + y ) 1, where (x, y) are the noralized coordinates at the receiver For siplicity, the deterinistic odel [8] is assued, which eans that the power received at a node is the su of the powers fro each of the node transissions This iplies that signals received fro different nodes are orthogonal The orthogonality can be approxiated, for exaple, with irect Sequence Spread Spectru (SSS) odulation, RAKE receivers and by allowing transitting nodes to delay their transission by a rando nuber of chips [9], [10] Let the noralized source and relay powers be denoted by P s and, respectively Let the relay transit power per unit area be denoted by = ρ We assue a continuu of nodes in the network, which eans that the we let the node density ρ becoe very large (ρ ) while is kept fixed We reark that the noralized relay transit power,, is actually the SNR received by a node at the reference distance away fro a single relay node We note that nonorthogonal transissions in fading channels produce siilarly shaped OLAs [8], therefore the A-OLA-T concept should work for the as well, although the theoretical results would have to be odified Lastly, we define ecoding Ratio (R) as = τ d /, because it can be shown to be the ratio of the receiver sensitivity (ie iniu power for decoding at a given data rate) to the power received fro 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 iniu distance between the nearest neighbors if the nodes were arranged in a unifor square grid The respective conditions for successful broadcast for both Basic OLA and OLA-T are upper bounds on, and the transit energy is optiized for each protocol when eets it upper bound The strategy in this paper is to assue a certain A-OLA-T property, naely the equal area property, which is true for the optiized = case, to also be true for the optiized > case Then, this property is used to derive the upper bound on for the > case III ALTERNATING OLA WITH TRANSMISSION THRESHOL (A-OLA-T) A A-OLA-T Concept Fig 1 illustrates the A-OLA-T concept with 3 alternating sets of OLAs Each broadcast is an OLA-T broadcast The grey areas in the left of Fig 1, are the OLAs in Broadcast 1, while the grey areas in the center and on the right, are the OLAs in Broadcast, and Broadcast 3, respectively Ideally these three sets of OLAs have no nodes in coon and their union includes all nodes In Fig 1, the sets of OLAs during Broadcasts 1,, and 3 coprise OLA 1,1 and OLA,1, OLA 1, and OLA,, and OLA 1,3 and OLA,3, respectively; these sets do not have any coon nodes and their union includes all the nodes in the network This increases the network longevity for broadcast applications because each node participates once in every three broadcasts, and therefore the load is shared equally For the two-set A-OLA-T, Broadcast 1 fixes the radii for Broadcast The trick then is to choose transission thresholds to ensure that the detection boundaries in Broadcast exceed (or atch up) with transission threshold boundaries in Broadcast 1 In [7], it was established that there exists a axiu value of, denoted by ax, and when > ax, network broadcast fails for A-OLA-T because the OLAs die out during Broadcast ax iplies a iniu value of for a given τ d, denoted by in Copared to Basic OLA, A-OLA-T with two sets extends the network longevity by about 17% when both OLA-based protocols operate in their iniu power configuration [7] This work ay be useful for future very large and very fine-grained onitoring applications, of the type that ay be enabled by sensor nodes that do energy harvesting B Equal Area Property Let the Ratio of Areas be the ratio of the total area of the Broadcast 1 OLAs to the total area of the network, and be given by L ( r d,k rb,k) Ψ = k=1 r d,l, (1) where r d,k and r b,k denote the outer and inner boundary radii, respectively, for the k-th OLA ring fored during the

Broadcast 1, and L is the nuber of OLAs in the OLA-T network In [7], it was shown that for the = case, Ψ = 1/ when = ax This iplies that the respective accuulated areas of the two sets of OLAs during Broadcasts 1 and are equal C Alternating Sets of OLAs, > In this section, we show that using alternating sets of OLAs ( > ) extends the life of the network even ore than for = To show this, we conjecture that the Equal Area Property applies to the > case Assuing that the conjecture is true iplies that Ψ = 1 for all broadcast sets, when the syste is in its lowest energy configuration, ie when = ax We confir the assuption nuerically in the next section Based on the assuption, we are able to derive an expression for ax, which in turn, allows us to quantify the relative transit energy consuption of -set A-OLA-T to Basic OLA The derivation of ax is sketched here and the details are in the appendices We use the closed-for expressions for OLA- T ring radii fro [] to put Ψ for Broadcast 1 solely in ters of R and Then setting Ψ = 1 allows an expression for R in ters of and Next, assuing the lowest energy configuration eans that R ust be equal to its lower bound (in [], the upper and lower bounds on R eet at the iniu energy configuration for = ) Solving this equality for yields the expression ( + 1 ) ax = ln () Next, we copute the broadcast life extension of A-OLA- T copared to Basic OLA By broadcast life, we ean the lifetie of the network if only broadcasts were transitted and if only radiated energy is considered At a first glance, it ight see that A-OLA-T increases the battery life of the sensors in the network by a factor of copared to Basic OLA This is true if A-OLA-T and Basic OLA use the sae However, this would not be a fair coparison since Basic OLA can achieve successful broadcast at a lower Since for a given protocol, all nodes use the sae aount of power in broadcasts, we assue the broadcast life of the network is inversely proportional to the tie-averaged power transitted by each node For Basic OLA, the tie-averaged power is For A-OLA-T with sets, the tie-averaged power is, since each node transits only every other broadcast The ratio of broadcast lives of Basic OLA to A-OLA-T is therefore Pr, and the Fraction of Life Extension (FLE), ay be defined as FLE = P r 1 (3) FLE can be evaluated for any powers that satisfy [ ax] 1 τ d, & [ ax] 1 τ d (4) Fro [8], ax = ln Substituting the values of ax in both the inequalities, we have [ ln 046τ d ( + 1 ) ] 1 τ d, and So increases with When the the iniu powers fro (4) are substituted, then (3) becoes FLE = P rin in 1 = ax 1 044 as, (5) ax where FLE represents the FLE achieved by A-OLA-T relative to Basic OLA when both protocols operate in their iniu power configurations IV NUMERICAL RESULTS First, the conjecture on the asyptotic convergence of the ratio of the accuulated areas of the utually exclusive sets of OLAs to 1 is verified nuerically in Fig for = 3 Fig is a plot of the ratio of areas versus k for the three successive broadcasts As seen in the figure, these curves overlap, ie, the ratio of areas for Broadcasts 1 and, Broadcasts and 3, and Broadcasts 3 and 1 grow in the exact sae way as a function of the OLA index Convergence of ratio of areas to 1 iplies that the widths of adjacent OLAs fro Broadcast 1,, and 3 becoe equal Fig Ratio of areas versus k for = 3 Next, we establish the network lifetie extensions using -A-OLA-T Fig 3 is a plot of the FLE versus the nuber of alternating sets,, on a logarithic scale We observe that as increases, the FLE increases (solid line), and for a large nuber of alternating sets, it reaches its asyptotic value (shown by dash-dot line) of around 045 This eans that -set A-OLA-T can offer a axiu life extension of about 44% when both protocols are optiized When =, FLE = 017, which is consistent with the findings in [7] Finally, it reains to check if infinite network broadcast can be achieved when the -set A-OLA-T is operating in

Fig 3 FLE as a function of the nuber of alternating sets, the iniu power configuration, ie, at = ax, which is given by () For our exaple, we use Matlab siulations and choose = 3 Let v d,k and v b,k, denote the outer and inner boundary radii for the k-th OLA ring fored during the Broadcast, respectively If u d,k and u b,k, denote the outer and inner boundary radii, respectively, for the k-th OLA ring fored during the Broadcast 3, and if ũ d,k+1 represents the decoding range of the (k + 1)-st OLA, then ũ d,k+1 v b,k+1 ust hold to guarantee infinite network broadcast The inner and outer boundaries have been siulated using the closed for expressions given by (7) It is rearked that even though the continuu assuptions of [] are used for these siulations, it has been shown in [] using Monte-Carlo siulations that the continuu and deterinistic assuptions can be approxiated well by networks of finite density with Rayleigh fading channels We test infinite network broadcast nuerically at = ax The shaded background in Fig 4 is a plot of the 3-set A-OLA-T noralized radii at ax for the 999-th and 1000-th levels as a function of noralized distance The white circle in the foreground is a agnified version of the region enclosed by the saller dotted circle The noralized Source power, P s was chosen to be 5 and using (), ax = 09038 We now explain the plot in the foreground Continuing to follow the notations fro the previous paragraph, Broadcast 3 boundary radii for the 999-th level, u b,999 and u d,999, are represented by the solid and dashed lines, respectively The dashed line (second fro the right) is the Broadcast inner boundary radii for the 1000-th level, v b,1000 The right-ost dotted line represents the decoding range of the 1000-th OLA, ũ d,1000 Fro Fig 4, we observe that ũ d,1000 > v b,1000, and so this is indicative of infinite network broadcast at ax It was observed that for > ax, Broadcast 3 OLAs die out (not shown in the paper) Practical Issues: The analysis in this paper has assued a Fig 4 3-set A-OLA-T radii growth in the iniu power case The 999-th and 1000-th levels are shown in the figure continuu of nodes and that all nodes transit orthogonal signals, neither of which is true in practice However, results based on these assuptions have been shown to be closely approxiated with high densities and liited orthogonality in fading channels [], [8]; in [], several exaples of un-noralized variables (ie relay powers in db, densities in nuber of nodes per, etc) are given that are consistent with the high density assuption Nevertheless, finite density ight ean that higher than iniu powers will be needed to ensure successful broadcast for both Basic OLA and A-OLA-T The additional power needed ight be called the density argin, and is a subject of ongoing research Finite density and ultipath fading will liit the nuber of sets that could be used by A-OLA-T to soe relatively low nuber Another practical issue is that radiated energy is not the only energy consued by a relay There is usually base of energy required by the electronics [11], and soeties, the energy required by the receiver electronics exceeds that of the transitter electronics [11] Since radiated and circuitconsued energies are added in a total energy odel of a node, then, the total energy Fraction of Energy Saved (FES) will be lower in coparison to Basic OLA than what is shown in this paper, since both protocols would have the sae circuitconsued energies V CONCLUSIONS In this paper, we proposed and analyzed a novel saesource broadcast strategy that extends the life of a static wireless ad hoc or sensor network by alternating between utually exclusive sets of Opportunistic Large Arrays (OLAs) in groups of broadcasts In this strategy, all participating nodes transit with the sae power We showed that the Alternating OLA with a Transission Threshold (A-OLA-T) with sets, >> 1 extends the network life by a axiu of 44% relative to the Basic OLA when both protocols operate in their iniu energy configuration Potential extensions of this work include an analysis of A-OLA-T for finite densities

of nodes, other path-loss exponents, fading environents, radiated versus non-radiated energy, and a consideration of the liitations of practical synchronization and SNR estiation A Ratio of Areas APPENIX We first derive a siplified expression for the ratio of accuulated OLA areas in a OLA-T broadcast to the total network area, denoted as Ψ in (1) Thus, the expression will apply to Broadcast 1 of A-OLA-T For siplicity of analysis, consider the ter rd,k r b,k rd,k, (6) r d,k 1 which is the ratio of the k-th OLA in OLA-T to the k-th stepsize Fro [], the closed-for expressions for OLA-T radii, which apply to Broadcast 1 in A-OLA-T, are given by rd,k = η 1A k 1 1 η A k 1, rb,k = ζ 1A k 1 1 ζ A k 1, (7) A 1 A A 1 A where A 1 = α(τ d ) α(τ b ), A = 1, A 1 A 0, { η i = [A i + α(τ b )] P s α(τ d ) P } s, τ d τ b ζ i = { [1 + α(τ b )] P s τ d + [A i α(τ d ) 1] P s τ b }, i {1, }, α(τ) = [β(τ) 1] 1, β(τ) = exp [ τ/ ] Substituting the closed-for expressions fro (7) into (6), we get rd,k r b,k rd,k = 1 α(τ b) r d,k 1 α(τ d ) (8) We observe that this ratio is independent of OLA index k Solving (8) for rd,k r b,k and substituting into (1), and noting L ( that r d,k rd,k 1) = r d,l, yields Ψ = 1 α(τ b) α(τ d ) We k=1 observe that the ratio of areas is invariant to the network size L When Ψ is evaluated at = Pr in for the A-OLA-T with two alternating sets [7], it is found that Ψ 05 B ax for -set A-OLA-T Still focussing on Broadcast 1, which is an OLA-T broadcast, set Ψ = 1 Substituting the definitions of α( ) into the expression for Ψ yields [ exp 1 R = ln 1 This expression tells us that for a given (ie a given data rate and relay power density) there will be exactly one value of transission threshold that will yield a ratio of areas of 1/ There is no guarantee, however, that the transission threshold is sufficiently high to ensure sustained OLA propagation (ie that the step sizes do not approach zero) That guarantee is ] provided by the following bound for OLA-T [] Fro [], the condition for a successful OLA-T broadcast takes the for of a lower bound on R given by { ln [ exp ] } R lower bound = ( 1) Here is where we ake our conjecture In [7], we found for the = case that the upper and lower bounds for R converged at the axiu possible value of, denoted ax Therefore, the value of that we get when we set R equal to R lower bound, is assued to be the axiu (corresponding to the lowest and consequently the lowest energy, since eventually every node transits in A-OLA-T) R = R lower bound, [ exp 1] [ ] exp = 1 1 Replacing exp with q, we can re-write the above as a quadratic equation in q as follows: q ( + 1)q + ( + 1) = 0, ( So, ax = ln the roots of which are q = +1 REFERENCES ) +1 [1] Y W Hong and A Scaglione, Energy-Efficient Broadcasting with Cooperative Transissions in wireless Sensor Networks, IEEE Trans Wireless Coun, vol 5, no 10, pp 844 55, Oct 006 [] A Kailas, L Thanayankizil, and M A Ingra, A Siple Cooperative Transission Protocol for Energy-Efficient 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