FOUNTAIN codes [1], [2] have been introduced to achieve

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1 Controlled Flooding of Fountain Codes Waqas bin Abbas, Paolo Casari, Senior Member, IEEE, Michele Zorzi, Fellow, IEEE Abstract We consider a multiho network where a source node must reliably deliver a set of data ackets to a given destination node. To do so, the source alies a fountain code and floods the encoded ackets through the network, until they reach their destination or are lost in the rocess. We model the robability that the destination can recover the original transmissions from the received coded ackets as a function of the network toology and of the code redundancy, and show that our analytical results redict the outcome of simulations very well. These results are emloyed to design distributed forwarding olicies that achieve a good tradeoff between the success robability and the total number of transmissions required to advance a acket towards the destination. We finally develo in detail the case where intermediate relays can inject additional redundancy in the network, rovided that they have successfully decoded the source ackets. Index Terms Fountain codes; flooding; restricted flooding; heuristic olicies; analysis; simulation. I. INTRODUCTION FOUNTAIN codes [], [2] have been introduced to achieve reliable communications in generic networks, where the acket loss robability may not be known a riori. The latter condition makes fountain codes articularly well suited to wireless networks [3], [4], where the ability to dynamically adjust the amount of redundancy offers a means to comensate for link outage events and communication failures [5]. In multiho wireless networks, and esecially in those conditions where a high degree of reliability is sought [6], fountain-coded ackets may be couled with some form of redundant network-layer transmission attern, e.g., flooding [7]. Flooded fountain codes introduce a double layer of redundancy: on one hand, each coded acket is generated by encoding over multile source ackets, and inherently carries information about all of them. As a result, each coded acket is useful not just for an intended relay, but rather it hels all nodes downstream increase their robability to correctly invert the fountain code. Furthermore, each coded source acket is retransmitted by a number of relays, i.e., those that articiate in the flooding rocess. This double layer of redundancy substantially increases the oortunities for each relay between the source and the destination to correctly receive the ackets, Manuscrit received: May 3, 26; revised: October 3, 26, and March 3, 27; acceted: Aril 2, 27. The associate editor coordinating the review of this manuscrit and aroving it for ublication was Ozan Koyluoglu. W. bin Abbas corresonding author, waqas.abbas@nu.edu.k is with the National University of Comuter and Emerging Sciences NUCES, Islamabad, Pakistan. P. Casari is with the IMDEA Networks Institute, Madrid, Sain. M. Zorzi is with the Deartment of Information Engineering, University of Padova, Italy. This work was mainly carried out when W. bin Abbas was with the Deartment of Information Engineering University of Padova, Italy. Part of this work was resented at the IEEE CAMAD 24 conference, Athens, Greece []. Digital Object Identifier: XX.XXXX/XXX... and thus be able to retransmit them. However, many relicas of each acket are circulated, ossibly leading to a waste of resources. Fountain codes have been widely considered in the literature as a flexible error control technique in both wired and wireless networks exeriencing erasure channels. For examle, the SYNAPSE++ rotocol [3] emloys a fountain code to reliably disseminate firmware over a wireless sensor network. Fountain codes have also been shown to imrove the erformance of unicast communications, rovided that early acknowledgment ackets are sent in order to revent the generation of useless redundancy [8]. In [6], the introduction of fountain codes in vehicular networks leads to a significant erformance imrovement comared to a simler aroach where data ackets are eriodically retransmitted. Delay-Tolerant Networks DTNs [9] and distributed storage systems [] also benefit from the distribution of redundancy that comes from the use of fountain codes. Therefore, it is imortant to design a rocedure to disseminate fountain-coded data through a network in an effective yet controlled method. Ideally, such method should exloit the double redundancy offered by coding and flooding in order to deliver ackets successfully, while at the same time avoiding excessive acket relication and the resulting energy wastage. In our recent work [], we analyzed the interlay between fountain codes and flooding, by considering both a full flooding and a restricted flooding olicy. Given the amount of redundancy generated at the source, the number of hos, and the number of relays that interact at each ho, we derive the average number of transmissions required to achieve a rescribed robability of success at the destination. The latter is emloyed to analyze the interlay between the number of transmissions and the code redundancy. In this aer, we extend the work in [] in several resects. First, while we maintain the analysis simle and tractable, we emloy a more realistic hysical layer model including ath loss and fading in our simulation results. We extend our analytical results to forecast the behavior of the network in different toologies and under different deloyment assumtions. Our forwarding schemes are evaluated based both on a simle error model and on a more realistic block fading model, which validate our analytical results. Second, we roose a family of simle olicies that imlement restricted flooding distributedly, based only on the local decisions of each node. We also formulate an adative olicy that aroaches the behavior of full flooding, but with a significantly lower retransmission overhead. Both olicies combine forwarding with ractical medium access control MAC-level considerations that hel reduce the comlexity and cost of acket forwarding. The remainder of this aer is organized as follows. In the next section we survey the related literature. In Section III

2 2 we introduce our network model: we analytically characterize the robability of success and the number of transmissions taking lace in the network under different acket forwarding schemes, and introduce ractical and distributed forwarding olicies; in Section IV, we evaluate the erformance of these olicies and comare them to the results obtained from the analytical model; in Section V we develo in detail one among several ossible extensions to our work. Finally, Section VI concludes the aer. II. RELATED WORK Several works in the literature have alied fountain codes to scenarios of interest, mostly considering end-to-end communications throughout a network. In [5], an analytical model is rovided for the throughut erformance of a fountain code transmitted through either a conventional or a cooerative multiho network, both in the delay tolerant and in the delayconstrained case, assuming a Nakagami fading model. The work in [2] derives an analytical exression for the average number of received ackets in delay-constrained networks, whereas [8] measures the comlexity of different fountain code relaying strategies. In [3], the authors consider a multiho scenario with a single node at each relay stage, and analyze the robability distribution of the transmission time in the resence of interference from randomly distributed nodes. However, in [5], [8], [2], [3], multiho communications take lace through a single, fixed ath determined a riori, rather than considering a flooding scenario as we do in this aer. In [4], erasure codes have been alied to multicast transmissions of short duration in a single-ho cellular scenario. Fountain codes have also been shown to imrove data dissemination [5], MAC layer communications [6] and reliable transort [4] in underwater networks. Random linear acket coding is alied to single-ho broadcast networks in [7]. Transmit ower adjustment olicies are roosed based either on the erformance of the worst link or on the average erformance of all links. In [8], in order to increase the reliability of acket delivery over multiho unicast connections, a node determines the lost ackets thanks to an imlicit acknowledgment rocedure and retransmits them at a later time if needed. In [9], restricted flooding is roosed, where a node transmits with robability after receiving the mth coy of a message, where is selected based on ercolation theory. Several aroaches have been roosed to mitigate the overhead coming from the excessive transmission redundancy. In [2], a network coding-based aroach is discussed to find the minimum forwarding robability required to achieve a given outreach robability over random grahs. The authors in [2], [22] discuss dynamic robabilistic broadcasting schemes in MANETs: the retransmission robability is adjusted based on the number of neighboring nodes the higher the number of neighboring nodes, the lower the rebroadcasting robability. In [23], adative robabilistic flooding is alied to ath discovery in multiath routing: the retransmission robability of advertisement messages is rogressively reduced to limit the flooding rocess. A similar scheme is also considered in [24] to reduce transmissions in a vehicular ad hoc network, where the rebroadcasting robability is tied to the vehicle seed. Recently, [25] roosed that a node tunes network coding based on acket recetion robability estimates for neighboring nodes, in order to maximize the aggregate number of source ackets decoded by all neighbors while limiting the dissemination delay. These estimates are evaluated based on the link quality correlation among the neighboring nodes. In [26], lower bounds for the required number of acket retransmissions at the MAC layer are derived in order to suort broadcast with and without alying network coding techniques. In [27], a reliable broadcast transmission aroach based on random linear network coding is considered, in which all source ackets are initially transmitted, and once the ACK/NACK storm has finished, the source transmits network-coded ackets to imrove the erformance of the worst receiver. In [28], the authors roosed a low-comlexity and energy-efficient fractional transmission scheme FTS broadcasting aroach that emloys fountain codes over a multiho wireless network. Based on ho distance, a fraction of the encoded ackets to be transmitted to a articular node is assigned among its neighboring nodes. If the sum of the received fractions exceeds one, then the assigned fractions can be adated to reduce the number of redundant transmissions. After successful decoding, a node starts transmitting additional encoded ackets. Unlike revious work, in this aer we study the interaction of flooding and fountain codes via a fundamental aroach. We start with a mathematically tractable model of the network erformance that is roven to match realistic simulations quite accurately. We emloy the analysis to identify desirable working oints in terms of retransmission overhead and robability of success, and finally roceed to roose heuristic olicies that can aroximate the desired robabilistic forwarding behavior, in terms of both robability of success and overhead. Unlike in such works as [8], [9], [23], our ractical olicies rely on decisions that are made by each node in a fully distributed manner. Moreover, unlike [2], [25] [27], our olicies do not rely on further network coding, hence the source is in control of the behavior of the flooding rocess. III. SCENARIO DESCRIPTION, ANALYTICAL MODEL, AND PRACTICAL FLOODING POLICIES We consider a multiho network scenario, where a source S has m ackets to send to a given destination node D. To rotect the communication from acket losses, the source encodes these m ackets into M > m ackets using a random fountain code over the Galois field of size η, F η, η 2. We define E = M m as the number of redundancy ackets injected into the network to favor the decoding rocess at the destination, and to combat acket loss over the network links. All ackets are forwarded towards the destination through a multiho network where we assume that there are r relay stages between S and D, and each stage i consists of N i relays. This way, the shortest ath joining S and D requires r + hos. In addition, we assume a Bernoulli error model, whereby the robability of acket error over each link is

3 3 constant and equal to. While this assumtion is required to make the roblem tractable, we will show in Section IV that it is a very good aroximation for more realistic link error models. In the following three subsections, we will model the forwarding rocess from S to D, by considering different tyes of flooding. We will derive the robability that D can decode the fountain code, as well as the number of transmissions that the network carries out to advance the fountain code ackets towards the destination. Secifically, Section III-A focuses on a baseline, unrestricted flooding aradigm; Section III-B describes distributed and tunable forwarding olicies that achieve a good tradeoff between the robability of decoding success at D and the number of transmissions in the network; Section III-C extends the analytical model to a form of restricted flooding, which is leveraged to rovide insight on the ractical olicies; finally, Section III-D comments on the fundamental interlay between the Galois field size emloyed for the fountain code and the network erformance. A. Unrestricted flooding model We start by assuming that the forwarding aradigm adoted in the network is a baseline full flooding, i.e., a multiho forwarding rocess where every unique acket is transmitted once by the source, and forwarded exactly once by each node that correctly receives it. Possible collisions among the transmissions by two or more relays of either the same or different stages can be avoided in several ways. Feasible measures include channel sensing, fine-tuned backoff times and transmission ielining [3], and tyically introduce additional delays. However, we note that alications relying on flooding to convey data through a network are most likely concerned about the robability of successful delivery than about the time it takes to deliver a acket. Therefore, in this aer, we focus mainly on the robability of success, discuss a model to measure it, and resent effective distributed ways to achieve successful acket delivery while limiting the number of transmissions in the network. In this light, a collision avoidance mechanism that introduces some delay is not a restraining assumtion. 2 For reference, a scheme of our network scenario is rovided in Fig.. We recall that, assuming at least l m ackets are received by D, the robability P e,d that D cannot decode the fountain code is uer-bounded by [29] P e,d η l m, l m It could be argued that allowing nodes at generic stage i to receive transmissions from farther stages rather than just i would hel imrove the robability of correct recetion at stage i and, as a consequence, at the destination as well. In our extensive set of simulations not shown here due to lack of sace we confirmed that this is not the case, excet in a very restricted set of scenarios with benign channels i.e., very low and very few nodes in each stage. The latter is tyical of standard multi-ho routing scenarios, which are not the focus of this aer. In all other cases, the gain is negligible because either a the robability of error from stage i 2 to stage i is very high or b the flooding rocess comensates for transmission errors and is thus the main reason behind successful recetion at stage i. 2 A ossible alternative for collision modeling would be to incororate the robability of collision into the link error robability. This entails the analysis of the interlay between MAC- and routing-level erformance, and is left as a future extension. S 2... N 2... N Stage Stage 2 Stage r Figure. Reference scenario for a multiho network with multile relay stages between a source S and a destination D. The robability of error over any link is equal to. where η is the size of the Galois field over which the code is designed, e.g., η = 2 for a binary fountain code. If l < m, we set P e,d =. Based on the assumtions above, we now derive the robability of decoding error at D. Assume for the moment that r =, i.e., there is only one relay stage between S and D, and the length of the ath from S to D is 2 hos. The event that a given encoded acket does not reach D is equivalent to the event that no relay receives the acket correctly, or that all relays that received the acket fail to forward it further to D. The robability of this event is q = N j = N j 2 N r j N j N j = + N, where we recall that is the link error robability. The argument of the summation means that if N j nodes out of N receive the acket correctly, all these nodes fail in forwarding it to D over the second ho. We note that the number of nodes that fail to receive a acket at relay stage follows a binomial distribution BN,,j with arameters N and, hence its average is N. Considering the degenerate cases where =,j = N all nodes fail to receive if the link error robability is identically and when =,j = all nodes are successful if the link error robability is identically, we note that BN,,j should be extended to yield in both cases. We therefore define B N,,j = if =,j = N or =,j =, and B N,,j = N j j N j otherwise. Assume now that r = 2, i.e., there are two relay stages between S and D, resectively, containing N and N 2 relays. The robability that a acket is correctly received by any given number of relays,2,...,n 2 of stage 2 deends on the number of nodes that successfully received the same acket at stage. Hence the robability that the acket does not reach D is q 2 = N N 2 j =j 2= D 2 B N,,j B N2, N j,j 2 N 2 j 2, 3 wherej,j 2 track the number of relays that failed the recetion of the acket at stages and 2, resectively. We note that, for

4 4 a given value of j, the robability of failing to advance a acket from stage to any node in stage 2 is N j. In the general case, there are r relay stages between S and D, and the exression for the robability of error can be given as q r = N N 2 j =j 2= N r j r= B N,,j B N2, N j,j 2 B N r, Nr jr,j r N r j r, 4 where j,j 2,...,j r are the numbers of relays that failed the recetion of the acket at the first, second, u to the rth stage, resectively, and therefore N r j r is the number of nodes which received the acket correctly at the r th relay stage. Finally, by virtue of the aroximate formula in, the robability that D fails to decode the fountain code is uerbounded by whenever D correctly receives only m or fewer coded ackets. Conversely, the robability of decoding failure is uer-bounded as in whenever at least m + ackets are received. Therefore, we have the following uer bound: P fail q r m B M, q r,k + k= M k=m+ B M, q r,k η k m which in non-degenerate cases can be exlicitly re-written as m M P fail q r qr M k q r k k k= M 6 M + qr M k q r k η k m, k k=m+ where k m is the number of extra ackets received by D. The formula in 4 does not admit a closed-form exression, and neither do 5 and 6. However, we will now illustrate a conveniently comact way to write 4, which will be leveraged to comute other metrics. We start by grouing the robabilities that a given number of relays fail acket recetion at a given stage into matrices. To do so, we define the following robabilities S,j = B N,,j, i,jk = B N i, Ni j,k, D,k = Nr k. 7 In articular, S,j is the robability that j nodes at the first relay stage fail to receive the acket froms; i,jk is the robability that k nodes at the ith relay stage fail to receive the acket given that j nodes failed to receive it at stage i ; D,k is the robability that the destination D fails to receive the acket, given that k nodes out of the available N r nodes at stage r failed to receive it at stage r. We now define the column vectors S = [ S, S, S,N Nmax N ] T, D = [ D, D, D,Nr Nmax N r ] T, and the matrix P i = i, i,ni Ni + N max N i i,ni N i Nmax N i N max+ i,ni 5, 8 where N max = max i N i, and we remark that i,ni N i = is the only non-zero element of the N i th row. Eq. 4 can now be rewritten as r q r = T S P i D, 9 i= where it is intended that P = I Nmax+, the N max + N max + identity matrix. The value of q r thus derived can be finally lugged into 6 to obtain an uer bound to the robability of decoding the fountain code at D. We illustrate the formulation above via a simle examle. Consider a scenario with r = 2 and N = N 2 =. In this case, S = [ ] T, D = [ ] T, and matrix P is given as [ B,, P 2 = B,, B,, B,, ] [ = Finally we have q 2 = T SP P 2 D = [ ][ ][ ] = j = j 2= N2 j ]. For convenience note that, if r = 2, 3 can be rewritten as N N 2 N q 2 = j N j j 2 N2 j k N j N2 j2 N2 j2 + N, 2 where the last term conveys the fact that if all nodes at stage fail to receive a acket from the source, the destination will also fail to receive it. For N = N 2 = and r = 2, 2 exands to q 2 = , 3 where the four terms are comuted for j,j 2 =,,,,, and,, resectively. As exected, equals 3, and both equal 3, i.e., the robability of error over a 3-ho ath, where the error robability over each link is. Along the same line of 9, we now comute the average number of transmissions that all nodes carry out to advance a given acket towards the destination regardless of whether the acket actually reaches the destination or not. Focus on relay stage i and assume that j i nodes failed the recetion of the acket at stage i. The conditional number of transmissions that take lace at stage i is equal to the number of nodes that received the transmission correctly. Its average over the distribution of the number of successful nodes at stage i can be found as u i ji j = N i j i= B N i, Ni ji,j i Ni j i = N i Ni ji, 4 Eq. 4 must be averaged over j i,j i 2,...,j to yield the unconditional average number of transmissions at the ith relay

5 5 stage. Call this quantityu i. To comute the average of 4, we can relicate the structure of 9 by first defining the column vectors t i = [ N i N i ] T. 5 Thus, u i = T i S j= P j ti, and the average total number of transmissions is found as r T avg = + u i. 6 i= We remark that u = N and u 2 = N 2 + N, after which no closed-form exression can be obtained for the remaining u i s. If we assume that each stage hosts the same number of relays N, then P i = P i, where P is an N + N + matrix, and we can simlify 9 to yield and T avg = + r i= T S Pi t i. B. Distributed Flooding Policies q r = T SP r D, 7 In many cases, tyically if the link error robability is sufficiently low, the multilicity of the flooding rocess is excessive, and leads to several acket relicas being uselessly transmitted, without noticeably increasing the success robability at the destination. It is therefore aroriate to design ractical flooding olicies that attemt to reduce this number of transmissions. Secifically, each node should decide locally whether to retransmit a given acket or not, without exlicit coordination mechanisms. We achieve this by allowing the nodes to overhear nearby traffic to understand how many relays have retransmitted a given acket at each stage. We design two olicies for this urose: the first is named Predetermined Restriction PR, and the second is Adative Restriction AR. PR is aimed at enforcing a maximum number of transmitters er stage. Conversely, the objective of AR is to kee the flooding rocess running by adating to the network conditions: if many successful retransmissions are detected, fewer nodes will tend to transmit; if overheard traffic suggests that a given acket is not being correctly advanced to subsequent stages, more nodes will act to suort the flooding rocess. For all olicies, we assume the resence of a MAC rotocol that avoids collisions between retransmissions by different relays. In ractice, this can be achieved with very high robability by having relays back off before erforming a retransmission, where the length of the backoff interval is drawn within a sufficiently long backoff window, or by loosely synchronizing subsequent relay stages in order to make transmission ielining ossible [3]. To fix ideas, in the following we assume the case of a backoff-based MAC rotocol. We also assume that each node listens to the channel during the backoff eriod and, when its own backoff timer exires, the decision to transmit is made based on the olicies described in the following subsections. Flooding with Predetermined Restriction PR: The PR flooding olicy rescribes that a node at a given stage should avoid retransmitting a acket if it overhears at least one retransmission of the same acket by any other node at the same stage, and the SNR of this transmission is greater than a rescribed value θ r. Therefore, we can aroach some desired average maximum number of relays er stage, ˆNres, by varying θ r. To formalize the above ideas, assume, without loss of generality, that the nodes are ordered and numbered increasingly in subsequent relay stages, i.e., those of the first stage are numbered from to S, those of the second stage from S + to S 2, and those of the ith stage from S i + to S i, where S k = k l= N l. As a acket is forwarded from stage i to stage i, the nodes at stage i start entering backoff eriods to relay the acket, and kee listening for retransmissions in the meantime. As a node, say j, is ready to retransmit a acket from stage i, it will check if there exists at least one additional relay S i + k S i, for which γ kj θ r, where γ kj is the SNR of node k s transmission as measured by node j. In this case the node will refrain from transmitting, otherwise it will relay the acket to the next stage. We note that, by increasing θ r, fewer overheard retransmissions will meet the SNR threshold; in turn, the robability that a node is silenced decreases, and the average number of relays er stage increases. Conversely, by decreasingθ r, a node can be silenced on average by a larger number of retransmitted ackets, hence the robability that the node will refrain from retransmitting increases. As a result, this will limit the average number of relays er stage. 2 Flooding with Adative Restriction AR: The AR olicy is designed to achieve a high robability of success while at the same time reducing the overhead of the full flooding mechanism. Focusing on a given acket with sequence number n, the AR olicy roceeds over the following two stes. Ste : Perform a relaying ste according to the PR olicy. At a generic stage i, this will result in some nodes having relayed the ackets, whereas other nodes will have refrained from doing so according to the rules of the PR olicy see Section III-B. For acket n, call C s i n the set of the nodes silenced at stage i: we note that the set can vary for different ackets, hence for different values of n. All silenced nodes in C s i n remain in a listening state, attemting to overhear acket n being forwarded further downstream by nodes at stage i+. Ste 2: For stage i, any node that hears acket n being transmitted at least once from stage i+ to stage i+2 exits the flooding rocess for acket n ermanently. Conversely, the nodes that could not hear acket n being forwarded from stage i+ to stage i+2 assume that the forwarding rocess might have been unsuccessful. Therefore, they break silence and relay acket n from stage i to stage i+. We note that this further transmission ste may result in additional nodes that receive the acket successfully at stage i+. In the meantime these nodes will have overheard other transmissions in their neighborhoods, and will also decide to transmit or not according to stes and 2 of the AR olicy. However, in no event will a node forward a received acket

6 6 more than once. Therefore, the number of transmissions carried out under the AR olicy will be always less than or equal to that of unrestricted flooding. Both the PR and AR olicies work based on the idea that the forwarding oerations could take advantage of some inherent MAC-level mechanisms that are tyically resent in MAC rotocols for wireless networks, including idle listening and backoff. Instead of restricting such oerations to the role of informing the MAC logic, we leverage on them to otimize the flooding erformance by achieving a good tradeoff between the robability of decoding success at D and the number of transmissions. Further insight on this asect is rovided in our erformance evaluation in Section IV. C. Restricted flooding model Both the PR and the AR olicies act by reducing the total number of relays er stage, either in a fixed PR or in an adative fashion AR. An analytical framework to assess the erformance of these olicies can be rovided by extending the model of Section III-A to include restricted flooding. Restricted flooding is generally defined as a flooding olicy where some nodes refrain from forwarding a acket even though they received it correctly. In this aer, we model restricted flooding as a limit N res on the maximum number of nodes that relay a acket at each stage. 3 To simlify the analysis, we assume that the number of successful nodes is known to all relays of a given stage, and that some form of arbitration takes lace, in order to make sure that u to N res relays transmit. Note that this assumtion was already relaxed in the design of our roosed heuristic olicies Section III-B. Under the assumtions above, the analysis carried out so far offers a straightforward way to model restricted flooding. Assume that we have the same number of relays at each stage, which is the case, e.g., in 7: the analysis can be easily extended to the more general case of Eq. 9. Restricting the maximum number of relays to N res means that whenever more than N res relays correctly receive a acket at a given stage, only N res of them will actually retransmit. This can be modeled by setting the first N N res + lines of matrix P in 7 to be equal to the vector res = [ N N res N N res N N res N ] T 8 to yield P res = res T. res T N N res + N N res +N N NN. 9 In the same vein, D becomes res D = [ D,N N res D,N N res D,N N res + D,N ] T, so that q res r = T S Pres r res D, and the robability that D 3 Note that, in this model, N res denotes the exact maximum number of nodes that will forward a correctly received acket at any stage, as oosed to the average number of nodes allowed to forward a acket at each stage in the PR and AR olicies, which was denoted as ˆN res in Section III-B. fails to decode the fountain code is P fail q res r. To comute the average number of transmissions in the network, we first define the column vector t res D = [ N N res N N res N N res ] T, 2 and finally the average number of transmissions is found as r Tavg res = + T S P res i t res D. 2 i= D. Selecting the Galois field size η Before roceeding to resent analysis and simulation results in Section IV, it is useful to choose the size of the Galois field over which the flooded fountain code is designed. We do this by evaluating the interlay between the Galois field size and P fail. In fact, a tyical fountain code acket contains an encoding vector that defines how many ackets have been linearly combined, and with which coefficients. Designing the fountain code over a small Galois field at the minimum, F 2 kees the encoding vector short at one bit er encoded acket; however, in this case Eq. dictates that the robability of successful decoding at the destination becomes lower for a fixed number of redundancy ackets E. Conversely, a large Galois field e.g., F 256 as tyically assumed in rateless code design rovides a higher robability of success for equal E, but at the same time it increases the size of each coded acket. Given that we consider the flooding of fountain codes through a network, where each acket is exected to be retransmitted several times by different nodes at each ho, we need to find a good tradeoff between the overhead yielded by the encoding vector size, and that yielded by the number of extra ackets. Assume that each acket has the following structure: a fixed overhead that reresents generic rotocol information e.g., Time-To-Live and version fields, flags, etc. of size α fix ; a sequence number that distinguishes different sequences of m source ackets, of size α seq ; and an encoding vector of m ξ bits, where ξ = log 2 η, and η is defined in ; finally, a ayload of L bits. We calculate the transmission overhead affecting a sequence of m source ackets as O tx = m+eα fix +α seq +mξ+e L m+eα fix +α seq +mξ +L. 22 Fig. 2 shows a lot of P fail q r vs. O tx for different values of ξ, E and. The acket arameters are α fix = 24 bits, α seq = 8 bits, L = 256 bits. For each value of, a set of markers of the same color is shown; the set is sanned from bottom-left to to-right by increasingξ from to 8. Three sets of curves are shown, resectively for E =, E = 4 and E = 8: a curly brace encomasses the horizontal san of each set. As exected, the robability of success increases with increasing ξ, and with increasing E for fixed ξ. However, both imrovements come at the cost of an increase in O tx. While the increase of O tx with E is exected and deends on the configuration of the code, the value of ξ can still be otimized. In articular, we observe that P fail q r remains almost constant after ξ = 4. Therefore, we will fix this value throughout the rest of this aer.

7 7 Pfailqr E = Increasing ξ from to 8 E = 4 E = 8 =.5 =.6 =.7 =.8 = O tx.5.6 Figure 2. P fail q r vs. O tx for different values of ξ, E and of the robability of link error. For each value of, a set of markers of the same color is shown; the set is sanned from bottom-left to to-right by increasing ξ from to 8. The results show no significant imrovement in P fail q r for ξ 4. A. Simulation scenario IV. NUMERICAL RESULTS We now resent some results related to the flooding of fountain codes through a multiho network. We will show that our analytical model, albeit based on a simle uniform Bernoulli error rocess, matches simulations based on a more realistic Rayleigh fading roagation model with good accuracy, rovided that an aroriate ma between the error robability of the Bernoulli model and the average robability of error in the Rayleigh fading scenario is used. The simulations are set u as follows. We reresent each relay stage as a set of nodes whose osition is drawn at random within a cell of size 2 m 6 m, where the centers of any two nearest cells are = 6 m aart. This scenario makes it ossible to test that our analysis is still valid even after removing one of the assumtions of Section III, namely that any two nodes belonging to subsequent relay stages are always connected. The source S and the destination D are located, resectively, at the center of the first and last cells. We assume that all nodes transmit with a ower P T. Therefore, if two nodes i and j are located at a distance d ij, the Signal-to- Noise Ratio SNR γ ij of the link between the two nodes can be comuted as γ ij = ρ ij P T d α ij /P N 23 where α is the ath-loss exonent, ρ ij is an exonentially distributed ower fading coefficient of average value which corresonds to a Rayleigh fading amlitude reresenting the fading realization on the link from i to j, and P N is the noise ower. We assume that a transmission from i to j is successful if and only if γ ij θ, where θ is a minimum SNR threshold. In order to comare the Rayleigh fading simulations with the Bernoulli link error model resented in Section III, we ma the arameter of the Bernoulli model to the average error robability comuted via a stochastic geometry argument. Consider two nodes i and j, where i belongs to stage l, l =,2,...,r and j to stage l +. Without loss of generality, let l =, i.e., focus on the first and second relay stages. Call x i,y i and x j,y j the coordinates of nodes i and j, and assume that they take values in [,X i ], [,Y i ], [, +X j ], [,Y j ], resectively. In our simulation scenario, X i = X j = 2 m, Y i = Y j = 6 m and = 6 m. The ma is obtained as follows: = Xi Yi +Xj Yj + dx i dy i dx j dy j [ e ρij ρij P T d k ij A i A j P N dρ ij ] < θ, 24 where A i = X i Y i, A j = X j Y j, e ρ /A i A j is the joint distribution of fading and of the locations of nodes i and j, and [ ] denotes the indicator function, which returns whenever the argument is true. We remark that 24 comutes the average value of the indicator function over all random arameters that concur to the comutation of the link error robability. Such an average is by definition the robability of a Bernoulli event [3, Section 3.], which is fully in line with our Bernoulli link error robability model. In the following, we set α = 2, P T = 34 dbm and P N = 7 dbm, we consider the network to be comosed of r = 5 relay stages, and we assume that there are exactly nodes in any stage, i.e., N = = N r =. Different robabilities of success are obtained by varying θ between and 2 db. All results are averaged over 2 random draws of the nodes ositions, and over different fading realizations for each osition. In the next Subsection, we will discuss the erformance of flooded fountain codes in the resence of both full and restricted flooding. In Subsection IV-C, we will discuss the erformance of our roosed ractical olicies, while in subsection IV-D, we will comare these olicies with the restricted flooding model. In Section V, we show the flexibility of our formulation by extending the analysis to the case of a heler node injecting additional redundancy in the network, and discuss the otimal lacement of that node. B. Performance of fountain codes under unrestricted flooding We start from Fig. 3, which shows P fail q r vs. in the case of unrestricted flooding, by comaring our analytical model against Rayleigh fading simulation outcomes, for two different values of E. We observe that the robability of decoding success at the destination for E = 8 remains ractically equal to for.75, after which it sharly falls and becomes ractically for.85 due to the excessive number of forwarding errors. The anticiated effect of a lower number of redundancy ackets, for E = 2, is that the robability of success is ractically only for.5, and the transition to is also smoother. In both cases, Fig. 3 shows a very good agreement between the simulations and the analytical model. A second metric of interest for characterizing the network behavior is the average number of nodes that successfully receive a given acket at each relay stage for different values of. This metric is deicted in Fig. 4. The results show that the number of successful nodes at each stage increases for

8 8 Pfailqr Analysis, E = 2 Analysis, E = 8 E = 2 E = 8 Simulation Figure 3. Success robability P fail q r vs. for m =, r = 5 and N =, for varying number of redundancy ackets, E. The robability that the destination can decode the fountain code, mainly driven by E, dominates the erformance for sufficiently low values of. Probability of decoding success at D.8.6 PR, ˆNres = 2.4 PR, ˆNres = 3 PR, ˆN res = 4 PR, ˆN res =.2 AR, ˆN res = 2 AR, ˆN res = Analysis, Unrestricted Average number of successful nodes Analysis, =.4 Analysis, =.6 Analysis, =.8 Simulation Relay stage Figure 4. Average number of nodes that received a given acket correctly as a function of the relay stage from to 5 for different values of. For sufficiently high link error robability =.8, in every stage there exist some nodes that fail to receive a given acket. Average number of transmissions Unrestricted ˆN res = ˆN res = 4 ˆN res = 3 ˆN res = Figure 5. Probability of decoding success atd as a function offor different combinations of ˆN res and N res, for the PR and the AR olicies, comared to the analysis for unrestricted flooding. Figure 6. Average number of transmissions as a function of for the PR olicy for different values of ˆN res, comared to the analysis for unrestricted flooding. decreasing and that, even for moderately high values of u to.6 in Fig. 4, ractically all N r = nodes in the last relay stage have correctly received the acket. Conversely, =.8 reduces the number of correct transmissions, and it is difficult for the flooding rocess to involve several nodes before the acket traverses the last relay stage. In all cases, the simulations match the analysis very well. One of the main conclusions from the revious results is that P fail q r remains equal to even for significantly high values of. This further motivates the design of ractical olicies that achieve the robability of success of unrestricted flooding, while requiring fewer transmissions. The following subsection resents the erformance evaluation of the PR and AR olicies, introduced in Section III-B and III-B2. C. Performance of the PR and AR olicies The behavior of the PR and AR olicies is simulated by assuming the Rayleigh fading roagation model discussed in Section IV-A, and comared to the analytical model for unrestricted flooding via the ma for given in 24. Fig. 5 shows a lot of the robability of correct decoding at D against obtained via the simulation of the PR and AR olicies, comared to the unrestricted flooding analysis. For the latter, this is equivalent to P fail q r. The PR and AR curves are lotted for different values of ˆNres. There are N = nodes in all relay stages, and the number of redundancy ackets transmitted by the source S is fixed to E = 4. Considering the PR olicy, we observe that increasing the ˆN res arameter by choosing θ r as exlained in Section III-B corresondingly increases the number of relays that are allowed to re-forward a acket at each stage. For examle, according to our simulation setu, settingθ r 4 db corresonds to N res = 8, whereas θ r 2 db corresonds to N res = 2. Increasing ˆN res by increasing θ r rogressively imroves the robability of success as a function of, until it finally matches that of unrestricted flooding for ˆN res =. Contrary to the PR olicy, the AR olicy is adative. Posing

9 9 Average number of transmissions Unrestricted ˆN res = 4 ˆN res = 3 ˆN res = 2 ˆN res = Figure 7. Average number of transmissions for a single acket carried out by the AR olicy as a function of for different values of ˆN res, comared to the number of transmissions of unrestricted flooding. restrictions on the number of forwarders via ˆN res makes the olicy react by allowing more silenced relays to retransmit, in case they fail to hear relays at the next stage re-forward a acket. Therefore, as we observe from Fig. 5, the success robability of the AR olicy is always comarable to that of unrestricted flooding. This roves the effectiveness of the olicy at comensating for forwarding errors. It is interesting to evaluate the behavior of the PR and AR olicies in terms of the number of transmissions erformed er acket carried through the network. This metric is deicted in Fig. 6 for PR, and shows the exected behavior that a higher value of ˆN res results in more relays being activated er stage, hence in more transmissions. For the same configurations shown in Fig. 6, we observe a steady increase in the number of transmissions. For ˆNres =, PR transmits ractically as many ackets as unrestricted flooding. A different behavior is observed for the average number of transmissions er acket of the AR olicy as a function of Fig. 7. When <.2, the low robability of error makes the choice of ˆN res = 4 very inefficient, as in this condition it is highly likely that the nodes will overhear other transmissions, refrain from transmitting, and hear the acket be correctly retransmitted by the nodes at the next stage. Instead, only a few relays will suffice to correctly convey a acket to D. In this resect choosing, e.g., ˆN res = 2 would achieve very good success robability with about half the transmissions comared to the case ˆN res = 4. The ranking among the curves is rogressively inverted when increases, as in this case more relays are needed to guarantee success. For examle, for =.5, ˆNres = 4 yields fewer transmissions than even ˆN res = which is too restrictive, and often causes several silenced nodes to transmit, after they fail to hear the relays at the next stage forward the ackets further on. Fig. 7 also suggests that ˆNres = 2 and ˆN res = 3 are good choices for almost all values of, excet erhas if <.2. Finally, note that the AR olicy erforms similar to unrestricted flooding in terms of robability of successful decoding at D Fig. 5, but requires much fewer transmissions to attain this, as confirmed by Fig. 7. Probability of decoding success at D.9 3,7 2, 2,9 2,8 Increasing N res, ˆNres Analysis, E = Analysis, E = 9 Analysis, E = 8 Analysis, E = 7 PR, E = PR, E = 9 PR, E = 8 PR, E = Average number of transmissions Figure 8. Interlay between the robability of decoding success at D and the average number of transmissions achieved by the PR olicy, comared to restricted flooding, for =.5 and for different values of N res, ˆNres, and E. Curves are sanned from left to right by increasing N res for restricted flooding or ˆN res for the PR olicy. D. Insight from the restricted flooding model The restricted flooding model described in Section III-C offers a good means to understand the behavior of the PR and AR olicy. In Fig. 8, we consider the interlay between the robability of decoding success at D and the number of transmissions erformed in the whole network. The grah shows a comarison between restricted flooding lines and the PR olicy markers, for =.5 and for different values of N res, ˆN res and E. The restricted flooding curves are sanned from left to right by increasing N res, the PR curves by increasing ˆN res : in both cases the effect is to increase the robability of decoding success at the rice of an increase in the number of transmissions. The figure marks four choices of the ˆN res,e airs that achieve a robability of success of at least.9. We observe that the PR olicy is an effective distributed imlementation of restricted flooding, and that it hels achieve the same erformance in terms of number of transmissions, at the rice of a very small decrease in the success robability, mainly due to the distributed imlementation. In any event, the mismatch becomes negligible for ˆN res 3. Fig. 9 comares the erformance of the AR olicy against the analysis of restricted flooding for =.5 and for different values of N res, ˆNres and E. The lot suggests that the AR olicy behavior tyically leads to an increased number of transmissions. However, this is comensated by the advantage of a higher robability of success. This is esecially the case for low values of E, which imlies that the tyical overhead achieved by the AR olicy is lower see also Fig. 2. Once the robability of success has achieved a value of about, a further increase of ˆNres yields a negligible imrovement and, as exected, causes the number of transmissions to increase. In any event, the increase is limited with resect to lain restricted flooding, as the AR olicy achieves a very high robability of success already for low values of E. For examle, for E = 2 the AR olicy already attains a robability of success close to, and requires fewer transmissions than lain restricted flooding in order to achieve the same result.

10 P hl fail q r,q v m k max j= k= M j qr M j q r j Eh k q Eh k v q v k + M E h j=j min k=k min M j qr M j q r j Eh k qv Eh k q v k η j+k m 28 Probability of decoding success at D.9 Analysis, E = Analysis, E = 9 Analysis, E = 8 AR, E = 4 AR, E = 3 AR, E = Average number of transmissions Figure 9. Interlay between the robability of decoding success at D and the average number of transmissions achieved by the AR olicy, comared to restricted flooding, for =.5 and for different values of N res, ˆNres, and E. Curves are sanned from left to right by increasing N res for restricted flooding. V. EXTENSION: HELPER NODES The study resented in this aer considers fountain-coded data flooded through a network u to a given destination, and rovides fundamental insights on the interlay between the robability of success and the number of transmissions required to achieve it. Two olicies are roosed to otimize the flooding rocess by reducing the total number of transmissions without decreasing the robability of success. The study lends itself to several extensions, in terms of both functionality and modeling. In the following subsection, we extend the study by allowing relays that successfully decoded the fountain-coded ackets from the source to become heler nodes, and inject additional redundancy. In general, this redundancy will be different from that sent by the source, and thus it will hel the nodes downstream decode the source ackets. A. Additional redundancy from intermediate heler nodes In the following, we assume that there exists one node that received enough ackets to decode the fountain code and reconstruct the original information sent by the source S. This node becomes able to inject additional redundancy ackets. We call this node a heler node, and refer to it via the subscrit h. We will now elaborate on the erformance of the network in the resence of a heler, rovide design guidelines for where this node should be laced, and discuss the utility yielded by the resence of additional heler nodes. We start by noting that node h must belong to one of the relay stages between the source S and the destination D. In articular, we assume that out of the r stages that searate S from D, there are u stages between S and h, and v stages between h and D, so that r = u + v +. After successful decoding, h transmits E h additional redundancy ackets, in order to increase the robability of successful decoding at D. For this analysis, we assume that only one heler node exists in the network. We start from the robabilities q u that a acket fails to reach node h after u relay stages, and q v that D fails to receive an extra redundancy acket transmitted by h; using the same formulation of 9, we can write u v q u = T S P i D, q v = T S P i D. 25 i= The robability of decoding failure at the relay stage of the heler node is found via q u as m M P fail q u qu M j q u j j j= M 26 M + qu M j q u j η j m j j=m+ i= The robability that at least one node successfully decodes the fountain code out of the N nodes belonging to the relay stage of node h is P fail q u N. Now, a fountain decoding failure occurs at D in one of the following two cases: i h fails to decode the fountain code and thus cannot hel: in this case, D may fail to decode after the M ackets transmitted by S according to the same arguments in Section III-A; ii h successfully decodes the fountain-coded ackets received from S and floods E h extra redundancy ackets towards D, but D still fails to decode. The total robability of failure is finally uer-bounded by Pfail q u Nu+ P hl fail q r,q v +P fail q u Nu+ P fail q r, 27 where P hl fail q r,q v is the robability of fountain decoding failure at D, given the robability q r that D fails the recetion of a acket from S, and the robability q v that D fails a recetion from h. As in 6, we distinguish between two cases, namely that D receives u to m ackets and that D receives more than m ackets. In both cases, the ackets may either come only from S or be a combination of ackets from S and h. We have the formula in Eq. 28, where k max = min{e h,m j}, j min = max{,m + E h } and k min = max{,m+ j}. In the comutation of the average number of transmissions Tavg hl in the resence of h, we need to account for the extra effort aid by the network to forward h s ackets. We have T hl avg = MT avg S D+E h T avg h D, 29 where T avg S D = + r t i and T avg h D = + v i= T S i i= T S j= P j i j= P j t i resectively

11 r = r = 2 r = 3 r = 4 r = 5 2,5,5 2,5,4 Pfailqr Relay stage Increasing E h from to =.5 =.6 =.7 =.8 =.9 Pfailq r res 2, 5, 3 Increasing N res.9 2,5, E = 5,E h = 5 E = 5,E h = 4 E = 5,E h = 3 E = 5,E h = T res avg Figure. Analysis and simulation of P fail q r vs relay stage in the case N i = N = i, for m =, r = 5, and E =, for varying number of redundancy ackets E h from the extra transmitting node. For higher values of the link error robability, a heler node located at stage r = 4 yields better robability of success, whereas for lower values of, r = 3 is a better choice. Figure. Interlay between P fail qr res and Tavg res for the case where only the source S transmits comared to the case with an extra heler node. For the former, we consider the analysis in Section III-C for different values of E; for the latter, we show simulation results for different N res,e,e h triles, for =.5. Curves are sanned from left to right by increasing N res. reresent the average number of transmissions from S to D and the average number of transmissions from h to D. Fig. lots the robability of successful decoding at the destination in the resence of a heler node, as a function of the relay stage where the heler is located, and for different values of the link error robability, where N =, r = 5 and E =. We observe that the actual value of the success robability increases with the number of redundancy ackets sent by h, E h, and that the best erformance for each value of deends on the relay stage where h is located. For examle, consider the case of =.7. Choosing to have h at stage 4 would make the success robability achieve a value of for E h = 8 ackets. However, this is not the case if the heler node h is laced at any other relay stage, esecially at the st or 5th stage. Similarly, for =.6 and, say, E h = 3, having the heler node at stage 3 achieves a robability of success very close to, whereas a lacement in any other stage would be less effective. While the osition of node h that maximizes the success robability deends also on N and E, the examle above shows our oint that h should not be located too close to S or D. The intuition behind the above result is that if h were too close to S, it would not benefit from the flooding of source ackets and the higher decoding robability that results. Conversely, if h were too close to D, any sufficiently high robability of error would make the acket flooding rocess die out before reaching h. In turn, h would not be able to reconstruct the source information and inject new redundancy ackets. Moreover, even if h could decode the message, extra redundancy ackets would not benefit from the flooding rocess on the way to the destination. Assuming that h is located at stage 4, in Fig. we show the interlay between the robability of decoding success at D and the number of transmissions in the network in the resence of h. We consider restricted flooding see Section III-C and lot different curves, each for a different value of E. All curves are sanned from left to right by increasing N res. In addition to these curves, we lot a set of oints described by the trilets N res,e,e h, which corresond to the cases where a heler node is resent. We observe that the inclusion of h hels reduce the average number of transmissions where the saving is larger for higher values of E. In fact, the best robability of success for the same total redundancy E+E h would be achieved if S sent all the redundancy itself. However, this would also lead to the largest ossible number of transmissions because of the flooding rocess. Having a heler node between S and D hels save considerable resources by reducing the total transmissions while not harming the robability of success. This demonstrates the feasibility of the heler node solution. We now consider the ossibility of having either limited or unlimited heler nodes, and to allow each heler to inject either a limited or an unlimited number of redundancy ackets. More secifically, in the same simulation scenario considered so far, we allow any intermediate node that correctly received a sufficient number of ackets from S to become a otential heler. When a heler injects additional redundancy ackets, these may be received by intermediate nodes and thereby generate additional helers in a sort of avalanche effect. Helers act only if D failed to decode the fountain code via the source ackets relayed by the network. In this case, we ick a heler node at random and let it transmit one redundancy acket each time. This acket will be flooded through the network between the heler and D and, if received by D, it will hel decoding the fountain coded ackets of S. The duration of this rocess and the set of heler nodes are regulated by the following four olicies: Limited number of helers, limited redundancy er heler: only nodes that decoded the fountain code using the ackets received from S can become helers no avalanche. Only one redundancy acket can be transmitted by each heler. 2 Unlimited number of helers, limited redundancy er

12 2 Probability of decoding success at D only Source.3 Lim. redund. / Lim. # helers.2 Lim. redund. / Unlim. # helers Unlim. redund. / Lim. # helers. Unlim. redund. / Unlim. # helers Unlim. redund. / random heler Conditional average # of transmissions Lim. redund. / Lim. # helers Lim. redund. / Unlim. # helers Unlim. redund. / Lim. # helers Unlim. redund. / Unlim. # helers Unlim. redund. / random heler Figure 2. Probability of decoding success atd againstfor different heler olicies. The best erformance is achieved when one or more heler nodes are allowed to send an unlimited number of redundancy ackets. N res = 6, E = 2. Figure 3. Conditional average number of transmissions er acket against, for the only cases where heler nodes are needed to achieve correct decoding at D. The most aggressive heler olicies exerience failure only if is sufficiently high to avoid that any helers actually exist. N res = 6, E = 2. heler: any node that decoded the fountain code using any set of ackets, including those sent by other helers, can become a heler itself avalanche allowed. Only one redundancy acket can be transmitted by each heler. 3 Limited number of helers, Unlimited redundancy er heler: heler avalanche not allowed, helers can transmit an unlimited 4 number of redundancy ackets, until the destination decodes the acket successfully. 4 Unlimited number of helers, Unlimited redundancy er heler: heler avalanche allowed, helers can transmit an unlimited number of redundancy ackets. We comare the four olicies above with the case of a single heler node chosen at random in the network, and allowed to transmit unlimited redundancy ackets, akin to the case discussed for Figs. and. Fig. 2 shows the robability of successful decoding at D for N res = 6 and E = 2. The leftmost curve corresonds to the case where only S transmits. Any heler node olicy rovides better erformance. Secifically, the olicies allowing only limited redundancy er heler start failing more often for.7, as transmission failures revent D from receiving the heler ackets; moreover, it becomes increasingly less likely that there are any helers at all. Conversely, the olicies that allow unlimited redundancy er heler achieve a very good erformance. In articular, the olicy allowing unlimited helers and redundancy er heler only fails in those cases where no heler is resent in the network. The cost of the imrovement in the robability of success is measured by the number of transmissions carried out to relay each acket in the network. For those cases where the heler nodes actually transmit, this metric is shown in Fig. 3. The olicies with limited redundancy er heler rogressively decrease their number of transmissions for.7, suorting the discussion above. Conversely, the olicies allowing unlim- 4 Allowing an unlimited number of redundancy acket transmissions er heler imlies the design of a stoing rule or rotocol, so that the destination can communicate that the fountain code has been correctly decoded. The design of such a rule is beyond the scoe of this aer. Average # of transmissions Lim. redund. / Lim. # helers Lim. redund. / Unlim. # helers Unlim. redund. / Lim. # helers Unlim. redund. / Unlim. # helers Unlim. redund. / random heler Figure 4. Average total number of transmissions er acket against, including both the cases where helers node are required and the cases where they are not. N res = 6, E = 2. ited redundancy kee increasing the number of transmissions until is sufficiently high to revent the generation of helers in the network. The total average number of transmissions in Fig. 4 taken over both the cases where helers act and those where they do not, further confirms the above discussion, and shows that the largest number of transmissions is reached for very high values of, as exected. We remark that a heler node injects redundancy ackets only when it has successfully decoded the source ackets. Removing the latter constraint would mean that the injected redundancy could only hel recover the set of source ackets originally encoded into the fountain-coded ackets received by the heler. The evaluation of this case is left as a future extension. VI. CONCLUSIONS We considered the flooding of fountain-coded ackets through a multiho network, and analytically determined the

13 3 robability that the destination can recover the original data ackets as a function of the number of nodes in the network, of the link error robability and of the amount of redundancy generated by the code. We showed that our model matches simulation results very well. Based on the conclusions drawn from the model, we designed ractical, distributed olicies that achieve the same success robability erformance while requiring fewer retransmissions to advance a acket through the network. We measured the erformance of these olicies and justified their behavior in light of an analytical model for flooding under a restriction on the number of relays er ho. Among several ossible extensions to this work, we considered the case of intermediate relays being able to decode the source ackets and thereby inject additional redundancy in the network. We thoroughly develoed and evaluated this case. 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Paolini, and M. Chiani, Performance versus overhead for fountain codes overf q, IEEE Commun. Lett., vol. 4, no. 2,. 78 8, Feb. 2. [3] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, 3rd ed. Academic Press, 998. Waqas Bin Abbas received the Bachelors and the Masters degree from National University of Comuter and Emerging Sciences NUCES, Islamabad, Pakistan in 28 and 22, resectively, and the Ph.D. in Information Engineering in 27 from the University of Padova, Italy. Currently, he is working as an Assistant Professor at NUCES, Islamabad, Pakistan. During Masters, his research was focused in underwater wireless communication, while during Ph.D., his research was mostly focused on energy efficiency in wireless networks. His current research interests include energy efficiency in 5G millimeter wave cellular networks, MIMO communication and multi-ho wireless networks.

14 4 Paolo Casari received the PhD in Information Engineering in 28 from the University of Padova, Italy. After being on leave at the Massachusetts Institute of Technology in 27, his research rogressively focused on underwater communications and networks. He collaborated to several funded rojects including CLAM FP7, RACUN EDA, and several US ARO, ONR and NSF initiatives. He was Technical Manager of the NAUTILUS and WISE- WAI rojects and is now the scientific coordinator of the EU H22 RECAP roject. In 25, he joined the IMDEA Networks Institute, Madrid, Sain, where he leads the Ubiquitous Wireless Networks grou. He served in the organizing committee of several conferences; he has been guest editor of a secial issue of the Hindawi Journal of Electrical and Comuter Engineering on Underwater Communications and Networking, and is currently co-guest editor of an IEEE Access secial issue in the same field. His research interests include many asects of underwater communications, such as channel modeling and simulation, network rotocol design and evaluation, localization, and field exeriments. Michele Zorzi [F 7] zorzi@dei.unid.it received his Laurea and Ph.D. degrees in electrical engineering from the University of Padova in 99 and 994, resectively. During academic year 992/993 he was on leave at the University of California San Diego UCSD. After being affiliated with the Diartimento di Elettronica e Informazione, Politecnico di Milano, Italy, the Center for Wireless Communications at UCSD, and the University of Ferrara, in November 23 he joined the faculty of the Information Engineering Deartment of the University of Padova, where he is currently a rofessor. His resent research interests include erformance evaluation in mobile communications systems, random access in mobile radio networks, ad hoc and sensor networks and IoT, energy constrained communications rotocols, 5G millimeter-wave cellular systems, and underwater communications and networking. He was Editorin-Chief of IEEE Wireless Communications from 23 to 25, Editor-in- Chief of IEEE Transactions on Communications from 28 to 2, and is currently the founding Editor-in-Chief of IEEE Transactions on Cognitive Communications and Networking. He was Guest Editor for several Secial Issues in IEEE Personal Communications, IEEE Wireless Communications, IEEE Network, and IEEE JSAC. He served as a Member-at-Large in the Board of Governors of the IEEE Communications Society from 29 to 2, and as its Director of Education from 24 to 25.