On the Broadcast Capacity of Multihop Wireless Networks: Interplay of Power, Density and Interference

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1 On the Broacast Capacity of Multihop Wireless Networks: Interplay of Power, Density an Interference Alireza Keshavarz-Haa Ruolf Riei Department of Electrical an Computer Engineering an Department of Statistics Rice University, 600 Main Street, Houston, TX 77005, USA Abstract In this paper we stuy the broacast capacity of multihop wireless networks which we efine as the maximum rate at which broacast packets can be generate in the network such that all noes receive the packets successfully within a given time. To asses the impact of topology an interference on the broacast capacity we employ the Physical Moel an Generalize Physical Moel for the channel. Prior work was limite either by ensity constraints or by using the less realistic but manageable Protocol moel [], [2]. Uner the Physical Moel, we fin that the broacast capacity is within a constant factor of the channel capacity for a wie class of network topologies. Uner the Generalize Physical Moel, on the other han, the network configuration is ivie into three regimes epening on how the power is tune in relation to network ensity an size an in which the broacast capacity is asymptotically either zero, constant or unboune. As we show, the broacast capacity is limite by istant noes in the first regime an by interference in the secon regime. In the secon regime, which covers a wie class of networks, the broacast capacity is within a constant factor of the banwith. I. INTRODUCTION There has been a growing interest to unerstan the funamental capacity limits of wireless networks [3] [8]. Results on network capacity are not only important from a theoretical point of view but also provie guielines for protocol esign in wireless networks. Hitherto, most research on network capacity has focuse on the capacity of unicast connections between ranom source an estination noes. In this paper we stuy the broacast capacity λ B (g) of multihop wireless networks, which we efine as the maximum rate of generation of broacast packets by a set of noes B in the network such that all noes receive the packets successfully. In this work, we consier two channel moels which are known in the literature as the Physical an the Generalize Physical Moel. For comparison purposes we will also refer to the simpler Protocol Moel. The Protocol Moel incorporates interference through simple istances, allowing to apply the usual graph theoretical approaches. The Physical Moel moels interference more accurately, but still assigns a constant transmission rate once successful transmission is guarantee. In this latter sense it is close to the Protocol Moel. The Generalize Physical Moel finally allows for a transmission rate that epens on the level of interference an the istance between sener an receiver an thus allows for a more precise assessment of the broacast capacity. To the best of our knowlege, only two papers so far stuy the capacity of wireless networks for broacasting. The first paper [] moels the locations of the noes via a Poisson point process an the channel via the Generalize Physical Moel. The author of [] stuies the broacast capacity λ B (g) for the case when B consists of one single generating noe, proviing an upper boun for λ B (g) an showing that this boun in achievable up to a constant factor uner the assumption that the Poisson intensity of the noes is fixe. The secon relate paper [2] computes the broacast capacity for an arbitrary connecte network but assuming the somewhat simpler an less realistic Protocol Moel for the channel. It shows that the broacast capacity is within a constant factor of the wireless channel capacity an that it oes not epen on raio range, number of noes, an area of the network. As the first contribution of this paper, we stuy the broacast capacity uner the Physical Moel. This channel moel has been use in earlier work such as [3], [9] but not for the stuy of broacast capacity. We evelop bouns of the broacast capacity λ B (g) for arbitrary connecte wireless networks an for an arbitrary set of generating noes B an arbitrary weight vector g. Surprisingly, we fin that the broacast capacity oes not change more than a constant factor when we vary the number of noes, the transmission power an the area of the network provie we keep a level of connectivity mae precise in the paper. Thus, we are able to confirm that the earlier similar results of [2] /07/$ IEEE This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2007 proceeings.

2 foun for the simpler Protocol Moel hol also uner the more realistic Physical Moels. For the special case of homogeneous networks with large number of noes, we fin that the broacast capacity is Θ(W ) where W enotes the wireless channel capacity in bits per secon. Here, we aopt the stanar notation from complexity theory where O(.), Ω(.), anθ(.) stan for asymptotic upper, lower, an tight bouns, respectively. As our secon contribution, we stuy the broacast capacity uner the Generalize Physical Moel which is wiely use in network capacity papers [], [5], [0]. In this moel the transmission rate is etermine by the Shannon capacity formula an thus epens on the receiving power an the interference of other signals. As a result, poor connectivity of some noes an high interference from simultaneous transmissions can limit the broacast capacity consierably, which we will quantify explicitly. Due to the strong impact of the geometry of noe arrangement in this moel, one nees to make some assumption about the topology. In this paper we focus on homogeneous networks which is a popular moel in network capacity papers. We fin uner these assumptions three power regimes for the broacast capacity in homogeneous networks with large number of noes. In the first regime, low transmission power causes the isolation of some noes in the sense of a limiting receiving rate from the rest of the noes. These noes then become bottlenecks for the broacast capacity which is asymptotically zero. In the secon regime, power is sufficiently elevate an tune to the size an ensity of the network to provie every noe in the network with a ata reception rate within a constant factor of the banwith. At the same time, the network size imposes simultaneous transmissions in orer to achieve a constant rate to all noes. This regime correspons to the most common settings in wireless networking an leas to a broacast capacity within a constant factor of the banwith. Here, interferences of simultaneous transmissions is the limiting factor for the broacast capacity. In the thir regime, the transmission power is so significant in relation to the network size that all noes receive a irect transmission from the broacast source with high rate. We will show that in this regime a multi-hop broacast oes not increase the throughput by more than a constant over irect single transmission from the source. Notably, no prior work ientifies nor stuies these three power regimes, to the best of our knowlege. More concretely, this paper applies to a broaer context than [] by not imposing restrictions on the asymptotic ensities an by treating also the Physical Moel. Inee, though not emonstrate ue to space constraints, the low power regime mentione above can be establishe also for a network moele by a Poisson point process, thus incluing the results of [] as a special case. The present work also goes beyon [2] which was base on the Protocol Moel by assuming the more realistic Physical an Generalize Physical Moels for wireless channels. The paper is organize as follows. In Section II we summarize existing work on the network capacity. We introuce a network moel an efine relevant terms in Section III. In Section IV we compute upper an lower bouns for broacast capacity which apply to arbitrary wireless networks uner the Physical Moel. Section V computes the broacast capacity of homogeneous networks uner the Generalize Physical Moel as the number of noes grows. Finally, we conclue in Section VI. All proofs are place in the Appenix. II. RELATED WORK Gupta an Kumar [3] stuie the per noe capacity for unicast connections between ranom sources an estinations in a static wireless network consisting of n noes istribute in a fixe area. For planar networks this per noe capacity ecreases at least as O(/ n) in an arbitrary network an even as O(/ n log(n)) in a ranom network as the number of noes grows [3]; extensions to three imensional space can be foun in [9]. Both stuies are base on the Protocol Moel an the Physical Moel. Later work [0] establishe that the same bouns still hol uner the more accurate Generalize Physical Moel. Also, [] showe how to achievable asymptotically a throughput within a constant factor of these bouns in ranom networks. In another approach, [5] showe that per noe capacity Ω(/ n) ia achievable using percolation theory techniques. For mobile wireless networks, Grossglauser an Tse [4] propose a mobility-base routing metho in which the number of retransmissions for unicast communication between source an estination is reuce to 2. Thus, mobility can increase the per noe capacity to Θ() provie that the packet elay is allowe to be arbitrarily large. Subsequent work analyze the capacity uner constraints on the elay or the mobility of the noes [2] [7]. Introucing a new irection in network capacity research, [] stuies the broacast capacity of static wireless networks. The paper computes an upper boun for broacast capacity using the Generalize Physical

3 Moel. The paper proves that the broacast capacity is asymptotically within a constant factor of this upper boun as the number of noes goes to infinity provie that the Poisson intensity of the noes remains fixe. However, the paper [] oes not aress the case where the noe intensity is allowe to ten to infinity as the number of noes grows. In particular, it leaves open the question whether the upper boun O(log n) is achievable when the intensity is n. Further, an equally important, it neglects the effects of interference when computing the upper boun. The present work goes consierably beyon the work of [], taking into account interference an allowing for more general asymptotic noe intensities. As we will see, there are three asymptotic regimes efine through power levels one of which covers [], one of which resolves the open question left for future stuy in [] an a thir one that establishes a novel performance regime. A relate stuy [2] computes the broacast capacity for an arbitrary network uner the Protocol Moel for the channel. The paper proves that the broacast capacity lies within a constant factor of the wireless channel capacity (W ) an that it oes not epen on the raio range, the number of noes in the network an the area of the network. Only the interference parameter has an effect on the broacast capacity. The present paper extens the prior work [2] by using the more realistic Physical an the Generalize Physical Moel for the channel. Interestingly, we are able to show that bouns similar to [2] hol uner these more realistic physical moels. Moreover, uner the Generalize Physical Moel an uner certain conitions on transmission power an network size, the broacast capacity comes within a constant factor of the banwith. Due to space constraints we are unable to continue the stuy of the broacast capacity of mobile networks initiate in [2] an leave it for future work. Also, there is another paper on broacast capacity [8] which stuies a simplifie channel an network moel. The paper shows that per noe broacast capacity is O(/n) for such a moel. Note that all the above mentione papers as well as this paper assume only point-to-point coing at the receivers. If the noes are allowe to cooperate an use sophisticate multi-user coing then a per noe capacity of a higher orer than escribe above can be achieve [9] [2]. A full iscussion of these results is beyon the scope of this paper. III. WIRELESS CHANNEL MODEL AND BASIC NOTIONS In this section we escribe the wireless network moel an efine several terms relevant to our analysis of broacast capacity. We consier a wireless network consisting of n wireless noes. Let X i for i =, 2,...,n enote the location of the ifferent noes. For simplicity we also use X i to refer to the i th noe itself. Note that in this paper we o all analysis in -imensional space, so X i R. A. Network Graph The following notions lie at the basis of many stuies an will become useful here for comparison an computation purposes. We enote by G(R) the geometric graph inuce by istance where the vertices of G(R) are the noes of the network an where two istinct noes X i an X j are ajacent in G(R) if an only if X i X j R. Note that increasing R can only a eges to the ones of G(R). A Minimum Connecte Dominating Set (MCDS) of a connecte graph is a connecte subset of noes with the minimum size an the following property: every noe of the graph is ajacent to at least one noe in the subset. MCDS of a network graph uses minimum number of noes to isseminate a broacast packet to all noes. For clarity, we inclue the istance parameter R in the notation for the above efine sets. For example MCDS(R max )isanmcdsof G(R max ).Weusethe symbol # to enote the size of a set. B. Homogeneous Networks We now recall a wireless network moel, calle homogeneous network, which is wiely employe in the stuy of network capacity. In this moel, the number of noes grows to infinity while the noes are istribute uniformly in a -imensional sphere or cube of volume V n. We point out that the volume V n is allowe to vary with the number of noes n. This moel is closely relate to moeling the noe locations via a homogeneous Poisson point process with intensity n V n per unit volume. Conitione on knowing that there are exactly ˆn points prouce by the process, these points are inepenently an uniformly istribute in V n. Note that ˆn is a Poisson ranom variable with mean n. For convenience, we introuce D as the iameter of V n. For a cube D = Vn / ()

4 C. Channel Moel Network capacity papers usually moel the wireless channel in three ways as we explaine in the introuction. Here, we escribe each moel in etails. Let T be the subset of noes simultaneously transmitting at some given time instant. For convenience we call the case, where T contains only the sener, an exclusive transmission. We call a network interference-free connecte whenever it is connecte uner the assumption of all transmissions being exclusive. ) Protocol Moel: A transmission from noe X i T to noe X j is moele as successful if X i X j R an if X k X j ( + )R for all X k T\{X i } where is the interference parameter. In this moel, the rate of a successful transmission is consiere to be constant, enote by W. We also efine R c as the minimal R such that the network becomes interference-free connecte uner the Protocol Moel with high probability (w.h.p.) as number of noes tens to infinity. Equivalently, R c is the minimal R such that G(R) is a connecte graph. It is well known [22] that in a homogeneous network R c behaves as follows as a function of the number n of noes, as n grows: ( Vn log(n) ) /. R c = R c (n) = (2) n 2) Physical Moel: We assume that all noes choose a common power level P for all their transmissions. A transmission from a noe X i, i T is successfully receive by a noe X j if PG ij SINR = N + k =i,k T PG β (3) kj where β is the threshol, N represents the ambient noise, an G ij enotes the signal loss, whence PG ij is the receiving power at the noe j from the transmitter i. We assume a low power ecay for the signal loss of the form G ij = X i X j α,whereα>is the signal loss exponent. For a successful transmission the rate is assume to be constant W. We introuce the parameter R max as the maximum possible istance between a transmitter an receiver to ensure a successful transmission uner the Physical Moel. Equation (3) implies that ( P ) /α R max = (4) Nβ 3) Generalize Physical Moel: In this moel the transmission rate W ij between a sener i an a receiver j is etermine using Shannon s formula for a wireless channel with aitive Gaussian white noise [23]. PG ) ij W ij = B log 2 (+ + k =i,k T PG (5) kj where B is the banwith of the wireless channel an N 0 /2 is the noise spectral ensity. While this moel assigns a more realistic transmission rate at large istance, it also results in a singularity for the signal loss G ij = X i X j α. Inee, accoring to (5) the receiving power an the rate are amplifie to unrealistic levels if sener an receiver are place very closely. Some papers have pointe out this rawback [24], [25]. We introuce R m as the maximum istance between any two noes such that the packets can be receive with rate B uner the Generalize Physical Moel. From (5) we have ( P ) /α R m = (6) D. Broacast Capacity We efine the broacast capacity for a subset B := {Y,Y 2,...} of noes that generate broacast packets. Doing so as flexibility an allows to cover cases where only a few noes may be require to broacast packets such as when using broacast backbones or a subset of control enable noes in istribute applications. Assume that the noe Y i generates packets at rate λ Yi 0. We say that the rate vector [λ Yi ] i is achievable if all noes of the network receive all generate broacast packets successfully within some given time T max <. We stuy the maximum achievable broacast rates for the case where pre-specifie fractions of the aggregate broacast rate are available to each noe of B. More precisely, given a vector of weights g =[g i ] #B i=, g i > 0 such that i g i =we stuy the broacast capacity λ B (g) :=sup{a : λ Yi = g i a, [λ Yi ] i is achievable} (7) Notably, the bouns we erive for the broacast capacity λ B (g) are inepenent of B an g. In equivalent terms, the bouns we erive apply to both, λ := sup B,g λ B (g) an λ := inf B,g λ B (g) which are the maximum broacast capacity which can be achieve by some noes, respectively which can be achieve by any noes. IV. BROADCAST CAPACITY UNDER THE PHYSICAL MODEL In this section we compute bouns for the broacast capacity of wireless networks using the Physical Moel escribe in Section III.

5 A. Broacast Capacity for Arbitrary Wireless Networks Here, we etermine upper an lower bouns for the broacast capacity that apply to any arbitrary connecte wireless network. The accuracy of these bouns varies with the network scenario. We first note the har upper boun W for the broacast capacity which hols for any network. Inee, since every noe must receive the broacast, its capacity cannot be higher than the maximum ata rate at which a noe can receive ata. We obtain the lower boun using the concept of the Minimum Connecte Dominating Set (MCDS) of a network. To this en, note that the network is interferencefree connecte uner the Physical Moel exactly when the graph G(R max ) is connecte. Inee, the maximal istance covere by a transmission is exactly R max. Finally, #MCDS(R max ) is well efine for an interference-free connecte network, since G(R max ) is connecte. Broacasting via exclusive transmissions over an MCDS we fin Theorem : Assume that the network is interferencefree connecte. Then, uner the Physical Moel, W #MCDS(R max )+ λ B(g) W. (8) Instea of consiering exclusive broacasts, one may note that for R < R max simultaneous transmissions on G(R) become feasible. Assuming that G(R) is connecte Theorem 2 erives a boun on λ B (g) for arbitrary topology by proviing a TDMA scheuling metho which consiers the location of the noes an which scheules the transmissions such as to reuce the interference between simultaneous transmissions. Allowing simultaneous transmissions Theorem 2 outperforms Theorem for large networks. Theorem 2: Assume that G(R) is connecte for some R = R max /ρ with ρ >. Necessarily, the network is then interference-free connecte. Then, uner the Physical Moel, W K(ρ) λ B(g) (9) where K(ρ) =(5 2 ) (2+ (β J >0 ) ) J Z /α (0) ρ α Note that the explicit lower boun on λ B (g) via (9) an (0) is improving as ρ grows. As the most important conclusion of theorems an 2, the broacast capacity is not affecte by more than a constant factor when changing transmission power, number of noes or volume V n as long as for R = R max /ρ the graph G(R) remains connecte with the same ρ. B. Broacast Capacity of Homogeneous Networks As a corollary of Theorems 2 an we are able to boun the broacast capacity of a homogeneous network within a constant factor of W. Theorem 3: Assume that the n noes are uniformly istribute in a -imensional cube with volume V n. Denote ρ = lim inf R max/r c (n). Ifρ> then w.h.p. n as n W K(ρ) λ B(g) W. () Clearly, if ρ< then the network is isconnecte w.h.p. an λ B (g) =0for infinitely many n. V. BROADCAST CAPACITY UNDER THE GENERALIZED PHYSICAL MODEL As the main ifference, in the Generalize Physical Moel a noe can transmit to any other noe with some rate which epens on various parameters while in the other two channel moels transmissions may or may not be successful. It oes not come as a surprise that this moel is harer to analyze an shows ifferent behavior in some cases. We will show that poor signal strength an high interference are the two main limiting factors of the broacast capacity. Clearly, alreay a single poorly connecte noe may form a bottleneck for the broacast capacity. In well connecte large networks, on the other han, the broacast capacity may suffer ue to high interference cause by retransmissions. Since the relative noe locations have such a strong impact uner this moel, we obtain stronger results when making simple assumptions of a homogeneous networks, a situation often aopte in capacity stuies. A. Broacast Capacity of Homogeneous Networks We allow for power to be ajuste to the number of noes, i.e., P may epen on n. Consequently, both R c = R c (n) an R m epen on n. We ientify three asymptotic regimes as n grows by consiering the rates of reception of the most isolate noes to their next neighbors an to some given arbitrary noe. Low Power Regime: P = O((V n log(n)/n) α/ ) In this case the transmission power is so limite that as n grows some istant noes cannot receive within a constant of the banwith, even without interfering signals. These noes then limit the broacast capacity as follows

6 Theorem 4: Assume the noes are uniformly istribute in -imensional cube with volume V n. Assume there exists a constant c such that R m c R c (n) for all n, i.e., P c α (V n log(n)/n) α/. Then, w.h.p. as n ( ( R m λ B (g) =Θ B log 2 +( ) α)) (2) R c An example of a network setting in the Low Power Regime is the extene network moel stuie in [] where V n = n with boune transmission power. Then λ B (g) =Θ(Blog 2 ( + P log(n) α/ )). Tune Power Regime: P = Ω((V n log(n)/n) α/ ) an P =O(Vn α/ ) In this regime, the transmission power is sufficiently large so that every noe can receive ata with rate at least proportional to the banwith from some noe. On the other han, the area covere by the network is so large such that noes cannot transmit their message to all noes irectly with such a rate. In this case the interference of simultaneous transmissions limits the capacity. Theorem 5: Assume the noes are uniformly istribute in -imensional cube with volume V n. Assume there exists positive constants c an c 2 such that c R c (n) R m c 2 D for all n, i.e., c α (V n log(n)/n) α/ P c α 2 α/2 Vn α/. Then, w.h.p. as n λ B (g) =Θ(B) (3) An example of a network setting in the Tune Power Regime is the ense network moel where V n = an transmission power is boune. Then, λ B (g) =Θ(B). The broacast capacity of this moel has been left as open problem for the future stuy in []. High Power Regime: P =Ω(Vn α/ ) In this case the transmission power is so elevate that a noe can transmit to all noes irectly with a rate that remains at least proportional to the banwith as n grows. In this regime, the area of the network ecreases or the transmission power increases as number of noes grows. Theorem 6: Assume the noes are uniformly istribute in -imensional cube with volume V n. Assume there exits a constant c 2 such that c 2 D R m for all n, i.e., P c α 2 α/2 Vn α/. Then w.h.p. as n ( λ B (g) =Θ B log 2 ( +( R m D )α)) (4) This regime is usually not stuie in the literature, as the assumptions seem unrealistic. We inclue it for completeness. Interestingly, Theorem 6 shows that in this regime a broacast via multi-hop oes not increase throughput by more than a constant factor as compare to a broacast by one exclusive transmission. VI. CONCLUSION AND FUTURE WORK We stuie the broacast capacity of wireless networks uner two physically motivate channel moels. First, in the Physical Moel the broacast capacity is shown to be within a constant of the channel capacity an little affecte by the number of noes or the volume occupie as long as the connectivity of network is maintaine properly. Secon, in the Generalize Physical Moel all noes receive a broacast at a rate epening on interference an istance to sener. Here, the broacast capacity can vary significantly epening on how the transmission power is set in relation to the network size an topology. Focussing on homogeneous networks we ientifie three asymptotic regimes via explicit power settings. ACKNOWLEDGMENT Financial support comes in part from NSF, grant number ANI , an from Texas ATP, project number REFERENCES [] R. Zheng, Information issemination in power-constraine wireless network, in INFOCOM, [2] A. Keshavarz-Haa, V. Ribeiro, an R. Riei, Broacast capacity in multihop wireless networks, in MobiCom, [3] P. Gupta an P. R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory, vol. 46, no. 2, pp , [4] M. Grossglauser an D. N. C. Tse, Mobility increases the capacity of a-hoc wireless networks, in INFOCOM, 200, pp [5] M. Franceschetti, O. Dousse, D. Tse, an P. Thiran, On the throughput capacity of ranom wireless networks, IEEE Transactions on Information Theory, [6] R. Negi an A. Rajeswaran, Capacity of power constraine a-hoc networks, in INFOCOM, [7] B. Liu, Z. Liu, an D. F. Towsley, On the capacity of hybri wireless networks, in INFOCOM, [8] S. Toumpis, Capacity bouns for three classes of wireless networks: asymmetric, cluster, an hybri, in MobiHoc, 2004, pp [9] P. Gupta an P. Kumar, Internets in the sky: The capacity of three imensional wireless networks, Communications in Information an Systems, vol., no., pp , 200. [0] A. Agarwal an P. R. Kumar, Capacity bouns for a hoc an hybri wireless networks, Computer Communication Review, vol. 34, no. 3, pp. 7 8, 2004.

7 [] S. R. Kulkarni an P. Viswanath, A eterministic approach to throughput scaling in wireless networks, IEEE Transactions on Information Theory, vol. 50, no. 6, pp , [2] N. Bansal an Z. Liu, Capacity, elay an mobility in wireless a-hoc networks, in INFOCOM, [3] M. J. Neely an E. Moiano, Capacity an elay traeoffs for a hoc mobile networks, IEEE Transactions on Information Theory, vol. 5, no. 6, pp , [4] S. N. Diggavi, M. Grossglauser, an D. N. C. Tse, Even one-imensional mobility increases the capacity of wireless networks, IEEE Transactions on Information Theory, vol. 5, no., pp , [5] G. Sharma, R. R. Mazumar, an N. B. Shroff, Delay an capacity trae-offs in mobile a hoc networks: A global perspective, in INFOCOM, [6] A. E. Gamal, J. Mammen, B. Prabhakar, an D. Shah, Optimal throughput-elay scaling in wireless networks: part i: the flui moel, IEEE/ACM Trans. Netw., vol. 4, no. SI, pp , [7] X. Lin, G. Sharma, R. R. Mazumar, an N. B. Shroff, Degenerate elay-capacity traeoffs in a-hoc networks with brownian mobility, IEEE/ACM Trans. Netw., vol. 4, no. SI, pp , [8] B. Tavli, Broacast capacity of wireless networks, IEEE Communications Letters, vol. 0, no. 2, pp , [9] P. Gupta an P. R. Kumar, Towars an information theory of large networks: an achievable rate region, IEEE Transactions on Information Theory, vol. 49, no. 8, pp , [20] M. Gastpar an M. Vetterli, On the capacity of wireless networks: The relay case, in INFOCOM, [2] G. A. Gupta, S. Toumpis, J. Sayir, an R. R. Müller, On the transport capacity of gaussian multiple access an broacast channels, in WiOpt, 2005, pp [22] R. B. Ellis, J. L. Martin, an C. Yan, Ranom geometric graph iameter in the unit ball, to appear Algorithmica, [23] T. M. Cover an J. A. Thomas, Elements of Information Theory. John Wiley & Sons, Inc., 99. [24] O. Dousse an P. Thiran, Connectivity vs capacity in ense a hoc networks, in INFOCOM, [25] O. Arpacioglu an Z. J. Haas, On the scalability an capacity of wireless networks with omniirectional antennas, in IPSN, [26] S. I. Resnick, A Probability Path. Boston: Birkhauser, 998. APPENDIX Proof of Theorem : Consier an arbitrary noe X i in the network. The maximum rate of transmission or reception of ata by X i is W, since each broacast packet must be either receive or generate by X i. For showing the lower boun, we esign a TDMA scheme which achieves it. We consier an MCDS of the graph G(R max ) as a backbone. Every broacast packet is transmitte from its generating noe to a neighbor MCDS noe, then the packet is retransmitte along MCDS noes. We simply have only one transmission at the time, so clearly all noes receive the packet successfully. This scheme gives us W/( + #MCDS(R max )) throughput. Proof of Theorem 2: Again, we esign a TDMA scheme which achieves the lower boun. We o so in three steps. Step : Divie the space into cells with iameter R such that the coorinates of their centers are R (i R,i 2 R,...,i ) for i,...,i Z (see Fig. ). Note that by esign an by assumption any two noes in the same cell are ajacent in G(R). Next, we buil a cell graph over the non-empty cells (colore grey in Fig. ). The vertices of the cell graph are the nonempty cells an two cells are connecte by an ege if there exist two noes, one in each cell, that are ajacent in G(R). Because G(R) is connecte it follows that the cell graph must be connecte. We then buil a spanning tree over the cell graph which we use to route broacast packets. Step 2: We assign color L(r i,r i2,...,r i ) to the cell R with center (i R,...,i ),wherer i = i(mo)k. The value of k is chosen large enough such that when two noes in ifferent cells with the same color transmit simultaneously, all of the noes closer than R to the seners can receive successfully. We boun the value of k as follows. First, we fin a lower boun for SINR for a given k. Recall that the simultaneous transmitters are in the ifferent cells with the same color. For example, consier a transmitter Y in cell (0,...,0) with a receiver X within istance R. Any simultaneous transmitter Z must lie in a cell (j k, j 2 k,...,j k) where J =(j,...,j ) Z := Z \{(0,...,0)}. The istance between the sener Z an the receiver X is then at least j j2 kr 2R, an since J, at least R( k 2) J. Thus, SINR = = One verifies now quickly that PR α N + J Z P (R( k 2) J ) α N P Rα +( k 2) α J Z ρ α β +( k 2) α J Z k (( β J Z J α ρ α ) /α +2) (5) ensures that the SINR becomes larger than β, guaranteeing successful simultaneous transmissions in cells of equal color. Notably, the lower boun of (5) is finite since α>. Step 3: For every pair of ajacent cells on the spanning tree of the cell graph we choose two noes which connect the cells to be relays. When a packet nees to

8 w.h.p. for large n. Proof: It has been prove in [22] that for any 0 <a<, G(aR c ) contains Ω(n a ) isolate noes w.h.p. for large n. Consier an isolate noe, the receiving rate for the noe is boune by B log 2 (+P (ar c ) α / ).This gives us Fig.. Illustrating the collision free TDMA scheme for broacasting in the case of a planar network. It uses k 2 colors to scheule the cells transmissions. be forware from cell S to ajacent cell S2, the relay in cell S forwars the packet to the relay in cell S2. The relay in S2 rebroacast the packet to all noes in S2. IfS2 is not a leaf vertex on the spanning tree of cell graph then the relays which connect it to other cells will forwar the packet an the process continues till the broacast packet has been isseminate to all noes. By geometry, it is easy to show that every cell has at most 5 2 ajacent cells. There are thus at most 5 2 relays in each cell. We ivie the time slot corresponing to each color into 5 2 equal time slots of length T. In the first time slot corresponing to every color, each noe Y i Bgenerates WTλ Yi /((5 2 )k λ B (g)) bits for broacast. If more than one Y i s are locate in the same cell they broacast the packets sequentially in some orer. In the remaining 5 2 time slots of any particular color, the relays of cells with that color transmit (to other noes in the cell or to the corresponing relay of an ajacent cell) any broacast ata that they have receive but not yet forware. Note that with this setup every relay can forwar packets at the rate W/(5 2 )k which establishes the esire lower boun for λ B (g). Proof of Theorem 3: The upper boun has been shown in Theorem. The lower boun follows quite easily from Theorem 2. Inee, it is known for homogeneous networks [22] that the graph G(R) is connecte for large n w.h.p provie R =(+ɛ)r c for some ɛ>0. Choosing ɛ ρ +ɛ small enough such that >, Theorem 2 provies the lower boun W/K( ρ +ɛ ). Now, letting ɛ 0, by left continuity of K(ρ) the lower boun becomes W/K(ρ). Proof of Theorem 4: We establish the upper boun the following lemma. Lemma : In a homogeneous network λ B (g) B log 2 ( + ( R m R c ) α ) (6) λ B (g) B log 2 ( + P (ar c) α )=B log 2 ( + ( R m ar c ) α ) (7) Then, letting a yiels (7). Next, we show a constant factor of the compute upper boun is achievable. We ivie the volume V n into cube cells with iameter br c for a constant b> (e.g. b =2). From [22] we know that G(bR c ) is connecte w.h.p. for large n. Then we apply the broacast scheme evelope in the proof of Theorem 2. We may set the parameter k to any integer larger than 3, i.e. here we choose k =4. Then the SINR of a receive broacast packets becomes SINR P (br c) α +( 4 2) α J Z R α = m(br c ) α +Rm(bR α c ) α ( 4 2) α J Z +(c /b) α ( 4 2) ( R m ) α α J Z J α br c Therefore, the throughput of this scheme is at least Θ(B log 2 ( + ( Rm R c ) α )). Proof of Theorem 5: Firstweshowthatλ B (g) =Θ(B) is achievable. We broacast the packets similarly to the proof of Theorem 4. Here, the SINR of broacast packets is at least SINR P (br c) α +( 4 2) α J Z J α = Rm α (br c ) α +( 4 2) α J Z (b/c ) α +( 4 2) α J Z So, the throughput in this case is at least Θ(B). To prove the upper boun we consier the set I of isolate noes of G(R) where R = a V n /n an a is a fixe constant. In the following lemmas we establish two properties. First, there exists p>0 such that #I p n for large n an w.h.p. (see lemma 2). Secon, the noes in I are so far from all other noes, that the sum of receiving rates is always boune by K B#I where K is a constant number an B is the banwith (see lemma 3). Consequently, since the sum of receiving rates of noes of I must be at least λ B (g)#i on average we fin λ B (g) KB =Θ(B) w.h.p.

9 Lemma 2: For some p>0, P [#I >np] as n. Proof: While we establish the lemma for homogeneous networks, we mention that a proof for the Poisson moel is easy to come by with. Assume that n noes are uniformly istribute in V n. To each noe X i we assign a ranom value ν i,whereν i is if X i is isolate in G(R), an0 otherwise. Thus, the ranom variable S = n i= ν i represents number of isolate noes. We also efine V n to be the set of points in V n which are within istance less than R from the bounary of V n. In abuse of notation we enote its volume also by V n.sincer = O(( Vn n )/ ), it follows that V n = O( Vn ). n We compute E(ν i )=P[ν i =]as follows E(ν i ) = P [ν i = X i / V n]p [X i / V n] +P [ν i = X i V n]p [X i V n] = ( π R ) n ( O( V n n )) + O( n ) = exp( π a )+O( n ) where π is the volume of unit raius sphere in - imensional space. Also, E(ν i ν j ) = P [ν i =,ν j = ] becomes E(ν i ν j )=P[ν i = ν j = X i,x j / V n]p [X i,x j / V n] + P [ν i = ν j = X i or X j V n]p[x i or X j V n] = P [ν i = ν j = X i,x j / V n]+o( n ) Splitting the two cases X i X j > 2R an X i X j 2R E(ν i ν j ) = P [ν i = ν j = X i,x j / V n, X i X j > 2R] P [ X i X j > 2R]] + O( n ) = ( π R ) n 2 ( π R ) n 2 + O( V n n ) V n = exp( 2π a )+O( n ) Setting p =exp( π a )/2 we have E(S) =2pn + O() an var(s) =E(S 2 ) (E(S)) 2 = O(n). Finally, P [ S E(S) > E(S)/2] var(s) (E(S)/2) 2 = O(n) Θ(n 2 ) (8) by the Markov inequality [26]. Lemma 3: The sum of receiving rates of the noes of I is boune by K B #I, wherek is a constant. For the proof, we consier two cases. Case (): Assume that X s is the only transmitter in the network. From equation (5) we have i I W si i I D/R B log 2 ( + P X i X s α ) f(k)b log 2 ( + P (kr) α ) where f(k) is the maximum number of the isolate noes in istance l from X s where kr < l (k +)R (note that X i X s >Rso we start from k =). Now consier the balls with raius R/2 aroun noes of I. Since the noes are in istance at least R from each other, the balls are isjoint an by simple geometric argument we boun the number of balls by f(k) < π (k++.5) R π (k.5) R π (.5R) =(k +3) (k ).So, clearly there is a constant K such that f(k) <K k. Then, continuing an applying Cauchy-Schwartz W si i I D/R D/R K B K B K k B log 2 ( + ( R m kr )α ) K k B log 2 ( Dα + R α m k α R α ) D/R D/R k 2 2 K 2 ( D R )2 D/R 0 log 2 2( Dα + R α m k α R α ) log 2 2( Dα + Rm α x α R α )x = K B K 2 ( D R )2 D R [2α2 +2ln( Dα + Rm α D α )+ln 2 ( Dα + Rm α D α )] log 2 (e) K B K 2 ( D R )2 K 3 ( D R )=BK K2 K 3 ( D R ) = K K2 K 3 Bn/a K K2 K 3 B#I/(pa ) where the constant K 2 epens on, an constant K 3 epens on c 2. Since this is a boun on the sum of receiving rates of noes of I, we result that in case () λ B (g) K K2 K 3 B pa Case (2): Now assume that there are more than one simultaneous transmitters. Denote T the set of transmitters. For every X s T, we consier X s as the closest transmitter to X s an we efine U(s) = X s X s /2.Also we efine Ball(s) to be the ball with raius U(s) aroun the noe X s. In lemma 4 we show that these balls are isjoint. Moreover, lemma 5 bouns the receiving rate of the noes in terms of the raii of the balls.

10 Lemma 4: The balls Ball(s) (X s T) are isjoint in -imensional space. Proof: Assume that two balls Ball(s ) an Ball(s 2 ) are not isjoint an U(s ) U(s 2 ). Then, X s2 X s < U(s )+U(s 2 ) 2U(s 2 ).ItshowsthatX s is closer than the closest transmitter to X s2. This is a contraiction. Lemma 5: Assume X s the closest transmitter to an arbitrary noe X i, then the receiving rate of X i is boune by B[ + α log 2 ( + 2U(s) )]. As a result if X X i X s i is not locate in the Ball(s), the rate is boune by [ + α log 2 (3)]B. Proof: The receiving rate of X i is boune by W si because the signal of X s is the strongest receiving signal at X i. So, we boun W si as the following W si B log 2 ( + P X i X s α + P X i X s α ) B log 2 ( + ( X i X s X i X s )α ) B log 2 ( + ( X i X s +2U(s) ) α ) X i X s B log 2 (2( + 2U(s) X i X s )α ) = B[ + α log 2 ( + 2U(s) X i X s )] Next, we boun the sum of receiving rates for the noes of I insie Ball(s) for an arbitrary X s T. i Ball(s) W si i Ball(s) U(s)/R B[ + α log 2 ( + 2U(s) X i X s )] f(k)b[ + α log 2 ( + 2U(s) kr )] U(s)/R K B k [ + α log 2 ( 3U(s) kr )] by applying Cauchy-Schwarz inequality we have K B U(s)/R U(s)/R k 2 2 [ + α log 2 ( 3U(s) kr )]2 K B K 2 ( U(s) U(s)/R R )2 [ + α log 2 ( 3U(s) 0 xr )]2 x K B K 2 ( U(s) R )2 K 4 ( U(s) R ) = BK K2 K 4 ( U(s) R ) where K 4 is a constant which is resulte from the integral. From the lemma 4 the balls Ball(s) (X s T)are isjoint. On the other han, we can easily show that all the balls are place insie the cube V n of sie size (2 +) V n an where V n is locate in its center. Therefore, we have π U(s) (2 +) V n (9) s T Combining this equation with the compute upper boun (2 +) V n W si BK K2 K 4 R s T i Ball(s) (2 +) = K K2 K 4 π a Bn Then, we boun the sum of receiving rates of the noes of I as follows W si = W si + W si i I s T i Ball(s) i/ Ball(s) (2 +) < K K2 K 4 π a Bn +[+αlog 2 (3)]B#I (2 +) K K2 K 4 π pa B#I +[+αlog 2 (3)]B#I Therefore, in case(2) we have λ B (g) < [K K2 K 4 (2 + ) /(π pa ) + + α log 2 (3)]B Proof of Theorem 6: The following lemma provies a lower boun. Lemma 6: Assume the network is containe in a boune set with iameter D, then π B log 2 ( + ( R m D )α ) λ B (g) (20) Proof: If the source noe of broacast transmit the packets to all noes irectly, then every noe in the network can receive with the rate larger than B log 2 ( + PD α )= B log 2 ( + ( Rm D )α ), because the istance between the source an any noe is less than D. For the proof of upper boun, we buil a set I similar to the proof of Theorem 5 an we consier the analogous two cases. In case (), we follow the same inequalities, but we write K 5 log 2 2( + ( Rm D )α ) (where K 5 is constant epens on c 2 ) instea of K 3. Continuing the argument we fin that λ B (g) is boune by Θ(B log 2 ( + Rm D )α ). In case (2), we use the same inequalities as before fining that λ B (g) is boune by Θ(B). This shows that using more than one simultaneous transmitters might even ecrease the broacast throughput in the thir regime.