ON THE CAPACITY OF WIRELESS SENSOR NETWORKS WITH OMNIDIRECTIONAL ANTENNAS

We establish some new results on the capacity of wireless sensor networks that employ single-user - detection, and we present the implications of our r esults on the scalability of such networks. In particular, we find bounds on the maximum achievable per-sensor end-to-end throughput, λe, and the maximum number of simultaneously successful wireless transmissions, Nt max , under a more general network scenario than previously considered . Furthermore, in the derivation of our results, we m ake no restrictions on the mobility pattern of the sensor and the destination nodes or on the number simultaneous transmissions and/or receptions that the nodes are capable of maintaining. In our derivation, we also analyze the effect of parameters such as the area of the network domai n, A, the path loss exponent, γ, the processing gain, G, and the SINR threshold, β. Specifically, we prove the following results for a wireless sensor network of N sensor nodes and M destination nodes that are equipped with omnidirec tional antennas:


INTRODUCTION
In recent years, there has been an increasing interest in certain class of wireless networks called wireless sensor networks.Typically, wireless sensor networks are deployed to collect information about an environmental variable by observing the information gathered at a set of destination nodes from a set of sensor nodes in the network.I Scalability of wireless sensor networks has been an important research topic in the recent years, because of the growing demand to support a large number of nodes in future sensor networks, which are envisioned to consist of thousands to millions of sensor nodes.Two important questions in this context are: (1) Are wireless sensor networks scalable?In other words, is it possible to support a large number of nodes in a wireless sensor network?(2) If there are scalable patterns of wireless sensor networks, what are the conditions that govern their scalability?
To answer these questions, one must first define what scalability is.In this work, we interpret scalability as: the ability of the network to operate with a non-vanishing per sensor end-to-end throughput, bounded end-to-end delay, bounded power consumption, bounded processing power and bounded memory at each node, as the number of nodes grows large.
In this paper, we focus on the throughput aspect of scalability.Our objectives are: (I) to obtain theoretical results that show the dependencies among the per-sensor end-to-end throughput capacity 4", the number of sensor nodes N, and other parameters of a wireless sensor network, and (2) to determine the implications of these results on the scalability of such networks.
One of the most well known studies that stimulated the research on the capacity of wireless sensor networks was published by Gupta and Kumar [1].In that paper, two network models were proposed to analyze the capacity of wireless networks.The first network model, the arbitrary network model, assumes that all N nodes in the network are static, there are no restrictions on nodes' locations, and the network domain (i.e., the region within which the nodes are located) is a circular disk of area 1 m 2 • Each node is capable of maintaining at most one transmission or one reception at any given time.There are no restrictions on the choice of transmission powers, traffic pattern, routing protocol, and spatial-temporal transmission scheduling policy.The second model is the random network model, which assumes constant.The first traffic pattern that they considered is the all-to-one model, where each sensor node has the same destination node which is located at the center of the network domain at all times.The second traffic pattern is the all-to-all model, where each sensor node has to broadcast the information that it generates to all other sensor nodes in the network.For these two traffic patterns, they concluded that 4( is 0( liN).
In [4], Chakrabarti et al. concluded that 4( is 0(log(N)1 N) for the random network model of [1], if the all-to-one traffic pattern and an adaptive transmission rate model are used instead of the random traffic pattern and the threshold SINR based reception model, respectively.
In [5], Toumpis and Goldsmith also used an adaptive rate model, and using numerical methods, they evaluated the effect of spatial reuse, multi-hop routing and power control for a particular placement of the nodes.They concluded that each of these schemes provides expansions of the capacity region that is defined by the set of achievable rates between the nodes.
In [6], Li et al. pointed out the effect of traffic pattern on ,1,( and concluded that only wireless networks with local traffic patterns can be scalable.
Usage of directional antennas at the transmitters or the receivers can provide significant increases in 4(, depending on how narrow the width of the antenna's main lobe and how small the side lobes of the antenna radiation pattern can be made.For example, in [7], using a sender-based interference model, Yi, Pei, and Kalyanaraman investigated the improvement in 4 e provided by the usage of directional antennas for arbitrary and random wireless networks. The ability of a node to maintain multiple simultaneous transmissions and/or receptions can also provide an increase in 4,.For example, in [8], using directional antennas, Peraki and Servetto studied random networks with and without multiple simultaneous transmission or reception capability, and they concluded that an improvement of at most 0(log2 (N» can be achieved over the e (1/ .JNlog( N») result of [1].
Deployment of a wired backbone can also provide an increase in 4" since it allows reducing average number of wireless transmissions per packet.For example, in [9], Liu et al. considered the benefit of deploying base stations connected to a wired backbone in the random network of [1].They concluded that if the number of base stations grows asymptotically faster than .IN, then aggregate throughput capacity increases linearly with the number of base stations.
The results of studies such as [1 ]- [9] are based on the assumption that the nodes are equipped with single-user detection based receivers, which are widely used in today's practical wireless networks.These types of receivers attempt to detect a particular transmission, while treating all other interfering transmissions as noise.On the other hand, studies such as [10]- [12] concluded that sizable gains in network throughput can be provided by not treating interference simply as noise, but rather employing more sophisticated receivers and implementing certain cooperation strategies among the nodes during encoding, transmission and decoding stages of each communication.
Revista da Sociedade Brasileira de Telecomunica(foes Volume 19, Numero 3, Dezembro de 2004 Our work has been motivated by the desire to relax some of the limitations of [I ]- (4], and to improve on their models.In particular, the radio propagation model of [I], [2] and [4], (i.e., PixY) becomes inappropriate as the transmitter receiver separation becomes small.Since the network domain in these studies has a fixed area, as N grows large, the intended transmitter-receiver pairs get arbitrarily close to each other, which leads to a very optimistic evaluation of the SINR values due to the assumed propagation model.To avoid this problem, we use a bounded propagation model, which we discuss in Section 2.3.In terms of mobility, in [I], [3] and [4] the nodes are immobile, while in (2] the mobility pattern is a very special one.In our work, we allow for a general mobility pattern of the nodes.Furthermore, by allowing a general traffic pattern, we relax the assumption of [I ]- [4] which states that the source-destination pairs never change over time, and the assumption of [3] and [4], which states that the traffic pattern is either all-to-one or all to-all.Moreover, in [1]- [4] each node can maintain either a single transmission or reception at any given time, whereas our work also considers the situtation when the nodes have the ability to maintain multiple transmissions and/or receptions at the same time.In addition, we also analyze the effect of parameters such as A, y, G and /3 on It e , none of which has been addressed in previous works.Above all, we evaluate the implications of our results on the scalability of wireless sensor networks.
The main contribution of this study is in the derivation of new results on the capacity and scalability of wireless sensor networks through the use of a more general network model and a more realistic bounded propagation model, as compared with the models of the previous studies.
The rest of the paper is organized as follows: Section 2 presents our network model, and Section 3 provides related definitions.In Section 4, we derive the upper bounds on the simultaneous transmission capacity and the per-sensor end to-end throughput capacity.In Section 5, we analyze the derived upper bounds, together with illustrative figures.Section 6 establishes the tightness of the scaling behavior of It e with respect to N. Section 7 discusses the implications of our results on scalability.Finally, we conclude in section 8.

SENSOR NETWORK MODEL
In this section, we explain the assumptions underlying our results.Some of these assumptions are quite general, allowing our upper bound results to hold even in wireless sensor networks that are configured with optimal settings of the parameters.Although this paper is primarily targeted at the sensor networks, the generality of our assumptions allows straightforward extensions of the results of this paper to many other types of wireless networks, such as wireless ad hoc networks.We will elaborate on some other consequences of these general assumptions after presenting our results.

3.network domain and nodes
We denote the total number of nodes in the network by N IOTal .There are two types of nodes in the network: sensor nodes and destination nodes.There are N sensor nodes and M destination nodes.Any given node is either a sensor node or a destination node or both of them, and the type of the node remains the same at all times.Two immediate consequences of these assumptions are that Nlotal::; N+M, and that N total =N+M if and only if the set of sensor nodes and the set of destination nodes are disjoint.
The task of each sensor node is to send the collected information to a destination node (of course, to a destination node other than itself, if the node itself happens to be a destination node, too).We assume that M::;N.Typically, this inequality is satisfied with "«" in practical wireless sensor networks.An immediate consequence of the inequality M::;N is that N total =N if and only if all destination nodes are also sensor nodes.Note that this case corresponds to the network models of [I] and [2].
Network domain is defined to be the space within which each node is constrained to reside.We denote this space by Q.We will assume that Q is a closed disk with a diameter D and an area A. There are no restrictions on the mobility pattern of the nodes within Q.

4.transmitter and receiver model
Each of the nodes is capable of being a transmitter and/or a receiver at any given time.All transmitters and receivers are equipped with omnidirectional antennas.There are no restrictions on the variation of transmission power during a transmission or on the number of simultaneous transmissions and/or receptions that a node is capable of maintaining.Hence, the assumption in [I ]- [4] that a node is capable of maintaining either one transmission or one reception at any given time is one of the many cases covered by our model.For the time being, we assume that all transmissions take place within the same communication bandwidth, but we will relax this assumption later.At the intended receiver of a transmitted signal, all of the remaining received signals are considered as interference.(;(t) is the power of the thermal noise present in the communication bandwidth at receiver i at time t.We assume that each of the receivers can receive information intended for itself reliably at a rate not larger than W max bits/s and only when the SINR at the receiver is greater than or equal to the SINR threshold, /3.Information received when the above condition does not hold is considered unreliable and, thus, discarded.In this work, we let /3 be any positive real number.In general, /3 is dependent on the modulation scheme, the required bit error rate of the received information, the required transmission rate, and the type of the error control code.The processing gain, G, is the factor by which the total received interference power is reduced at each of the receivers.In this work, we let G be any positive real number.Typically G> I for wideband communication systems, such as spread spectrum COMA, and is taken to be I for narrowband communications.

propagation model
For any given two given transmissions, let i and j be the indices that represent the transmissions.Let P/ (t) be the On capacity of wireless sensor networks with omnidirecitional antennas power transmitted by transmitter j at time t.Let dji(t) be the distance' between transmitter j and receiver i at time t.Let p'F (t) denote the power received by receiver i from transmitter j at time t.We will assume that p,J; (t) = P/ (t)a(dj;(t», (a.l)where a(x) is the attenuation function.One of the most commonly used expressions for a(x) is l/xY, where r~ °is the path loss exponent. 4However, this expression becomes inappropriate as x becomes smaller than 1, as it results in receiving power that is larger than the transmitted power.In fact, the received power in the formula approaches infinity as x tends to zero.Of course, this is unrealistic and is an artifact of the inappropriateness of the expression for small distances.'The inappropriateness of the l/x r formula was also noticed in some previous works on connectivity, and to obtain more meaningful results at small distances, while approximating the conventional model at large distances, the following alternative propagation model was proposed in those studies (see, e.g., [15] and [16]): (l+JY We call this propagation model, the power law decaying propagation model,6 and we will use this model in our calculations.

traffic pattern
We make no restrictions on the temporal variation of the destination of each of the sensor nodes, the selection of the internlediate nodes that are involved in routing the information originated by the sensor nodes, and the segmentation of information, so that different segments can possibly be transmitted over different paths and at different times.Note that, together with the assumptions that we stated in Section 2.1, these imply that the class of traffic , In this paper, all distance measures and area measures are in units of "meter" and ••meter."respectively." In free space y = 2, but in realistic mobile radio channels, y can take values between 1.6 and 6 (see [13] and [14]).
, The precise reason for this problem can be explained as follows: Consider the free space case.In the derivation of the received power expression, a unity gain point source in free space is assumed and the flux of the transmitted power P" per unit surface area of the sphere with radius x around the source is calculated.The resulting power flux density expression, P, / 4Jrx', has the unit Watts/meter.The wave-front of the transmission occupies only part of the aperture of the receiving antenna, so that the power captured by the aperture results from only that part of the wave-front that is seen by the aperture.To quantify this partial aperture area occupied by the wave-front, effective aperture area, A" is defined as the ratio of the available power at the terminals of the receiver antenna to the power flux density at the location of the receiver antenna.In general, A, depends on the physical characteristics of the receiver antenna and the distance x between the transmitter and the receiver antennas.In [I], [2] and [4], A, is assumed to be independent of x and is taken to be 4ltmeter, so that the received power expression simplifies to P, / r.In fact.as x---+O, the power flux density approaches infinity, and with the constant A, assumption.the received power also approaches infinity.This shows that A.. should not be taken as a constant for small values of x.In fact, A, should approach zero as x approaches zero, so that the received power, A, P, / 4Jrx', never exceeds P,.
o The corresponding A, for this model in free space is equal to 41tr/(l+x)' meter.Note that this expression converges to 4lt meter as x becomes large, which is also the assumed aperture area in the conventional model.patterns that our model covers is indeed a very large class: in particular it contains all many-to-many type traffic patterns for which the set of sensor nodes is at least as large as the set of destination nodes.Hence, this class contains the two specific traffic patterns considered in [3], as well.
On the other hand, as in [1]- [4], we assume that intermediate nodes that act as relays do not jointly encode and transmit chunks of information that are gathered from distinct transmissions.
Finally, we denote the average number of hops between the originating sensor node and the destination of a bit by H , and we let it be any real number larger than or equal to 1, since each bit has to be transmitted over at least one hop to reach its destination.

DEFINITIONS
We define a transmission at an arbitrary time t to be a successful transmission, if the SINR at the intended receiver of the transmission at time t is not smaller than p.We denote the number of simultaneously successful transmissions at time t by Nt.We define the simultaneous max transmission capacity of the sensor network, N t , as the maximum value of Nt over all the placements of the nodes, the choices of the transmitters, their intended receivers, and the transmission powers.Next, we define the simultaneous transmission capacity of the network domain, NP as the maximum value of Nt over all the placements of the nodes, the choices of the transmitters, their intended receivers, and the transmission powers, given that there are no restrictions on the number of nodes in the network.An immediate max consequence of these definitions is that N t :::: NP' Let b i (1) be the total amount of bits of information generated by sensor node i and received by its destinations during a T second time interval [0,11.We define the end-to end throughput of sensor node i, .-1.;, as follows:

T
We also define the per-sensor average end-to-end throughput as the arithmetic mean of .-1.;'s, i.e.,

N
A.:=-LA; .N ;~I Next, we propose two achievability definitions: an end to-end throughput ~ is said to be achievable by all sensor nodes, if there exist a mobility pattern of the nodes, a traffic pattern, a spatial-temporal transmission scheduling policy, and a temporal variation of transmission powers, so that .-1.; ~~ for all 1SiSN.Likewise, an end-to-end throughput ~ is said to be achievable on average, if there exist a mobility pattern of the nodes, a traffic pattern, a spatial-temporal transmission scheduling policy, and a temporal variation of transmission powers so that A~ ~.Note that if ~ is achievable by all sensor nodes, then it is also achievable on average, and if ~ is not achievable on average, then it is not achievable by all sensor nodes, either.Hence, we will sensor end-to-end throughput capacity, An is the supremum of all end-to-end throughputs that are achievable by all sensor nodes.The per-sensor average end-to-end throughput capacity, ..1,,,, is the supremum of all end-to-end (2) throughputs that are achievable on average.An immediate where step (a) follows from (a.I), and step (b) follows from consequence of these definitions is that ..1,,, 2: A c • dividing both sides by p"i (t) = p'i (t)a(d ii (t» .In general, for 0:::: x ::; y + :, y:?: 0 , and .:::?: 0 , In this subsection, we prove the following theorem that a(x):?:a(y)a(.::),O::::x::::y+.::,y:?:O,and .:::?:O.provides upper bounds on N t max and NP:

DERIVATION OF THE UPPER BOUNDS
(3) Theorem 1: Let t be an arbitrary time instant.Then, Now, let l'i(t) be the distance between receiver j and (i) the simultaneous transmission capacities of the sensor receiver i at time t.Then, from the triangle inequality, network and the network domain have the following upper  where steps (a), (c), and (d) follow from the fact that a(lF(t» = a(lit» for every i, j, t, and step (b) follows from the fact that x+ l/x ?: 2 for every positive real number x.
From (7), we observe that the problem of obtaining an upper bound on Nt can be reduced to finding a lower bound on the summation term on the left-hand side of inequality (7).This term involves the sum of the attenuation function evaluated at the inter-receiver distances defined by the Nt (Nt -1 )/2 pairs of receivers.To find the lower bound, we make use of Lemma 1, which is derived in the next subsection.

interpoint distance sum inequality
In this subsection, we derive a lemma that gives an upper bound on the sum of the square of the nearest, the second nearest, ... , and the (n_1)5t nearest neighbor distances from each of the n points that are arbitrarily located in a disk of diameter D.  .:!.

'],Jr'
Proof of Lemma 1: The proof involves a spherical geometric approach, which is used to solve similar problems in [17].Let Bi(x) denote the disk of diameter x, whose center is at point i.
Let us first consider the disks in R I.All disks in R 1 are non overlapping. 8This can be proven by contradiction: Suppose that there exist two points i and j such that Bi(un) and B/ un) are overlapping.This, by the definition of overlapping, implies (Uil+Ujl )/2>lij.Without loss of essential generality, suppose Un ?:Ujl.Then Un >Iij' However, this would contradict the definition of Un.Therefore, our original assumption, i.e., the existence of two overlapping disks in R" is invalid.
Let A(X) denote the area of a region X.If X is a disk with diameter a, then A(X) = lra 2 14.Next, we find a lower bound 8 Two disks are defined to be overlapping if the distance between the centers of the disks is smaller than the sum of the radii of the two disks.In (L1.6), A(B(D»=lrD 2  Next, we show that any arbitrarily chosen point within B(D) does not belong to more than m overlapping disks from R m • The proof is again by contradiction: Suppose there is a point in B(D) that belongs to more than m overlapping disks from R m • Take the largest m+ 1 of these overlapping disks.Consider the largest disk.Let b be the point at the center of this largest disk.Then all other m disks belong to Simi due to the result proved in the previous paragraph.However, this contradicts the fact that the cardinality of Sbm is m -I.Therefore, our original assumption, i.e., the existence of a point belonging to more than m overlapping disks from R m , is invalid.
Since any chosen point within B(D) can belong to at most m overlapping disks from R m , then for every 2 ~ m ~ n -I , we have In this subsection, we derive Lemma 2, which in combination with Lemma 1, provides a necessary condition for Nt simultaneously successful transmissions.Next, by using this necessary condition and another lemma, Lemma 3, we complete the derivation of the upper bound on Nt, NP and Nt"'ax.
In Lemma 1, by setting n = Nt and the location of the points as the location of the receivers at time t, Uim(t) becomes the Euclidean distance between receiver i and the m 1h closest receiver to receiver i at time t.Hence, we obtain the following inequality: Also, from (7), we obtain the following necessary condition for Nt simultaneously successful transmissions at time t: 111=1 1=1 (1 +un" (t)r f3 To incorporate the constraint of ( 8) into (9), we use the following lemma: Lemma 2: For n2"1, let xi'x~, ...,x",C be nonnegative real numbers satisfying the following inequality: Let b be a nonnegative real number.Then Proof of Lemma 2: For b=O, (L2.2) is satisfied with equality.Thus, we consider the case when b>O. Define the column vector x and the multivariate functionf(x) as follows: We use Kuhn-Tucker Theory from [18] to find the minimum value off (x) subject to the constraint in (L2.l).
We define the constraint function, g(x) as follows: " Let y = [YI y~ .... y" r be the column vector at whichftakes its minimum value.Then, from Kuhn-Tucker Theory, there exists B ~ 0, such that the following conditions are satisfied: (Vf(x)-B Vg(X»)Ix=y ~O,  " [-b ] Since y 2" 0, we need to determine whether the constraint is binding or not.Namely, we need to determine whether or not there exist any components of y that are zero.To do this, we compare the values off in all possible cases.Let y has k zero components, i.e., k := I{i : Yi = 0, 1 :

1O) as h(x).
Taking the partial derivative of h(x) with respect to x and using Bernoulli's Inequality from [19], we obtain,

D
Next, in Lemma 2, we set n = Nt, b = Y, Xi = Uim(t), C=mD11c1 for every l::;i::;l1, and l::;m::;n-l, so that (L2.I) and ( 8) become identical.Also the left-hand side of (L2.2) and the inner summation in (9) become identical.Hence, we obtain the following lower bound on the left hand side of ( 9 The quantity (J'2 is proportional to the average number of successful transmissions per unit area.Combining (11) and ( 12), the necessary condition for Nt simultaneously successful transmissions at time t becomes:
Recall that NP is the maximum value of Nt over all the placements of the nodes, the choice of the transmitters, their intended receivers, and the transmission powers, given that there are no restrictions on the value of N. Since there have been no restrictions on any of these parameters during the derivation of ( 17), hence, the right-hand side of ( 17) is also max an upper bound on NP, which is not less than N t .This completes the proof of (T 1.1) in Theorem I.
Finally, we prove (T1.2) and part (ii) of Theorem I as follows: Suppose there is a single receiver node and Nt transmissions intended for this node at time t.Then, in (7), lij(t) is equal to zero for every i, j, and t.Thus, Nt will be no more than I+GI/3.This shows that none of the nodes can receive more than I+GI/3 simultaneously successful transmissions intended for itself.This completes the proof of part (ii) of Theorem 1. Now, (T 1.2) follows from part (ii) and the fact that the total number of nodes in the network is equal to Ntotal .
• upper bounds on per-SENSOR end-to-end throughput capacity In this subsection, firstly, we prove the following theorem: Theorem 2: A e and A,n have the following upper bounds: Proof of Theorem 2: Let t be an arbitrary time instant, and define the total infonnation transmission rate of the network at time t, C(t), as follows: where Wi (t) is the transmission rate of the {" successful transmission at time t.By the definition of H in section 2-4, each bit of information delivered to its destination is transmitted in H hops on the average.Therefore, the time average of C(t) over the time interval [0,(0) is not less than max Hr.:,A; = HNA.Also, since Wi (t) :s W max and Nt:S Nt , max we find that C(t) :s W max N t • Thus, average of C(t) over any time interval cannot exceed W max multiplied by a time max invariant upper bound on N t .Since the quantity Uyin (T 1.1) is such an upper bound, this implies HNA::;WIrulP y ' (18) On the other hand, due to part (ii) of Theorem I, we know that no destination node can receive more than I+GI/3 transmissions successfully.Since each transmission cannot occur at a rate more than W max , this implies that at the given time t, the total rate at which information is being delivered to the destinations cannot exceed MW max (1+GI/3).Hence, the total average rate at which information is delivered to the destinations over the time interval [0,(0), which is equal to NA, cannot exceed MW max (1+GI/3).This implies N A::; MW max (l+GI /3).(19) Hence, (18) and (19)   It is worth emphasizing that, because of the generality of the network model underlying the derivation of ( 20) and (21), they are applicable even when the mobility pattern of the nodes, the spatial-temporal transmission scheduling policy, the temporal variation of transmission powers, the sensor-destination pairs, and the possibly multi-path routes between them are optimally chosen as to maximize A. Similarly, (20) and ( 21) are applicable even when the nodes are capable of maintaining multiple transmissions and/or receptions simultaneously.

imply that
Recall that ,1,,, is the supremum of all end-to-end throughputs ..10 for which there exist: a mobility pattern of the nodes, a traffic pattern, a spatial-temporal transmission scheduling policy, and a temporal variation of transmission powers, so that A 2..10.There have been no restrictions on these parameters during the derivation of ( 20) and ( 21).
Hence, the right-hand sides of ( 20) and ( 21) are also upper bounds on ,1"", which is not less than 4(,.This completes the proof of (T2.1) and (T2.2).The proof of (T2.3) is identical with the proof of (T2.1) except that the expression on the right-hand side of (T 1.2) is used instead of U y.

_
So far, there have been no restrictions on the number of simultaneous transmissions and/or receptions that a node is capable of maintaining.If, as in [1]- [4], there is also the additional restriction that a node cannot transmit and receive simultaneously and that a node is capable of maintaining at most one transmission or one reception at any given time, then the upper bounds on 4(' and ,1"" can be further tightened.We will henceforth refer to this case as the half-duplex restricted case.In this case, no destination node can receive more than a single transmission, and thus following the same derivation method that was used between ( 18) and (19) shows that ks.Wmax MIN.Combining this inequality with (18), and following the same derivation method that was used after (19) in the proof of Theorem I lead to the following upper bound on 4(' and ,1"" for the half duplex restricted case: Finally.we show that dividing the communication bandwidth into several sub-channels of smaller bandwidth does not change the terms other than W max in all of the results on 4(' and ,1"" that we have presented so far.An assumption behind those results is that all transmissions are taking place in the same communication bandwidth.If the communication bandwidth is partitioned into several sub channels of smaller bandwidth, then there still is an upper bound on the transmission rate in each of these sub channels.All of the upper bounds on simultaneous transmission capacities of Q and the network are still valid for each of these sub-channels individually.Therefore, if there are K sub-channels and the transmission rate of the e h max sub-channel is no more than W k , then all of the upper bounds on 4(' and ,1"" are still valid if W max is replaced with L~~I W/'''l\ .

ANALYSIS OF THE UPPER BOUNDS
In this section, we analyze how the upper bounds on NP, N,lllax, 4", and ,1"" are affected as various parameters of the network are varied individually.Firstly, we analyze the asymptotic and limiting behavior of the upper bound Uy in Theorem I, to draw the following conclusions about NP :

NQ . 0 GI{J
eqUIvalent to lack of space Also, since the area of the network domain is A = !dJ"/4,D can be replaced with (4Alm ll2 .Doing so, we can also if y= 2. Regardless of the value of y, this also implies that NP cannot grow with the area of Q super-linearly.Linear growth is not possible when y :s 2, and can only be possible when Y> 2.  In Figure 3, U y is plotted as a function of A and y, for G= f3 = 10.This figure illustrates the growth trend of Uy as y and/or A increase.It is possible to observe the linear and the sub-linear growth of U y with A when y> 2 and 0 < y:s 2, respectively.The figure also illustrates the equivalence of the lack of attenuation (y= 0) and the lack of space (A = 0).
One should also notice the quadratic growth of U y with r Secondly, we analyze the upper bounds on N,lllax.Inequality (T 1.1) of Theorem 1 shows that N,max is O ( I) with respect to N. Since N,max :s NP, all of the above asymptotic results are valid for N,max, too.
However, from (T1.2) and the facts that N total :SN+M and M :SN, we find that N,lllax :s 2N(l +GIf3).Therefore, for a given N, G, and {J, the upper bound on N,max in (Tl.l) loses its tightness beyond some finite values of D and r Existence of an upper bound on N,max independent of D and yalso shows that N,max is 0(1) with respect to A and r The reason is that beyond some finite values of A or y, the network domain provides sufficient space and attenuation, so that the upper bound on the number of simultaneous receptions per-node, i.e., I+GI{J, becomes the limiting factor.
. that ~: and It", are O(Glj3 ). 9 We also observe that A e and A,,, additional attenuation provide considerable increase in Ae are upper bounded by W max ( 1+Glj3 ) I ( H N) when the and A,,,, where the behavior of A e and A,,, resembles the network domain lacks attenuation or space.Due to (T2.2), asymptotic behavior of UY' and beyond this region the A e and A,,, cannot exceed W max M(l+Glj3)IN, which is behavior A" and It", changes into 0( 1) with respect to A and independent of A and r So, the upper bound in (T2.2) r becomes more restrictive than the upper bound in (T2.1)Next, we demonstrate the above results through an beyond some finite values of A or y, and thus A and A,,, are example.Consider the half-duplex restricted case.For this e O( 1) with respect to A and r Similar behavior is also case, we have shown that A" and A,,, cannot exceed the right hand side of ( 22).Now, we normalize this quantity with observable in the half-duplex restricted case; for example, if respect to W max , and we denote the resulting expression by the set of sensor nodes and the set of destination nodes are Au.In Figure 4, Au is plotted as a function of A and r The disjoint and H = 1 ,10 then beyond some finite values of A or other parameters for this example are: G=j3=lO, N=250, y, the network domain provides sufficient space and M=NI2, and H =1. This figure illustrates the variation in attenuation so that at any given time, there is a placement of the growth trend of Au as a function of A for various values the nodes for which M simultaneously successful of y.Also, it demonstrates the presence of a region of (A,y) transmissions can be established between M of the N pairs where the limitation of A e and A,,, is due to shortage of sensors and the M destinations.However, no more space and attenuation.For the (A,y) pairs outside of this transmissions can be scheduled, since there are no region, shortage of inactive destination nodes becomes the remaining inactive destination nodes, and thus A e and A,,, dominant limitation, and thus WmaxM IN becomes the cannot exceed W max MIN, which can be observed from (22).dominant upper bound on A e and A,,, (this can also be In general, there is a region of (A,y) pairs for which the observed from (22».dominant upper bound on A e and A,,, is In Figure 5, parameter values are the same except that N From (T2.1) and (T2.2), it can observed that this region is is now an independent variable (so is M, since M=NI2 in contained within the region bounded by the A axis, the y this example) and y= 3. The light green region consists of axis, and the set of (A,}?pairs for which the (A,N) pairs where the limitation of A e and A,,, is due to ) .Since Uy is an increasing function of A shortage of space.For the (A,N) pairs outside of this region, and y, this region will expand as M (and thus N, since namely inside the darker blue region, shortage of inactive M SN) increases.This shows that the limitation of A e and A,,, destination nodes is the dominant limitation, and thus due to shortage of space and attenuation is more WmaxMIN is the dominant upper bound on A e and A,,,.The figure also demonstrates that if the area of the network pronounced when M (and thus N) is large compared to U y.
domain is kept constant and the number of sensor nodes is Additionally, we have shown that Uyis 0(A min (yl2.1})when increased, then Au decays as 00 IN), so that A" and A,n yt-2, 0(A 1l0g(A» when y= 2, and also 0(y\ These vanish as N grows large.However, if the area also increases observations support the claim that for large M (and thus N) with N, we observe that it can be possible to keep Au at a there is a region of (A,y) pairs where additional space and constant level so that it does not rule out the possibility of achieving a non-vanishing per-sensor end-to-end throughput as the number of sensor nodes grows large.We will In the previous section, we have shown that A and "1,,, are c O( liN).Next, to prove that they are also 0(1/N), we will show that they are Q(1/N).We do this by constructing a TDMA scheme that assigns each of the sensor nodes a separate time slot of constant duration.In such a scheme, there are N slots in each cycle and each of the sensor nodes transmits directly to its destination in the slot assigned to itself, with a transmission power large enough to satisfy the signal to noise ratio requirement.Assuming (is an upper bound on the power of noise in the used communication bandwidth, a transmission power of jJ( / a(D) guarantees successful reception.
Although this simple scheme takes no advantage of spatial reuse, it allows each of the sensor nodes transmitting 1INfraction of the time.Thus, assuming that each transmission satisfying the signal-to-noise ratio requirement can occur with rate W, an node end-to-end throughput of WIN is achievable by all sensor nodes.This shows that A e and "1,,, are Q( liN).As a result, A e and "1,,, are 0(1/N).

IMPLICATIONS OF THE RESULTS ON SCALABILITY
In this section, we consider the following scalability problem: we are increasing the number of sensor nodes in the network indefinitely, and we want to achieve a desired per-sensor end-to-end throughput, say Ao. ~ is not achievable if no other parameter is increased as a function of N, since due to (T2.1)A e and "1,,, are no more than W max U yI( H N), which is O( liN).So, one or more of the parameters from W max , Y, GIjJ, or A must increase with N and N must be increasing according to a function of Uy, so that W max U I( H N ) 2: ~ (note that H cannot be indefinitely reduced to compensate for increasing N, because H ~ 1 , as every bit of information has to be transmitted for at least one hop).This shows that HN must be O(WmaxU y ).
For practical systems, y is a property of the wireless channel and it cannot increase with N. W max cannot increase indefinitely with N, because of the presence of noise and because of the maximum transmission power constraints.These limit reliable information transmission to rates that do not grow with N. On the other hand, GIjJ depends on the implementation of the communication system and increasing it for a given system bandwidth usually requires decreasing W max .For example, it is shown in [13] that in spread spectrum CDMA, for a given system bandwidth, symbol transmission rate is inversely proportional to the processing gain.Likewise, reducing f3 requires a proportional decrease in the symbol transmission rate to satisfy a given bit error rate requirement.Therefore, increasing GljJ will not compensate for increasing N. So, the only way of achieving Ao would be increasing A as N increases.Hence, N must be increasing as a function of A. We have shown that U y is 0(Aminlyn.I) when yic 2 and 0(Allog(A» when y=2.Therefore, unless N is 0(A min {)12.11)when yic 2 and O(A /log(A» when y= 2, ~ is not achievable.Also, H must be e( I) with respect to N due to the following reasoning.We know that H 2: I, which implies that H is Q(1).To see why H must be O( I) with respect to N, observe that ~ cannot exceed 2W max ( I+G/f3 ) I H due to (T2.3) and the fact that M'SN.Since increasing GIjJ requires a proportional reduction in W max , as is the case in spread spectrum CDMA, we find that compensating for indefinitely growing H by increasing GljJ is not possible.
On the other hand, ~ is not achievable unless the upper bound in (T2.2) is at least as big as ~.Hence, (T2.2) implies that if M is not Q(N), then ~ is not achievable, because the above argument in the previous paragraph shows that Wmax( 1+GIjJ) cannot grow indefinitely with N.
The above results can also be stated in terms of the sensor density, p := N I A. From the above 0(-) results, dividing N and the asymptotic upper bounds on N by A, we obtain the following result: unless pis 0(AminlyI2.I,O) when y t 2 and O(1/log(A» when y = 2, ~ is not achievable. In other words, ~ is not achievable if p grows with N indefinitely when y > 2, if plog(A) grows with N indefinitely when y = 2, and if pA l•yn grows with N indefinitely when y< 2. In any case, ~ is not achievable if p grows with N indefinitely.Also, when y 'S 2, unless p decays down to 0 as N --> 00, ~ is not achievable.Our observations in this and the previous two paragraphs prove the following corollary regarding practical systems: Corollary: (A necessary condition Jar the scalability oj practical systems) A desired per-sensor end-to-end throughput is not achievable as N --> 00, unless H is e( I) with respect to N, M and A grow with N, such that M is Q(N).and theJollowing equivalent conditions are sati•~fied: • N is O(Amint y1 2 .I}) when yt 2 and O(A /log(A » when y= 2. • pis 0(AminlyI2.I,O}) when yt2 and 00 !log(A» when y=2. )'E {O, 1,2,3 }.We know that normalized A,. and "1,,, are no more than Au.Also, it follows from the definition of Au that Au is a decreasing function of N and an increasing function of A when Au < MIN = 0.5.Therefore, each of these curves separates a region of (A,N) pairs where a normalized end-to-end throughput of 0.1 is not achievable and another region where it may be achievable on average or by all sensor nodes.For example, when y= 2, and (A,N)= (3,400), the normalized end-to-end throughput 0.1 is not achievable, whereas it may be achievable on average or by all sensor nodes for (A,N)=(3,lOO).The corollary tells us that for the sequence of (A,N) pairs forming each of the curves in Figure 6, N is 0( 1), e(A 1/2), e(A/log(A», and 0(A) when yis 0, 1,2 and 3, respectively.Equivalently, for the sequence of (A.p) pairs associated with each of these curves, pis eOIA), 0(1IA I12 ), 0(1/log(A», and eo) when y is 0, 1,2 and 3, respectively.

CONCLUSIONS
In this paper, we have studied the capacity of single-user detection based wireless sensor networks through the use of a more general network model than the models used in the literature.
Instead of the propagation model used in the previous studies, we used the bounded power law decaying propagation model, which was proposed in other studies on connectivity such as [15] and [16], to obtain more realistic results for small transmitter-receiver distances, while approximating the conventional model at large distances.Using this model, we concluded that N,max cannot exceed NP, which 1 ! is independent of N, but depends on A, y, G, and fJ.The analysis of the upper bound on NP in Theorem I has revealed that NP is O(A min I y/2 .1 1 ) for yt 2 and is O(Allog(A» for y= 2. The analysis has also shown that NP is O( y2) and O(GIfJ ).
Additionally, since the network model that we have used is quite general, our results in this paper do not only hold for the network scenarios of [1]- [4], but also hold for networks whose nodes move with any mobility pattern or are capable of maintaining any number of simultaneous transmissions and/or receptions.Hence, we have been able to show that the maximum achievable per-sensor end-to end throughput is 0( liN), even when the mobility pattern of the sensor and the destination nodes, the spatial-temporal transmission scheduling policy, the temporal variation of transmission powers, the sensor-destination pairs, and the possibly multi-path routes between the sensors and the destinations are optimally chosen.Furthermore, this result holds even the communication bandwidth is partitioned into sub-channels of smaller bandwidth.
Moreover, our results are valid for any nonnegative value of y, 12 This allowed us to characterize the behavior of NP , N,ma" lie' and Il.,I1 under low attenuation conditions.In particular, it allowed us to show that lack of attenuation and lack of space are equivalent, where N,ma, and NP cannot exceed 1+GIfJ.Also, in these equivalent cases, lie and Il.,I1 cannot exceed W max ( 1+GIfJ )/( H N ).
We have also shown that no node can receive more than 1+GIfJ simultaneously successful transmissions intended for itself.This allowed us to show that N,max, lif' and Il.,I1 are 0(1) with respect to A and yfor a given N. Together with (T2.1) and (T2.2), this also allowed us to justify that the limitation of lie and Il.,I1 due to shortage of space and attenuation is more pronounced when M and N are large.
Finally, we have studied the implications of our results on the scalability of wireless sensor networks.We have shown that as N becomes large, unless one or more of the parameters from W max , Y, GlfJ, or A grows with N, and HN is O( W max Uy), a desired per-sensor end-to-end throughput is not achievable.Regarding scalability of practical systems, we have concluded that M must be n(N) and H must be 0( 1) with respect to N.Moreover, we have concluded that A is the only remaining parameter whose growth can compensate for increasing N. Above all, we have proved that as N ---> ex, a desired per-sensor end-to-end throughput is not achievable, unless A also grows with N, and N is O(AllI;n Iy/2 .1 1 ) when yt 2 and is O(A/log(A» when y= 2. In summary, in this paper, we analyzed the capacity of single-user-detection based wireless sensor networks through the use of a more general network model, and we determined several necessary conditions for the scalability of such networks.This was performed by considering only one of the fundamental requirements for scalability, which is the requirement of a non-vanishing per-sensor end-to-end throughput as the number of sensor nodes grows large.An interesting extension of this work would be to determine the additional necessary conditions that result from other fundamental requirements for scalability, such as bounded end-to-end delay, bounded power consumption, bounded processing power, and bounded memory consumption at the nodes. Y -(1+ y+:)Y -(1+ y+.::+ y.::)Y (1+ yf(1+.:yupper boundS on simultaneous transmission capacity Therefore, for a(x) as defined by (a.2), (a) N,-I N, GN the SINR at receiver i at time t.Then ¢:::} I I a(lij(t))(Pji(t) + PuCt)) ::::_T i=l j=i+l P SINR(t)= P:'(t) Ib) N,-I N, GN I N ' => I I2a(lij(t))::::-' (,(t)+-L.p,.l' (t) I ~ i=l )=;+1 f3 G j~1 (e) N,-] N, GN Ii"i) ¢:::} I I (a(lji(t»+a(lij(t»)::::_T (1) ;=1 j=i+l /3 ... From the definition of a successful transnusslon, Nt N, N, GN simultaneously successful transmissions can take place at ¢:::} I I a(lji(t» ::::_T 1=1 j~1 13 time t if and only if (j*i) Id' N, N, GN SINR, (t) :?: 13 , I:::: i :::: NT ' ¢:::} I I a(lij(t)) ::::_T , I N, pii(t) i=1 j=1 13 (j:t:;l => G I P' l i(t)::::.....!:.....-f3 -(,(t), l::::i::::N T , i=1 l ) wherec .=1._..J3

•
~ (l+u;JI/(t)f ;::::?; (I+...Q... C;;;:)Y (10) -F 'J iV , Next, we define d:= D I C1 111 The quantity d is the diameter of the network domain divided by a constant approximately equal to 0.625.Combining this definition with (9) and (10), we obtain the following necessary condition for Nt simultaneously successful transmissions at time t: (l+dJif f3 ",) 2N I+d U -I G ~_,t f --du::;-, j(x)dx::; I~,~" j(m), whenever a and b are integers and.f (x) is a continuous and non-increasing function of x on [a, b+l], and step (b) follows from changing the variable of the integration by defining u=l+d(xIN t )ll1.Next, we define the communication density at time t, 0; as follows: (J':= IN: I d .(12)

DFigure 3 -
Figure 3 -Upper bound on the simultaneous transmission capacity of the network domain as a function of the area of the network domain and the path loss exponent • lim u;: = (r-I ;(r-21 (I +Q.) D->= D-

Fig. 4 - 5 -
Fig. 4 -Upper bound on the normalized per sensor end-to-end Fig. 5 -Upper bound on the normalized per sensor end-to throughput capacity as a function of the area of the end network domain and the path loss exponent throughput capacity as a function of the area of the .

•Figure 6
Figure 6 illustrates this corollary.In this figure, G=jJ=lO, H = 1, M=NI2 and the curves are obtained by plotting the (A,N) pairs, for which Au = 0.1 and
i=1 c 2Next, let us consider the disks in Rill for every 2::; m ::; n -1 .In this case, there can be overlaps between some pairs of disks in R m .Consider two overlapping disks, B i ( Uim) and B/ujm) centered at the points i and j, respectively.Now we show that if Uim ?: Ujll/ then jE Sit", where Sill/:= {aik: 1 :' S k :' S Revista da Sociedade Brasileira de Telecomunicac;oes Volume 19, Numero 3, Dezembro de 2004 m-I }.