A Multi-Sink Multi-Hop Wireless Sensor Network Over a Square Region: Connectivity and Energy Consumption Issues Flavio Fabbri

Chiara Buratti

WiLAB, DEIS WiLAB, DEIS University of Bologna, ITALY University of Bologna, ITALY Email: [email protected] Email: [email protected]

Abstract—In this Paper we present a novel mathematical approach to evaluate the degree of connectivity of a multisink Wireless Sensor Network, where sink and sensor nodes are uniformly distributed over a given region. We consider both unbounded and bounded domains, specifically squares, and the impact of border effects is also shown. Random fluctuations as well as a distance-dependent deterministic path-loss are accounted for in our radio channel model. In particular, we deal with randomly shaped wireless footprints, rather than with the less realistic (yet widely adopted) disk model, which is a special case of our channel model. Nodes are organized in a tree-based topology with trees rooted at the sinks (multihop communications). The approach allows the computation of the probability that a randomly chosen sensor is not isolated, from which the probability that a certain amount of nodes are connected can be easily derived. The mean energy spent by the network is also accounted for. The model provides guidelines to optimally design the tree-based topology, taking into account connectivity and energy consumption issues. Index Terms—Wireless Sensor Networks, Connectivity, Energy Consumption, Multi-Hop, Multi-Sink.

I. I NTRODUCTION Wireless Sensor Networks (WSNs) are a recent field of application of wireless networks [1], [2]. The sensor nodes deployed in the monitored area need to communicate the sensed data to one or more sinks, responsible for collection of information from the area. Communication can take place through multi-hop paths, owing to the short transmission ranges typical of sensor nodes. A high degree of connectivity is required in order to have a minimum amount of information gathered by the sink(s) guaranteed. Also, a suitable metric to assess connectivity needs to be employed. Many papers in the literature based on random graph theory, continuum percolation and geometric probability [3]– [5] devoted their attention to connectivity issues of networks. In particular, wireless ad hoc and sensor networks have recently attracted a growing attention [6]–[9]. Connectivityrelated issues of WSNs are addressed in [8], [9]. In [8], while considering channel randomness, the Authors restrict the analysis to a single-sink scenario. In [9], instead, the problem of deploying multiple sinks in a multi-hop limited WSN is addressed. However, the work presents a deterministic

Roberto Verdone WiLAB, DEIS University of Bologna, ITALY Email: [email protected]

approach to distribute the sinks on a given region, rather than considering a more general uniformly random deployment. To the best of our knowledge, no one has so far introduced any connectivity model for WSNs while jointly considering the following aspects: presence of both sensors and sinks, random deployment of nodes, multi-hop communication, bounded scenarios, channel fluctuations and energy consumption. In this paper we mathematically derive the probability that sensor nodes uniformly distributed in the monitored area are connected to at least one sink, where multiple sinks are also uniformly distributed over the same region. Starting from such a result, we also derive the probability that all nodes, or a subset of them, are connected. Such derivation is performed assuming a link power loss which takes both dependance on distance and random channel fluctuations into account and considering border effects due to finiteness of the deployment region. The latter is assumed to be a square as it often happens (see, e.g., [10]), because of its simplicity. Nonetheless, rectangular networks exhibit very similar connectivity properties unless one side is much greater than the other [11]. The work is based on previous papers [11], [12] devoted to single hop networks. The analysis is first carried out in the case of single-hop communication (i.e., every sensor transmits the sensed data directly to a sink). Then, the multi-hop case (i.e., sensors may also act as routers) is considered assuming tree-based topologies of various heights and widths. This is performed under a specific assumption: sensor nodes are split into N groups (that we call levels) obtained through a random procedure which lets nodes belonging to each level be all uniformly distributed in the bounded region; the nodes are then connected through a hierarchical architecture, where nodes at a given level need to connect to nodes at a lower level to reach a sink (sinks belonging to the lowest level, in our formalism). As an example, it takes 3 hops to a node belonging to level 3 to reach the sink: two nodes (one belonging to level 2 and the other belonging to level 1) will act as relays. This assumption, that we denote as a-priori level partitioning, accounts for networks where a node belongs to one out of N categories of devices, each one having different physical features. The expression

2

”a-priori” stems from the fact that the partitioning procedure occurs independently from the nodes positions and thus before a particular realization of the deployment random process is observed. Just to give a practical example, in IEEE802.15.4 [13] networks, devices (such as the 13192 Evaluation Boards by Freescale [14]) operating on a peer-to-peer topology, can be either full function (FFD) or reduced function (RFD): hence, since RFD devices may only talk to FFD ones, if the latter belongs to level k, the former will necessarily belong to level (k−1). We emphasize that the nodes are so grouped with fixed densities a-priori: in fact, regardless of whether we are dealing with two diverse boards or with the same board running two different pieces of software, both the hardware (in the first case) and the software (in the second) remain the same for the entire operational time of the network (e.g., the software may not be re-compiled on-the-fly). Hence, although it is not the optimal situation from a connectivity perspective (not all possible paths to the sinks are exploitable), the a-priori partitioning assumption is noteworthy because it is widely adopted in practice. Moreover, connectivity models for twodimensional N-hop networks under more general conditions are still being studied [15]. Finally, the mean energy consumed by the network is evaluated, and the tradeoff between connectivity and energy consumption is shown. The paper is organized as follows. Section II describes the link model and reports a connectivity model for infinite networks, which represents the starting point of our analysis. Then, in Section III the bounded region is introduced and the full and partial connectivity probabilities are derived for the single-hop case. In Section IV the multi-hop case is considered. In Section V the mean energy consumption is examined. Numerical results are found in Section VI while Section VII reports conclusions. II. L INK M ODEL AND C ONNECTIVITY N ETWORKS

IN I NFINITE

The first scenario that we consider consists of an infinite bi-dimensional plane with nodes of two different types both distributed according to a homogeneous Poisson Point Process (PPP): sensors and sinks, with densities ρs [m−2 ] and ρ0 [m−2 ], respectively. We also denote as ρtot the sum ρs + ρ0 . The link model that we exploit accounts for the power loss due to propagation effects including both a distance-dependent path loss and the random channel fluctuations caused by possible obstructions. Specifically, a direct radio link between two nodes is said to exist if L < Lth , where L is the power loss and Lth represents the maximum loss tolerable by the communication system. In that case, the two nodes are said to be ”audible”. The threshold Lth depends on the transmit power and the receiver sensitivity. The power loss in decibel scale at distance d is expressed in the following form L = k0 + k1 ln d + s,

(1)

where k0 and k1 are constants, s is a Gaussian r.v. with zero mean, variance σ 2 , which represents the channel fluctuations.

This channel model was also adopted in [12] and [8]. By suitably setting k1 , it is possible to accommodate an inverse square law relationship between power and distance (k1 = 8.69), or an inverse fourth-power law (k1 = 17.37), as examples. By solving (1) for the distance d with  L = Lth, we can define the 0 −s transmission range TR = exp Lth −k , as the maximum k1 distance between two nodes at which communication can still take place. Such range defines the connectivity region of the sensor. Note that by adopting independent r.v.’s s for separate links, we have different values of TR for every nodes pair. This means that any sensor observes a different realization of the r.v. TR depending on the direction of the potential interlocutor, thus acquiring a jaggy wireless footprint. In other words, unlike many papers dealing with connectivity issues in the literature [6], [7], we do not use circles to predict sensor connectivity. However, by setting σ = 0, we neglect the channel fluctuations and may still define  an ideal  transmission 0 , which is the range, as a reference, as TRi = exp Lthk−k 1 radius of the circular deterministic footprint. From [12], the number, Nr1 ,r , of audible sinks within a range of distances r1 and r from a generic sensor node (r ≥ r1 ), is Poisson distributed with mean µr1 ,r , given by µr1 ,r = πρ0 [Ψ(a1 , b1 ; r) − Ψ(a1 , b1 ; r1 )], where Ψ(a1 , b1 ; r)

=

r2 Φ(a1 − b1 ln r)

−

e

2a1 b1

+

2 b2 1

Φ(a1 − b1 ln r + 2/b1 ),

(2) (3)

and th − k0 )/σ, b1 = k1 /σ and Φ(x) = R x a1 √= (L −u2 /2 (1/ 2π)e du. From (2) and by letting r1 = 0 −∞ and r → ∞ (see [11]), the mean value, µ0,∞ , of the total number of audible sinks for a generic sensor on the infinite plane is obtained. Its non-isolation probability is simply the probability that the number of audible sinks is greater than zero: q∞ = 1 − e−µ0,∞ . III. C ONNECTIVITY IN B OUNDED S INGLE - HOP N ETWORKS When moving to networks of nodes located in bounded domains, two important changes happen. First, even with ρ0 unchanged, the number of sinks that are audible from a generic sensor will be lower due to geometric constraints (a finite area contains (on average) a lower number of audible sinks than an infinite plane). Second, the mean number of audible sinks will depend on the position (x, y) in which the sensor node is located in the region that we consider. The reason for this is that sensors which are at a distance d from the border, with d ∼ TRi , have smaller connectivity regions and thus the average number of audible sinks is smaller. These effects, known in literature as ”border effects” [6], are accounted for in our model. The result (2) can be easily adjusted to show that the number of audible sinks within a sector of an annulus having radii r1 and r and subtending an angle 2θ, is once again Poisson distributed with mean µr1 ,r;θ = θρ0 [Ψ(a1 , b1 ; r) − Ψ(a1 , b1 ; r1 )],

(4)

3

0 ≤ θ ≤ π. If the annulus extends from r to r + δr, and θ = θ(r), this mean value becomes µr,r+δr;θ = θ(r)ρ0

δΨ(a1 , b1 ; r) δr, 0 ≤ θ ≤ π. δr

q = Fcon (ρ0 , S).

(5)

Consider now a polar coordinate system whose origin coincides with a sensor node. As a consequence of (5), if a region is located within the two radii r1 and r2 and its points at a distance r from the origin are defined by a θ(r) law, then the number of audible sinks in such aRregion is again Poisson r 1 ,b1 ;r) distributed with mean µr1 ,r2 ;θ(r) = r12 θ(r)ρ0 dΨ(adr dr, i.e., from (3) and after some algebra, Z r2 µr1 ,r2 ;θ(r) = 2θ(r)ρ0 rΦ(a1 − b1 ln r)dr. (6) r1

Now consider a square SA of side S meters and area W = S 2 , sensors and sinks uniformly distributed on it with densities ρs and ρ0 , respectively. Equation (6) is suitable for expressing the mean number of audible sinks from an arbitrary point (x, y) of SA, provided that such point is considered as a new origin and that the boundary of SA is expressed with respect to the new origin as a function of r1 , r2 and θ(r). In order to apply equation (6) to our scenario and obtain the mean number, µ(x, y), of audible sinks from the point (x, y), we need to set the origin of a reference system in (x, y), partition SA in eight subregions (Sr,1 . . . Sr,8 ) by means of circles whose centers lie in (x, y) (see [11], Fig. 1). Thank to the properties of Poisson r.v.’s, we can sum the contribution of each region and obtain an exact expression for 8 Z r2,i X µ(x, y) = 2θi (r) · ρ0 · r · Φ(a1 − b1 ln r)dr, (7) i=1

For the sake of simplicity, we define the function Fcon (·, ·) to be equal to the right side of (9), so that

Several results may be derived from (10). First, we compute the probability, Z, that the network is fully connected (i.e. every sensor can directly reach at least one sink). Assume that we have k sensors in SA with positions (x1 , y1 ), (x2 , y2 ), . . . , (xk , yk ). By indicating with F the event of full connectivity and with Ns the number of sensors in a scenario, we have Prob{F |Ns = k; (x1 , y1 ), . . . , (xk , yk )} =

q(x, y) = 1 − e−µ(x,y) .

q(xi , yi ),

i=1

2k

. . . fXk ,Yk (xk , yk )dx1 dy1 . . . dxk dyk Z Z  = q(x1 , y1 )fX1 ,Y1 (x1 , y1 )dx1 dy1 · . . . Z Z  ...· q(xk , yk )fXk ,Yk (xk , yk )dxk dyk ,

where 

fXi ,Yi (xi , yi ) =

1/W, 0,

(xi , yi ) ∈ SA otherwise

is the p.d.f. of the position of the ith node. Note now that the same assumption (i.e., uniform distribution) holds for all nodes, thus we have Z Z k Prob{F |Ns = k} = q(x, y)fX,Y (x, y)dxdy (12) "

1 = W

(8)

Having assumed that sensor nodes are uniformly and randomly distributed in SA, if we now want to predict the probability that a randomly chosen sensor node is not isolated, we need to average q(x, y) on SA. In fact, the probability that a randomly chosen sensor node is not isolated (which is an ensemble measure) and the average non-isolation probability over a single realization coincides due to the ergodicity of stationary Poisson processes (see [16], page 104). This was also verified by simulation. Recalling that we have considered the lower half of the first quadrant, which is one eighth of the totality, we have Z S/2 Z x 8 q= q(x, y)dydx. (9) W 0 0

k Y

(11) where we assumed that sensors connect to the sink independently from each others, which is a realistic assumption in networks that are not capacity-limited. Now, by deconditioning with respect to the nodes positions, we have Z Z Y k Prob{F |Ns = k} = . . . q(xi , yi )fX1 ,Y1 (x1 , y1 ) . . . | {z } i=1

r1,i

which is the mean number of sinks in SA that are audible from (x, y), where r1,i , r2,i , θi (r) are reported in [11], Tables 1-2. If we assume a single-hop network, a sensor potentially located in (x, y) is isolated (i.e. there are no audible sinks from its position) with probability p(x, y) = e−µ(x,y) and it is non-isolated with probability

(10)

"

8 = W

Z

S/2

0

Z

Z

S/2

−S/2

x

Z

S/2

q(x, y)dxdy −S/2

#k

q(x, y)dydx

0

= qk .

#k

(13)

(14)

Since Ns is Poisson distributed with mean ρs W , we can decondition (14) with respect to Ns and obtain Z = Prob {F } = =

+∞ X

k=1

q¯k ·

+∞ X

k=1

Prob {F |Ns = k}Prob {Ns = k}

e−ρs W (ρs W )k . k!

(15)

Equation (15) represents the probability that a sensor network performs at best (full connectivity), but the event F turns out

4

to be a strict requirement for most of them. In other words, for many applications it is sufficient to guarantee that a certain amount of sensors can transmit their data to the sinks. For this reason, it is of interest to compute the probability of the event, Cj , of having at least a number, j, of connected sensor (partial connectivity). We first consider the event Cj∗ of having exactly j connected sensors. When Ns = k, the probability of having j connected sensors is   k j ∗ Prob {Cj |k} = q¯ (1 − q¯)k−j , (16) j
k! j ≤ k, where the binomial coefficient kj = j!(k−j)! accounts for all the possible ways to group j sensors out of k. Note that for the events Cj and Cj∗ the following holds: ∗ Cj = {Cj∗ ∪ Cj+1 ∪ . . . ∪ Ck∗ }.

(17)

Thus, if we consider the event Cj we need to add contributions similar to (16) for all j, j ≤ k, to obtain k   X k l Prob {Cj |Ns = k} = q¯ (1 − q¯)k−l , (18) l l=j

j ≤ k. Once again, by deconditioning (18) with respect to Ns we have Prob {Cj } = =

+∞ X k=j

Prob {Cj |k}Prob {k}

+∞ X k   X k k=j l=j

l

q¯l (1 − q¯)k−l ·

e−ρs W (ρs W )k . (19) k!

Note that the outer sum in (19) starts at j instead of 1, because when k < j it gives no contribution (i.e., the probability of having j connected sensors in a network with less than j sensors is zero). For this reason, we want to highlight that Prob {Cj } of (19) depends also on ρs : in fact, the probability of having at least j connected sensors is affected, besides q¯, also by how many sensors we have at all in the network (i.e., either connected or not). In order to emphasize this, we introduce a new notation, Zm ¯ (j), and, after some simple algebra, we have Zm ¯ (j) = Prob {Cj } +∞ X k   X k m ¯ k q¯l (1 − q¯)k−l −m ¯ =e · , l k!

(20)

k=j l=j

with m ¯ = ρs W being the average number of sensors in SA. Thus, Zm ¯ (j) of (20) has the meaning of probability of having at least j connected sensors in a network with (on average) m ¯ sensors. IV. C ONNECTIVITY

IN

B OUNDED M ULTI - HOP N ETWORKS

Now we wish to extend our analysis to the case of multi-hop wireless sensor networks. Each sensor is allowed to forward its data to another sensor instead of trying to communicate directly with the sinks, with the constraint of a fixed maximum number of hops.

We assume that each node a-priori belongs to one out of N different levels, meaning that an ith level node can send its data only to an (i − 1)th level node, hence, it will take i hops to such a node to communicate with a sink (which is considered a zeroth level node according to this formalism). This approach is justified by the fact that in some classes of sensor networks each node has a certain probability px,0 to be a coordinator (zeroth level node), px,1 to belong to the 1st PN level, . . . , px,N to belong to the N th level, with i=0 px,i = 1. Thus the parental relations between nodes are in some sense pre-assigned. If ρtot is the overall nodes density and ρs is the overall sensor nodes density (i.e. ρtot = ρ0 + ρs ), we have for the generic ith level density ρi = ρtot · px,i , 0 ≤ i ≤ N , with PN PN i=0 ρi = ρtot and i=1 ρi = ρs . We also assume that nodes at each level are uniformly distributed in SA. We now want to find the probability q¯1 that a randomly chosen sensor is connected and that it is one hop away from the sink. In terms of the Fcon function introduced in (10), we can write q¯1 = px,1 · Fcon (ρ0 , S), (21) where the two factors account for the events of belonging to the 1st level and being actually connected to a sink, respectively. Note that q¯1 of (21) has the same meaning of q¯ in (9) when N = 1. If we consider multi-hop paths, we can define the probability q¯i that a randomly chosen sensor has a connection to the sink through a path containing at most i hops. In other words, it must be a connected 1st level sensor, or a connected 2nd level sensor, . . . , or a connected ith level sensor. As an example, the probability q¯2 may be written as q¯2 = px,1 · Fcon (ρ0 , S) + px,2 · Fcon (ρs · q¯1 , S) = q¯1 + px,2 · Fcon (ρs · q¯1 , S),

(22)

where px,2 · Fcon (ρs · q¯1 , S) is the probability that the sensor belongs to level 2 and has a connection to any 1st level sensor which is, in turn, connected to a sink. As for q¯3 , the chain is one hop longer, thus we need to write q¯3 = q¯2 + px,3 · Fcon (ρs · px,2 · Fcon (ρs · q¯1 , S), S).

(23)

In general, for an N -level network we have the recursive expression q¯N = q¯N −1 + px,N · Fcon (ρs · px,(N −1) · Fcon (. . . ρs · px,2 · Fcon (ρs · q¯1 , S) . . . , S), S), (24) with (21) providing expression for q¯1 . We can now introduce the probability, Z (N ) , of having all sensors connected in an N -level network by following the same reasoning as in the 1-hop case (see equations (14-15)). We recognize that, once the parameters of the network W and ρs are fixed, the only difference between the 1-hop and the multi-hop case resides in how the non-isolation probability is computed, i.e., we have q¯ for the 1-hop case and q¯N for the multi-hop case. In virtue of this, we can generalize (15) as Z(x) =

+∞ X

k=1

xk ·

e−ρs W (ρs W )k , k!

(25)

5

where we preserved the structure and set the non-isolation probability as variable. Recalling (16-20), we find that the same holds for (20), which yields +∞ X k   X k m ¯ k xl (1 − x)k−l −m ¯ Zm · , (26) ¯ (j; x) = e l k! k=j l=j

where we set, once again, the non-isolation probability as variable. Thus, for Z (N ) we can simply use (25) with x = q¯N , getting Z

(N )

= Z(¯ qN ) =

+∞ X

k=1

k q¯N ·

e−ρs W (ρs W )k . k!

(27)

(N )

Similarly, we also compute the probability, Zm ¯ (j), of having at least j connected sensors in an N -level network with (on average) m sensors by using (26) with x = q¯N and obtain (N )

Zm ¯N ) ¯ (j; q ¯ (j) = Zm +∞ k   l XX k m ¯ k q¯N (1 − q¯N )k−l −m ¯ . (28) =e · k! l k=j l=j

The way in which the densities ρi (i ≥ 1) are defined can follow, as an example and without loss of generality, the simple partitioning criterion ρi+1 /ρi = η,

0 ≤ i < N,

(29)

where η is a constant (i.e., level densities follow an exponential growth, which is kind of a ’natural’ law in hierarchical networks). Note that (29) holds only for i < N : in fact, if we fix ρ0 P and ρs , the N th level nodes must have density N −1 ρN = ρs − j=1 ρj in order for the sensor densities to sum up to ρs . Moreover, by fixing ρ0 , ρs and η (or equivalently ρ0 , ∆ = ρs /ρ0 and η), there are no longer degrees of freedom and the number N of levels in the network is also consequently assigned. V. E NERGY C ONSUMPTION We assume that each node consumes energy when transmitting and receiving packets, whereas we neglect the energy spent by the node in idle or sleep states. We also assume that the sinks do not have energy consumption problems, thus we do not consider the energy spent by them. The mean energy spent in the network for each transmission towards the sink is given by E=

N X i=1

[Erx + Etx · i + Erx · (i − 1)] · (q i − q i−1 ), (30)

where Erx is the energy spent to receive a packet, Etx is the energy spent to transmit a packet, and q¯i is given by (21), (22), (23) and (24). (q i −q i−1 ) is the probability that a generic node belongs to level i of a connected tree. The energy spent in the network to deliver a packet from a source node to the final sink, instead, depends on the level at which the source node is located. In particular, if the source node is at level one, the packet can reach the sink through

a single transmission; if, instead, the node is at level two its packet must be (i) transmitted by the source node, (ii) received by the level one node and (iii) transmitted by the latter node to the final sink; therefore two transmissions and one reception are needed. We also consider the energy spent by each node to receive the triggering packet coming from its parent in the tree (tree formation). According to the Freescale devices data sheets [14], we set the energy spent to transmit a bit equal to 0.3 µJ/bit and the energy spent to receive a bit equal to 0.33 µJ/bit. Moreover, we set the packet size equal to 20 Bytes, therefore Etx = 48µJ and Erx = 52.8µJ. VI. N UMERICAL R ESULTS Fig. 1 shows q for different sink densities as a function of Lth , proportional to the transmit power if the receiver sensitivity is fixed: clearly, as such density grows, for a fixed transmit power it is more likely for a sensor to reach at least a sink and thus q also grows. For example, if we want a randomly chosen sensor to be connected with 90% probability, we need Lth ≈ 95 dB when ρ0 W = 150, Lth ≈ 98 dB when ρ0 W = 100, Lth ≈ 103 dB when ρ0 W = 50 and Lth ≈ 115 dB when ρ0 W = 10. Also note the comparison to the curve for q∞ obtained with no consideration of border effects: the error becomes non negligible for transmission ranges which are of the same size as the side S of the domain (e.g., TRi (Lth = 115 dB) ≈ 316 m), a typical case for WSNs. In Figs. 2 and 3 connectivity results related to multi-hop WSNs are reported. The criterion of a-priori partitioning is used in accordance with (29). Observe that for N = 5, η ranges from 1.9 to 2.3. This means that when η = 2.3 the network has 4 levels or, equivalently, 5 levels with the 5th (N ) being empty. q¯N and Zm ¯ (j) are plotted as functions of η, respectively. They show arches and local optima which depend on the loss threshold Lth , η and N . In particular, from Fig. 2, we conclude that a large value of N is opportune only if Lth (and, consequently, the transmit power) is large enough: in fact, when N = 5 (η ranging from 1.9 to 2.3) we have global optima for Lth = 95 dB and Lth = 100 dB but only local optima for Lth = 85 dB and Lth = 90 dB. Finally, in Figure 4 we show the mean energy spent, E, as a function of η and N for different values of Lth . As we can see E increases by increasing N , since (on average) more transmissions and receptions are needed to reach the sink. Therefore, for large values of Lth a tradeoff between connectivity and energy consumption should be found: in fact, large N improves connectivity but also increases energy consumption. Moreover, the evaluation of the energy consumption behavior is useful to select the optimum values of η and N , for a desired degree of connectivity. As an example, when we set Lth = 90 dB, we obtain approximatively the same maximum of q¯N for N = 4 and N = 3; however, the consumed energy is notably larger for N = 4. VII. C ONCLUSION In this paper we have introduced a novel mathematical model to study the connectivity of multi-sink WSNs over

6 1

q (ρ0 W =50) q (ρ0 W =10) q ∞ (ρ0 W =10)

0.5 0.4 0.3 0.2 0.1 0 60

70

80

90

100

110

0.6

(N )

Z2500 (2250)

0.4

0.2 (N )

Z2500 (2300) 0

120

N=2

0.7 0.6

(N )

Z2500 (2200)

0.8

N=3

q (ρ0 W =100)

N=4

q (ρ0 W =150) sim

0.8

N=5

q (ρ0 W =150)

Probability of partial connectivity

Average non-isolation probability

1 0.9

2

4

6

Lth [dB]

Fig. 1. q as a function of Lth for different values of ρ0 , with S = 1000 m, k0 = 40, k1 = 13.03, σ = 3.5.

(N)

η

(N)

8

10

12

(N)

Fig. 3. Z2500 (2200), Z2500 (2250) and Z2500 (2300) as functions of η with Lth = 95 dB, ρ0 = 50/S 2 , ρs = 2500/S 2 (∆ = 50), S = 1000 m, k0 = 40, k1 = 13.03, σ = 3.5

Energy consumption [J]

0.6 q¯N (Lth = 85 dB) q¯N (Lth = 90 dB) q¯N (Lth = 95 dB)

0.4

q¯N (Lth = 100 dB) q¯N (Lth = 90 dB)(sim)

0.2

N=2

N=3

N=4

0.0006

N=5

N=2

N=3

N=4

0.0007 0.8

N=5

Average non-isolation probability

1

E(Lth = 85 dB) E(Lth = 90 dB)

0.0005

E(Lth = 95 dB) 0.0004

E(Lth = 100 dB)

0.0003

0.0002

0.0001 0

2

4

6

η

8

10

12

Fig. 2. q¯N as a function of η with ρ0 = 50/S 2 , ρs = 2500/S 2 (∆ = 50), S = 1000 m, k0 = 40, k1 = 13.03, σ = 3.5.

squared regions. Both the single-hop and multi-hop cases have been considered. The practical outcome of this approach is the possibility i) to set the proper power level of nodes and their density, given a requirement in terms of connectivity; ii) to select the optimum height/width of a tree; iii) to evaluate the tradeoff between connectivity and energy consumption. The work is based on realistic assumptions as far as link connectivity and scenario are concerned. The mathematical model does not require any approximation, once that starting assumptions have been accepted. Simulations have been performed to validate the numerical results and, where not shown, they simply overlap to the mathematical results, when same assumptions (e.g. uniform distribution of nodes, etc) apply. R EFERENCES [1] C.-Y. Chong and S. P. Kumar, “Sensor networks: evolution, opportunities, and challenges,” vol. 91, no. 8, pp. 1247–1256, Aug. 2003. [2] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,” IEEE Commun. Mag., vol. 40, no. 8, pp. 102–114, Aug. 2002. [3] B. Bollobs, Random Graphs, s. e. Cambridge University Press, Ed., 2001. [4] R. Meester and R. Roy, Continuum Percolation, C. U. Cambridge University Press, Ed., 1996. [5] M. D. Penrose, “On the spread-out limit for bond and continuum percolation,” Annals of Applied Probability, vol. 3, pp. 253–276, 1993. [6] C. Bettstetter and J. Zangl, “How to achieve a connected ad hoc network with homogeneous range assignment: an analytical study with consideration of border effects,” in Mobile and Wireless Communications Network, 2002 4th International Workshop on, Sep. 2002, pp. 125–129.

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Fig. 4. Average energy consumption E [J] as a function of η with ρ0 = 50/S 2 , ρs = 2500/S 2 (∆ = 50), S = 1000 m, k0 = 40, k1 = 13.03, σ = 3.5.

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