Jean-Marie Gorce

CITI / INSA-Lyon , ARES / INRIA Rhone-Alpes 69621,Villeurbanne CEDEX, France Email: [email protected]

CITI / INSA-Lyon , ARES / INRIA Rhone-Alpes 69621,Villeurbanne CEDEX, France Email: [email protected]

Abstract—Connectivity, as an important property of Wireless Sensor Networks (WSNs), was studied by many previous researchers using the geometric disk model which rests on the switched link model. Basically, shadowing and fading have so heavy impact on links that there are many unreliable links in WSNs. In this paper, we quantify the effect of the long but unreliable hops on the connectivity of WSNs. In our model, each node connects to another node in the network with a certain probability at a given time, and this probability depends on techniques of transmission, channels etc. Based on this model, an overall expression of mean node degree is deduced. Finally, both analytic and simulation results indicate that unreliable long hops can greatly enhance the connectivity of WSNs.

I. I NTRODUCTION Wireless Sensor Networks (WSNs) attract numerous attentions because of their broad applications [1]. Connectivity is a key issue for WSNs regarding its tight relation to performances such as reliability and life-time [2], [3]. Many works [4]–[7] which study the connectivity of ad-hoc network are based on a perfect geometric disk model, i.e. all links are reliable and occur only when the communication distance is lower than a threshold: the radio range. This assumption has been the foundation of many works [8], [9] in which numerous wireless network connectivity bounds were provided. Recently, a realistic radio link model was advocated to studying the impact of log-shadowing on WSNs [10]–[13]. But these works still relied on the switched link model,i.e. the transmission between two nodes x and x0 succeeds if and only if the signal to noise ratio (SNR) γ(x, x0 ) at the receiver is above a minimal value γmin . In our study, an unreliable link model is employed as a more realistic model, i.e. for each pair of nodes, there is a probability of link success. The probability tends to 1 for a short link and tends to 0 when the distance between two nodes tends to infinity. With such a realistic model, the communication range becomes undefined and is replaced by a reception probability depending on the distance. This probability relies on various parameters such as propagation properties of channel, the radio transmission technique (type and order of modulation, coding. . . ) and packet size. Flat fading is also considered because it greatly contributes to the unreliability of wireless links leading to a fairly large deviation from the usual link quality prediction.

This paper is organized as follows: Section II introduces some definitions of previous related works studying the connectivity under the switched-link assumption. the mean node degree definition is extended to unreliable radio links in this section. Then, the mean node degree is deeply studied in section III for block-fading channels. The accuracy of theory analysis is evaluated using extensive simulations in section IV. Some conclusions and perspectives are drawn in section V. II. C ONNECTIVITY This section firstly introduces some previously published definitions and connectivity properties of WSNs based on the switched link model. Then they are extended to the case of unreliable links. A. Graph model and Connectivity In this paper, we suppose that the nodes of a WSN are independent and are randomly distributed according to a random point process of density ρ, over the space D. The number of nodes N in a subarea A (A ∈ D) then follows the Poisson distribution: (ρ · SA )n −ρ·SA P (n nodes in SA ) = P (N = n) = e , (1) n! with E[N]=ρ · SA , where SA is the surface of the subarea A. A WSN is modeled as a graph G(L, V ) having V nodes and a set of radio links L. For the switched link model, each link l(x, x0 ) ∈ L is a binary variable, i.e. 1 or 0. With the unreliable links model, each link l is a probability. G(L, N ) is defined as a model of subarea A having N nodes, G(L, N ) ∈ G(L, V )1 . The connectivity is an important feature for WSNs. A graph G(L, N ) is said connected if at least one multi-hop path exists between each pairs of nodes in the graph. Note that all nodes can communicate with a unique sink if and only if the corresponding graph is connected. In [14], the connectivity is defined as the probability of having an infinite connected component in G. In [9] the network is scaled down to a finite disk area, and the connectivity is assessed using the range R(n) which allows to make the graph asymptotically connected (i.e. for n → ∞). In [10], the connectivity is also studied in a finite disk but defined as a 1 In this work and other referenced works in this paper, radio links are assumed to be symmetrical, and thus associated graphs are undirected

subregion of a whole infinite network at a constant density. This definition is substantially different, because the nodes outside the disk can help for the connectivity of nodes inside the disk. Then, connectivity is assessed through the probability Pcon (A) that the nodes inside a subarea A are connected one to each other. In this paper,the connectivity is defined in as the probability Pcon (A). The connectivity is first bounded by a no node isolation probability according to: Pcon (A) ≤ P (no node iso). Secondly, the probability that no nodes are isolated in a subarea A is obtained by P (no node iso) = exp(−ρ·SA ·P (iso)) [10], [15], where P (iso) is the isolation probability relying on the mean node degree µ0 (defined in II-B) as P (iso) = exp(−µ0 ). For small P (iso) and large number of nodes N , the fundamental property is: Pcon (A) ≤ exp(−ρ · SA · e−µ0 )

(2)

B. Mean node degree The node degree µ(x) is defined as the number of links of a node x, the mean value being referred to as µ0 . The mean node degree under unreliable links differs from that under reliable links, because, for a probabilistic graph, the neighborhood is different at each transmission. The instantaneous node degree µ(x, τ ) of a node x is defined as the number of simultaneous successful transmissions at a given time τ . The instantaneous mean node degree is the expected value over a region A, i.e. µ0 (τ ) = hµ(x, τ )ix , where h·i stands for the expectation. The mean node degree is the value averaged over time µ0 = hµ0 (τ )iτ = hµ(x, τ )ix,τ . The expectation degree ofR a node x relies on the radio links according to: µ(x, τ ) = x0 ∈V l(x, x0 ) · fx (x0 )dx0 , where fx (x0 ) is the probability density function of having a node in x0 and l(x, x0 ) is the link probability between nodes x and x0 . Because the nodes are uniformly distributed, fx (x0 ) = ρ, and the process is ergodic, spatial and time expectations then converge to the same value given by: Z ∞ µ0 = hµ(x, τ )ix,τ = 2π · ρ l(dxx0 = s) · sds, (3) s=0

where l(dxx0 = s) is equal to the transmission probability. In slow varying channels, occurring in fixed WSNs, when the transmission packets are short, the channel can be assumed to be constant within a packet duration. Under such an assumption, referred to as pseudo-stationarity, the channel is called a block-fading channel. The link probability with block-fading is given by successful transmission probability i.e. l(dxx0 = s) = PS (tr|¯ γ (dxx0 = s)): Z ∞ PS (tr|¯ γ (dxx0 = s)) = PS (tr|γ) · fγ (γ|¯ γ (dxx0 = s))dγ γ=0

with PS (tr|γ) = (1 − BER(γ))

(4) Nb

(5)

where Nb is the number of bits per frame and BER(γ) is the bit error rate. This BER(γ) depends on modulation, coding, and more generally on transmitting and receiving

techniques (diversity, equalization, ...). It also relies on the channel impulse response. 2 In(5), fγ (γ|¯ γ ) is the pdf of γ having a mean SNR γ, representing the fast fluctuations of received power. III. T HEORETICAL M EAN N ODE D EGREE D ERIVATION In this section the theoretical mean node degree expression is derived in block-fading channels. The case of a simple AWGN channel is considered first. Then, the results are extended to block-fading channels. A. Normalized mean node degree According to (3), (4)and (5) , the connectivity depends on several system parameters: the node density ρ, the transmission power and the noise level (all involved in γ¯ ). It is obvious that the connectivity of a network can be improved by either increasing the transmission power or the node density. In order to clearly explain the impact of unreliability links on connectivity, we introduce the definitions of normalized node density and normalized mean node degree. Let d1 be the distance at which the received power is unitary: γ¯ (d1 ) = 1. The normalized node density n ¯ 1 is then defined as the mean number of nodes located inside a disk of radius d1 : n ¯ 1 = π · ρ · d21

(6)

It is important to note that d1 depends physically on the pathloss exponent α, the reception noise N0 and the transmission power P0 . The mean SNR γ¯ (s) is now expended according to a path loss model [16] as a function of d1 : γ¯ = (s/d1 )−α . A variable change from s to γ¯ in (3) leads to: Z µ0 2 ∞ −(1+ 2 ) α · P (tr|¯ = γ¯ γ )d¯ γ (7) S n ¯1 α γ¯ =0 In (7), µ0 /¯ n1 is defined as normalized mean node degree and it only relies on the attenuation parameter α and PS (tr|¯ γ ). B. Derivation in AWGN channel Without fading, the mean SNR γ¯ is merged in its instantaneous value γ i.e. PS (tr|γ) = PS (tr|¯ γ ). Let us now focus on the instantaneous success probability PS (tr|γ), which is directly related to the bit error rate (BER). A closed-form of the BER is found in [17] for coherent detection in AWGN: p (8) BER(γ) = 0.5erf c( k · γ) R∞ 2 with erf c(x) = √2π √x e−u du, the complementary error function. k relies on the modulation kind and order, e.g. k = 1 for Binary Phase Shift Keying (BPSK). The framebased success probability is given in (5). 2 In

this work selective fading is not considered.

The frame based mean node degree for Nb bits frames is denoted by µn . Plugging the exact success probability (5) into (7), we get: Z µn 2 ∞ −(1+ 2 ) N α · (1 − BER(γ)) b dγ = γ (9) n ¯1 α γ=ε Note that this integral can be divergent for γ → 0, because the success transmission probability in (5) tends toward 2−Nb instead of 0. This can be easily counteracted by the use of a power detection threshold ε as a lower integration bound. (a)

C. Nakagami-m block-fading channels This section now aims at extending the previous results to the case of block-fading channels described in section II-B. 1) radio link: We propose the use of the Nakagami-m distributions [16], [17] which are often used for modeling fading in various conditions from AWGN (m → ∞) to Rayleigh (m = 1). The SNR’s pdf is given by: fγ(m) (γ|¯ γ) =

mm · γ m−1 −m · γ exp( ) m Γ(m) · γ¯ γ¯

(10)

where Γ(m) is the gamma function, and m drives the strength of the diffuse component. 2) Frame-based mean node degree: The success probability is given by (4). The mean node degree in block fading, namely µf , is derived from (7) as: Z Z µf 2 ∞ −(1+ 2 ) ∞ α = γ¯ PS (tr|γ) · f (γ|¯ γ )dγd¯ γ (11) n ¯1 α γ¯ =¯γr γ=0 In order to evaluate the relative influence of short and long hops, we introduce a reliability threshold γ¯r as a lower bound for the integral relative to γ¯ in (11). The normalized mean node degree is plotted in Fig.1(a) as a function of dr /d1 = (1/¯ γr )α , the distance at which the mean power is equal to the reliability threshold γ¯r . These curves are provided for 4 pairs (α, m). For α = 2, the same asymptotic value is reached for any m value, even in Rayleigh conditions. It means that the neighborhood stretches with increasing fading, making the links less reliable. Fig.1(b) shows the same curves as a function of the reliable probability limit Prel (tr|¯ γr ) according to (4). The connectivity leakage owing to a stringent Prel is seen, especially for strong fading. With α = 2, the maximal connectivity is equal to 0.2, whatever m. A constraint of Prel = 0.9 is fulfilled having a mean node degree going down to 0.11 and 0.02 for m = 10 and m = 1, respectively. The capability of managing unreliable links is thus a very important feature for WSNs roll-out in strong fading environments. In this section the asymptotic mean node degree was provided for AWGN channel and fading channel. And the effect of fading on reliability has been assessed. The close-form formulas and integral theoretical analysis are gotten in [18]. The leading conclusion is that unreliable links may contribute significantly to improve the connectivity of a WSN.

(b) Fig. 1. Normalized mean node degree estimation for a BPSK modulation (k=1), with Nb = 1000 bits, plotted as a function of the reliability range, dr /d1 in (a), and the limit transmission error probability Perr = 1 − Prel (tr|¯ γr ) in (b). In (a), the curves are numerically obtained from (11). In (b), the connectivity loss owing to a reliability threshold is plotted.

IV. S IMULATIONS AND A NALYSES The theoretical analyses of the previous section are now validated by extensive simulations. A. Simulation setup The whole simulation area D is a disk. In order to avoid boundary problem, the mean node degree and the connectivity were evaluated in a subarea A (surface SA ), A ∈ D. Such, A is considered as a subpart of an infinite network [10]. From a practical point of view, it is enough to chose D as a disk having a radius i.e.RD = 2 · RA . In the simulations, the nodes are deployed uniquely in Poisson distribution, and the average number of nodes in A is kept constant. We make the normalized node density n ¯ 1 vary by changing the transmission power. Note that the radius of the unitary received power area (d1 ) changes accordingly. This choice is equivalent to modifying the node density, but with our approach, we keep the mean number of simulated nodes constant. The simulations following that are not explained specially employ BPSK modulation and coherent receiver. a path-loss coefficient α = 2 and a frame size of 1000 bits. The mean number of nodes in region A is 50. For each transmission power level, 100 random sets of nodes were generated. For

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Fig. 2. A peculiar realization of graphs is given for Nakagami 10 2(a) and Nakagami 1 2(b) conditions with about 30 nodes in the smaller disk. Mean node degree = 5.53

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each set, the pairwise connection matrix was independently computed 20 times. To verify the theoretical results, the following simulations are carried out. The first simulation is to mensurate the influence of density of nodes to the mean node degree; the second simulation is performed to access the effect to the mean node degree of attenuation coefficient (α) and different strength of fading channels corresponding to different Nakagami m parameter values; the third simulation studies the connectivity and the effect of the reliability threshold on connectivity. B. Simulation results and analysis In this section, the simulation results are shown and are analyzed. Fig.2 shows that: in despite of the same mean node degree, in heavy fading channels, the long hops have a important role in making the network better connected. 1) Relation of mean node degree and normalized node density: The mean node degree is firstly investigated in Fig.3 . The average simulation results are provided for Nakagami channels using different attenuation coefficient α. The relationship between n ¯ 1 and mean node degree is linear for all conditions. For α = 2 the mean node degree is not dependent on m which fits the theoretical results [18]. This group of curves further emphasizes that fading weakly impacts the mean node degree. As stated above, this result will be experienced by a WSN, only if it is able to deal with opportunistic transmissions. For α = 3 and α = 3.5 the mean node degree’s

loss corresponds to the connectivity loss coefficient in [18]. 2) Mean node degree in fading channels: In order to verify the theoretical result of (11), the corresponding simulations are carried out. Fig.4 provides the simulation results for α = 2 and α = 3.5 in different channels: Nakagami 1 and Nakagami 10. These results agree with the theoretical results Fig.1(a). This group of curves reveals that in heavy fading channels (m = 1), long but unreliable hops proves to be very important to preserve the connectivity. 3) Connectivity probability: Figure 5 represents the mean connectivity experienced in our simulations. As found in previous section, the no isolated node probability relying on the mean node degree is not affected by fading whatever m. Basically, the non isolated node probability acts as an upper bound for the connectivity: a network cannot be connected if just one node is isolated. These figures show that the more heavy the fading strength is the tighter bound gets when there is no threshold at the receiver, i.e. all unreliable links can be applied. In fact, fading indeed changes the graph from a pure geometric graph to a random graph. The mean number of links is constant whatever m, but fading favors longer links which are more efficient to maintaining the network connectivity. The conclusions are twofold: • The mean path-loss function is sufficient to predicting the mean node degree and thus the non isolated node probability, • Second order variations of the received power due to fading do not change the mean node degree but even improve the connectivity. These figures also show that the effect of the threshold on connectivity increases with the strength of fading. V. C ONCLUSION This research is based on the framework proposed by Bettstetter et al., as described in section II. The main novelty of our works rests on introducing a more realistic model: unreliable link model. For this purpose, we propose the definition of instantaneous node degree. The main advantage over the usual definition rests on modeling random transmission losses.

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ACKNOWLEDGMENT This work was done in the framework of the ARC IRAMUS of INRIA. R EFERENCES

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appears indeed challenging. This is an important issue for WSNs, because improving the connectivity means reducing the number of active nodes, thus increasing the life-time of WSNs. Basically, increasing the connectivity by using an opportunistic transmission over several paths is no difference from a distributed extension of the well-known diversity principle usually used in mobile communications to enhance individual radio links.

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(c) γ ¯r = 5 Prel = 35.45% (m=1) Prel = 42.49% (m=10) Fig. 5. Connectivity under block-fading, Nakagami m = 10 and m = 1 γ ¯r = 0,¯ γr = 3 and γ ¯r = 5 conditions. The connectivity and the non isolated node probabilities have been assessed by simulation. The theoretical no isolated node probability according to (2)

Furthermore, several parameters have been introduced in the model such as fading strength, modulation kind and frame size. In section III, we first defined a normalized node density bringing together noise levels, mean path-losses, transmission powers and node densities. Then, the trade-off between connectivity and link reliability was widely studied for fading channel. According to these theoretical analysis, we also observed that fading weakly reduces the mean node degree but spreads the neighbors over a wider area, making the links less reliable. In section IV, Simulation results have exhibited fading has a positive effect on connectivity and have validated the theoretical analysis. Indeed, for a same mean node degree value, connectivity in fading channels is closer to the no-nodeisolation probability. This work advocates long hops routing as discussed in [2] and opportunistic routing. These results indicate that developing protocols which are able to exploit unreliable links

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