Connectivity of Wireless Sensor Networks with Unreliable Links Ruifeng Zhang

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.

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)) , , 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  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  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 ,  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 . 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 . 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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Fig. 4. Normalized mean node degree in Nakagami channels as function of the reliability range dr /d1

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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 . 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

1 Pnoiso Theory Pcon Nakagami m=1 Pcon Nakagami m=10 Pnoiso Nakagami m=1 Pnoiso Nakagami m=10

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(a) γ ¯r = 0 Prel = 0% (m=1) Prel = 0% (m=10) 1 Pnoiso Theory Pcon Nakagami m=1 Pcon Nakagami m=10 Pnoiso Nakagami m=1 Pnoiso Nakagami m=10

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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)

## Connectivity of Wireless Sensor Networks with ...

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