Mean node degree in fading channels with opportunistic communications Ruifeng Zhang and Jean-Marie Gorce and Herv´e Parvery ARES INRIA & CITI, INSA-Lyon, F-69621 Villeurbanne Many works concerning WSNs focus on energy minimization. An usual approach consists in deactivating redundant nodes. A fundamental constraint is to preserve sensing coverage and connectivity. The connectivity is mainly studied with the perfect switched radio link model. Unreliable long-hops are thus not considered, the neighborhood of a node being selected from near nodes. we show in this paper that the connectivity could be greatly enhanced if all unreliable links were used. We quantify the increase connectivity in AWGN and block-fading channels, which enhances the potential of opportunistic routing in WSNs. Keywords: wireless sensor networks, connectivity, mean node degree, fading, Nakagami-m channel

1 Introduction Energy preserving is the most important feature to provide Wireless Sensor Networks (WSNs) with a long life time. Scheduling sleeping periods for some nodes has been proved efficient for this purpose. This approach however fails in practice if taking care of network connectivity is missing. This is the reason why connectivity of WSNs has been widely studied in the literature. In previous works, the radio links are often considered in an ideal manner, thanks to two fundamental assumptions : i) the switched link model and ii) the disc model. The former means that the radio link between two nodes A and B is a binary variable set to 1 if the path between them exists, otherwise set to 0. The later means that the radio link quality depends only on the distance between the emitter and the receiver. Under these assumptions the corresponding graph is a pure geometric graph. The effect of relaxing the disc model assumption has been studied in some recent works, showing how uncertainty may increase the connectivity. In these works however, the switched link assumption is still assumed. From a practical point of view, the switched link assumption can be easily achieved for stationary networks with an appropriate carrier sense mechanism, for instance : the power detection level can be set high enough such as only reliable links are detected. In this case, the PHY layer provides itself a switched link like system. If the power detection threshold is not high enough, the MAC layer can compensate for by itself implementing a threshold function. The data link or the network layers can also be exploited to achieve a switched link model by implementing algorithms selecting only reliable links over time. Thus, an efficient cross-layer policy can be devoted to make the WSN a truly switched link network. Unfortunately, the price to be paid for assessing this comfortable model may be very high in terms of energy consumption when compared to an approach that would be able to exploit all possible links, even unreliable. Such an approach may be said opportunistic since all links are used when they are available, even for a short time. We study in this work how the connectivity can be improved thanks to unreliable long-hops. We first give a short outline of the network model. We then propose an exact formulation of the mean node degree in fading channels.

2 Network model 2.1 Graph model As described in [Bet04, BH05], the nodes are assumed independently and randomly distributed, according to a random point process of density ρ , over the space R2 . The number of nodes N in a subarea A

Ruifeng Zhang and Jean-Marie Gorce and Herv´e Parvery (surface SA ) then follows a poisson distribution with E[N] = ρ · SA . Boundaries are disregarded by considering a circular sub-area A in an infinite plan [Bet04]. A WSN is modeled as a graph G(L,N) having N vertexes (nodes) and a set of edges L(ni , n j ) (radio links). With switched links, edges are binary variables and the graph is called a pure geometric random graph. In the case of opportunistic communications under unreliable links, edges are probabilistic. The general graph is complete, each pair of nodes having a given probability to communicate.† . Each time a packet is transmitted across the WSN, a new peculiar realization of the probabilistic graph is experienced, i.e. g(L, N, τ ). Such a realization is itself a binary graph but not further a perfect geometric random graph.

2.2 Connectivity The connectivity is defined in [Bet04] as the probability, Pcon (Γ), that a graph G(L, N) ∈ Γ associated with a WSN randomly distributed on an area A is connected. For a probabilistic graph, the connectivity of G(L, N) is itself a probability, defined as the probability that the network is connected at a given time, i.e. the probability that a realization g(L, N, τ ) is connected. The connectivity probability Pcon (Γ) is thus : Pcon (Γ) = hPcon (G(L, N))i, where h.i stands for expectation. Connectivity is closely related to the mean node degree [Bet04]. The connectivity is firstly bounded by the no node isolation probability according to : P(con) ≤ P(no node iso). Secondly, the probability that no nodes are isolated in a subarea A is derived as P(no node iso) = exp (−ρ · SA · P(iso)), where P(iso) is the isolation probability relying on the mean node degree µ0 as P(iso) = exp(−µ0 ). True for small P(iso) and large nodes number Nnd , the fundamental property is then : ¡ ¢ P(con) ≤ exp −ρ · SA · e−µ0 . (1)

3 Mean node degree 3.1 Analytic formulation The mean node degree under unreliable links differs from the case of reliable links, because for a probabilistic graph, the neighborhood differs at each transmission. The instantaneous degree of a given node x is defined as the number of simultaneous successful transmissions µ0 (x, τ ). The instantaneous mean node degree of the network is the expected value over A, i.e. µ0 (τ ) = hµ0 (x, τ )ix . The mean node degree is the average value over timeµ¯ 0 = hµ0 (τ )iτ = hµ0 (x, τ )ix,τ . Under the disc model assumption, the radio link relies on the emitter-receiver distance r only. If the nodes are uniformly and spatially distributed, independent, and if the receiver is randomly chosen over A having (SA = π · R2lim ), the mean node degree is given by

µ¯ 0 = 2π · ρ

Z Rlim r=0

PS (tr/γ¯(r)) · r · dr,

(2)

where PS (tr/γ¯(r)) is the transmission success probability at a given mean SNR γ¯(r). A more tractable formulation is obtained when considering the mean SNR γ¯(r) as a log-slope function of distance : γ¯ = γ¯0 · (r/r0 )−α . Without loss of generality, r0 is set to 1 and a normalized density n¯ 0 is defined as the mean nodes number in the unit SNR disk (r1 = rγ¯=1 ), noting that r1 relies on the path-loss (α ), the receiver noise N0 , and the transmission power. After an appropriate variable change, (2) leads for an infinite area A to : 2 · n¯ 0 · Jα ,k (γ¯lim ), γ¯lim →0 α

µ¯ 0 (0) = lim

with

Jα ,k (γ¯lim ) =

Z ∞

γ¯lim

2

γ¯−(1+ α ) · PS (tr/γ¯) · d γ¯.

(3)

The mean node degree only depends on the normalized node distribution and the attenuation coefficient α . µ¯ 0 (0) is the bound value for opportunistic communications when all nodes may contribute to the reception of a packet. In a non-opportunistic approach, neighbors are selected among those having reliable links, e.g. thanks to an appropriate threshold power detection. The mean node degree comes down to :

µ¯ 0 (γ¯th ) = †

2 · n¯ 0 · Jα ,k (γ¯th ) α

(4)

For large distances a link probability tends toward 0, but at the same time the number of nodes tends toward infinity for infinite WSNs. The trust in a null probability for a long distance single-hop transmission should be therefore questionable in an opportunistic framework

Mean node degree with fading channels

3.2 Radio link model The instantaneous radio link relies directly on the bit error rate (BER)which may be written as BER(γ ) = Q(k · γ ) in Additive White Gaussian Noise (AWGN) with k = 1 for BPSK [WG03]. The success probability of a frame relies on the instantaneous SNR γ , which is assumed constant during the frame : PS (tr/γ )

= (1 − BER(γ ))Nb ∼ 1 − Nb · BER(γ ); for a high SNR

(5) (6)

Then, in block-fading channels, the averaged frame error rate (FER) relies on SNR variations according to Z

PS (tr/γ¯) =

4 4.1

γ

PS (tr/γ ) · f (γ /γ¯) · d γ

(7)

Results AWGN channels

Computing the mean node degree from (3) in AWGN relies on computing Jα ,k (γ¯lim ) thanks to (5). This can be easily achieved numerically. A closed-form is also derived under the high SNR approximation from (1) N =1 (6). Firstly, (3) is solved exactly for single bit frames providing µ¯ 0 = µ¯ 0 b after cumbersome computations as : µ ¶ (1) ´ ³ √ α −4 −γ p µ¯ 0 α − α2 2/α 2α th √ = γth · 1 − 0.5 · er f c( k · γth ) + kγth e − k (Γ(ξ ) + Γinc (ξ , k · γth )) (8) n¯ 0 (4 − α ) π with ξ = 3α2α−4 , and Γ and Γinc the well-known complete and incomplete gamma functions. Then, the approximated frame-based mean node degree in AWGN is obtained as (N ) (1) −2/α µ¯ 0 b ∼ Nb · µ¯ 0 − (Nb − 1) · n¯ · γth

(9)

The accuracy of this approximation is illustrated in Fig.1 providing both exact and approximated curves. The analytical approximation according to (9) is good for a success probability higher than 0.35. The exact numerical curve seems to reach a ceil albeit this function diverges for a very large distance (not seen in this figure). Indeed, PS (tr/γ ) relying on BER converges to limγ →0 Ps (tr/γ ) = 2−Nb for low SNR (random reception), corresponding to a rare but theoretically possible success. This is in practice a totaly random and unreliable reception : the node has no mean to know which packet is good ! On the opposite, the connectivity achieved at the ceil value can be exploited when combined with an error detection process. Figure 1 also illustrates that a standard power threshold granting a given success probability (e.g. 90% ) would lead to a sub-optimal mean node degree (t1 reference point in the figure) : in this example, the corresponding connectivity is aboout µ¯ 0 = 0.15 · n¯ 0 . Unreliable communications allow to increase the connectivity up to 30% (µ¯ 0 = 0.2 · n¯ 0 ). This may increase significantly the WSN life-time. But the counterpart is that some links are unreliable and then only opportunistic protocols can make a noticeable profit from these unreliable long hops.

4.2

block-fading channels

The Nakagami-m distribution is used in literature for modeling fading channels allowing to model various fading conditions from AWGN (m → ∞) to Rayleigh (m = 1). The mean node degree depends on Jα ,k , which is now derived as : Jα ,k (γ¯lim , γth ) =

Z ∞ γ¯lim

2

γ¯−(1+ α ) ·

Z ∞ γth

PS (tr/γ ) · f (γ /γ¯) · d γ · d γ¯.

(10)

where γth stands for the instantaneous power detection threshold. γ¯lim can be interpreted either as the mean SNR corresponding to the radius of A or as the mean power detection threshold (if defined at the MAC

Ruifeng Zhang and Jean-Marie Gorce and Herv´e Parvery

F IG . 1: Mean node degree in AWGN (BPSK, Nb = 1000, α = 2). Upper frame : link success probability. Lower frame : µ¯ 0 as a function of rth /r1 in exact form (dashed), approx., see (9) (black) and corresponding to the switched link model (grey).

F IG . 2: Mean node degree in Nakagami-m channel. Upper frame : link success probability. Lower frame : µ¯ 0 as a function of rlim /r1 . Exact numerical computation without (plain) and with (dashed) optimal power detection threshold. Approximated with (11) (dot).

layer). Introducing the Nakagami-m pdf and after cumbersome computations, the mean node degree is found to be : (m,α ) (∞,α ) (11) µNb = Closs (m, α ) · µNb (∞,α )

where µNb

is the mean node degree in AWGN channel given by (9), and Closs (m, α ) is a connectivity loss 2

coefficient equals to Closs (m, α ) = m− α · Γ(m + α2 ) · Γ(m)−1 . A noticeable result is that for α = 2 one have Closs = 1. This means that the mean node degree is not influenced by fading, in this case : the packet loss in the near range is compensated for by receiving packets, by chance, at a far distance. To give a better insight of how fading acts the curves of Fig.1 are extended to Nakagami-m channels. But while the plots of Fig.1 are function of the relative power detection range rth /r1 , this cannot further be made in block-fading channels because a power detection threshold γth doesn’t further corresponds to a fixed range rth , due to power variations. A mean range rlim can be complementarily defined relying on the mean received power γ¯lim . Fig.2 shows the evolution of µ¯ 0 as a function of rlim /r1 . Plain curves are obtained with the exact formulation of the success probability Ps (tr/γ ) given in (5) , with an infinite detection range (i.e. γth = 0). Whatever m, the relative mean node degree reaches an identical ceil value. The only difference is that for high m values the ceil is reached further : nodes having a significant contribution to the connectivity are spread over. Dashed lines are obtained with a power detection threshold fixed at γ˜th = 0.45 : the connectivity loss is very low and due to unreliable long hop transmissions only. Finally, dotted curves are obtained by approximating Ps (tr/γ ) according to (6) and then using the closed-form expression of (11). Note that this approximation appears valid only because γth ≥ γ˜th . Finally, even with a same mean node degree, the fading parameter m appears greatly influencing the graph connectivity. Upper frame in Fig.2 emphasizes how the connected nodes are spread over the network while m decreases. Fig.2 also exhibits how is it difficult to work with reliable links in fading channels. The links with a reliability of 90% are brought together around the emitter (r/r1 < 0.15 for m = 1). The overhead of using reliable links appears therefore very high.

´ erences ´ Ref [Bet04] Christian Bettstetter. On the connectivity of ad hoc networks. The Computer journal, 47(4) :432– 447, 2004. [BH05] Christian Bettstetter and Christian Hartmann. Connectivity of wireless multihop networks in a shadow fading environment. Wireless Networks, 11 :571–579, 2005. [WG03] Zhengdao Wang and Georgios Giannakis. A simple and general parametrization quantifying performance in fading channels. IEEE trans on Communications, 51(8) :1389–1398, August 2003.