The Number of Isolated Nodes in a Wireless Network with a Generic Probabilistic Channel Model

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(1)IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 595. PAPER. The Number of Isolated Nodes in a Wireless Network with a Generic Probabilistic Channel Model Chao-Min SU†,†† , Nonmember, Chih-Wei YI†a) , Member, and Peng-Jun WAN††† , Nonmember. SUMMARY A wireless node is called isolated if it has no links to other nodes. The number of isolated nodes in a wireless network is an important connectivity index. However, most previous works on analytically determining the number of isolated nodes were not based on practical channel models. In this work, we study this problem using a generic probabilistic channel model that can capture the behaviors of the most widely used channel models, including the disk graph model, the Bernoulli link model, the Gaussian white noise model, the Rayleigh fading model, and the Nakagami fading model. We derive the expected number of isolated nodes and further prove that their distribution asymptotically follows a Poisson distribution. We also conjecture that the nonexistence of isolated nodes asymptotically implies the connectivity of the network, and that the probability of connectivity follows the Gumbel function. key words: connectivity, isolated nodes, multihop wireless networks, wireless channel models. 1.. Introduction. Wireless ad hoc and sensor networks dispense with the need for fixed infrastructures, such as cables or base stations, which means they can be flexibly deployed with minimal cost for various tasks. However, without the aid of infrastructures, connectivity is a major concern [1]–[4]. In the past, many theoretical studies on network connectivity were based on disk graph models, in which two nodes have a link if and only if the distance between them is no more than their transmission radii [5]; even so, the disk graph models were criticized for oversimplifying wireless channels. In this work, using a generic probabilistic channel model, we study the connectivity problem by investigating the number of isolated nodes in a randomly deployed wireless network. In many applications of multihop wireless networks, e.g., wireless sensor networks and mobile ad hoc networks, wireless nodes are distributed in a random manner, so it is natural to represent wireless nodes by a set of random points. A good tutorial on modeling wireless networks and techniques is presented [6]. Nodes are called isolated if they do not have links to other nodes. The nonexistence of isolated nodes is a prerequisite for connectivity. In [7], it was proved that if n nodes with transmission raManuscript received April 10, 2012. Manuscript revised October 4, 2012. † The authors are with the Department of Computer Science, National Chiao Tung University, Hsinchu City 30010, Taiwan. †† The author is also with Ta Hwa University of Science and Technology, Hsinchu County 30740, Taiwan. ††† The author is with the Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA. a) E-mail: [email protected] DOI: 10.1587/transcom.E96.B.595. dius (ln n + ξ) /πn, in which ξ is a tunable parameter that remains constant during the analysis, are independently distributed over a unit-area square, then the network asymptoti cally has no isolated nodes with probability exp −e−ξ . Furthermore, in [8], it was proved that the network is asymptotically connected if ξ → ∞ and asymptotically disconnected if ξ → −∞. However, neither node failures nor link failures were considered in these works. In [9], assuming that each node has the same probability to be active independently, p1 , and that each link has the same probability to be up independently, p2 , the authors proved that the total number of isolated active nodes asymptotically follows a Poisson dis−ξ tribution with mean p1 e , if the transmission radius is given by (ln n + ξ) /πp1 p2 n. In [10]–[12], the impact on connectivity due to link failures caused by shadowing or fading was investigated. An analytical procedure for the computation of the node isolation probability and coverage in an ad hoc network in the presence of channel randomness was presented [13]. In [14], the node isolation probability under a specific channel model, the Nakagami fading model, was derived. However, the majority of previous analytical works were not based on practical channel models. In this paper, our study is based on a generic probabilistic model that can capture the behaviors of the most widely used models, including the disk graph model, the Bernoulli link model, the Gaussian white noise model, the Rayleigh fading model, and the Nakagami fading model. In this generic probabilistic model, we let f (r) be the probability of the event that two nodes have a link if they are separated by a distance of r. It is realistic to assume that f (r) is a decreasing function and that there are two constants 0 ≤ R1 ≤ R2 < ∞ such that f (r) = 1 for 0 ≤ r ≤ R1 and f (r) = 0 for r ≥ R2 . The values of R1 , R2 , and f (r) are set to define the generic model. The nonexistence of isolated nodes is a prerequisite for network connectivity and is also a good indicator for it. We will first derive the expected number of isolated nodes. In addition, we will further prove that the probability distribution of the number of isolated nodes asymptotically follows a Poisson distribution. The equations given in this work will allow users to control the expected number of isolated nodes by tuning the node density or even the transmission power. Thus, the desired level of connectivity can be achieved. These results can be used to obtain the connectivity features of the network. In addition, they are of practical value for researchers or designers in this field. We run exten-. c 2013 The Institute of Electronics, Information and Communication Engineers Copyright .

(2) IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 596. sive simulations to verify our probabilistic analysis. Moreover, we conjecture that the nonexistence of isolated nodes asymptotically implies connectivity. This conjecture is also verified through simulations. A similar work focused on the connectivity of wireless networks with arbitrary wireless channel models [15]. This was built upon the wireless model, in which a total of n nodes are randomly, independently, and uniformly distributed in a unit square, and each node has uniform transmission power. The authors proved that the probability of having a connected network and the probability of having no isolated nodes asymptotically converges to the same value as n tends to infinity. In contrast, our work uses a different network model, which is not scaling. Furthermore, besides the node isolation probability, we also investigate the distribution of the number of isolated nodes. The rest of this paper is organized as follows. In Sect. 2, we give our main results. In Sect. 3, lemmas needed to prove the main results are given. In Sect. 4, we derive the expected number of isolated nodes. In Sect. 5, we give the asymptotic distribution of the number of isolated nodes. In Sect. 6, the channel models used in the simulations are introduced, and simulation results are given to verify our theoretical theorems. We summarize the paper in Sect. 7. 2.. Main Results. The nonexistence of isolated nodes is a precondition for network connectivity. The probability of nonexistence of these isolated nodes is considered as a tight upper bound for the probability of connectivity [16]. It was proved that if there are no isolated nodes in a random geometric graph, then the graph is asymptotically almost surely connected. In this paper, we assume that wireless nodes are represented by a Poisson point process over a deployment region D = [0, l]2 with density λ (l). For convenience, we will suppress the parameter l in λ (l) from this point on. To avoid tedious arguments on boundary effects and to simplify the calculations, we will apply the torus convention described, for example, in [17]–[19]. Hence, instead of the Euclidean distance, the toroidal distance [20], [21] is applied. The toroidal distance between nodes u and v is denoted by d (u, v). In addition, scaling disk models, like that used in [8] and [9], are not adopted. Instead, the transmission radius of the nodes is independent of the number of nodes, and the node density is a function of the deployment region. As stated before, f (r) is the probability of the event that two nodes have a link if they are separated by r. Let a = ∞ 2 2 f (r) 2πrdr, λ = ln l + ln ln l + ξ /a, and ω = ξ + ln a. 0 Here, ξ is a tunable parameter that remains constant during the analysis and λa = ln l2 +ln ln l2 +ξ is the expected number of nodes with which a node has links. Our first result gives the expected total number of isolated nodes in the network. Theorem 1: The expected total number of isolated nodes in the network is e−ω . Based on Theorem 1, the expected number of isolated. nodes can be controlled by tuning the parameter ω that depends only on ξ and a. The ratio of a to πR22 is the conditional probability that two nodes have a link between them if they are within the distance R2 . Specifically, if f (r) = 1 for r ∈ [0, R2 ], the ratio is equal to 1. In other words, any two nodes within a distance R2 always have a link, and this is the traditional r-disk graph. Different channel models have different settings for R1 , R2 , and f (r). Accordingly, ∞ a = 0 f (r) 2πrdr also differs. This shows variation in the expected number of isolated nodes. The greater the value of a, the lesser the expected number of isolated nodes. In some applications, it is tolerable to have a certain percentage of nodes isolated. With the knowledge of the expected number of isolated nodes, this can be achieved by choosing a proper ξ. We further give the probability distribution of the number of isolated nodes in the following theorem. Theorem 2: The total number of isolated nodes is asymptotically Poisson with mean e−ω . According to Theorem 2, the probability of the event that there are k isolated nodes in the network is asymptoti−kω cally equal to e /k! exp (−e−ω ). Specifically, the event of the nonexistence of isolated nodes is asymptotic with probability exp (−e−ω ), and is called the Gumbel function. This probability is an upper bound for the probability of connectivity. In addition, we conjecture that, even considering link failures, a network without isolated nodes is almost surely connected. If this conjecture is true, then (1) Pr (the network is connected) ∼ exp −e−ω . This conjecture will be verified through simulations in Sect. 6. 3.. Preliminaries. In this section, lemmas for proving the above theorems are given. The proofs of these lemmas are given in the Appendix. Let Gr (x1 , · · · , xk ) denote the r-disk graph over x1 , · · · , xk in which there is an edge between two nodes if and only if their distance is at most r. For any positive integers k and m with 1 ≤ m ≤ k, let Ckm denote the set of (x1 , · · · , xk ) ∈ Dk satisfying the condition that G2R2 (x1 , · · · , xk ) has exactly m connected components. The space Dk can be partitioned into Ck1 , Ck2 · · · , Ckk . We use X1 , X2 , · · · and X1 , X2 , · · · to denote independently and uniformly distributed random points in D. Let. Pλ = X1 , X2 , · · · , XPo(λ|D|) denote a Poisson point process with density λ over D. where Po (·) is the Poisson random variable, and Xk = X1 , X2 , · · · , Xk denotes a random kpoint process over D. Specifically, Xk is independent of Pλ . Let Bi for i = 1, 2, · · · denote the event that the node Xi is isolated in the network over Pλ , and let B i for i = 1, · · · , k denote the event that Xi is isolated in the network over ∞ Xk ∪ Pλ . Recall that ξ is a constant, a = 0 f (r) 2πrdr, λ = ln l2 + ln ln l2 + ξ /a, and ω = ξ + ln a. Thus, we have the following lemmas..

(3) SU et al.: THE NUMBER OF ISOLATED NODES IN A WIRELESS NETWORK WITH A GENERIC PROBABILISTIC CHANNEL MODEL. 597. Lemma 3: For any integer k ≥ 2 and (x1 , x2 , · · · , xk ) ∈ Ckk , ⎛ ⎞ X1 = x1 ⎟⎟ ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜ .. ⎟⎟⎟ = e−kλa . Pr ⎜⎜⎜⎜ B i (2) . ⎟⎟⎠ ⎜⎝. i=1 Xk = xk Lemma 4: For any positive integer k, ⎛ k ⎞ ⎜⎜ ⎟⎟⎟ k k −kλa ⎜ ⎜⎜⎝⎜ λ e dxi ⎟⎟⎠⎟ ∼ e−ω . (x )∈C 1 ,··· ,xk. kk. (3). i=1. Lemma 5: For any two integers k ≥ 2 and 1 ≤ m ≤ k − 1, there is a positive constant c such that, for any (x1 , · · · , xk ) ∈ Ckm , ⎛ ⎞ X1 = x1 ⎟⎟ ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ ≤ e−(m+c)λa . (4) Pr ⎜⎜⎜ B i . ⎟⎟⎠ ⎜⎝ i=1 Xk = xk Lemma 6: For any two integers k ≥ 2 and 1 ≤ m ≤ k − 1, ⎛ k ⎞ ⎜⎜ ⎟⎟⎟ k −(m+c)λa ⎜ ⎜ ⎜⎜⎝ λ e dxi ⎟⎟⎟⎠ = o (1) . (5) (x )∈C 1 ,x2 ,··· ,xk. km. i=1. We write g (λ) = o (h (λ)) if limλ→∞ g (λ) /h (λ) = 0. Generally speaking, Lemmas 3 and 4 are for estimating the probability of the existence of k widely spaced isolated nodes, and Lemmas 5 and 6 are for estimating the probability of the existence of k isolated nodes among which some are close to others. The density λ and a together will affect the probability of the isolated nodes. 4.. To calculate Pr (X is isolated | X = x), we apply the partition technique of the Riemann integral. The disk B (x, R2 ) is divided into k annuli by k concentric circles with centers at x and radii r1 < r2 < · · · < rk = R2 , respectively. For convenience, let r0 = 0. For 1 ≤ i ≤ k, the annulus with radii ri−1 and ri is called the ith annulus. Let

(4) ri = ri − ri−1 . The area of the ith annulus can be approximated by 2πri

(5) ri . If a node is in the ith annulus, the event that X has a link to that node is approximately the probability f (ri ). Let Ni denote the number of nodes in the ith annulus. Then, X does not have links with Pr X =x nodes in the ith annulus ∞ All links between X and = Pr N = j nodes in the ith annulus fail i j=0 · Pr Ni = j | X = x ∞ j j (λ2πri

(6) ri ) −λ2πri

(7) ri (1 − f (ri )) e = j! j=0 = e− f (ri )λ2πri

(8) ri .. The Expected Number of Isolated Nodes. Briefly, the expected number of isolated nodes equals the total number of nodes in the network multiplied by the probability of a typical node being isolated. To derive the expected number of isolated nodes, we use Palm theory [22]. Let X be a point randomly located in D and independent of Pλ , and let B (x, r) denote a disk with center x and radius r. Let h (Y, X) for Y ⊆ X be the indicator function such that h (Y, X) = 1 if # (Y) = 1, where # (·) is the cardinality function, and the node in Y is isolated in X; otherwise, h (Y, X) = 0. Then, we have The expected number of isolated nodes in Pλ = Pr (X is isolated). The introduction of the random point X in Eq. (10) is to ensure that there is at least one node in D. The equality from Eqs. (9) to (10) is based on Palm theory, and λl2 is the expected total number of nodes in the network. The probability Pr (X is isolated in {X } ∪ Pλ ) can be given by Pr X is isolated in X ∪ Pλ = Pr X is isolated | X = x Pr X = x dx x∈D 1 = 2 Pr X is isolated | X = x dx. (12) l x∈D. (6) (7). (13). Therefore, Pr X is isolated X = x ⎛ ⎞ For all 1 ≤ i ≤ k, ⎜⎜⎜ ⎟⎟⎟ ⎜. = Pr ⎜⎜⎜⎝ X does not have links with X = x⎟⎟⎟⎟⎠ nodes in the ith annulus k X does not have links with Pr X =x = lim nodes in the ith annulus k→∞ i=1. = lim. k . k→∞. =e. −λ. e− f (ri )λ2πri

(9) ri = lim e−λ. i=1 R2 f (r)2πrdr 0. !k i=1. f (ri )2πri

(10) ri. k→∞. = e−λa .. (14). {X}⊆Pλ. =. . E [h ({X} , Pλ )]. (8). {X}⊆Pλ. ⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢ = E ⎢⎢⎢⎣ h ({X} , Pλ )⎥⎥⎥⎦ {X}⊆Pλ 2 = λl E h X , X ∪ Pλ = λl2 Pr X is isolated in X ∪ Pλ .. (9). Putting Eqs. (11), (12), and (14) together, we have The expected number of isolated nodes in Pλ =λ e−λa dx ∼ e−ω .. (15). x∈D. (10) (11). The last equality holds due to Lemma 4. Thus, Theorem 1 is proved..

(11) IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 598. 5.. ⎞ ⎛ X1 = x1 ⎟⎟ ⎜⎜⎜ k ⎟⎟⎟ ⎜ ⎜ .. ⎟⎟⎟ Pr ⎜⎜⎜⎜ B i = λl2 . ⎟⎟⎠ ⎝⎜ i=1 (x1 ,x2 ,··· ,xk )∈Dk Xk = xk ⎞ ⎛ ⎞ ⎜⎜⎜ X1 = x1 ⎟⎟⎟ ⎛⎜ ⎟⎟⎟ ⎟⎟ ⎜⎜⎜ k ⎜⎜⎜ ⎟ . .. ⎟⎟⎟ ⎜⎜⎝ dxi ⎟⎟⎟⎠ · Pr ⎜⎜⎜ ⎟⎠ ⎜⎝ i=1 Xk = xk ⎛ ⎞ ⎞ X1 = x1 ⎟⎟ ⎛ k ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎜⎜ . k ⎜ ⎟ ⎜ .. ⎟⎟⎟ ⎜⎜⎝ Pr ⎜⎜ Bi dxi ⎟⎟⎟⎠ =λ ⎟⎠ ⎝⎜ i=1 (x1 ,x2 ,··· ,xk )∈Dk i=1 Xk = xk ⎛ ⎞ X1 = x1 ⎟⎟ ⎜⎜⎜ k k ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ ... λk Pr ⎜⎜⎜⎜ B i = ⎟⎟⎠ ⎜⎝ (x )∈C ,x ,··· ,x 1 2 k ki i=1 i=1 Xk = xk ⎞ ⎛ k ⎜⎜⎜ ⎟⎟⎟ ⎜ dxi ⎟⎟⎟⎠ . (21) · ⎜⎜⎝ k . Asymptotic Distribution of the Number of Isolated Nodes. To prove Theorem 2, we apply Brun’s sieve in the form, for example, used in [23]. This is used to derive the asymptotic distribution of the number of isolated nodes. For completeness, we give Brun’s sieve here. Theorem 7 (Brun’s Sieve): Assume m (n) is a non-negative random integer variable. Let B1 , · · · , Bm(n) be events and let Y be the number of Bi that hold, and let S ( j) = Pr Bi1 ∧ · · · ∧ Bi j . (16) {i1 ,··· ,i j }⊆{1,··· ,m(n)} Suppose there is a constant μ such that for every fixed j, " # 1 E S ( j) ∼ μ j . j!. (17). Then Y is also asymptotically Poisson with mean μ. Let Y be the number of Bi events that hold. In other words, Y is the total number of isolated nodes. To prove Theorem 2 by applying Brun’s sieve, we need to show that for every fixed k, ⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥ ⎢ ⎥ Pr Bi1 ∧ · · · ∧ Bik ⎥⎥⎥⎥ E ⎢⎢⎢⎢ ⎣ ⎦ {i1 ,··· ,ik }⊆{1,··· ,Po(λl2 )} 1 −ω k . (18) e ∼ k! Again, by applying Palm theory to prove Eq. (18), we have ⎡ ⎤ ⎢⎢⎢ ⎥ ⎥⎥⎥⎥ ⎢⎢⎢ E ⎢⎢ Pr Bi1 ∧ · · · ∧ Bik ⎥⎥⎥ ⎣ ⎦ {i1 ,··· ,ik }⊆{1,··· ,Po(λl2 )} k λl2 Pr B 1 ∧ · · · ∧ B k . (19) = k! Comparing Eqs. (19) with (18), we can see that if k λl2 Pr B 1 ∧ · · · ∧ B k ∼ (e−ω )k for all k = 1, 2, · · · then the proof is complete. The case of k = 1 was proved in the previous section. Thus, here we only need to prove that, for any integer k ≥ 2, k λl2 Pr B 1 ∧ · · · ∧ B k ∼ e−ω k .. (20). As described in Sect. 3, for any positive integers k and m with 1 ≤ m ≤ k, Ckm denotes the set of (x1 , · · · , xk ) ∈ Dk satisfying the condition that G2R2 (x1 , · · · , xk ) has exactly m connected components and the space Dk can be partitioned into Ck1 , Ck2 · · · , Ckk . Therefore, we use a divide and conquer strategy to prove this statement. We have . k λl2 Pr B 1 ∧ · · · ∧ B k. i=1. For the integral over Ckm with 1 ≤ m ≤ k − 1, according to Lemmas 5 and 6, we have ⎛ ⎞ ⎞ X1 = x1 ⎟⎟ ⎛ k ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎜⎜⎜ . k ⎟ ⎜ .. ⎟⎟⎟ ⎜⎜⎝ Pr ⎜⎜ Bi dxi ⎟⎟⎟⎠ λ ⎟⎠ ⎝⎜ i=1 (x1 ,x2 ,··· ,xk )∈Ckm i=1 Xk = xk ⎛ ⎞ k ⎜⎜⎜ ⎟⎟⎟ e−(m+c)λa ⎜⎜⎜⎝ dxi ⎟⎟⎟⎠ = o (1) . (22) ≤ λk (x1 ,x2 ,··· ,xk )∈Ckm. i=1. For the integral over Ckk , according to Lemmas 3 and 4, we have ⎛ ⎞ ⎞ X1 = x1 ⎟⎟ ⎛ k ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎜⎜⎜ . k ⎟ ⎜ .. ⎟⎟⎟ ⎜⎝⎜ Pr ⎜⎜ Bi dxi ⎟⎟⎠⎟ λ ⎜⎝ ⎟⎠ (x1 ,x2 ,··· ,xk )∈Ckk i=1 i=1 Xk = xk ⎛ ⎞ k ⎜⎜⎜ ⎟⎟⎟ e−kλa ⎜⎜⎝⎜ dxi ⎟⎟⎠⎟ ∼ e−ω k . (23) = λk (x )∈C 1 ,x2 ,··· ,xk. kk. i=1. If we combine Eqs. (21), (22), and (23) we have k λl2 Pr B 1 ∧ · · · ∧ B k ∼ e−ω k . 6.. (24). Simulation Results. The generic probabilistic channel model used in this work is a generalization of many widely used channel models. Channel models can be described by setting R1 , R2 , and f (r). In this section, we begin with a brief introduction to several well-known channel models: the log-distance path loss model, the Bernoulli link model, the Gaussian white noise model, the Rayleigh fading model, and the Nakagami fading model. Extensive simulation results are given to verify our theoretical results. For the sake of succinctness, similar figures for different channel models will not be included. 6.1 Channel Models and Simulation Parameters Let S ref (dBm) be the signal strength measured at a reference.

(12) SU et al.: THE NUMBER OF ISOLATED NODES IN A WIRELESS NETWORK WITH A GENERIC PROBABILISTIC CHANNEL MODEL. 599. distance d0 from the transmitter. Based on the log-distance path loss model [24], the signal strength received at a distance d from the transmitter will be S ref − 10α log (d/d0 ) (dBm) where α is the path loss exponent, whose value is between 2 and 6 depending on the environment. Let S thr (dBm) be the minimum received signal strength for decoding a signal. Hence, the transmission radius denoted as R can be derived from the equation S thr = S ref − 10α log. R . d0. (25). In the simulation, we assume that d0 = 1, S ref = 35 dBm, S thr = 15 dBm, and α = 2. Thus, for the log-distance path loss model, we have R1 = R2 = 10. We should note that f (r) is not needed here. In a realistic system, links may be down due to the environment or barriers between nodes. To characterize the uncertainty of the existence of links, the Bernoulli link model assumes that two nodes within each other’s transmission range may have a link with probability p, depending on the specific propagation environment. In other words, R1 = 0 and f (r) = p. Let RB denote the R2 in the Bernoulli link model. To achieve a fair comparison, the average node degree should be kept the same. The degree of a node, or ’node degree’, is the number of connections it has to other nodes. A node of degree 0 is isolated. Thus, we have πR2 = pπR2B . In the simulation, we set p = 0.8, which can be set according to the environment, and therefore, R1 = 0, R2 = 11.2, and f (r) = 0.8. To model background noises, the Gaussian white noise model assumes that the signal strength received at a distance r from the transmitter is given by S ref − 10α log (r/d0 ) − N, where N is a log-normal random variable with mean μ = 0 and standard deviation σ. Therefore, r f (r) = Pr N ≤ S ref − S thr − 10α log d0 ⎛ ⎛ r ⎞⎞ ⎜⎜⎜ S ref − S thr − 10α log d0 ⎟⎟⎟⎟⎟⎟ 1 ⎜⎜⎜ ⎟⎠⎟⎠ , = ⎜⎝1 + erf ⎜⎝ (26) √ 2 σ 2 where erf (·) is the error function. Let RG denote the R2 for the Gaussian white noise model. To have the same mean node degree, RG is given by πR2 = 0. RG. ⎛ ⎛ r ⎞⎞ ⎜ S ref −S thr −10α log d0 ⎟⎟⎟⎟⎟⎟ 1 ⎜⎜⎜ ⎟⎠⎟⎠ 2πrdr. ⎜⎝1+erf ⎜⎜⎜⎝ √ 2 σ 2 (27). In the simulation, we set σ = 8, and therefore, f (r) = √ 1 2 , R1 = 0, and R2 = 13.186. 1 + erf 5 − 5 log r / 2 2 In the Rayleigh fading model [25], the amplitude of a signal will vary according to the Rayleigh distribution. The cumulative distribution function (CDF) of the Rayleigh distribution with standard deviation σ is x2 R (x) = 1 − exp − 2 , (28) 2σ. where 2σ2 = 10(S ref −10α log(r/d0 ))/10 is the average received signal power, and for successful reception, the received signal power x2 must be at least 10S thr /10 . Therefore, ⎞ ⎛ $ % S ref −S thr −10α log r ⎟ ⎜⎜⎜ d0 ⎟ S thr ⎟⎟⎟ 10 f (r) = Pr x2 ≥ 10 10 = exp ⎜⎜⎝⎜−10− ⎠⎟ α S ref −S thr r 10− 10 . (29) = exp − d0 Let RR denote the R2 for the Rayleigh fading model. To have the same mean node degree, RR is given by RR α S ref −S thr r 2 πR = exp − 10− 10 2πrdr. (30) d0 0 In the simulation, f (r) = exp −r2 /100 , R1 = 0, and R2 = 38.1 for the Rayleigh fading model. The Nakagami fading model is described by two parameters: μ is the shape parameter denoting the severity of fading, and ω is the scale parameter equal to the average received power. We have $ % S thr f (r) = Pr received power ≥ 10 10 S thr ω = 1 − G 10 10 ; μ, , (31) μ where G is the CDF of the gamma distribution and ω = 10(S ref −10α log(r/d0 ))/10 . Let RN denote the R2 for the Nakagami fading model. To have the same mean node degree, RN must satisfy . RN. πR2 = 0. ⎛ ⎛ S ref−10α log r ⎜⎜⎜ ⎜⎜⎜ S10thr d0 1 ⎜⎜⎜1−G ⎜⎜⎜10 ; μ, 10 10 ⎝ ⎝ μ. ⎞⎞ ⎟⎟⎟⎟⎟⎟ ⎟⎟⎟⎠⎟⎟⎟⎠ 2πrdr. (32). In we set μ = 2, and therefore, f (r) = 2 the simulation, r2 /50 + 1 e−r /50 , R1 = 0, and R2 = 28.3. 6.2 Network Snapshots A schematic of a wireless network with the Gaussian white noise model is depicted in Fig. 1. In the network, nodes are generated by a Poisson point process with mean density λ = 0.02 in a square region D = [0, 100]2 . The solid lines represent the links between the nodes under the Euclidean metric; and the dotted lines denote additional links between nodes under the toroidal metric that connects pairs of nodes lying on opposite sides. 6.3 Expected Number of Isolated Nodes To verify Theorem 1, i.e., the theorem of the expected total number of isolated nodes, we depict the average number of isolated nodes w.r.t. ξ in networks with l = 500. We are interested in the condition where the network is close to being.

(13) IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 600. Fig. 1 A schematic of a wireless network with the Gaussian white noise model.. Fig. 3 The probability distribution functions of the total number of isolated nodes.. Fig. 2 The number of isolated nodes in a network with the Gaussian white noise model.. connected, i.e., there are few isolated nodes in the network. Thus, we choose the corresponding domain and range for the parameter ξ. Figure 2 illustrates the outcome for the Gaussian white noise model. The red solid line represents the expected number of isolated nodes given by Theorem 1, and the blue dotted line represents the average number of isolated nodes given by the simulations. We can see that there is only a small gap between the two curves. The rounding in the mathematical derivation for the theorem results in the gap between the expected number and the average number of isolated nodes. Even so, Theorem 1 accurately approximates the number of isolated nodes in a network. 6.4 Distribution of the Number of Isolated Nodes To verify Theorem 2, i.e., the theorem of the probability distribution of the total number of isolated nodes, network instances are generated with ξ = −5.8 and l = 500, 1000. As Fig. 2 shows, there is about one isolated node on average in the network when ξ = −5.8. This helps us understand the features of the network in such circumstances. The node densities corresponding to l = 500 and l = 1000 are 0.029 and 0.339, respectively. The probability distribution functions of the total number of isolated nodes are depicted in Fig. 3 for the Gaussian white noise model. The blue solid. Fig. 4 The CDFs of Diso , Dcon , and Dth for the log-distance path loss model.. line marked by triangles denotes the asymptotic probability distribution, i.e., the Poisson distribution with parameter e−ω . The black dash-dot line marked by stars and the red dash line marked by circles denote the experimental probability distribution corresponding to l = 500 and l = 1000, respectively. The results show that Theorem 2 can accurately capture network behavior. 6.5 Network Connectivity Lastly, we investigate the conjecture that a random network without isolated nodes is almost surely connected. In the simulation, nodes are added into the network one by one. After each node is added, isolated nodes are counted and network connectivity is verified. Let Diso denote the node density the first time that the network has no isolated nodes in the simulation, let Dcon be the node density the first time that the network becomes connected in the simulation, and let Dth denote the theoretical node density for nonexistence of isolated nodes. Figure 4 depicts the CDFs of Diso , Dcon , and Dth based on 400 network instances over the log-distance path loss model for l = 200, l = 500, and l = 800. In addition, Figs. 5,.

(14) SU et al.: THE NUMBER OF ISOLATED NODES IN A WIRELESS NETWORK WITH A GENERIC PROBABILISTIC CHANNEL MODEL. 601. The simulation results show that the node degree and the connectivity are closely related. It can infer that a simple model, e.g., the disk graph model, can capture the behaviors of network connectivity, and that theoretical analysis can provide insights for designing a real network. 7.. Fig. 5. The CDFs of Diso , Dcon , and Dth for the Bernoulli link model.. Conclusions. In this work, we derive the expected total number of isolated nodes and the asymptotic probability distribution of the total number of isolated nodes in a wireless network, based on a generic probabilistic wireless channel model. The probabilistic model studied in this work was a generalization of many widely used channel models, including the log-distance path loss model, the Bernoulli link model, the Gaussian white noise model, the Rayleigh fading model, and the Nakagami fading model. We presented the exact equation for the expected number of isolated nodes, and proved that the distribution of the number of isolated nodes asymptotically follows a Poisson distribution. In future work, we would like to prove the conjecture that a random network with a generic channel model without isolated nodes is almost surely connected. Acknowledgments. Fig. 6 The CDFs of Diso , Dcon , and Dth for the Gaussian white noise model.. This work of C.-W. Yi was supported in part by Sinica under Grant No. AS-102-TP-A06, by NSC under Grants No. NSC101-2628-E-009-014-MY3 and No. NSC100-2218-E009-002, by ITRI under Grants No. 101-EC-17-A-05-011111, No. 101-EC-17-A-03-01-0619, No. B352BW1100 and No. B301EA3300, by the MoE ATU plan, and by DLink, Inc. References. Fig. 7. The CDFs of Diso , Dcon , and Dth for the Rayleigh fading model.. 6, and 7 depict the CDFs for the Bernoulli link model, the Gaussian white noise model and the Rayleigh fading model, respectively, using the same value of l. The simulation results support our conjecture. It is worth noting that, under the same average node degree condition, there are no significant differences between the behaviors of the different models. In fact, the node degree is one of the fundamental indicators to measure the connectivity of a wireless network.. [1] O. Dousse, P. Thiran, and M. Hasler, “Connectivity in ad-hoc and hybrid networks,” Proc. 21st Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2002), pp.1079–1088, June 2002. [2] O. Dousse and P. Thiran, “Connectivity vs. capacity in dense ad hoc networks,” Proc. 23rd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2004), pp.476– 486, March 2004. [3] G. Zhang, J. Li, Y. Chen, and J. Liu, “Effect of mobility on the critical transmitting range for connectivity in wireless ad hoc networks,” Proc. 19th International Conference on Advanced Information Networking and Applications (AINA 2005), pp.9–12, March 2005. [4] X. Liu, “Coverage with connectivity in wireless sensor networks,” Proc. 3rd International Conference on Broadband Communications, Networks and Systems (BROADNETS 2006), pp.1–8, Oct. 2006. [5] E.N. Gilbert, “Random plane networks,” J. Society for Industrial and Applied Mathematics, vol.9, no.4, pp.533–543, Dec. 1961. [6] M. Haenggi, J.G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” IEEE J. Sel. Areas Commun., vol.27, no.7, pp.1029–1046, Sept. 2009. [7] H. Dette and N. Henze, “The limit distribution of the largest nearestneighbour link in the unit d-cube,” J. Applied Probability, vol.26, no.1, pp.67–80, March 1989..

(15) IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 602. [8] P. Gupta and P.R. Kumar, “Critical power for asymptotic connectivity in wireless networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, ed. W.M. McEneaney, G. Yin, and Q. Zhang, pp.547–566, Birkhauser, Boston, MA, March 1998. [9] C.W. Yi, P.J. Wan, C.M. Su, K.W. Lin, and S.C.H. Huang, “Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with unreliable nodes and links,” Discrete Mathematics, Algorithms and Applications (DMAA), vol.2, no.1, pp.107–124, March 2010. [10] M. Haenggi, “Link modeling with joint fading and distance uncertainty,” Proc. 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, pp.1–6, April 2006. [11] S. Mukherjee and D. Avidor, “Connectivity and transmit energy considerations between any pair of nodes in a wireless ad hoc network subject to fading,” IEEE Trans. Veh. Technol., vol.57, no.2, pp.1226–1242, March 2008. [12] R. Zhang and J.M. Gorce, “Connectivity of wireless sensor networks with unreliable links,” Proc. 2nd International Conference on Communications and Networking in China (CHINACOM 2007), pp.866–870, Aug. 2007. [13] D. Miorandi, E. Altman, and G. Alfano, “The impact of channel randomness on coverage and connectivity of ad hoc and sensor networks,” IEEE Trans. Wireless Commun., vol.7, no.3, pp.1062–1072, March 2008. [14] A. Babu and M.K. Singh, “Node isolation probability of wireless adhoc networks in Nagakami fading channel,” Int. J. Computer Networks and Communications, vol.2, no.2, pp.21–36, March 2010. [15] X. Ta, G. Mao, and B.D.O. Anderson, “On the connectivity of wireless multi-hop networks with arbitrary wireless channel models,” IEEE Commun. Lett., vol.13, no.3, pp.181–183, March 2009. [16] C. Bettstetter and C. Hartmann, “Connectivity of wireless multihop networks in a shadow fading environment,” Wireless Netw., vol.11, no.5, pp.571–579, Sept. 2005. [17] P. Hall, Introduction to the Theory of Coverage Processes, John Wiley & Sons, New York, 1988. [18] M.D. Penrose, “The longest edge of the random minimal spanning tree,” The Annals of Applied Probability, vol.7, no.2, pp.340–361, May 1997. [19] H. Zhang and J. Hou, “On deriving the upper bound of α-lifetime for large sensor networks,” Proc. 5th ACM international symposium on Mobile ad hoc networking and computing (MobiHoc 2004), pp.121– 132, New York, May 2004. [20] D.A. Griffith, Advanced Spatial Statistics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [21] C. Bettstetter, “On the minimum node degree and connectivity of a wireless multihop network,” Proc. 3rd ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc 2002), pp.80–91, New York, June 2002. [22] M. Penrose, Random Geometric Graphs, Oxford University Press, New York, 2003. [23] P.J. Wan and C.W. Yi, “On the longest edge of Gabriel graphs in wireless ad hoc networks,” IEEE Trans. Parallel Distrib. Syst., vol.18, no.1, pp.111–125, Jan. 2007. [24] T.S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed., Prentice Hall PTR, Upper Saddle River, NJ, 2002. [25] A.F. Molisch, Wireless Communications, Wiley-IEEE Press, West Sussex, England, 2005.. ⎛ X1 = x1 ⎜⎜⎜ k ⎜⎜ .. Pr ⎜⎜⎜⎜ B i . ⎜⎝ i=1 Xk = xk. For any (x1 , · · · , xk ) ∈ Ckk , since B (x1 , R2 ) , · · · , B (xk , R2 ) are pairwise disjoint, we have. (A· 1) . Appendix B: Proof of Lemma 4 This can be proved by straightforward calculation. ⎛ k ⎞ ⎜⎜ ⎟⎟⎟ k −kλa ⎜ ⎜ ⎜⎜⎝ λ e dxi ⎟⎟⎟⎠ (x )∈C 1 ,··· ,xk. kk. i=1. k 1 2 2 ln l2 + ln ln l2 + ξ e−k(ln l +ln ln l +ξ) = a ⎞ ⎛ k ⎜⎜⎜ ⎟⎟⎟ 2 2 ⎜ · ⎜⎝⎜ l − (i − 1) πR2 ⎟⎟⎠⎟ i=1. ∼. 1 −ξ e a. k. = e−ω k .. (A· 2) . Appendix C: Proof of Lemma 5 First, we prove the inequality for k = 2 and m = 1. Consider the case in which d (x1 , x2 ) ≥ R2 /2. Let B2 be the event that X2 does not have links to nodes in B (x2 , R2 ) − B (x1 , R2 ). Then, . X1 = x1 Pr B1 ∧ B2 X2 = x2 X = x1 . (A· 3) ≤ Pr B1 X1 = x1 Pr B2 1 X2 = x2 Since it is known from Eq. (14) that Pr B 1 X1 = x1 = e−λa , we only need to show that there is a positive constant c1 such that X = x1 Pr B2 1 (A· 4) ≤ e−c1 λa . X =x 2. 2. Let ρ = d (x1 , x2 ). For any ρ ∈ [R2 /2, R2 ] and r ∈ [0, R2 ], let θ (ρ, r) denote the angle of the arc of ∂B (x2 , r) not contained in B (x1 , R2 ). See Fig. A· 1. Since θ (ρ, r) is increasing w.r.t. ρ and f (r) ≥ 0 for r ∈ [0, R2 ], we have R2 f (r) θ (ρ, r) rdr 0 R2 1 ≥ f (r) θ R2 , r rdr. (A· 5) 1 2 2 R2 Let. Appendix A: Proof of Lemma 3. ⎞ ⎟⎟⎟ ⎟⎟⎟ −λa k ⎟⎟⎟ = e = e−kλa . ⎠⎟. R2 c1 =. 1 2 R2. f (r) θ. R2 0. . 1 2 R2 , r. . rdr. f (r) θ (ρ, r) rdr. .. Applying the same approach to deriving the probability.

(16) SU et al.: THE NUMBER OF ISOLATED NODES IN A WIRELESS NETWORK WITH A GENERIC PROBABILISTIC CHANNEL MODEL. 603 k . ≤ lim. k→∞. =e. −λ. = e−λ. Fig. A· 1 θ (ρ, r) is the angle of the arc of ∂B (x2 , r) not contained in B (x1 , R2 ).. e−( f (ri )+ f (ri +ρ)− f (ri ) f (ri +ρ))λ2πri

(17) ri. i=1 R2 ( f (r)+ f (r+ρ)− f (r) f (r+ρ))2πrdr 0 R2 0. f (r)2πrdr−λ. Then, 0. R2. 1R 2 2. = e−c1 λ. R2 0. = e−c1 λa .. 1 4 R2 1 8 R2. f. . 3 4 R2. 1 − f 18 R2 2πrdr. R2. (A· 6). .. We should note that the inequality still holds for annuli not fully contained in B (x2 , R2 ). Thus, . X1 = x1 Pr B1 ∧ B2 X2 = x2. (A· 10). f (r + ρ) (1 − f (r)) 2πrdr 1 4 R2 1 8 R2 1 4 R2. f (r + ρ) (1 − f (r)) 2πrdr f. 1 8 R2. 3 1 R2 1 − f R2 2πrdr 4 8 (A· 11). Thus, if 0 ≤ d (x1 , x2 ) ≤ R2 /2, we have . X1 = x1 Pr B1 ∧ B2 ≤ e−(1+c2 )λa . X2 = x2. (A· 12). If we choose c = min (c1 , c2 ), the lemma for k = 2 is proved. For any k ≥ 3 and m = 1, since&there are always two overlapping disks in the component i=1,··· ,k B (xi , R2 ), it is trivial to see that the inequality is still correct. For any k ≥ 3 and 2 ≤ m ≤ k − 1, if (x1 , · · · , xk ) ∈ Ckm , then {x1 , · · · , xk } is partitioned & into m sets K1 , K2 , · · · , Km such that for each j = 1, · · · , m, x∈K j B (x, R2 ) is a maximal component. Let n j = K j be the number of elements in K j .. Suppose K j = x j1 , · · · , x jn j . Then,. · (1 − f (ri )) j (1 − f (ri + ρ)) j (A· 7). (A· 8). f (r) 2πrdr. = c2 a.. Therefore, if R2 /2 ≤ d (x1 , x2 ) ≤ R2 , we have X = x1 Pr B 1 ∧ B 2 1 ≤ e−(1+c1 )λa . X2 = x2 Now, consider the case in which 0 ≤ d (x1 , x2 ) ≤ R2 /2. For this case, we only consider nodes in B (x1 , R2 ) and divide B (x1 , R2 ) by h concentric circles with center at x1 and radii r1 < r2 < · · · < rh = R2 as illustrated in Fig. A· 2. Since f (r) is a decreasing function, we have X1 and X2 do not have links X1 = x1 Pr with nodes in the ith annulus X2 = x2 ∞ (λ2πri

(18) ri ) j −λ2πri

(19) ri e ≤ j! j=0 = e−( f (ri )+ f (ri +ρ)− f (ri ) f (ri +ρ))λ2πri

(20) ri .. . ≥. f (r)θ(R2 /2,r)rdr f (r)2πrdr. R2. ≥. An annulus with center at x1 .. −λ. .. then this case is also proved. For any r ∈ [R2 /8, R2 /4], we have f (r + ρ) ≥ f (3R2 /4) and 1 − f (r) ≥ 1 − f (R2 /8). Let. 0. ≤e. ( f (r+ρ)− f (r) f (r+ρ))2πrdr. 0. c2 =. Pr ( X is isolated| X = x) in Sect. 4, we have R2 X = x1 = e−λ 0 f (r)θ(ρ,r)rdr Pr B2 1 X2 = x2. 0. R2 Since 0 f (r) 2πrdr = a, if we can prove that there is a positive constant c2 such that R2 ( f (r + ρ) − f (r) f (r + ρ)) 2πrdr ≥ c2 a, (A· 9). . Fig. A· 2. R2. ⎛ ⎜⎜⎜ X = x j1 ⎜⎜⎜ j1 . . Pr ⎜⎜⎜⎜ B ⎜⎝ x ∈K i . i j X jn j = x jn j. ⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ = e−λa if n j = 1 ⎟⎟⎠. ⎛ ⎜⎜⎜ X = x j1 ⎜⎜⎜ j1 . . Pr ⎜⎜⎜⎜ B ⎜⎝ x ∈K i . i j X jn j = x jn j. ⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ≤ e−(1+c)λa if n j > 1. ⎟⎟⎠. (A· 13). and. (A· 14).

(21) IEICE TRANS. COMMUN., VOL.E96–B, NO.2 FEBRUARY 2013. 604. Since there is at least one component that contains more than one node, we have ⎛ ⎞ X1 = x1 ⎟⎟ ⎜⎜⎜ k ⎟⎟⎟ ⎜⎜ .. ⎟⎟⎟⎟ Pr ⎜⎜⎜⎜ B i . ⎜⎝ ⎟⎠. i=1 Xk = xk ⎛ ⎞ ⎜⎜⎜ X j1 = x j1 ⎟⎟⎟ m ⎜⎜⎜ ⎟⎟⎟ .. Pr ⎜⎜⎜⎜ Bi = ⎟⎟⎟⎟ ≤ e−(m+c)λa . (A· 15) . ⎜ ⎟⎠ ⎝ xi ∈K j j=1 X jn j = x jn j . Thus, the lemma is proved. Appendix D: Proof of Lemma 6. First, consider m = 1. This can be validated by straightforward calculation. ⎛ k ⎞ ⎜⎜ ⎟⎟⎟ k −(1+c)λa ⎜ ⎜ ⎜⎜⎝ e dxi ⎟⎟⎟⎠ λ (x1 ,x2 ,··· ,xk )∈Ck1. i=1. ⎞ ⎛ k ⎜⎜ ⎟⎟⎟ k 2⎜ 2 ⎜ π (2 (i − 1) R2 ) ⎟⎟⎟⎠ e−(1+c)λa ≤ λ l ⎜⎜⎝ i=2. k 2 2 = O (1) ln l2 + ln ln l2 + ξ l2 e−(1+c)(ln l +ln ln l +ξ) . = o (1) .. Chih-Wei Yi received his Ph.D. degree in Computer Science from the Illinois Institute of Technology in 2005, and B.S. degree in Mathematics and M.S. degree in Computer Science and Information Engineering from the National Taiwan University in 1991 and 1993, respectively. He is currently an Associate Professor with the Department of Computer Science, National Chiao Tung University. He is a member of the IEEE and ACM. He had been a Senior Research Fellow of the Department of Computer Science, City University of Hong Kong. He was bestowed the Outstanding Young Engineer Award by the Chinese Institute of Engineers in 2009. He received the Young Scholar Best Paper Award from IEEE IT/COMSOC Taipei/Tainan Chapter in 2010. His research focuses on wireless ad hoc and sensor networks, vehicular ad hoc networks, intelligent transportation systems, network coding, and algorithm design and analysis.. (A· 16). We write g (λ) = O (h (λ)) if there are λ0 , c0 such that |g (λ) /h (λ)| ≤ c0 for all λ ≥ λ0 . Next, consider 2 ≤ m ≤ k − 1. If (x1 , · · · , xk ) ∈ Ckm , then {x1 , · · · , xk } can be partitioned into m sets K1 , K2 , · · · , Km such that for each j = 1, · · · , m, & ) x∈K j B (x, R2 is a maximal connected component. Let n j = K j be the number of elements in K j , and suppose. K j = x j1 , · · · , x jn j . For fixed k and m, the number of mpartitions of {x1 , · · · , xk } are constant. Then, ⎛ k ⎞ ⎜⎜ ⎟⎟⎟ k −(m+c)λa ⎜ ⎜⎜⎝⎜ λ e dxi ⎟⎟⎠⎟ (x1 ,x2 ,··· ,xk )∈Ckm i=1 ⎛ ⎛ nj ⎞⎞ m ⎜ ⎜⎜⎜ ⎟⎟⎟⎟⎟⎟ ⎜⎜⎜ n j −cλa −λa ⎜ ⎜⎜⎝λ e ⎜⎜⎝ dxi ⎟⎟⎟⎠⎟⎟⎟⎠ = O (1) e j=1. = o (1) .. Chao-Min Su received his M.S. and B.S. degree from the National Taiwan University. Currently, he is a Ph.D. candidate in Computer Science at the National Chiao Tung University. His research interests are in wireless ad hoc and sensor networks, and sensor applications.. x j1 ,··· ,x jn j ∈Cn j 1. j=1. (A· 17). The last equality holds because of at least one n j > 1. Thus, the lemma is proved. . Peng-Jun Wan received his Ph.D. degree from University of Minnesota, M.S. degree from The Chinese Academy of Science, and B.S. degree from Tsinghua University. He is currently a Professor in Computer Science at Illinois Institute of Technology, and at City University of Hong Kong. His research interests include wireless networks, optical networks, and algorithm design and analysis..

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