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Link probability, node degree and coverage in three-dimensional networks
Ad Hoc Networks 37 (2016) 153–159
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Link probability, node degree and coverage in three-dimensional networks Luiz Filipe M. Vieira∗, Marcelo G. Almiron, Antonio A.F. Loureiro Department of Computer Science, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
a r t i c l e
i n f o
Article history: Received 16 September 2014 Revised 3 August 2015 Accepted 20 August 2015 Available online 29 August 2015 Keywords: Link probability Coverage Connectivity Three dimensional networks Applications K-coverage
a b s t r a c t Underwater Sensor Networks (UWSNs) and Aeronautical Ad Hoc Networks (AANETs) are Mobile Ad Hoc Networks (MANETs) that need to consider the 3D nature of the network. Underwater Sensor Network (UWSN) presents the opportunity for many applications such as coast surveillance, and 4D monitoring (space and time) for ocean and biology studies. In AANET, aircrafts can communicate among each other and enhance situation awareness. In this work we study the fundamental properties of 3D Networks: link probability, node degree and network coverage. This work was motivated by fundamental problems currently faced in the deployment of UWSNs and AANETs. Link probability is important in Medium Access Control (MAC) protocols development. Node degree is useful in scheduling duty cycles. Determining network coverage is important for sensing applications, i.e., how to guarantee sensing coverage. Since energy is a limited resource, utilizing duty cycles is a major technique to improve network lifetime. We present analytical results for link probability, node degree and network coverage for 3D MANETs assuming deployments following the random uniform distribution. These results may be applied to a variety of scenarios, platforms and applications. In addition, we describe applications that would benef it from our results. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In this work we investigate fundamental properties of 3D Networks, such as Underwater Sensor Networks (UWSNs), Mobile Ad Hoc Networks (MANETs), and Aeronautical Ad Hoc Networks (AANETs). UWSN has the potential for many applications, such as coast surveillance, ocean monitoring, 4D monitoring (space and time) for ocean and biology studies [1]. AANETs can enhance aircrafts situation awareness. All these applications can benefit from network fundamental properties. First, sensing applications require the deployment of sensor nodes. This can be planned such that they
∗
Corresponding author. Tel.: +553134095860. E-mail addresses: [email protected], [email protected] (L.F.M. Vieira), [email protected] (M.G. Almiron), [email protected] (A.A.F. Loureiro).
http://dx.doi.org/10.1016/j.adhoc.2015.08.011 1570-8705/© 2015 Elsevier B.V. All rights reserved.
would benefit from network coverage and connectivity in the specified deployment region. Second, while in operation, it would be interesting to guarantee sensing coverage, even when some sensor nodes are operating in a duty cycle that has sleeping and awakening periods. Finally, the application performance can be influenced by the Medium Access Control (MAC) contention, which in turn, depends on the network node degree. Those are examples that demonstrate why it is important to investigate fundamental properties such as link probability, network coverage and node degree. The model presented in this work can be extended to any 3D MANET. UWSN and AANET are practical examples of the importance of studying 3D MANETs. Previous proposals on MANETs focus on 2D deployments as they were concerned with terrestrial applications. We present this novel study focusing 3D MANETs. This work was motivated by fundamental problems currently faced in the deployment of UWSNs, more specifically,
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intrusion detection applications. We are interested in answer the following question: given a region where nodes are deployed and that this region is to be protected, how many sensors should be deployed so that every point in this region is cover by one sensor? This question can be answered by solving the coverage problem. Coverage is important for sensing and surveillance applications. Every point in a surveillance region should be coverage by at least one sensor. These are important issues in network management and protocol design for scheduling duty cycles and controling transmission power. A proper adjustment of the duty cycle can extend the lifetime of the network. By knowing the node degree, some nodes can sleep, or reduce their transmission power and still guarantee message delivery. 1.1. Contributions We summarize our main contributions as follows: • We provide closed form analysis on the link probability between two distinct nodes i and j. This is the basic building block that allows us to compute node degree and realize further analysis; • We derive the formulae for node degree for both cases: a single node and for the entire network. By having an estimate of the expected node degree, it is possible to tune applications or protocols for a better performance. One such example is the case of the MAC protocol; • We describe how to compute the network coverage for two possible cases: considering border effects or not. This important result allows us to determine the required number of sensor nodes that guarantee sensing coverage (when considering the sensing radius) and network coverage (when considering the communication radius); • We solve the coverage problem. • We describe how to use these results into a network protocol to illustrate the results’ usefulness. The results and theories present here can be used to answer the formulated question in surveillance applications. In addition, it may be applied to a variety of scenarios, platforms and applications, other than the one mention here. To demonstrate the usefulness of the results, we mention one in the design of MAC protocols. The rest of the paper is organized as follows: in Section 2 we describe the MANET physical layer. In Section 3, we compute the analytical probability that two distinct nodes i and j have a common link. In Section 4, we compute the expected node degree. In Section 5, we present the analytical result on one-coverage. In Section 6, we describe some practical applications that can benefit from our results. In Section 7, we discuss the related work. Finally, we present our conclusion in Section 8. 2. Physical layer In terrestrial 3D MANETs, the signal propagation is described by the power received (PL) in dB, at a distance d, which is given by:
PL(d) = PL(d0 ) + 10α log (d/d0 )
where PL(d) is the path loss at distance d, PL(d0 ) is the known average path loss over distance d0 and α is the path loss exponent [2]. In UWSNs, unlike terrestrial networks, radio signal is not used due to the high attenuation. Instead, current research proposes acoustic channels [4], which have low bandwidth [3] and large propagation latency. The speed of sound underwater is about five orders of magnitude lower than the speed of light in the air. An acoustic data transmission consumes more energy than terrestrial microwave data communications. Moreover, high latency makes the whole network vulnerable to congestion due to packet collisions. We are only interested on those transmissions that arrive at the receiver at a sufficient power. This can be modeled as a sphere of radius r.1 Therefore, in this study, we consider the ball unit model. Coverage problems can concern sensing or communication. Sensing coverage is interested on how well the volume is been covered by sensors while communication coverage investigates the network connectivity. As long as the sensor or communication range fits the ball unit model, the results described in the paper can be considered, being applicable to both coverage problems. 3. Link probability In this section, we compute the analytical probability that two distinct nodes i and j have a common link. We assume the simple ball model. This is a simplified model as only path loss is taken into account. In the following, we derive a general expression for the link probability. After that, we instanciate this result to obtain expressions for the link probability in two specific cases, namely, the uniform and left triangular distributions. Two nodes have a common link if they are within each other transmission range. Nodes are deployed into a volume of size 1 × 2 × 3 . Let the location of node i be represented by Cartesian coordinates (Xi , Yi , Zi ), where 0 ≤ Xi ≤ 1 , 0 ≤ Yi ≤ 2 and 0 ≤ Zi ≤ 3 . Consider random variables Xi , Yi and Zi independent and identically distributed (i.i.d.). Let r be smaller than any of the dimensions 1 , 2 and 3 . Lemma 1. Let i and j be two distinct nodes. Let UiGj = (Gi − G j )2 be the squared distance between these two nodes in the orthogonal projection onto the generic G-axis. Consider also Gi and Gj as being i.i.d. on (0, ) with probability density function (p.d.f.) fG (·). We have
Pr UiGj ≤ u = 2 +
√ √ − u gi + u
0
gi
√ − u gi
fGi G j (gi , g j ) dg j dgi
fGi G j (gi , g j ) dg j dgi .
where fGi G j (gi , g j ) = fG (gi ) fG (g j ) is the joint p.d.f. of Gi and Gj , and 0 ≤ u ≤ 2 . Proof. Let us consider a new random variable AGij = |Gi − G j |. √ √ Clearly, Pr (UiGj ≤ u) = Pr (AGij ≤ u), where u ≤ . Then, we 1 One should notice that this is not the case in shallow water, where multipath propagation is considerable.
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one-dimensional space, we can instantiate this result to the orthogonal projections of the three-dimensional deployment region. Theorem 1 shows how to compute the link probability between any two nodes in the network. Let n, r, 1 , 2 , 3 represent a network with n nodes, each with a communication range r, deployed in a region of dimensions 1 × 2 × 3 . Theorem 1. In a network n, r, 1 , 2 , 3 , the occurrence probability of link i, j between any two distinct nodes i and j is
Pr (i, j) =
r2 0
r2 −u
r2 −u−v 0
0
f (u)g(v)h(w) dw dv du
where f(u), g(v) and h(w) are the p.d.f.s of random variables UiXj = (Xi − X j )2 , UiYj = (Yi − Y j )2 and UiZj = (Zi − Z j )2 , respectively. Proof. A link i, j between nodes i at (Xi , Yi , Zi ) and j at (Xj , Yj , Zj ) exists if the nodes are within each other communication range. In other words, if their Euclidean distance d(i, j) is smaller or equal to r. The distance d(i, j)2 is given by UiXj + UiYj + UiZj . Thus, the occurrence probability of link i, j
Fig. 1. Computing Pr (Gi ≤ G j ≤ Gi + a).
need to obtain an expression for Pr (AGij
≤ a), where 0 ≤ a ≤ .
We observe that the probability Pr (AGij ≤ a) can be divided into two terms using the law of total probability as follows:
Pr AGij ≤ a = Pr (G j ≤ Gi + a | Gi ≤ G j ) Pr (Gi ≤ G j ) + Pr (Gi ≤ G j + a | G j < Gi ) Pr (G j < Gi ).
(1)
Because Gi and Gj are i.i.d., we know that fGi G j (gi , g j ) is symmetric with respect to the identity over the domain [0, ]2 , and consequently Pr (G j ≤ Gi + a | Gi ≤ G j ) = Pr (Gi ≤ G j + a | G j < Gi ). Additionally, since Gi and Gj are i.i.d., we have Pr (Gi ≤ G j ) = Pr (G j < Gi ) = 1/2. Thus, we can rewrite (1) as
Pr AGij ≤ a = Pr (G j ≤ Gi + a | Gi ≤ G j ) =
(2)
The probability in (2) can be computed by calculating integrals over two non-overlapping intervals, when Gi + a ≤ and Gi + a > , and adding them up (see Fig. 1). Then, we have
+
−a gi +a
0
gi
−a gi
fGi G j (gi , g j ) dg j dgi
fGi G j (gi , g j ) dg j dgi .
(3)
√ ≤ u) = Pr (AGij ≤ u), the result of this lemma √ is obtained replacing a by u in (3), and this concludes the proof. Since
Pr (UiGj
In Lemma 1 we presented the distribution of squared distances between two nodes randomly deployed in a one-dimensional region. Since we considered a generic2 2
and UiZj , respectively. We know from Lemma 1 how to compute the distribution of each one of these three random variables. The derivative functions f (u) = F (u), g(v) = F (v) and h(w) = H (w) represent the p.d.f.s of these random variables. Additionally, since i and j are deployed independently, UiXj ,
UiYj and UiZj are independent. Then, we can compute Pr (UiXj + UiYj + UiZj ≤ r2 ) by
r2
0
r2 −u
r2 −u−v 0
0
UiZj ≤ r2 ).
= 2 Pr (Gi ≤ G j ≤ Gi + a)
Let F(u), G(v) and H (w) be the distributions of UiXj , UiYj
f (u)g(v)h(w) dw dv du.
This conclude the proof, since Pr (i, j) = Pr (UiXj + UiYj +
Pr (G j ≤ Gi + a ∩ Gi ≤ G j ) Pr (Gi ≤ G j )
Pr AGij ≤ a = 2
is Pr UiXj + UiYj + UiZj ≤ r2 .
We just assume that random variables are i.i.d. in [0, ].
3.1. The uniform and left triangular cases In this section we instantiate Theorem 1 in two particular cases, namely, the uniform and left triangular distributions. We start considering the former case, where nodes are uniformly deployed at random, represented as U. Here, nodes’ positions are denoted by (Xi , Yi , Zi ), considering Xi ∼ U (0, 1 ), Yi ∼ U (0, 2 ) and Zi ∼ U (0, 3 ). Corollary 1. In a network n, r, 1 , 2 , 3 , the occurrence probability p of link i, j between any two distinct nodes i and j, uniformly distributed, is:
p=
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
Proof. Let nodes i and j be positioned at (Xi , Yi , Zi ) and (Xj , Yj , Zj ), respectively, where Xi and Xj are i.i.d. with p.d.f. fX (x) = 1/1 , Yi and Yj are i.i.d. with p.d.f. fY (y) = 1/2 and, Zi and Zj are i.i.d. with p.d.f. fZ (z) = 1/3 .
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Consider random variables UiXj = (Xi − X j )2 , UiYj = (Yi −
Y j )2 , and UiZj = (Zi − Z j )2 . Applying Lemma 1, we obtain
Pr (UiXj ≤ u) =
√ −u + 21 u 1 2
Similarly, we obtain
Pr (UiYj ≤ v) = and
Pr (
UiZj
≤ w) =
−v + 22
,
0 ≤ u ≤ 1 2 .
(4)
,
0 ≤ v ≤ 2 2 ,
(5)
√
2 2
v
√ −w + 23 w 3 2
0 ≤ w ≤ 3 2 .
,
(6)
In addition, the p.d.f. of UiXj is f (u) = (1 u−0.5 − 1)/1 2 . Anal-
ogously, the p.d.f.s of UiYj and UiZj are g(v) = (2 v−0.5 − 1)/2 2
and h(w) = (3 w−0.5 − 1)/3 2 , respectively. Since UiXj , UiYj and UiZj are independent, their joint
p.d.f. is given by t (u, v, w) = f (u)g(v)h(w). Then, applying Theorem 1, we finally obtain
r2 r2 −u r2 −u−v 0
0
0
2. The distribution of UiZj is
1
8 5 − r6 + r (1 + 2 + 3 ) 6 15
1
=
t (u, v, w) dw dv du
In the following, we instantiate Theorem 1 for the left triangular distribution, defined as follows. Definition 1 (Left Triangular Distribution). Z has a Left Triangular distribution with parameter 3 , if its density function is
3 2
.
This distribution can be used to model different depths in a UWSN where nodes are more likely to be near the surface of the water. Corollary 2 presents the link probability of a scenario with nodes deployed uniformly in the plane and triangularly in the third dimension. Corollary 2. Let i and j be two nodes deployed at positions (Xi , Yi , Zi ) and (Xj , Yj , Zj ), respectively, where Xi , X j ∼ U (0, 1 ), Yi , Y j ∼ U (0, 2 ) and Zi , Zj follow a left triangular distribution over the range [0, 3 ]. The probability of existence of a link between i and j, namely p, is
p=
1
1 21 22 43 +
1 9
−π
36
r8 −
1 2 π −
2 3
33
32 (1 + 2 )r7 + 315
1 2 6 r + 3 3
(1 + 2 ) +
16
15
1 2 23
23 (1 + 2 ) +
2
w2 − 6wl3 +
√ 3 w8l3
33 4
, and;
3. The p.d.f. of UiZj is
2(3 − z)
Pr UiZj ≤ w =
1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
fZ (z) =
Fig. 2. Occurrence probability of a link as a function of the communication range.
32 3 5 r 45 3
16 r + π 1 2 33 r3 . 9 4
Proof. Analogous to proof of Corollary 1 with the following considerations: 1. Zi and Zj are i.i.d. with p.d.f. fZ (z) = 2(l3 − z)/l3 ; 2
h(w) =
3
4l3 w−1/2 + 2w − 6l3 3l3
4
2
.
Fig. 2 shows the occurrence probability p of a link i, j, where 1 = 2 = 3 = 10 as a function of the communication range r for two different distributions considering experimental and theoretical results. UnifE stands for the curve obtained in experimental results via simulation using a Uniform distribution. UnifT identifies the curve obtained using theoretical analytical functions using a uniform distribution. The same way, TriangE represents the curve obtained via experimental results using a triangular distribution and, finally, TriangT stands for the curve generated via theoretical analytical functions using a triangular distribution. 4. Node degree Let random variable Li, j be the number of links connecting nodes i and j. Li, j is either 1 or 0. Let Di = j Li, j be the node degree of node i. Theorem 2. Consider a network n, r, 1 , 2 , 3 . Then, we have 1. The expected node degree is (n − 1) p, and; 2. The expected number of links is n(n − 1) p/2, where p is the link probability. Proof. From Theorem 1, the expected link occurrence E(Li, j ) is equal p. By definition, E(Di ) = E( i Li, j ). By linearity of ex pected values, E( i Li, j ) = i E(Li, j ), no matter if Li, j ’s are independent or not. Since each node may have n − 1 links, we have E(Di ) = (n − 1) p. On the other hand, there are potentially n(n − 1)/2 links between n nodes. Considering p as the link probability, the expected number of links is n(n − 1) p/2.
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Let us denote by Ci the volume covered by i nodes. In the following, we will compute the expected volume covered by n nodes in terms of expected volume covered by a single node. We can compute E[Cn ] recursively, thinking about the nodes as being positioned one by one, as a stochastic process. We know by definition that C1 = φ , which implies that E[C1 ] = E[φ ]. Now, suppose we know the expected coverage for i nodes, and we add another node at random. We can write the updated expected coverage as follows,
E[Ci+1 ] = E[Ci ] + E[Ai+1 ],
Fig. 3. Expected node degree as a function of radio radius and number of nodes.
Fig. 3 shows the expected node degree as a function of radio radius and number of nodes for 1 = 2 = 3 = 10. 5. Expected coverage
5.1. one-coverage
E[Cn ] =
1−
1−
− 16 r6 +
− 12 π r4 (2 1 + 3 1 + 2 3 ) + 1 2 2 2 3 2
(
8 5 r 1 + 2 15 1 2 2 2 3 2 4 3
+ 3 )
π r3 1 2 3
n 1 2 3 .
Proof. We denote with φ the volume covered by a single node. It is easy to see that φ = 43 π r3 , under our ball model, whenever the node’s sensory volume is properly contained in R. Obviously, when the node is close enough (for distances less than r) to the border this value is lower. In order to obtain the expected coverage of a single node we use an interesting fact observed by [7], when the sensor detection range equals the communication range: the link probability p is equal to the expected coverage of a single node divided by the volume of the deployment region, that is
p=
E[φ ] . V
(7)
From (7), we derive that the expected coverage of a single node is E[φ ] = p 1 2 3 .
E[Ci ] V − E[Ci ] =1− . V V
(9)
Replacing the result of (9) in (8), we obtain the following recurrence relation
E[Ci+1 ] = E[Ci ] +
E[Ci ] 1− E[φ ]. V
Solving this, we have
E[Cn ] = 1 − (1 − p)n V, which concludes the proof, since we know that V = 1 2 3 and
p=
Theorem 3. Let n, r, 1 , 2 , 3 be a network with n nodes deployed uniformly at random in a region R = 1 2 3 , where each node have a sensory region determined by a ball of radius r. Assuming that min {1 , 2 , 3 } ≥ 2r, the expected one-coverage volume of all nodes is
where Ai+1 is the one-coverage volume contributed by the last placed node. Considering ρ i as being the proportion of the volume contributed by the ith node to the size of the covered region, we have E[Ai ] = E[ρi φ ]. Clearly, ρ i and φ are correlated, but we can use E[ρ i ]E[φ ] as a good approximation for E[Ai ]. As nodes are uniformly distributed, ρi+1 is expected to be the proportion of the uncovered volume to the whole deployment region,
E[ρi+1 ] =
Given n nodes deployed in a region R with volume V = 1 2 3 , we are interested in computing the expected network k-coverage, that is, the volume of R covered by at least k nodes. This kind of characterizations are useful in several and diverse sensing applications. In the following we preceed to derive an expression for the simplest and more common case, namely, the expected one-coverage. Based on the aforementioned result, we derive an expression for the expected k-coverage. In this section we extend the results presented by [6], obtaining results for three dimensional networks.
(8)
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
5.2. k-coverage Theorem 4. Let n, r, 1 , 2 , 3 be a network with n nodes deployed uniformly at random in a region R = 1 2 3 , where each node have a sensory region determined by a ball of radius r. Assuming that min {1 , 2 , 3 } ≥ 2r, the expected k-coverage volume of all nodes is
E[Ckn ] = pk−1
n−k
t Nk−1 (1 − p)t E[C(n−k)−t+1 ] ,
t=0
where
p=
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
Proof. Let us define generalizations for the random variables j considered in the proof of Theorem 3. We define Ci to be the size of the j-covered volume after i nodes have been deployed. Note that C0i = V and C1i = Ci . Additionally, by definij
j
tion, we have Ci = 0 whenever i < j. We consider also Ai as being the extra volume contributed by the ith placed node to
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the size of the j-covered volume, and ρi as being the proporj
tion of to φ . By definition, we have j Ai
j E[Cij ] = E[Ci−1 ] + E[Aij ].
(10)
j
We can approximate Ai , similarly as we did with Ai in the
proof of Theorem 3, by E[ρi ]E[φ ]. By definition, we have that j
ρij
is expected to be the proportion of the volume that is exactly covered by j − 1 out of i − 1 nodes to the total volume V, that is,
E[ρij ] =
j−1 j E[Ci−1 − Ci−1 ]
V
=
j−1 j E[Ci−1 ] − E[Ci−1 ]
V
.
With this, and using the expression in (10), we can write j j−1 E[Cij ] = (1 − p)E[Ci−1 ] + pE[Ci−1 ],
(11)
where p = E[φ ]/V is the link probability. The recurrence relation in (11) have a simple solution for the case i = j, E[Cii ] = pi−1 E[φ ]. In order to find an expression, we expand (11) until the case i = j. With this approach, we obtain
E[Ckn ] = pk−1
n−k
t Nk−1 (1 − p)t E[C(n−k)−t+1 ] ,
t=0
where
Nij
j Ni
⎧ 1 ⎪ ⎨
is defined recursively by
i = ⎪i(i + 1)/2 ⎩ i−( j−2) Nij−1 + t=1 t
j j j j
= 0, = 1, = 2, > 2.
6. Applications In this section we describe how the previous analysis can be useful in the design of protocols and applications. We describe an uncoordinate node scheduling for MAC protocols. MAC protocols have to cope with channel contention in order to work properly. For example, in underwater networks, due to the fact that high-frequency waves are rapidly absorbed by water, one must use acoustic channel. This channel is characterized by low and range-dependent bandwidth. Also, slow acoustic signal propagation (around 1500 m/s). Usually higher link degree indicates higher degree of channel contentions. More nodes are competing for the shared communication medium. This results in poor link performance in contention-based MAC protocols. In addition, in most scenarios, energy is a limiting resource. A solution to both problems is to have nodes sleep periodically. A decentralized solution works as follows. Each node independently alternates between active and sleep modes. The transition between states is purely stochastic. The time periods of active and sleep are exponentially distributed random variables with means μa and μs , respectively. The probability of a node being in sleep mode is ps = μs /(μa + μs ). Similarly, the probability of a node being in active mode is pa = μa /(μa + μs ). In a network with n nodes, npa nodes are expected to be active at any time. Thus, the
expected coverage and node degree can be estimated by replacing n by npa in the respective equations. This approach, although stochastic, is deterministic to set the values of parameters μa and μs for a desired network coverage. As previously described by [6], this is not possible without our analytical results. The above algorithm was proposed in [8], and also described in [6] for 2D sensor networks. Here we demonstrate the utility of our results for 3D MANETs. 7. Related work Yen and Yu [7] described the fundamental properties of a 2D wireless ad hoc networks. In their work they consider a wireless Ad Hoc network as n, r, 1 , 2 -network with n nodes, with transmission radius r, placed in a 1 × 2 rectangle area. The occurrence probability of a link (i, j) between two distinct nodes i and j is given by Eq. 12. We improve that work for 3D MANETs and considere the effects of the 3D environment. We investigate further into how these results apply to certain applications. 1 4 r 2
− 43 1 r3 − 43 2 r3 + π r2 1 2 1 2 2 2
(12)
Yen et al. [6] showed how to obtain the k-coverage in wireless sensor network for networks with and without boundaries. They describe results for coverage and studied node scheduling. Our work is more general since we are studying 3D MANETs. Kumar et al. [9] studied the problem to determine the number of sensors that achieves two goals: having nodes sleep to increase network lifetime and guarantee k-coverage. They considered three deployments scenarios: unit square, random uniform, and poisson. They obtained the critical value for the k-coverage of every point. Our results can be applied into their application that randomly selects a node to sleep and guarantees one-coverage for sensing applications. Coverage and Connectivity in 3D Networks have been explored, [10], but do not provide the mathematical results we presented. [13] provide a k-coverage scheduling protocol. In [14], the authors evaluate a coverage compensation for Wireless Sensor Networks. In [15], Fei et al. gave a bio-inspired coverage scheduling. In [16] and [17] the authors consider irregular sensing ranges. In [18], it is presented a coverage scheme based on local information. Currently, most of the proposals in the literature do not provide the same mathematical analysis. Mansoor and Ammari [11] summarizes the work conducted in the domain of coverage and connectivity in 3D Wireless Sensor Networks(WSNs). As described by the authors, 3D WSNs require more complex and computationally intensive analysis. They present different coverage planning and placement strategies. We provide new mathematical analysis. In the theory of Random Graphs, we study graphs that are generated by some random process. A random graph is a graph in which properties such as the number of graph vertices, graph edges, and connections between them are determined in some random way [12]. Percolation theory characterizes the connectedness of random graphs, focusing on the
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asymptotically behavior. Our work differ in that since we analyze a 3D deployment, and focus in network properties with application, such as coverage. 8. Conclusion In this work, we considered the fundamental properties of 3D Networks. We showed how to compute the link probability, node degree, and expected coverage. Those are necessary and useful in real applications. Furthermore, we were able to derive analytical results for those properties. The results presented are useful for many applications. In particular, we showed one such an application: reducing MAC contention. The approach of increasing duty cycle reduces energy consumption and increases network lifetime. Estimating the necessary network sensing coverage can reduce overall costs by reducing the number of nodes in the deployment and reducing maintenance costs during network lifetime.
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[12] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, Wiley, 1996. [13] X. Fei, A. Boukerche, F.R. Yu, A pomdp based k-coverage dynamic scheduling protocol for wireless sensor networks, in: GLOBECOM, IEEE, ISBN, 2010. [14] X. Fei, A. Boukerche, A performance evaluation of a coverage compensation based algorithm for wireless sensor networks, in: Proceedings of the 11th International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems, MSWiM ’08, ACM, New York, NY, USA, 2008, pp. 109–116. 978-1-60558-235-1. [15] X. Fei, S. Samarah, A. Boukerche, A bio-inspired coverage-aware scheduling scheme for wireless sensor networks, in: Proceedings of the 2010 IEEE International Symposium on Parallel Distributed Processing, Workshops and Phd Forum (IPDPSW), 2010, pp. 1–8. [16] X. Fei, A. Boukerche, R.B. Araujo, Irregular sensing range detection model for coverage based protocols in wireless sensor networks, in: Proceedings of the 28th IEEE Conference on Global Telecommunications, GLOBECOM’09, IEEE Press, Piscataway, NJ, USA, 2009, pp. 5699– 5704. 978-1-4244-4147-1. [17] A. Boukerche, X. Fei, A coverage-preserving scheme for wireless sensor network with irregular sensing range, Ad Hoc Net. 5 (8) (2007) 1303– 1316. [18] A. Boukerche, X. Fei, R.B. de Araujo, An optimal coverage-preserving scheme for wireless sensor networks based on local information exchange, Comput. Commun. 30 (14–15) (2007) 2708–2720.
References [1] J. Kong, J.-H. Cui, D. Wu, M. Gerla, Building underwater ad-hoc networks and sensor networks for large scale real-time aquatic applications, in: Proceedings of the IEEE Military Communications Conference (MILCOM’05), Atlantic City, NJ, 2005, pp. 1535–1541. [2] T.S. Rappaport, Wireless communications: principles and practice, second, Prentice Hall, 2002. [3] M. Stojanovic, On the relationship between capacity and distance in an underwater acoustic communication channel, ACM SIGMOBILE Mob. Comput. Commun. Rev. 11 (2007) 34–43. 1559–1662. [4] M. Ainslie, P. Dahl, C.d. Jong, R. Laws, Practical spreading laws: the snakes and ladders of shallow water acoustics, in: Proceedings of the 2nd International Conference and Exhibition on Underwater Acoustics, Rhodes, Greece, 2014. [5] M. Stojanovic, J. Preisig, Underwater acoustic communication channels: propagation models and statistical characterization, IEEE Commun. Mag. 47 (1) (2009) 84–89. 0163–6804. [6] L.-H. Yen, C.W. Yu, Y.-M. Cheng, Expected k-coverage in wireless sensor networks, Ad Hoc Net. 4 (5) (2006) 636–650. [7] L.-H. Yen, C.W. Yu, Link probability, network coverage, and related properties of wireless ad hoc networks, in: Proceedings of the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS’04), 2004, pp. 525–527. [8] M. Liu, C. Hsin, Network coverage using low duty-cycled sensors: random and coordinated sleep algorithms, in: Proceedings of the Third International Symposium on Information Processing in Sensor Networks (IPSN’04), Berkeley, CA, 2004, pp. 433–442. [9] S. Kumar, T.H. Lai, J. Balogh, On k-coverage in a mostly sleeping sensor network, in: Proceedings of the 10th Annual International Conference on Mobile Computing and Networking (MobiCom ’04), Philadelphia, PA, USA, 2004, pp. 144–158. 1-58113-868-7. [10] S.M.N. Alam, Z.J. Haas, connectivity in three-dimensional networks, in: Proceedings of the 12th Annual International Conference on Mobile Computing and Networking, MobiCom ’06, ACM, New York, NY, USA, 2006, pp. 346–357. 1-59593-286-0. [11] U. Mansoor, H. Ammari, Coverage and connectivity in 3d wireless sensor networks, in: H. Ammari (Ed.), The Art of Wireless Sensor Networks, Signals and Communication Technology, Springer Berlin Heidelberg, 2014, pp. 273–324. 978-3-642-40065-0.
Luiz F. M. Vieira is an Assistant Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate and M.S. degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of California Los Angeles (UCLA). His research interest is in Computer Networking.
Marcelo G. Almiron is a post-doc. He received his Ph.D. degree in Computer Science from the Universidade Federal de Minas Gerais.
Antonio A. F. Loureiro is a Full Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of Britich Columbia (UVC). His research interest is in Computer Networking.
Contents lists available at ScienceDirect
Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc
Link probability, node degree and coverage in three-dimensional networks Luiz Filipe M. Vieira∗, Marcelo G. Almiron, Antonio A.F. Loureiro Department of Computer Science, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
a r t i c l e
i n f o
Article history: Received 16 September 2014 Revised 3 August 2015 Accepted 20 August 2015 Available online 29 August 2015 Keywords: Link probability Coverage Connectivity Three dimensional networks Applications K-coverage
a b s t r a c t Underwater Sensor Networks (UWSNs) and Aeronautical Ad Hoc Networks (AANETs) are Mobile Ad Hoc Networks (MANETs) that need to consider the 3D nature of the network. Underwater Sensor Network (UWSN) presents the opportunity for many applications such as coast surveillance, and 4D monitoring (space and time) for ocean and biology studies. In AANET, aircrafts can communicate among each other and enhance situation awareness. In this work we study the fundamental properties of 3D Networks: link probability, node degree and network coverage. This work was motivated by fundamental problems currently faced in the deployment of UWSNs and AANETs. Link probability is important in Medium Access Control (MAC) protocols development. Node degree is useful in scheduling duty cycles. Determining network coverage is important for sensing applications, i.e., how to guarantee sensing coverage. Since energy is a limited resource, utilizing duty cycles is a major technique to improve network lifetime. We present analytical results for link probability, node degree and network coverage for 3D MANETs assuming deployments following the random uniform distribution. These results may be applied to a variety of scenarios, platforms and applications. In addition, we describe applications that would benef it from our results. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In this work we investigate fundamental properties of 3D Networks, such as Underwater Sensor Networks (UWSNs), Mobile Ad Hoc Networks (MANETs), and Aeronautical Ad Hoc Networks (AANETs). UWSN has the potential for many applications, such as coast surveillance, ocean monitoring, 4D monitoring (space and time) for ocean and biology studies [1]. AANETs can enhance aircrafts situation awareness. All these applications can benefit from network fundamental properties. First, sensing applications require the deployment of sensor nodes. This can be planned such that they
∗
Corresponding author. Tel.: +553134095860. E-mail addresses: [email protected], [email protected] (L.F.M. Vieira), [email protected] (M.G. Almiron), [email protected] (A.A.F. Loureiro).
http://dx.doi.org/10.1016/j.adhoc.2015.08.011 1570-8705/© 2015 Elsevier B.V. All rights reserved.
would benefit from network coverage and connectivity in the specified deployment region. Second, while in operation, it would be interesting to guarantee sensing coverage, even when some sensor nodes are operating in a duty cycle that has sleeping and awakening periods. Finally, the application performance can be influenced by the Medium Access Control (MAC) contention, which in turn, depends on the network node degree. Those are examples that demonstrate why it is important to investigate fundamental properties such as link probability, network coverage and node degree. The model presented in this work can be extended to any 3D MANET. UWSN and AANET are practical examples of the importance of studying 3D MANETs. Previous proposals on MANETs focus on 2D deployments as they were concerned with terrestrial applications. We present this novel study focusing 3D MANETs. This work was motivated by fundamental problems currently faced in the deployment of UWSNs, more specifically,
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intrusion detection applications. We are interested in answer the following question: given a region where nodes are deployed and that this region is to be protected, how many sensors should be deployed so that every point in this region is cover by one sensor? This question can be answered by solving the coverage problem. Coverage is important for sensing and surveillance applications. Every point in a surveillance region should be coverage by at least one sensor. These are important issues in network management and protocol design for scheduling duty cycles and controling transmission power. A proper adjustment of the duty cycle can extend the lifetime of the network. By knowing the node degree, some nodes can sleep, or reduce their transmission power and still guarantee message delivery. 1.1. Contributions We summarize our main contributions as follows: • We provide closed form analysis on the link probability between two distinct nodes i and j. This is the basic building block that allows us to compute node degree and realize further analysis; • We derive the formulae for node degree for both cases: a single node and for the entire network. By having an estimate of the expected node degree, it is possible to tune applications or protocols for a better performance. One such example is the case of the MAC protocol; • We describe how to compute the network coverage for two possible cases: considering border effects or not. This important result allows us to determine the required number of sensor nodes that guarantee sensing coverage (when considering the sensing radius) and network coverage (when considering the communication radius); • We solve the coverage problem. • We describe how to use these results into a network protocol to illustrate the results’ usefulness. The results and theories present here can be used to answer the formulated question in surveillance applications. In addition, it may be applied to a variety of scenarios, platforms and applications, other than the one mention here. To demonstrate the usefulness of the results, we mention one in the design of MAC protocols. The rest of the paper is organized as follows: in Section 2 we describe the MANET physical layer. In Section 3, we compute the analytical probability that two distinct nodes i and j have a common link. In Section 4, we compute the expected node degree. In Section 5, we present the analytical result on one-coverage. In Section 6, we describe some practical applications that can benefit from our results. In Section 7, we discuss the related work. Finally, we present our conclusion in Section 8. 2. Physical layer In terrestrial 3D MANETs, the signal propagation is described by the power received (PL) in dB, at a distance d, which is given by:
PL(d) = PL(d0 ) + 10α log (d/d0 )
where PL(d) is the path loss at distance d, PL(d0 ) is the known average path loss over distance d0 and α is the path loss exponent [2]. In UWSNs, unlike terrestrial networks, radio signal is not used due to the high attenuation. Instead, current research proposes acoustic channels [4], which have low bandwidth [3] and large propagation latency. The speed of sound underwater is about five orders of magnitude lower than the speed of light in the air. An acoustic data transmission consumes more energy than terrestrial microwave data communications. Moreover, high latency makes the whole network vulnerable to congestion due to packet collisions. We are only interested on those transmissions that arrive at the receiver at a sufficient power. This can be modeled as a sphere of radius r.1 Therefore, in this study, we consider the ball unit model. Coverage problems can concern sensing or communication. Sensing coverage is interested on how well the volume is been covered by sensors while communication coverage investigates the network connectivity. As long as the sensor or communication range fits the ball unit model, the results described in the paper can be considered, being applicable to both coverage problems. 3. Link probability In this section, we compute the analytical probability that two distinct nodes i and j have a common link. We assume the simple ball model. This is a simplified model as only path loss is taken into account. In the following, we derive a general expression for the link probability. After that, we instanciate this result to obtain expressions for the link probability in two specific cases, namely, the uniform and left triangular distributions. Two nodes have a common link if they are within each other transmission range. Nodes are deployed into a volume of size 1 × 2 × 3 . Let the location of node i be represented by Cartesian coordinates (Xi , Yi , Zi ), where 0 ≤ Xi ≤ 1 , 0 ≤ Yi ≤ 2 and 0 ≤ Zi ≤ 3 . Consider random variables Xi , Yi and Zi independent and identically distributed (i.i.d.). Let r be smaller than any of the dimensions 1 , 2 and 3 . Lemma 1. Let i and j be two distinct nodes. Let UiGj = (Gi − G j )2 be the squared distance between these two nodes in the orthogonal projection onto the generic G-axis. Consider also Gi and Gj as being i.i.d. on (0, ) with probability density function (p.d.f.) fG (·). We have
Pr UiGj ≤ u = 2 +
√ √ − u gi + u
0
gi
√ − u gi
fGi G j (gi , g j ) dg j dgi
fGi G j (gi , g j ) dg j dgi .
where fGi G j (gi , g j ) = fG (gi ) fG (g j ) is the joint p.d.f. of Gi and Gj , and 0 ≤ u ≤ 2 . Proof. Let us consider a new random variable AGij = |Gi − G j |. √ √ Clearly, Pr (UiGj ≤ u) = Pr (AGij ≤ u), where u ≤ . Then, we 1 One should notice that this is not the case in shallow water, where multipath propagation is considerable.
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one-dimensional space, we can instantiate this result to the orthogonal projections of the three-dimensional deployment region. Theorem 1 shows how to compute the link probability between any two nodes in the network. Let n, r, 1 , 2 , 3 represent a network with n nodes, each with a communication range r, deployed in a region of dimensions 1 × 2 × 3 . Theorem 1. In a network n, r, 1 , 2 , 3 , the occurrence probability of link i, j between any two distinct nodes i and j is
Pr (i, j) =
r2 0
r2 −u
r2 −u−v 0
0
f (u)g(v)h(w) dw dv du
where f(u), g(v) and h(w) are the p.d.f.s of random variables UiXj = (Xi − X j )2 , UiYj = (Yi − Y j )2 and UiZj = (Zi − Z j )2 , respectively. Proof. A link i, j between nodes i at (Xi , Yi , Zi ) and j at (Xj , Yj , Zj ) exists if the nodes are within each other communication range. In other words, if their Euclidean distance d(i, j) is smaller or equal to r. The distance d(i, j)2 is given by UiXj + UiYj + UiZj . Thus, the occurrence probability of link i, j
Fig. 1. Computing Pr (Gi ≤ G j ≤ Gi + a).
need to obtain an expression for Pr (AGij
≤ a), where 0 ≤ a ≤ .
We observe that the probability Pr (AGij ≤ a) can be divided into two terms using the law of total probability as follows:
Pr AGij ≤ a = Pr (G j ≤ Gi + a | Gi ≤ G j ) Pr (Gi ≤ G j ) + Pr (Gi ≤ G j + a | G j < Gi ) Pr (G j < Gi ).
(1)
Because Gi and Gj are i.i.d., we know that fGi G j (gi , g j ) is symmetric with respect to the identity over the domain [0, ]2 , and consequently Pr (G j ≤ Gi + a | Gi ≤ G j ) = Pr (Gi ≤ G j + a | G j < Gi ). Additionally, since Gi and Gj are i.i.d., we have Pr (Gi ≤ G j ) = Pr (G j < Gi ) = 1/2. Thus, we can rewrite (1) as
Pr AGij ≤ a = Pr (G j ≤ Gi + a | Gi ≤ G j ) =
(2)
The probability in (2) can be computed by calculating integrals over two non-overlapping intervals, when Gi + a ≤ and Gi + a > , and adding them up (see Fig. 1). Then, we have
+
−a gi +a
0
gi
−a gi
fGi G j (gi , g j ) dg j dgi
fGi G j (gi , g j ) dg j dgi .
(3)
√ ≤ u) = Pr (AGij ≤ u), the result of this lemma √ is obtained replacing a by u in (3), and this concludes the proof. Since
Pr (UiGj
In Lemma 1 we presented the distribution of squared distances between two nodes randomly deployed in a one-dimensional region. Since we considered a generic2 2
and UiZj , respectively. We know from Lemma 1 how to compute the distribution of each one of these three random variables. The derivative functions f (u) = F (u), g(v) = F (v) and h(w) = H (w) represent the p.d.f.s of these random variables. Additionally, since i and j are deployed independently, UiXj ,
UiYj and UiZj are independent. Then, we can compute Pr (UiXj + UiYj + UiZj ≤ r2 ) by
r2
0
r2 −u
r2 −u−v 0
0
UiZj ≤ r2 ).
= 2 Pr (Gi ≤ G j ≤ Gi + a)
Let F(u), G(v) and H (w) be the distributions of UiXj , UiYj
f (u)g(v)h(w) dw dv du.
This conclude the proof, since Pr (i, j) = Pr (UiXj + UiYj +
Pr (G j ≤ Gi + a ∩ Gi ≤ G j ) Pr (Gi ≤ G j )
Pr AGij ≤ a = 2
is Pr UiXj + UiYj + UiZj ≤ r2 .
We just assume that random variables are i.i.d. in [0, ].
3.1. The uniform and left triangular cases In this section we instantiate Theorem 1 in two particular cases, namely, the uniform and left triangular distributions. We start considering the former case, where nodes are uniformly deployed at random, represented as U. Here, nodes’ positions are denoted by (Xi , Yi , Zi ), considering Xi ∼ U (0, 1 ), Yi ∼ U (0, 2 ) and Zi ∼ U (0, 3 ). Corollary 1. In a network n, r, 1 , 2 , 3 , the occurrence probability p of link i, j between any two distinct nodes i and j, uniformly distributed, is:
p=
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
Proof. Let nodes i and j be positioned at (Xi , Yi , Zi ) and (Xj , Yj , Zj ), respectively, where Xi and Xj are i.i.d. with p.d.f. fX (x) = 1/1 , Yi and Yj are i.i.d. with p.d.f. fY (y) = 1/2 and, Zi and Zj are i.i.d. with p.d.f. fZ (z) = 1/3 .
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Consider random variables UiXj = (Xi − X j )2 , UiYj = (Yi −
Y j )2 , and UiZj = (Zi − Z j )2 . Applying Lemma 1, we obtain
Pr (UiXj ≤ u) =
√ −u + 21 u 1 2
Similarly, we obtain
Pr (UiYj ≤ v) = and
Pr (
UiZj
≤ w) =
−v + 22
,
0 ≤ u ≤ 1 2 .
(4)
,
0 ≤ v ≤ 2 2 ,
(5)
√
2 2
v
√ −w + 23 w 3 2
0 ≤ w ≤ 3 2 .
,
(6)
In addition, the p.d.f. of UiXj is f (u) = (1 u−0.5 − 1)/1 2 . Anal-
ogously, the p.d.f.s of UiYj and UiZj are g(v) = (2 v−0.5 − 1)/2 2
and h(w) = (3 w−0.5 − 1)/3 2 , respectively. Since UiXj , UiYj and UiZj are independent, their joint
p.d.f. is given by t (u, v, w) = f (u)g(v)h(w). Then, applying Theorem 1, we finally obtain
r2 r2 −u r2 −u−v 0
0
0
2. The distribution of UiZj is
1
8 5 − r6 + r (1 + 2 + 3 ) 6 15
1
=
t (u, v, w) dw dv du
In the following, we instantiate Theorem 1 for the left triangular distribution, defined as follows. Definition 1 (Left Triangular Distribution). Z has a Left Triangular distribution with parameter 3 , if its density function is
3 2
.
This distribution can be used to model different depths in a UWSN where nodes are more likely to be near the surface of the water. Corollary 2 presents the link probability of a scenario with nodes deployed uniformly in the plane and triangularly in the third dimension. Corollary 2. Let i and j be two nodes deployed at positions (Xi , Yi , Zi ) and (Xj , Yj , Zj ), respectively, where Xi , X j ∼ U (0, 1 ), Yi , Y j ∼ U (0, 2 ) and Zi , Zj follow a left triangular distribution over the range [0, 3 ]. The probability of existence of a link between i and j, namely p, is
p=
1
1 21 22 43 +
1 9
−π
36
r8 −
1 2 π −
2 3
33
32 (1 + 2 )r7 + 315
1 2 6 r + 3 3
(1 + 2 ) +
16
15
1 2 23
23 (1 + 2 ) +
2
w2 − 6wl3 +
√ 3 w8l3
33 4
, and;
3. The p.d.f. of UiZj is
2(3 − z)
Pr UiZj ≤ w =
1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
fZ (z) =
Fig. 2. Occurrence probability of a link as a function of the communication range.
32 3 5 r 45 3
16 r + π 1 2 33 r3 . 9 4
Proof. Analogous to proof of Corollary 1 with the following considerations: 1. Zi and Zj are i.i.d. with p.d.f. fZ (z) = 2(l3 − z)/l3 ; 2
h(w) =
3
4l3 w−1/2 + 2w − 6l3 3l3
4
2
.
Fig. 2 shows the occurrence probability p of a link i, j, where 1 = 2 = 3 = 10 as a function of the communication range r for two different distributions considering experimental and theoretical results. UnifE stands for the curve obtained in experimental results via simulation using a Uniform distribution. UnifT identifies the curve obtained using theoretical analytical functions using a uniform distribution. The same way, TriangE represents the curve obtained via experimental results using a triangular distribution and, finally, TriangT stands for the curve generated via theoretical analytical functions using a triangular distribution. 4. Node degree Let random variable Li, j be the number of links connecting nodes i and j. Li, j is either 1 or 0. Let Di = j Li, j be the node degree of node i. Theorem 2. Consider a network n, r, 1 , 2 , 3 . Then, we have 1. The expected node degree is (n − 1) p, and; 2. The expected number of links is n(n − 1) p/2, where p is the link probability. Proof. From Theorem 1, the expected link occurrence E(Li, j ) is equal p. By definition, E(Di ) = E( i Li, j ). By linearity of ex pected values, E( i Li, j ) = i E(Li, j ), no matter if Li, j ’s are independent or not. Since each node may have n − 1 links, we have E(Di ) = (n − 1) p. On the other hand, there are potentially n(n − 1)/2 links between n nodes. Considering p as the link probability, the expected number of links is n(n − 1) p/2.
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157
Let us denote by Ci the volume covered by i nodes. In the following, we will compute the expected volume covered by n nodes in terms of expected volume covered by a single node. We can compute E[Cn ] recursively, thinking about the nodes as being positioned one by one, as a stochastic process. We know by definition that C1 = φ , which implies that E[C1 ] = E[φ ]. Now, suppose we know the expected coverage for i nodes, and we add another node at random. We can write the updated expected coverage as follows,
E[Ci+1 ] = E[Ci ] + E[Ai+1 ],
Fig. 3. Expected node degree as a function of radio radius and number of nodes.
Fig. 3 shows the expected node degree as a function of radio radius and number of nodes for 1 = 2 = 3 = 10. 5. Expected coverage
5.1. one-coverage
E[Cn ] =
1−
1−
− 16 r6 +
− 12 π r4 (2 1 + 3 1 + 2 3 ) + 1 2 2 2 3 2
(
8 5 r 1 + 2 15 1 2 2 2 3 2 4 3
+ 3 )
π r3 1 2 3
n 1 2 3 .
Proof. We denote with φ the volume covered by a single node. It is easy to see that φ = 43 π r3 , under our ball model, whenever the node’s sensory volume is properly contained in R. Obviously, when the node is close enough (for distances less than r) to the border this value is lower. In order to obtain the expected coverage of a single node we use an interesting fact observed by [7], when the sensor detection range equals the communication range: the link probability p is equal to the expected coverage of a single node divided by the volume of the deployment region, that is
p=
E[φ ] . V
(7)
From (7), we derive that the expected coverage of a single node is E[φ ] = p 1 2 3 .
E[Ci ] V − E[Ci ] =1− . V V
(9)
Replacing the result of (9) in (8), we obtain the following recurrence relation
E[Ci+1 ] = E[Ci ] +
E[Ci ] 1− E[φ ]. V
Solving this, we have
E[Cn ] = 1 − (1 − p)n V, which concludes the proof, since we know that V = 1 2 3 and
p=
Theorem 3. Let n, r, 1 , 2 , 3 be a network with n nodes deployed uniformly at random in a region R = 1 2 3 , where each node have a sensory region determined by a ball of radius r. Assuming that min {1 , 2 , 3 } ≥ 2r, the expected one-coverage volume of all nodes is
where Ai+1 is the one-coverage volume contributed by the last placed node. Considering ρ i as being the proportion of the volume contributed by the ith node to the size of the covered region, we have E[Ai ] = E[ρi φ ]. Clearly, ρ i and φ are correlated, but we can use E[ρ i ]E[φ ] as a good approximation for E[Ai ]. As nodes are uniformly distributed, ρi+1 is expected to be the proportion of the uncovered volume to the whole deployment region,
E[ρi+1 ] =
Given n nodes deployed in a region R with volume V = 1 2 3 , we are interested in computing the expected network k-coverage, that is, the volume of R covered by at least k nodes. This kind of characterizations are useful in several and diverse sensing applications. In the following we preceed to derive an expression for the simplest and more common case, namely, the expected one-coverage. Based on the aforementioned result, we derive an expression for the expected k-coverage. In this section we extend the results presented by [6], obtaining results for three dimensional networks.
(8)
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
5.2. k-coverage Theorem 4. Let n, r, 1 , 2 , 3 be a network with n nodes deployed uniformly at random in a region R = 1 2 3 , where each node have a sensory region determined by a ball of radius r. Assuming that min {1 , 2 , 3 } ≥ 2r, the expected k-coverage volume of all nodes is
E[Ckn ] = pk−1
n−k
t Nk−1 (1 − p)t E[C(n−k)−t+1 ] ,
t=0
where
p=
1
1
8 5 r (1 + 2 + 3 ) − r6 + 6 15 1 2 2 2 3 2 1 4 − π r4 (2 1 + 3 1 + 2 3 ) + π r3 1 2 3 . 2 3
Proof. Let us define generalizations for the random variables j considered in the proof of Theorem 3. We define Ci to be the size of the j-covered volume after i nodes have been deployed. Note that C0i = V and C1i = Ci . Additionally, by definij
j
tion, we have Ci = 0 whenever i < j. We consider also Ai as being the extra volume contributed by the ith placed node to
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the size of the j-covered volume, and ρi as being the proporj
tion of to φ . By definition, we have j Ai
j E[Cij ] = E[Ci−1 ] + E[Aij ].
(10)
j
We can approximate Ai , similarly as we did with Ai in the
proof of Theorem 3, by E[ρi ]E[φ ]. By definition, we have that j
ρij
is expected to be the proportion of the volume that is exactly covered by j − 1 out of i − 1 nodes to the total volume V, that is,
E[ρij ] =
j−1 j E[Ci−1 − Ci−1 ]
V
=
j−1 j E[Ci−1 ] − E[Ci−1 ]
V
.
With this, and using the expression in (10), we can write j j−1 E[Cij ] = (1 − p)E[Ci−1 ] + pE[Ci−1 ],
(11)
where p = E[φ ]/V is the link probability. The recurrence relation in (11) have a simple solution for the case i = j, E[Cii ] = pi−1 E[φ ]. In order to find an expression, we expand (11) until the case i = j. With this approach, we obtain
E[Ckn ] = pk−1
n−k
t Nk−1 (1 − p)t E[C(n−k)−t+1 ] ,
t=0
where
Nij
j Ni
⎧ 1 ⎪ ⎨
is defined recursively by
i = ⎪i(i + 1)/2 ⎩ i−( j−2) Nij−1 + t=1 t
j j j j
= 0, = 1, = 2, > 2.
6. Applications In this section we describe how the previous analysis can be useful in the design of protocols and applications. We describe an uncoordinate node scheduling for MAC protocols. MAC protocols have to cope with channel contention in order to work properly. For example, in underwater networks, due to the fact that high-frequency waves are rapidly absorbed by water, one must use acoustic channel. This channel is characterized by low and range-dependent bandwidth. Also, slow acoustic signal propagation (around 1500 m/s). Usually higher link degree indicates higher degree of channel contentions. More nodes are competing for the shared communication medium. This results in poor link performance in contention-based MAC protocols. In addition, in most scenarios, energy is a limiting resource. A solution to both problems is to have nodes sleep periodically. A decentralized solution works as follows. Each node independently alternates between active and sleep modes. The transition between states is purely stochastic. The time periods of active and sleep are exponentially distributed random variables with means μa and μs , respectively. The probability of a node being in sleep mode is ps = μs /(μa + μs ). Similarly, the probability of a node being in active mode is pa = μa /(μa + μs ). In a network with n nodes, npa nodes are expected to be active at any time. Thus, the
expected coverage and node degree can be estimated by replacing n by npa in the respective equations. This approach, although stochastic, is deterministic to set the values of parameters μa and μs for a desired network coverage. As previously described by [6], this is not possible without our analytical results. The above algorithm was proposed in [8], and also described in [6] for 2D sensor networks. Here we demonstrate the utility of our results for 3D MANETs. 7. Related work Yen and Yu [7] described the fundamental properties of a 2D wireless ad hoc networks. In their work they consider a wireless Ad Hoc network as n, r, 1 , 2 -network with n nodes, with transmission radius r, placed in a 1 × 2 rectangle area. The occurrence probability of a link (i, j) between two distinct nodes i and j is given by Eq. 12. We improve that work for 3D MANETs and considere the effects of the 3D environment. We investigate further into how these results apply to certain applications. 1 4 r 2
− 43 1 r3 − 43 2 r3 + π r2 1 2 1 2 2 2
(12)
Yen et al. [6] showed how to obtain the k-coverage in wireless sensor network for networks with and without boundaries. They describe results for coverage and studied node scheduling. Our work is more general since we are studying 3D MANETs. Kumar et al. [9] studied the problem to determine the number of sensors that achieves two goals: having nodes sleep to increase network lifetime and guarantee k-coverage. They considered three deployments scenarios: unit square, random uniform, and poisson. They obtained the critical value for the k-coverage of every point. Our results can be applied into their application that randomly selects a node to sleep and guarantees one-coverage for sensing applications. Coverage and Connectivity in 3D Networks have been explored, [10], but do not provide the mathematical results we presented. [13] provide a k-coverage scheduling protocol. In [14], the authors evaluate a coverage compensation for Wireless Sensor Networks. In [15], Fei et al. gave a bio-inspired coverage scheduling. In [16] and [17] the authors consider irregular sensing ranges. In [18], it is presented a coverage scheme based on local information. Currently, most of the proposals in the literature do not provide the same mathematical analysis. Mansoor and Ammari [11] summarizes the work conducted in the domain of coverage and connectivity in 3D Wireless Sensor Networks(WSNs). As described by the authors, 3D WSNs require more complex and computationally intensive analysis. They present different coverage planning and placement strategies. We provide new mathematical analysis. In the theory of Random Graphs, we study graphs that are generated by some random process. A random graph is a graph in which properties such as the number of graph vertices, graph edges, and connections between them are determined in some random way [12]. Percolation theory characterizes the connectedness of random graphs, focusing on the
L.F.M. Vieira et al. / Ad Hoc Networks 37 (2016) 153–159
asymptotically behavior. Our work differ in that since we analyze a 3D deployment, and focus in network properties with application, such as coverage. 8. Conclusion In this work, we considered the fundamental properties of 3D Networks. We showed how to compute the link probability, node degree, and expected coverage. Those are necessary and useful in real applications. Furthermore, we were able to derive analytical results for those properties. The results presented are useful for many applications. In particular, we showed one such an application: reducing MAC contention. The approach of increasing duty cycle reduces energy consumption and increases network lifetime. Estimating the necessary network sensing coverage can reduce overall costs by reducing the number of nodes in the deployment and reducing maintenance costs during network lifetime.
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[12] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, Wiley, 1996. [13] X. Fei, A. Boukerche, F.R. Yu, A pomdp based k-coverage dynamic scheduling protocol for wireless sensor networks, in: GLOBECOM, IEEE, ISBN, 2010. [14] X. Fei, A. Boukerche, A performance evaluation of a coverage compensation based algorithm for wireless sensor networks, in: Proceedings of the 11th International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems, MSWiM ’08, ACM, New York, NY, USA, 2008, pp. 109–116. 978-1-60558-235-1. [15] X. Fei, S. Samarah, A. Boukerche, A bio-inspired coverage-aware scheduling scheme for wireless sensor networks, in: Proceedings of the 2010 IEEE International Symposium on Parallel Distributed Processing, Workshops and Phd Forum (IPDPSW), 2010, pp. 1–8. [16] X. Fei, A. Boukerche, R.B. Araujo, Irregular sensing range detection model for coverage based protocols in wireless sensor networks, in: Proceedings of the 28th IEEE Conference on Global Telecommunications, GLOBECOM’09, IEEE Press, Piscataway, NJ, USA, 2009, pp. 5699– 5704. 978-1-4244-4147-1. [17] A. Boukerche, X. Fei, A coverage-preserving scheme for wireless sensor network with irregular sensing range, Ad Hoc Net. 5 (8) (2007) 1303– 1316. [18] A. Boukerche, X. Fei, R.B. de Araujo, An optimal coverage-preserving scheme for wireless sensor networks based on local information exchange, Comput. Commun. 30 (14–15) (2007) 2708–2720.
References [1] J. Kong, J.-H. Cui, D. Wu, M. Gerla, Building underwater ad-hoc networks and sensor networks for large scale real-time aquatic applications, in: Proceedings of the IEEE Military Communications Conference (MILCOM’05), Atlantic City, NJ, 2005, pp. 1535–1541. [2] T.S. Rappaport, Wireless communications: principles and practice, second, Prentice Hall, 2002. [3] M. Stojanovic, On the relationship between capacity and distance in an underwater acoustic communication channel, ACM SIGMOBILE Mob. Comput. Commun. Rev. 11 (2007) 34–43. 1559–1662. [4] M. Ainslie, P. Dahl, C.d. Jong, R. Laws, Practical spreading laws: the snakes and ladders of shallow water acoustics, in: Proceedings of the 2nd International Conference and Exhibition on Underwater Acoustics, Rhodes, Greece, 2014. [5] M. Stojanovic, J. Preisig, Underwater acoustic communication channels: propagation models and statistical characterization, IEEE Commun. Mag. 47 (1) (2009) 84–89. 0163–6804. [6] L.-H. Yen, C.W. Yu, Y.-M. Cheng, Expected k-coverage in wireless sensor networks, Ad Hoc Net. 4 (5) (2006) 636–650. [7] L.-H. Yen, C.W. Yu, Link probability, network coverage, and related properties of wireless ad hoc networks, in: Proceedings of the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS’04), 2004, pp. 525–527. [8] M. Liu, C. Hsin, Network coverage using low duty-cycled sensors: random and coordinated sleep algorithms, in: Proceedings of the Third International Symposium on Information Processing in Sensor Networks (IPSN’04), Berkeley, CA, 2004, pp. 433–442. [9] S. Kumar, T.H. Lai, J. Balogh, On k-coverage in a mostly sleeping sensor network, in: Proceedings of the 10th Annual International Conference on Mobile Computing and Networking (MobiCom ’04), Philadelphia, PA, USA, 2004, pp. 144–158. 1-58113-868-7. [10] S.M.N. Alam, Z.J. Haas, connectivity in three-dimensional networks, in: Proceedings of the 12th Annual International Conference on Mobile Computing and Networking, MobiCom ’06, ACM, New York, NY, USA, 2006, pp. 346–357. 1-59593-286-0. [11] U. Mansoor, H. Ammari, Coverage and connectivity in 3d wireless sensor networks, in: H. Ammari (Ed.), The Art of Wireless Sensor Networks, Signals and Communication Technology, Springer Berlin Heidelberg, 2014, pp. 273–324. 978-3-642-40065-0.
Luiz F. M. Vieira is an Assistant Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate and M.S. degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of California Los Angeles (UCLA). His research interest is in Computer Networking.
Marcelo G. Almiron is a post-doc. He received his Ph.D. degree in Computer Science from the Universidade Federal de Minas Gerais.
Antonio A. F. Loureiro is a Full Professor at the Computer Science Department, Universidade Federal de Minas Gerais (UFMG). He received his undergraduate degree at the Universidade Federal de Minas Gerais in Belo Horizonte, and Ph.D. degree in Computer Science from the University of Britich Columbia (UVC). His research interest is in Computer Networking.