A STOCHASTIC ANALYSIS OF A GREEDY ROUTING SCHEME IN ...
SIAM J. APPL. MATH. Vol. 70, No. 7, pp. 2214–2238
c 2010 Society for Industrial and Applied Mathematics
A STOCHASTIC ANALYSIS OF A GREEDY ROUTING SCHEME IN SENSOR NETWORKS∗ H. P. KEELER† AND P. G. TAYLOR† Abstract. We address the stochastic characteristics of a recently proposed greedy routing scheme. The behavior of individual hop advancements is examined, and asymptotic expressions for the hop length moments are obtained. The change of the hop distribution as the sink distance is varied is quantified with a Kullback–Leibler analysis. We discuss the effects of the assumptions made, the inherent dependencies of the model, and the influence of a sleep scheme. We propose a renewal process model for multiple hop advancements and justify its suitability under our assumptions. We obtain the renewal process distributions via fast Fourier transform convolutions. We conclude by giving future research tasks and directions. Key words. multihop, geometric routing, sleep scheme AMS subject classifications. 90B15, 68M12, 60K20 DOI. 10.1137/080725684
1. Introduction. Recent developments in the fields of wireless communications and computing have produced the early examples of sensor networks. Sensor networks consist of large collections of small electronic devices known as sensor nodes. Each node has the ability to collect, process, and transmit data via neighboring nodes to a main communication station known as a sink. Although this research area is still in its infancy, sensor networks show great promise for applications in a number of areas where remote data retrieval is required, including medical, security, and environmental monitoring [1, 5]. Sensor nodes are usually scattered over a region known as the sensor field. Each node is equipped with a limited power source, which results in the service lifetime of sensor networks being a crucial consideration. To maximize this lifetime, stochastic power schemes have been suggested that entail having a random subset of the sensor nodes existing in a low energy-consuming sleep state at any given time [14]. During its sleep state it may or may not be possible for the node to collect energy depending on the surrounding environmental conditions [5]. Since a stochastic sensor network has a random subset of nodes awake at any given time, the network topology continually varies, resulting in the need for more innovative data routing methods. Many routing methods proposed for sensor networks are instances of geometric or position-based routing algorithms [7, 15]. The function of these algorithms is based on the assumption that each node knows its geographical position in relation to the sink, and the location of neighboring nodes within its transmission radius. These assumptions are reasonable when sensor nodes have a global positioning system [5]. Alternatively, algorithms have been proposed to calculate their geographical positions [4, 9] and to discover their neighbors [13, 17], thus enabling geographic routing in sensor networks. ∗ Received
by the editors May 30, 2008; accepted for publication (in revised form) January 13, 2010; published electronically April 21, 2010. This research was supported by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. http://www.siam.org/journals/siap/70-7/72568.html † Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia ([email protected], [email protected]). 2214
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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A commonly used geometric routing method is to have the source node look only at nodes that are closer to the sink in a region known as the total feasible region. The message is forwarded to the node that is closest to the sink, and this step is repeated until the message finally reaches the target sink. This approach can serve either as a simple greedy routing method by itself or as the basis for more sophisticated routing methods [3, 6, 8, 10, 19]. Here, we restrict ourselves to analyzing the simple greedy routing method suggested by Zorzi and Rao [19] known as Geographic Random Forwarding (GeRaF), while noting that the analysis can apply to other greedy-based routing methods. We extend the work of Zorzi and Rao by investigating the distribution of the number of hops required for a message to reach a sink. In particular, under a simplifying assumption, Zorzi and Rao effectively used renewal models to bound mean message advancements. We extend this work by deriving bounds on the distributions of multihop advancement and discuss their accuracy. Furthermore, we examine Zorzi and Rao’s simplifying assumption by developing a mathematical model and explaining the inherent stochastic properties of the problem. We discuss two forms of dependence that arise in the multihop advancement problem and how they stochastically affect the advancement of a message over multiple hops. We devise approximations for hop distributions and moment expressions. We discuss the effects of a simple energy-saving sleep scheme and its immediate influence on the aforementioned model dependencies. In conclusion, we discuss the various future research directions in developing and examining routing models in sensor networks. 2. Mathematical model. We present a mathematical model of a sensor network consisting of an ensemble of randomly positioned sensor nodes. We assume that the nodes are scattered according to a two-dimensional homogeneous Poisson process over a finite region and that at any given time a random number of nodes are in sleep mode while the remaining are in awake mode. In the model presented here we consider only the transmission of data and not the actual data-collection range, otherwise known as the coverage of the network. We assume that nodes communicate data radially and that a node’s transmission radius clearly cuts off at some distance, which implies that a node can relay data to another node only when it is within the forwarding node’s transmission radius. The transmission radius is set to the constant r. Let the constant α be the density of nodes scattered over a sensor field. We set p as the probability of a node being awake at any given time and assume that separate transmission attempts sample independent thinnings of the underlying Poisson process. It follows that the number of awake nodes in an area A is a homogeneous Poisson random variable NA and has the probability mass function (2.1)
P(NA = n) =
(λA)n −λA e , n!
where the new density parameter λ = pα represents the mean number of awake nodes per unit area. In our model there is a positive probability that no nodes, regardless of their state, lie within the feasible region of a forwarding node. Hence, there is a positive probability that a message will never reach the sink. This event never occurs in the model of Zorzi and Rao [19]. In their model the locations of the nodes are resampled at each hop, and if there is no node in the feasible region in one sample, there is
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H. P. KEELER AND P. G. TAYLOR
Aγ (u)
2φγ (u)
2ψγ (u) u
r
XS
X0 γ
Fig. 3.1. Feasible region of nodes located a distance of u or less from the sink.
a chance that there will be a node in the region in the next sample. In fact, with probability one, there will eventually be a forwarding node, and the message will reach the sink. As α approaches infinity and p approaches zero with λ held constant, our model will closely approximate that of Zorzi and Rao. We informally describe our definition of greedy routing by first introducing some notation. Let the random points X0 and XS denote the respective locations of the source node and the target sink. Under greedy routing the next forwarding node is located at X1 , which is the neighboring node of point X0 that is closest to the sink. This procedure is repeated until the message reaches the sink. More formally, let the point M (X) be the next message hop from a node located at point X. Let N (X) be the set of neighboring nodes of a node at point X with each neighboring node location n ∈ N (X) iff |n − X| ≤ r, where |.| is the Euclidean metric. For the integer i ≥ 0, greedy routing [2] is defined such that M (XS ) = XS and for all Xi = Xs , M (Xi ) = Xi+1 iff Xi+1 ∈ N (Xi ) and (2.2)
|Xi+1 − XS | =
min
n∈N (Xi )
|n − XS |.
Because the underlying node distribution is a Poisson process, the minimum is unique with probability one. For more details on routing definitions in a stochastic geometry setting, see the recent monograph [2, page 110], where greedy routing is called “best hop” routing. 3. Single hop distribution. Let the parameter γ be the distance between a forwarding node and the sink (refer to Figure 3.1). The sink distance is set to γ = when the forwarding node is the source of the message. The total feasible region of a node is the area formed by the intersection of the two circles of radii r and γ centered at the source node and the sink, respectively, thus forming a lens. After a single hop, let the random variable U be the distance between the sink and the new forwarding node chosen by the greedy routing algorithm. To obtain the distribution of U , we employ a nearest neighbor approach. Consider the feasible region of the source node where potential receiving nodes can be located at some distance u or less from the sink (shaded region in Figure 3.1). For > r, the probability that no receiving nodes exist in this feasible region of area A (u) is equivalent to the probability that U is strictly greater than u. The complement of this probability yields the distribution ⎧ ⎨ 1 − e−λA (u) , − r ≤ u < , (3.1) F (u) = 1, u ≥ , ⎩ 0, u < − r.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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Since there is a positive probability that no nodes lie within the feasible region, F (u) is a mixed discrete-continuous distribution, with a discontinuity at u = . We differentiate the distribution where it is absolutely continuous to give the probability density f (u) = λA (u)e−λA (u) ,
(3.2)
− r ≤ u < ,
where the prime denotes differentiation with respect to u. We need an expression for the general feasible area function Aγ (u) as a function of u for a given sink distance γ. Define the angles of the two intersecting sectors as 2φγ and 2ψγ , and see that 2 r + γ 2 − u2 φγ (u) = arccos (3.3) , 2rγ 2 u + γ 2 − r2 ψγ (u) = arccos (3.4) . 2uγ It follows that (3.5)
Aγ (u) = r2 φγ (u) + u2 ψγ (u) − uγ sin ψγ (u),
γ − r ≤ u ≤ γ,
and the derivative is given by Aγ (u) = 2uψγ (u).
(3.6)
Let C = γ − U represent the distance advanced by the message toward the sink when the originating node is at a distance γ, and let F¯γ (c) denote the distribution of C, namely, ⎧ ⎨ e−λAγ (γ−c) , 0 < c ≤ r, (3.7) F¯γ (c) = 1, c > r, ⎩ 0, c ≤ 0, which is discontinuous at c = 0. The first moment of C is r (3.8) E(C) = r − e−λAγ (γ−c) dc, 0
and the second moment is (3.9)
2
2
E(C ) = r − 2
r
ce−λAγ (γ−c) dc.
0
3.1. Asymptotic analysis. The form of the area function (3.5) implies that it is not feasible to obtain analytic expressions for integrals such as those in the moment equations (3.8) and (3.9). As an alternative, we present an asymptotic result for these moment expressions. Theorem 3.1. For γ > r, under greedy routing the first hop moment E(C) ∼ r −
(3.10)
Γ(5/3) 2/3
,
(λa0 )
and the second hop moment (3.11)
E(C 2 ) ∼ r2 − 2r
Γ(5/3) 2/3
(λa0 )
+
Γ(7/3) 4/3
,
(λa0 )
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H. P. KEELER AND P. G. TAYLOR
as the node density λ → ∞, where Γ(·) is the gamma function, and
2r a0 = γ(γ − r)
1/2
4(γ − r) . 3
Proof. Expand the area expression at u = γ − r such that (3.12)
Aγ (u) ∼ a0 (u − γ + r)3/2 + a1 (u − γ + r)5/2 + O(u)7/2 as u → γ − r,
and introduce the change of variable t = u − γ + r = r − c, with a slight abuse of notation, which leads to the area function expansion (3.13)
Aγ (t) ∼ a0 t3/2 + a1 t5/2 + O(t7/2 ) as
t → 0,
where the first two terms a0 and a1 are, respectively,
(3.14) (3.15)
1/2 2r 4(γ − r) a0 = , γ(γ − r) 3 1/2 2r (−3γ 2 + 9γr + r2 ) a1 = . γ(γ − r) 15γr
Consider integrals of the form (3.16)
b
I(λ) =
tk e−λA(t) dt,
a
where A(t) > A(a) for all t ∈ (a, b), and asymptotically (3.17)
A(t) ∼ a0 (t − a)μ
as t → a.
Applying a generalized version of Laplace’s method (see Wong [18, page 58]) to this integral gives (3.18)
I(λ) ∼
Γ(τ ) μ(λa0 )τ
as λ → ∞,
where (3.19)
τ=
2(k + 1) . 3
With a = 0 and μ = 3/2, result (3.10) readily follows from (3.18), and, similarly, for the second moment expression r (3.20) E(C 2 ) = r2 − 2 (1 − t)e−Aγ (t) dt, 0
result (3.11) follows.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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1 Analytic Solution First Order Second Order
0.9 0.8 0.7
Fγ(u)
0.6 0.5 0.4 0.3 0.2 0.1 0
9
9.2
9.4
9.6
9.8
10
u
Fig. 3.2. Approximation of distribution of U (γ = 10 and λ = 3).
1 Numerical Asymptotic 0.9
0.8
E(C)
0.7
0.6
0.5
0.4
1
2
3
4
5
λ
6
7
8
9
10
Fig. 3.3. Approximation of E(C) by the generalized Laplace method ( = 10).
For all numerical results, unless stipulated otherwise, the length values will be rescaled by the transmission radius; thus each node has a unit transmission radius. The two-term area expansion (3.12) appears to give reasonably accurate results regardless of the value of γ. This accuracy carries over when the above expansion is substituted into (3.1) to yield an approximation to the hop advancement distribution, which is almost indistinguishable from the true distribution (see Figure 3.2) when γ = 10 and λ = 3. The closed-form moment approximations (3.10) and (3.11) agree well with numerical results for a practical range of λ (see Figures 3.3 and 3.4). The approximations start to break down for λ ∼ 1, particularly for the second moment (in Figure 3.4). However, the approximations give sufficiently accurate results for the first and second moments of C. 3.2. Effect of sink distance. The area function increases with respect to the parameter γ, which implies that hop lengths increase stochastically with γ. This area
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H. P. KEELER AND P. G. TAYLOR
0.9 Numerical Asymptotic 0.8
0.7
2
E(C )
0.6
0.5
0.4
0.3
0.2
1
2
3
4
5
λ
6
7
8
9
10
Fig. 3.4. Approximation of E(C 2 ) by the generalized Laplace method ( = 10).
behavior can be verified geometrically by observing that the arc of the intersecting sector of radius u flattens as γ increases; hence the total feasible region approaches a semicircle (refer to Figure 3.1). A simple geometrical argument can be used to obtain the area of this semicircle, which is the supremum of the area function. Alternatively, we consider the derivative of the area function (3.6) with respect to c = γ − u, that is, c2 − r 2 , (3.21) Aγ (γ − c) = 2 (γ − c) arccos 1 + 2γ (γ − c) and in the limit as γ approaches infinity we obtain
(3.22) A∞ (c) = −2 r2 − c2 , which is the derivative of the supremum of the area function; thus
0 ≤ c ≤ r. (3.23) A∞ (c) = r2 arccos(c/r) − c r2 − c2 , This is the feasible area function under the Most Forward within Radius routing model proposed by Takagi and Kleinrock [16] provided that messages travel only forward. Kleinrock and Silvester [11] stated earlier that the difference in the two area functions is negligible provided that γ is sufficiently large. The difference in the hop distribution parameterized by two different values, γ1 and γ2 , of γ can be expressed in terms of a goodness-of-fit measure. We choose the Kullback–Leibler divergence [12], which measures the difference between the “true” probability measure P and a proposed probability measure Q over some sample space Ω. In general, the Kullback–Leibler divergence, defined as dQ log (3.24) D(P, Q) = − dP, dP Ω is nonnegative, and zero when the two measures are identical. Consequently, the mixed discrete-continuous hop distribution leads to the expression ¯ ¯ r Fγ2 (0+ ) fγ2 (c) + ¯ ¯ dc + Fγ2 (0 ) log ¯ fγ2 (c) log ¯ , (3.25) D(γ1 , γ2 ) = Fγ1 (0+ ) fγ1 (c) 0
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
0.18
2221
=3 =6 =∞
0.16 0.14
D(, γ)
0.12 0.1 0.08 0.06 0.04 0.02 0
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Fig. 3.5. Kullback–Leibler divergence quantifying the change in the distribution (λ = 3).
where the probability of a routing void or zero advancement is (3.26)
F¯γ (0+ ) = lim F¯γ ( )
(3.27)
=e
→0 −λAγ (γ)
.
We evaluated the integral in (3.25) numerically to determine how the hop distribution changes when γ1 = and γ2 = γ are varied (refer to Figure 3.5). We observed that D(, γ) is virtually zero for the majority of γ, and then it increases markedly for γ ≤ 3. This implies that the algebraic dependence of F¯γ (c) on γ significantly affects the distribution of the remaining hop lengths only after this point. The negative consequence of this is that hop lengths cannot be assumed to be independent of the current sink distance for γ ≤ 3. Conversely, the hops up to this point can be modeled reasonably well with the first hop distribution. Moreover, for finite ≥ 3, the divergence D(, γ) is approximately equal to D(∞, γ), which suggests that the simpler area expression (3.23) may be used. 4. Multihop distribution. 4.1. Dependence between successive hops. We introduce some notation to describe the locations of the forwarding nodes. Specifically, we introduce indexing for each sink distance so that U0 = and U1 = U , and, after i hops, the remaining distance to the sink is Ui . Let the random variable Θi be the angle between the ith node and the previous node in relation to the sink. We note that Ui is a global coordinate (in relation to the sink) and Θi is a local coordinate (in relation to the previous node). We assign the point Xi = (Ui , Θi ) to the ith forwarding node. The source (or zeroth) node corresponds to the point X0 = (U0 , 0) = (, 0). A message travels i hops along
i = (X0 , X1 , . . . , Xi ). a path that corresponds to a sequence of random points X The decrease in the sink distance between ith and (i−1)th hops is Ci = Ui−1 −Ui . As we saw above, Ci depends on the forwarding node’s sink distance Ui−1 . We refer to this dependence of the distribution on the sink distance simply as the sink dependence. Zorzi and Rao [19] pointed out that each Ci+1 is stochastically dominated by Ci in the sense that, for i ≥ 0, there exists the stochastic ordering (4.1)
P(Ci+1 > c) ≤ P(Ci > c),
c ∈ (0, r).
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H. P. KEELER AND P. G. TAYLOR
RA X1 X0
U1 Θ1 RB
Fig. 4.1. No awake nodes in the intersection region RB during the first message relay.
This follows directly from the observation in section 3.2 that the feasible area function increases with respect to the sink distance. The Kullback–Leibler analysis suggests that the sink dependence is negligible for the majority of hop advancements. However, after the first hop there is another source of dependence between the random variables Ci . To simplify our explanation of this phenomenon we assume that all nodes within the forwarding node’s feasible region are in their awake state. Then there can be no nodes closer to the sink in the feasible region of the original node other than the forwarding node. Hence, at the second hop there is a region where there can be no potential forwarding nodes (region RB in Figure 4.1). If this intersection region did contain a node, then this node would have been chosen as the new forwarding node at the previous hop instead of the one that was chosen. This implies that C2 is dependent on both U1 and Θ1 . We refer to this dependence on both the sink distance and the sink angle as path dependence and to the hop model that includes path dependence simply as the dependent model. Conversely, the independent model includes only the sink dependence. We have assumed for the purpose of our discussion of path dependence that all nodes within the forwarding node’s feasible region were in their awake state. However, the analysis can be extended to the situation when either a stochastic or deterministic power scheme is in operation. Then, there is a positive probability that asleep nodes found within the intersection of feasible regions when a message is received will be awake when the transmitting node is choosing the next forwarding node. Specifically, if α is the actual density of nodes and p is the probability of each node being awake, an event which is sampled independently at each time step, then the nodes in the intersection area will have a density α(1 − p)p, whereas the nodes in the nonintersecting area will have a density αp. Hence, the path dependence decreases as a new ensemble of possible forwarding nodes is sampled, implying the dependent model will more closely resemble the independent model. The path dependence will be reduced further as the number of nodes increases and the probability of them being awake decreases. In the limit as α → ∞ and p → 0 with λ = αp held constant, the dependence disappears entirely, and we have the independent model of Zorzi and Rao [19]. Future work lies in investigating the values of p and α for which the dependent model closely resembles the more tractable, independent model. We use F and G to denote the distributions of Ui under the independent and dependent models, respectively. The distribution of Ui+1 under the independent model is dependent only on the sink distance of the current forwarding node, and thus we can write (4.2)
Fi (ui+1 ) = PI (Ui+1 ≤ ui+1 |Ui = ui ),
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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while under the dependent model the distribution is dependent on the entire message path, and thus we write (4.3)
i = xi ), Gi (ui+1 ) = PD (Ui+1 ≤ ui+1 |X
where the subscripts I and D indicate probability measures under the two models.
i = xi , We stress that the distributions (4.2) and (4.3) depend on Ui = ui and X respectively, but for convenience, we have omitted this in the subscript notation. Further shorthand notation is used to denote the feasible area under the independent
i (ui+1 ). and dependent models, respectively, as Ai (ui+1 ) and A The area expression under the independent model is always given by the original area equation (3.5), and hence, the distribution and probability density of Ui+1 are obtained by setting = ui in (3.1) and (3.2). Under the dependent model, if the feasible area function is given after i hops, the conditional sink distribution immediately follows: ⎧ ⎨ 1 − e−λAi (ui+1 ) , ui − r ≤ u < ui , (4.4) Gi (ui+1 ) = 1, u ≥ ui , ⎩ 0, u < ui − r. and its probability density is defined on the region where it is absolutely continuous (4.5)
i (ui+1 )e−λAi (ui+1 ) . gi (ui+1 ) = λA
For Ci , we adopt notation similar to that used for the sink distance random ¯ denote the hop distributions under the two models. The variables; hence F¯ and G complement of the sink distribution yields the conditional hop distribution under both the independent and dependent models, the latter being ⎧ ⎨ e−λAi (ui −ci+1 ) , 0 < ci+1 ≤ r, ¯ (4.6) Gi (ci+1 ) = 1, ci+1 > r, ⎩ 0, ci+1 ≤ 0. and its probability density is defined on the region where it is absolutely continuous (4.7)
(ci+1 )e−λA i (ui −ci+1 ) . g¯i (ci+1 ) = λA i
Let Ii (ui+1 ) ⊂ R2 be the feasible region of the ith forwarding node as a function of ui+1 under the independent model. The set representing the feasible region under the dependent model follows by excluding the intersections of previous feasible regions; thus Di (ui+1 ) = Ii (ui+1 ) \ ∪j=i−1 j=0 Ij (uj+1 ). Methods of calculating the area of D1 (u2 ) and D2 (u3 ) are given in the appendix (sections C.1 and C.2). We note that analytic area expressions after two hops are intractable given the number of possible geometric configurations.
1 = (X0 , X1 ), the feasible region for the second hop For any two node path X under the dependent model is clearly a subset of the region under the independent model; hence (4.8)
1 (u1 − c2 ) ≤ A1 (u1 − c2 ), A
c2 ∈ [0, r].
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H. P. KEELER AND P. G. TAYLOR
This inequality allows us to compare the hop distributions (3.7) and (4.6) under the two models, which leads to the stochastic ordering (4.9)
1 = x1 ) ≤ PI (C2 > c|X
1 = x1 ), PD (C2 > c|X
c ∈ (0, r),
on the conditional distributions after one hop. Hence, for any two node path, C2 under the dependent model is stochastically dominated by C2 under the independent model. We observed that there is a stochastic ordering (4.1) on single hops stemming from the sink dependence. Furthermore, there is a conditional stochastic ordering on the second hop owing to the path dependence (4.9). We now investigate the possibility of a stochastic ordering for the multihop case. Subsequently, under the dependent model we introduce the distribution
i ). Gi (ui+1 , θi+1 ) = PD (Ui+1 ≤ ui+1 , Θi+1 ≤ θi+1 |X The respective angles −ψui (ui+1 ) and ψui (ui+1 ) denote the lower and upper values of the θi+1 domain. The distribution Gi (ui+1 , θi+1 ) is absolutely continuous on the domain Di (ui+1 ). However, it has a mass at the point Xi corresponding to the event when there are no nodes in the feasible region. In the dependent case, this means that there will never be a closer node in the feasible region, and hence, the sink is unreachable. We denote the density function of Gi (ui+1 , θi+1 ) by gi (ui+1 , θi+1 ), which is defined on the region where the distribution is absolutely continuous. Under the dependent model the joint probability density of Ui and Θi is (4.10)
gi (ui+1 , θi+1 ) = λDi (ui+1 , θi+1 )ui+1 e−λAi (ui+1 ) .
The spatially dependent density function is λDi (ui+1 , θi+1 ) = λIDi (ui+1 , θi+1 ) and the indicator function of the dependent feasible region is 1, (ui+1 , θ0i ) ∈ Di , IDi (ui+1 , θi+1 ) = 0 otherwise. The global angular coordinate θ0i is simply the angle between the source node and the ith forwarding node in relation to the sink. The indicator function gives a zero joint probability density in the regions where there is zero node density owing to the path dependence. Under the independent model we introduce a similar distribution:
i ). Fi (ui+1 , θi+1 ) = PI (Ui+1 ≤ ui+1 , Θi+1 ≤ θi+1 |X The distribution Fi (ui+1 , θi+1 ) is absolutely continuous on the domain Ii (ui+1 ), and, similarly to the dependent model, it has a mass at the point Xi . However, even if there is zero advancement, that is, (ui+1 , θi+1 ) = (ui , θi ), it is still possible that (ui+2 , θi+2 ) = (ui+1 , θi+1 ) since under the independent model we effectively resample node positions at every hop so there is a possibility that there will be a node in the feasible region at the next resampling. We also note that Fi (ui+1 , θi+1 ) depends
i−1 . Denote the only on the current point Xi and not on the previous points X corresponding density function by fi (ui+1 , θi+1 ), which is defined on the region where the distribution is absolutely continuous.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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Write the distribution and the density for the remaining distance to the sink for the dependent model, respectively, as ψu (ui+1 ) i Gi (ui+1 ) = Gi (ui+1 , θi+1 )dθi+1 , −ψui (ui+1 )
gi (ui+1 ) =
ψui (ui+1 )
−ψui (ui+1 )
gi (ui+1 , θi+1 )dθi+1 ,
and similarly for the independent model: ψui (ui+1 ) Fi (ui+1 ) = Fi (ui+1 , θi+1 )dθi+1 , −ψui (ui+1 )
fi (ui+1 ) =
ψui (ui+1 )
−ψui (ui+1 )
fi (ui+1 , θi+1 )dθi+1 .
The stochastic dominance inequality after the first hop (4.9) is equivalent to (4.11)
G1 (u2 ) ≤ F1 (u2 ),
u1 − r ≤ u2 < u1 .
This result leads to a stochastic bound after two hops between the respective models. Lemma 4.1. For i = 2 hops, the sink distances Ui under the two models behave such that (4.12)
PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ).
Proof. This is equivalent to the inequality (4.13) PD (Ui ≤ ui ) = G0 (ui ) + (4.14) (4.15)
≤ F0 (ui ) +
u− 0
max(ui ,u0 −r) u0 max(ui ,u0 −r)
ψu0 (u1 ) −ψu0 (u1 )
g0 (u1 , θ1 )G1 (ui )dθ1 du1
F1 (ui )dF0 (u1 )
= PI (Ui ≤ ui ),
where we denote u− i as the left limit: u− i = lim (ui − ). →0
The zero-advancement probability (that is, the upper limit u0 ) is not included in the expression (4.13), as such an event under the dependent model implies that it is not possible for any further advancement toward the sink in future hops. The Riemann–Stieltjes-type integral (4.14) has the upper limit u0 instead of u− 0 , as the message may not advance during one hop and then advance during the next under the independent model. Under the independent model, the joint density and distribution of the first hop are identical to those under the dependent model. Thus the proof immediately follows; see Appendix B for details. It is tempting to conjecture that the two-hop stochastic order (4.12) can be extended to the multihop case, which would lead to the following conjecture.
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H. P. KEELER AND P. G. TAYLOR
Conjecture 1. For any number of hops i, the sink distances Ui under the two models behave such that PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ). The conjecture hinges upon the truth of the inequality (4.16)
PD (Ui ≤ ui ) = G0 (ui ) +
u− 0
+
u− 0
u− i−1
u0
max(ui ,u0 −r) ui
+
ψui−2 (ui−1 )
g0 (u1 , θ1 ) . . . Gi−1 (ui ) . . . dθ1 du1
f0 (u1 , θ1 )F1 (ui )dθ1 du1 + · · ·
max(ui ,ui−1 −r)
ψui−2 (ui−1 ) −ψui−2 (ui−1 )
f0 (u1 , θ1 ) . . . Fi−1 (ui ) . . . dθ1 du1
F1 (ui )dF0 (u1 ) + · · ·
... max(ui ,u0 −r)
g0 (u1 , θ1 )G1 (ui )dθ1 du1 + · · ·
−ψui−2 (ui−1 )
...
max(ui ,u0 −r) −ψu0 (u1 ) u0
≤ F0 (ui ) +
ψu0 (u1 )
ψu0 (u1 )
+
max(ui ,ui−1 −r)
−ψu0 (u1 )
−ψu0 (u1 )
u− i−1
max(ui ,u0 −r)
u− 0
ψu0 (u1 )
...
−ψu0 (u1 )
max(ui ,u0 −r)
≤ F0 (ui ) +
max(ui ,u0 −r)
ψu0 (u1 )
u− 0
max(ui ,ui −r)
Fi−1 (ui )dFi−2 (ui−1 ) . . . dF0 (u1 )
= PI (Ui ≤ ui ), where we observe that fi (ui+1 , θi+1 ) is independent of θi . For i ≥ 2, we observe that gi (ui+1 , θi+1 ) is greater than or equal to fi (ui+1 , θi+1 ) on the region where it is positive, but the range of θi over which fi (ui+1 , θi+1 ) is integrated into (4.16) is larger. The trade-off between these two competing influences is delicate.
2 (u3 ) can be calculated; see section C.1 for details. For The area function A three or fewer hops, the conjecture can be tested via regular numerical integration methods. The results depicted in Figure 4.2 provide support to the suggestion that PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ), but we have not been able to prove it. For higher hop numbers, deriving an analytic area expression is intractable, and regular numerical integration is too slow. Numerical investigation reveals that the difference between the two distributions is small for large λ values. This implies that the independent model is a good approximation for the dependent model at high node density. However, it is possible that even a small difference in the two models will accumulate over many hops. We stress that it is significantly easier to calculate the distributions using the independent model compared to the dependent model. If our conjecture is true, the independent model will always provide a stochastic lower bound for the hop advancement distributions irrespective of how close the two distributions are. 4.2. Distribution of the total number of hops. We give some motivation for calculating the distribution of Ci in general. Let the random variable N represent the total number of hops required for a message to reach the sink. We define the
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2227
1.2 Independent Dependent 1
P(U3≤ u3)
0.8
0.6
0.4
0.2
0
7
7.5
8
8.5 u
9
9.5
10
3
Fig. 4.2. Comparison of PD (Ui ≤ ui ) and PI (Ui ≤ ui ) (for = 10 and λ = 3).
random variable (4.17)
Zn =
n
Ci
i=1
to represent the distance advanced by a message in n hops assuming the parametric dependence discussed in section 3.2. We are interested in a means of quickly calculating the distribution of Zn owing to its inherent relation with N . This is encapsulated in the following simple result. Proposition 4.1. For a sink distance , the number of message hops N has the distribution (4.18) (4.19)
PD (N = 1) = 1 PD (N ≤ n) = 1 − PD (Zn−1 < − r)
∀n ≥ 1
if ≤ r, if > r.
Proof. Clearly, when ≤ r, only a single hop is needed to reach the sink. For > r, P(N > n) = PD (Un−1 > r). Observing that Un−1 = − Zn−1 , we see that P(N > n) = PD (Zn−1 < − r). The distribution of the random variable N arguably presents more critical information on the performance of the sensor network. For instance, if transmission errors are introduced at a given relay node with probability pE , then it is elementary to calculate the probability that errors are introduced along a path. However, the sink dependence of γ on F¯γ (c) implies that high dimensional integrals need to be evaluated in order to calculate the distribution of Zn ; more details will be given in the next section. It is possible, however, under the independent model, to obtain bounds for the average number of hops required to reach a sink given an initial sink distance (see Zorzi and Rao [19]). We develop this idea further by defining
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2228
H. P. KEELER AND P. G. TAYLOR
n the general random variable Yn = i=1 Ci (γ), where Ci (γ) is a sequence of independent and identically distributed random variables which represent hop advancements
with a fixed sink distance γ. Let the random variable Sn = ni=1 Ci (r), noting that, for any γ > r, Ci (r) is stochastically smaller than Ci (γ). Thus, Sn is stochastically smaller than Yn , which provides a stochastic lower bound z ∈ (0, nr). P(Sn > z) ≤ PI (Zn > z),
n Let the random variable Tn = i=1 Ci (), which is the sum of n random variables that have the distribution of the first (that is, the stochastically largest) hop. The stochastic dominance inequality (4.20)
P(Sn > z) ≤ PI (Zn > z) ≤ P(Tn > z),
z ∈ (0, nr),
easily follows. Both Sn and Tn form renewal processes, and their distributions can be calculated as the n-fold convolutions of the hop distributions: P(Yn ≤ z) = F¯γ (c1 ) ∗ F¯γ (c2 ) ∗ · · · ∗ F¯γ (cn ).
(4.21)
The nature of the probability distributions motivates us to use numerical methods to perform the convolutions. We approximate hop distribution F¯γ (c) with a discrete distribution by partitioning the hop interval into nc subintervals of width ΔC = r/nc . Let the random variable J represent the jth subinterval in which a hop value C(γ) may lie. We approximate the probability mass function of J by the difference relation (4.22)
P(J = j) = pJ (j) ≈ F¯γ ((j + 1/2)ΔC) − F¯γ ((j − 1/2)ΔC),
where the integer j ∈ [0, nc ]. A fast Fourier transform is applied to the probability mass function (4.22), which is raised to the power of n, transformed back, and summed to give an approximation to the distribution of Yn . The final results (in Figure 4.3) show the difference in the distributions of multihop bounds S5 and T5 , the difference of which grows over multiple hops. 4.3. Integration approach. The Kullback–Leibler analysis (Figure 3.5) reveals that the single hop distribution varies markedly in the parameter range γ ≤ 3 and only slightly for large γ. This implies that the sink dependence needs to be included in our model for this parameter range. It follows that under the independent model the advancement after two hops has the distribution z F¯1 (z − c1 )dF¯0 (c1 ), z ∈ [0, 2r]. (4.23) PI (Z2 ≤ z) = 0
This integral is easily numerically evaluated and readily extends to the n-hop case: (4.24) PI (Zn ≤ z) = (4.25)
min(z,r)
min(z−c1 ,r)
dF¯1 (c2 ) . . . min(z−n−2 n−1 i=1 ci ,r) ¯ Fn−1 z − ci dF¯n−2 (cn−1 ), 0
0
dF¯0 (c1 )
0
z ∈ [0, nr].
i=1
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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1.4 F (z) S
FT(z)
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
z
Fig. 4.3. Comparison of multihop distributions of S5 and T5 (λ = 3 and = 10).
1.4 Sink Indpendent γ = 10 Sink Dependent
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1 z
1.5
2
Fig. 4.4. Comparison of multihop distributions (λ = 3, = 2, and a fixed γ = 10).
Unfortunately, calculating the above distribution involves evaluating high dimensional integrals. Furthermore, the dependent model involves integrating over the θi domains, which further increases the integral dimensionality. However, we calculate the simple two-hop distribution (4.23) to offer some insight into the accuracy of the sink-independent renewal process model. We compare the renewal process model to our two-hop sink-dependent result for an initial sink distance = 2 to illustrate where the former approach fails. With this initial sink distance, the sink-dependent model differs markedly from a renewal model with fixed γ = 10 (refer to Figure 4.4). However, the distributions are relatively close when a fixed γ = 2 is used for the renewal model (see Figure 4.5). This confirms the observations about the Kullback divergence in Figure 3.5. Hence, in the range γ ≥ 3 we can use a renewal model with γ = . However, if γ ≤ 3, the sink-dependent model is not approximated well by the renewal model.
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2230
H. P. KEELER AND P. G. TAYLOR
1.4 Sink Indpendent γ = 10 Sink Dependent
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1 z
1.5
2
Fig. 4.5. Comparison of multihop distributions (λ = 3, = 2, and a fixed γ = 2).
5. Future work and conclusions. We presented and examined a multihop model that is based on homogeneous spatial Poisson processes. For the initial hop, we obtained closed-form asymptotic expressions of the first and second moments that gave accurate results. We examined the first hop distribution and the effects that the “sink” dependence has on message advancement via Kullback–Leibler analysis. After the first hop, we examined the emergence of “path” dependence with a focus on the effects of a sleep scheme. We examined in more detail the simplifying assumption of Zorzi and Rao by deriving intersection area expressions. The exclusion and inclusion of the path dependence gave rise, respectively, to the “independent” and “dependent” multihop forwarding models. We established a stochastic order that relates the distributions of two-hop advancements under the independent and dependent models. We conjectured that a similar order holds for the multihop case. We supported this conjecture with numerical calculation of the three-hop distributions, which further showed that the hop advancements under the dependent model are stochastically dominated by those under the independent model. Further investigation is needed for the n-hop case. However, the number of integration dimensions soon grows to unmanageable numbers. One approach to tackling such unwieldy dimensionality is to use quasi–Monte Carlo methods, which we plan to implement as a future research task. As it is, under the dependent model, we could calculate the area functions after two and three hops. Deriving expressions beyond this is intractable due to the large number of geometrical configurations. However, we believe that only a “two-hop memory” model is needed to accurately describe the n-hop case. Furthermore, for sufficiently large λ, perhaps only the location of the previous node is needed; hence further work, both simulation and model-based, is needed to shed light on these issues. Our node deployment model incorporates the fact that nodes are awake at each transmission with a simple probability p, independently of their awake state at other transmission times. The inclusion of more complex sleep schemes into the model would result in the awake probability becoming a function of a number of variables. The nature of the sleep scheme (stochastic, deterministic, or a combination of both)
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2231
will directly affect the complexity of the awake parameter. A simple example is a sleep scheme based on a continuous or discrete time Markov chain, which results in the awake probability becoming a function of time. Consequently, work lies in developing and studying models that include more complex awake probabilities due to sleep schemes, and quantifying their effect upon the sensor network’s performance. Future research lies in deriving local node density functions under a sleep scheme, which would have a direct effect on the Poisson model. It would be interesting to examine the range of the underlying node density α and the sleep parameter p that allows the dependent model to be closely approximated by the independent model. Under this setting, the multihop distribution can be quickly calculated with renewal models, as the Kullback–Leibler analysis showed that the sink distance has little effect on hop behavior for approximately γ ≥ 3. Although the final few hops are not represented well by the renewal process model with γ = , the difference may not be significant overall. Replacing the constant radius assumption so that the transmission radius of a node varies randomly is a more realistic model extension, which also leads to some intriguing research directions. For example, one might need to assume that nodes can forward data only when they are within mutual transmission radius of each other. It follows that not choosing or detecting potential forwarding nodes, owing to them having insufficiently small radii, induces another form of node thinning. This will have an effect on the path dependence and the performance of routing. Alternatively, a simple “one-way” communication model may be sufficient under the random radius assumption. Another meaningful model extension is to let the node density be a spatially dependent function. A model of this kind would reflect the need to examine more realistic node placement scenarios, which inspires various suggestions for node density functions. For example, the node density may increase as one nears the sink, which would allow nodes to better accommodate the convergence of messages approaching the sink. However, ideally suggestions for node density functions would still need to be amenable to analytic and asymptotic methods. Finally, future work lies in applying our stochastic analysis to other routing schemes. These schemes may already exist or are yet to be proposed, such as one that chooses to minimize the intersection regions between nodes during each hop, thus reducing the path dependence. Appendix A. Asymptotic expansions. Consider the angle function ψγ (u) = arccos
u2 + γ 2 − r 2 2uγ
= arccos W (u) which we wish to expand at u = γ − r. The function W (γ − r) = 1; thus, we expand the function arccos x at x = 1 by observing 2 −1/2
−(1 − x )
−1/2 1 = −2 (1 − x) 1 − (1 − x) 2 1 3 −1/2 −1/2 1/2 3/2 ≈ −2 + (1 − x) + (1 − x) (1 − x) , 4 32 −1/2
−1/2
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2232
H. P. KEELER AND P. G. TAYLOR
which, integrated, leads to arccos x ≈ 2
1/2
1/2
2(1 − x)
1 3 3/2 5/2 (1 − x) − (1 − x) + . 12 160
Expand the function W (u) at u = γ − r, so that W (u) ≈ 1 −
γ+r γ+r r 2 3 (u − γ + r)+ 2 (u − γ + r) + 3 (u − γ + r) . γ(γ − r) 2 (−γ + r) γ 2 (−γ + r) γ
After some work we arrive at the approximation ψγ (u) ≈ b0 (u − γ + r)1/2 + b0 (u − γ + r)3/2 + b2 (u − γ + r)5/2 , where the expansion terms are b0 =
2r γ(γ − r)
1/2 ,
1/2 2 2r r − 3rγ − 3γ 2 b1 = , γ(γ − r) 12(γ 2 r − γr2 ) 1/2 4 2r 3r + 25r2 γ 2 − 10r3 γ + 30γ 3 r − 5γ 4 . b2 = γ(γ − r) 160γ 2(γ − r)2 r2 The feasible area function is given by Aγ (u) = 2
u
γ−r
wψγ (w)dw;
hence the area expansion is Aγ (u) ≈ a0 (u − γ + r)3/2 + a1 (u − γ + r)5/2 + a2 (u − γ + r)7/2 , and the expansion terms are
1/2 2r 4(γ − r) a0 = , γ(γ − r) 3 1/2 2r (−3γ 2 + 9γr + r2 ) a1 = , γ(γ − r) 15γr 1/2 2r (−15γ 4 − 30γ 3 r − 45γ 2 r2 + 10γr3 + 9r4 ) . a2 = γ(γ − r) 840γ 2 (γ − r)r2
Appendix B. Integral inequality (4.14). Under the independent model, the joint density and distribution of the first hop are identical to those under the dependent model; that is, g0 (u1 , θ1 ) = f0 (u1 , θ1 ) and F0 (ui ) = G0 (ui ). This fact and
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
X1 X0
2233
u1 Xs u01 X01
Fig. C.1. The nature of the intersection region depends on the u2 interval.
inequality (4.11) immediately lead to
u− 0 max(ui ,u0 −r)
ψu0 (u1 )
−ψu0 (u1 )
u− 0
g0 (u1 , θ1 )G1 (ui )dθ1 du1 ψu0 (u1 )
= max(ui ,u0 −r)
≤ ≤
u− 0
max(ui ,u0 −r) u0 max(ui ,u0 −r)
−ψu0 (u1 )
ψu0 (u1 )
−ψu0 (u1 )
f0 (u1 , θ1 )G1 (ui )dθ1 du1 f0 (u1 , θ1 )F1 (ui )dθ1 du1
F1 (ui )dF0 (u1 ).
Appendix C. Feasible area expressions. C.1. After one hop. The source and the current forwarding nodes are located at the points X0 and X1 , respectively, and the sink is located at the point XS . Symmetry allows us to assume that X1 is located above the baseline which runs from X0 to XS (as shown in Figure C.1). The location of X1 is represented by the sink distance u1 and the sink angle θ1 . The point closest to the sink where the transmission circumferences of the source and the current forwarding nodes intersect is also of importance. Denote this point by X01 , and let u01 be the distance from this point to the sink (see Figure C.1). To calculate u01 we observe that an isosceles triangle with two r-sides is formed by the points X0 , X1 , and X01 (see Figure C.2). The third side of this triangle is the distance separating X0 and X1 ; hence 1/2 h01 = u21 + 2 − 2u1 cos θ1 . Let δ1 represent the angle X1 X0 X01 , given by h01 δ1 = arccos , 2r and let β1 be the angle X1 X0 XS , given by 2 h01 + 2 − u21 β1 = arccos . 2h01
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2234
H. P. KEELER AND P. G. TAYLOR
h01 δ1 r
u1 r u01
Fig. C.2. The intersection of transmission circumferences.
Represent the last angle, XS X0 X01 , by η1 = δ1 − β1 , thus giving the expression 1/2 u01 = 2 + r2 − 2r cos η1 . Consider the point of distance u from the sink that lies on the transmission circumference of a node that has a sink distance γ. Recall that the angle that is formed by connecting this point to the node via the sink is given by the expression 2 u + γ 2 − r2 ψγ (u) = arccos . 2uγ We place an emphasis on this angle function, as it appears in the kernels of the integral area expressions for both the independent and dependent cases, and we introduce the function Δψ(u2 ) to describe the angular width of the intersection of the source and current feasible regions. That is, Δψ(u2 ) represents the angle between the top and bottom edges of the intersection region at a distance u2 from the sink (refer to Figure C.3). Subsequently, the intersection area function u2 A1\0 (u2 ) = w2 Δψ(w2 )dw2 −r
allows us to calculate the feasible area under the dependent model. Hence, we need an analytic form of Δψ(u2 ) over the entire u2 domain. The intersection area is zero on the interval [u1 − r, − r], and hence, Δψ(u2 ) is zero on this interval (refer to Figure C.1). The boundary of the intersection region is formed by arcs from the two circles of radius r and a sector of radius u2 . The shape of this boundary naturally affects the value of Δψ(u2 ). Thus, we observe that the rest of the domain can be divided into two intervals, [ − r, u01 ] and [u01 , u1 ].
u2 Δψ(u2 )
Fig. C.3. The angle function Δψ(u2 ) describes the intersection region.
On the first interval, [−r, u01 ], the intersection area is zero when the point X01 is located above the baseline, which extends from X0 to XS . Conversely, the intersection
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2235
area on this interval is positive if δ1 > β1 . When this condition is met, we observe on the first interval that the upper and lower arcs, which form the feasible region boundary, coincide with the transmission circumference of the source node. Hence, on the first interval we have the intersection angle expression Δψ(u2 ) = 2ψ (u2 )I− 01 ,
− r ≤ u2 ≤ u01 ,
− where I− 01 is an indicator function for when X01 is below the baseline. That is, I01 = 1 when X01 is below the baseline or, equivalently, when δ1 > β1 . We observe on the second interval, [u01 , u1 ], that part of the feasible region boundary coincides with the transmission circumference of the current forwarding node. Hence, on the second interval we obtain the intersection angle expression
Δψ(u2 ) = ψ (u2 ) + ψu1 (u2 ) − θ1 ,
u01 ≤ u2 ≤ u1 .
To perform the integration step we note that our expression for the feasible area under the independent model gives us the general solution to the integral Aγ (u) = 2
u
γ−r
wψγ (w)dw
= r2 φγ (u) + u2 ψγ (u) − rγ sin φγ (u), where we recall the angle functions
r 2 + γ 2 − u2 , 2rγ 2 u + γ 2 − r2 ψγ (u) = arccos . 2uγ φγ (u) = arccos
Thus, on the first interval, [ − r, u01 ], we have the area expression A1\0 (u2 ) = 2
u2
−r
w2 ψ (w2 )dw2 I− 01
= A (u2 )I− 01 . On the second interval, [u01 , u1 ], we have the slightly more complicated area expression A1\0 (u2 ) = =
u2
−r
w2 [ψ (w2 ) + ψu1 (w2 ) − θ1 ] dw2 + A (u01 )I− 01
1 A (u2 ) + Au1 (u2 ) + θ1 u201 − u22 2 1 A (u01 )[2I− + 01 − 1] − Au1 (u01 ) . 2
Consequently, for i = 1 the feasible area function under the dependent model is given by
1 (u2 ) = Au1 (u2 ) − A1\0 (u2 ). A
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2236
H. P. KEELER AND P. G. TAYLOR
X2
X1
X01 X0
X02
Fig. C.4. The intersection region R0∩2\1 (u3 ) can consist of up to four arcs.
C.2. After two hops. After two hops, an area expression for the intersection region R0∩2\1 (u3 ) = I0 (u3 ) ∩ I2 (u3 ) \ I1 (u3 ) is needed to calculate the area function under the dependent model (see one example of the intersection region in Figure C.4). Again, the angle function Δψ(u3 ) describes the angular width of the area under consideration, thus giving the general integral expression A0∩2\1 (u3 ) =
u3
u0 −r
w3 Δψ(w3 )dw3 .
We assume again that the first forwarding node is above the baseline, which runs from the source node to the sink. We outline a more general approach to find an analytic expression for Δψ(u3 ). In relation to the baseline, we refer to the boundary of the feasible region that is above the baseline as simply the top path. We consider a point on the top path which is a distance u3 from the sink. Let ψT (u3 ) be the angle between the baseline and the line connecting the sink to this point. Likewise, we refer to the boundary below the baseline as the bottom path, and we let ψB (u3 ) be its corresponding angle in relation to the sink, thus leading to the angle expression Δψ(u3 ) = ψT (u3 ) − ψB (u3 ). The region R0∩2\1 (u3 ) is enclosed by the three transmission circles and a sector of radius u3 . This knowledge can be used to obtain general expressions for ψT (u3 ) and ψB (u3 ). We consider the positioning of the intersection points of the transmission circumferences. For the top path, if the intersection point X01 is above the baseline, then ψT (u3 ) = min[θ1 − ψu1 (u3 ), ψu0 (u3 )]. A similar approach applied to the bottom path gives
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2237
ψB (u3 ) = max[θ1 + θ2 − ψu2 (u3 ), −ψu0 (u3 )]. Combining these results gives the angular width function Δψ(u3 ) = min [θ1 − ψu1 (u3 ), ψu0 (u3 )] − max [θ1 + θ2 − ψu2 (u3 ), −ψu0 (u3 )] , which allows the calculation of the area of R0∩2\1 (u3 ). Thus, we have a method of calculating the feasible areas after one and two hops under the dependent model:
2 (u3 ) = Au2 (u3 ) − A2\1 (u3 ) − A0∩2\1 (u3 ). A Arguably, a simpler but slightly more computationally exhaustive method is to use a crude Monte Carlo method. Alternatively, all the different geometrical configurations could be laboriously listed and their intersection expressions derived. REFERENCES [1] I. F. Akyildiz, S. Weilian, Y. Sankarasubramaniam, and E. Cayirci, A survey on sensor networks, IEEE Communications Magazine, 40 (2002), pp. 102–114. [2] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, Volume II: Theory, NOW Publishers, Delft, The Netherlands, 2009. [3] P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia, Routing with guaranteed delivery in ad hoc wireless networks, in Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, ACM, New York, 1999, pp. 48–55. [4] S. Capkun, M. Hamdi, and J.-P. Hubaux, GPS-free positioning in mobile ad hoc networks, in Proceedings of the 34th Annual Hawaii International Conference on System Sciences, IEEE, Washington, DC, 2001, p. 10. [5] C. Chong and S. P. Kumar, Sensor networks: Evolution, opportunities and challenges, Proc. IEEE, 91 (2003), pp. 1274–1256. [6] K. Fabian, W. Roger, Z. Yan, and Z. Aaron, Geometric ad-hoc routing: Theory and practice, in Proceedings of the 22nd Annual Symposium on Principles of Distributed Computing, ACM, Boston, 2003, pp. 63–72. [7] H. Frey, Scalable geographic routing algorithms for wireless ad hoc networks, IEEE Trans. Networks, 18 (2004), pp. 18–22. [8] S. Giordano and J.-Y. Le Boudec, Self-organizing wide-area routing, in Proceedings of SCI 2000/ISAS 2000, Orlando, FL, 2000. [9] R. Iyengar and B. Sikdar, Scalable and distributed GPS free positioning for sensor networks, in Proceedings of the IEEE International Conference on Communications, Vol. 1, IEEE, Washington, DC, 2003, pp. 338–342. [10] B. Karp and H. T. Kung, Greedy perimeter stateless routing for wireless networks, in Proceedings of the 6th Annual ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom 2000), ACM, New York, IEEE, Washington, DC, 2000, pp. 243–254. [11] L. Kleinrock and J. Silvester, Optimum transmission radii for packet radio networks or why six is a magic number, in Proceedings of the IEEE National Telecommunications Conference, IEEE, Washington, DC, 1978, pp. 4.31–4.35. [12] S. Kullback, Information Theory and Statistics, Wiley, New York, 1959. [13] R. Madan and S. Lall, An energy-optimal algorithm for neighbor discovery in wireless sensor networks, Mobile Networks and Applications, 11 (2006), pp. 317–326. [14] A. Sinha and A. Chandrakasan, Dynamic power management in wireless sensor networks, IEEE Design and Test of Computers, 18 (2001), pp. 62–74. [15] I. Stojmenovic, Position-based routing in ad hoc networks, IEEE Communications Magazine, 40 (2002), pp. 128–134. [16] H. Takagi and L. Kleinrock, Optimal transmission ranges for randomly distributed packet radio terminals, IEEE Trans. Communications, 32 (1984), pp. 246–257.
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[17] S. Vasudevan, J. Kurose, and D. Towsley, On neighbor discovery in wireless networks with directional antennas, in Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, Vol. 4, IEEE, Washington, DC, 2005, pp. 2502– 2512. [18] R. Wong, Asymptotic Approximations to Integrals, Academic Press, New York, 1989. [19] M. Zorzi and R. R. Rao, Geographic random forwarding (GeRaF) for ad hoc and sensor networks: Multihop performance, IEEE Trans. Mobile Computing, 2 (2003), pp. 337–348.
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c 2010 Society for Industrial and Applied Mathematics
A STOCHASTIC ANALYSIS OF A GREEDY ROUTING SCHEME IN SENSOR NETWORKS∗ H. P. KEELER† AND P. G. TAYLOR† Abstract. We address the stochastic characteristics of a recently proposed greedy routing scheme. The behavior of individual hop advancements is examined, and asymptotic expressions for the hop length moments are obtained. The change of the hop distribution as the sink distance is varied is quantified with a Kullback–Leibler analysis. We discuss the effects of the assumptions made, the inherent dependencies of the model, and the influence of a sleep scheme. We propose a renewal process model for multiple hop advancements and justify its suitability under our assumptions. We obtain the renewal process distributions via fast Fourier transform convolutions. We conclude by giving future research tasks and directions. Key words. multihop, geometric routing, sleep scheme AMS subject classifications. 90B15, 68M12, 60K20 DOI. 10.1137/080725684
1. Introduction. Recent developments in the fields of wireless communications and computing have produced the early examples of sensor networks. Sensor networks consist of large collections of small electronic devices known as sensor nodes. Each node has the ability to collect, process, and transmit data via neighboring nodes to a main communication station known as a sink. Although this research area is still in its infancy, sensor networks show great promise for applications in a number of areas where remote data retrieval is required, including medical, security, and environmental monitoring [1, 5]. Sensor nodes are usually scattered over a region known as the sensor field. Each node is equipped with a limited power source, which results in the service lifetime of sensor networks being a crucial consideration. To maximize this lifetime, stochastic power schemes have been suggested that entail having a random subset of the sensor nodes existing in a low energy-consuming sleep state at any given time [14]. During its sleep state it may or may not be possible for the node to collect energy depending on the surrounding environmental conditions [5]. Since a stochastic sensor network has a random subset of nodes awake at any given time, the network topology continually varies, resulting in the need for more innovative data routing methods. Many routing methods proposed for sensor networks are instances of geometric or position-based routing algorithms [7, 15]. The function of these algorithms is based on the assumption that each node knows its geographical position in relation to the sink, and the location of neighboring nodes within its transmission radius. These assumptions are reasonable when sensor nodes have a global positioning system [5]. Alternatively, algorithms have been proposed to calculate their geographical positions [4, 9] and to discover their neighbors [13, 17], thus enabling geographic routing in sensor networks. ∗ Received
by the editors May 30, 2008; accepted for publication (in revised form) January 13, 2010; published electronically April 21, 2010. This research was supported by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. http://www.siam.org/journals/siap/70-7/72568.html † Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia ([email protected], [email protected]). 2214
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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A commonly used geometric routing method is to have the source node look only at nodes that are closer to the sink in a region known as the total feasible region. The message is forwarded to the node that is closest to the sink, and this step is repeated until the message finally reaches the target sink. This approach can serve either as a simple greedy routing method by itself or as the basis for more sophisticated routing methods [3, 6, 8, 10, 19]. Here, we restrict ourselves to analyzing the simple greedy routing method suggested by Zorzi and Rao [19] known as Geographic Random Forwarding (GeRaF), while noting that the analysis can apply to other greedy-based routing methods. We extend the work of Zorzi and Rao by investigating the distribution of the number of hops required for a message to reach a sink. In particular, under a simplifying assumption, Zorzi and Rao effectively used renewal models to bound mean message advancements. We extend this work by deriving bounds on the distributions of multihop advancement and discuss their accuracy. Furthermore, we examine Zorzi and Rao’s simplifying assumption by developing a mathematical model and explaining the inherent stochastic properties of the problem. We discuss two forms of dependence that arise in the multihop advancement problem and how they stochastically affect the advancement of a message over multiple hops. We devise approximations for hop distributions and moment expressions. We discuss the effects of a simple energy-saving sleep scheme and its immediate influence on the aforementioned model dependencies. In conclusion, we discuss the various future research directions in developing and examining routing models in sensor networks. 2. Mathematical model. We present a mathematical model of a sensor network consisting of an ensemble of randomly positioned sensor nodes. We assume that the nodes are scattered according to a two-dimensional homogeneous Poisson process over a finite region and that at any given time a random number of nodes are in sleep mode while the remaining are in awake mode. In the model presented here we consider only the transmission of data and not the actual data-collection range, otherwise known as the coverage of the network. We assume that nodes communicate data radially and that a node’s transmission radius clearly cuts off at some distance, which implies that a node can relay data to another node only when it is within the forwarding node’s transmission radius. The transmission radius is set to the constant r. Let the constant α be the density of nodes scattered over a sensor field. We set p as the probability of a node being awake at any given time and assume that separate transmission attempts sample independent thinnings of the underlying Poisson process. It follows that the number of awake nodes in an area A is a homogeneous Poisson random variable NA and has the probability mass function (2.1)
P(NA = n) =
(λA)n −λA e , n!
where the new density parameter λ = pα represents the mean number of awake nodes per unit area. In our model there is a positive probability that no nodes, regardless of their state, lie within the feasible region of a forwarding node. Hence, there is a positive probability that a message will never reach the sink. This event never occurs in the model of Zorzi and Rao [19]. In their model the locations of the nodes are resampled at each hop, and if there is no node in the feasible region in one sample, there is
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H. P. KEELER AND P. G. TAYLOR
Aγ (u)
2φγ (u)
2ψγ (u) u
r
XS
X0 γ
Fig. 3.1. Feasible region of nodes located a distance of u or less from the sink.
a chance that there will be a node in the region in the next sample. In fact, with probability one, there will eventually be a forwarding node, and the message will reach the sink. As α approaches infinity and p approaches zero with λ held constant, our model will closely approximate that of Zorzi and Rao. We informally describe our definition of greedy routing by first introducing some notation. Let the random points X0 and XS denote the respective locations of the source node and the target sink. Under greedy routing the next forwarding node is located at X1 , which is the neighboring node of point X0 that is closest to the sink. This procedure is repeated until the message reaches the sink. More formally, let the point M (X) be the next message hop from a node located at point X. Let N (X) be the set of neighboring nodes of a node at point X with each neighboring node location n ∈ N (X) iff |n − X| ≤ r, where |.| is the Euclidean metric. For the integer i ≥ 0, greedy routing [2] is defined such that M (XS ) = XS and for all Xi = Xs , M (Xi ) = Xi+1 iff Xi+1 ∈ N (Xi ) and (2.2)
|Xi+1 − XS | =
min
n∈N (Xi )
|n − XS |.
Because the underlying node distribution is a Poisson process, the minimum is unique with probability one. For more details on routing definitions in a stochastic geometry setting, see the recent monograph [2, page 110], where greedy routing is called “best hop” routing. 3. Single hop distribution. Let the parameter γ be the distance between a forwarding node and the sink (refer to Figure 3.1). The sink distance is set to γ = when the forwarding node is the source of the message. The total feasible region of a node is the area formed by the intersection of the two circles of radii r and γ centered at the source node and the sink, respectively, thus forming a lens. After a single hop, let the random variable U be the distance between the sink and the new forwarding node chosen by the greedy routing algorithm. To obtain the distribution of U , we employ a nearest neighbor approach. Consider the feasible region of the source node where potential receiving nodes can be located at some distance u or less from the sink (shaded region in Figure 3.1). For > r, the probability that no receiving nodes exist in this feasible region of area A (u) is equivalent to the probability that U is strictly greater than u. The complement of this probability yields the distribution ⎧ ⎨ 1 − e−λA (u) , − r ≤ u < , (3.1) F (u) = 1, u ≥ , ⎩ 0, u < − r.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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Since there is a positive probability that no nodes lie within the feasible region, F (u) is a mixed discrete-continuous distribution, with a discontinuity at u = . We differentiate the distribution where it is absolutely continuous to give the probability density f (u) = λA (u)e−λA (u) ,
(3.2)
− r ≤ u < ,
where the prime denotes differentiation with respect to u. We need an expression for the general feasible area function Aγ (u) as a function of u for a given sink distance γ. Define the angles of the two intersecting sectors as 2φγ and 2ψγ , and see that 2 r + γ 2 − u2 φγ (u) = arccos (3.3) , 2rγ 2 u + γ 2 − r2 ψγ (u) = arccos (3.4) . 2uγ It follows that (3.5)
Aγ (u) = r2 φγ (u) + u2 ψγ (u) − uγ sin ψγ (u),
γ − r ≤ u ≤ γ,
and the derivative is given by Aγ (u) = 2uψγ (u).
(3.6)
Let C = γ − U represent the distance advanced by the message toward the sink when the originating node is at a distance γ, and let F¯γ (c) denote the distribution of C, namely, ⎧ ⎨ e−λAγ (γ−c) , 0 < c ≤ r, (3.7) F¯γ (c) = 1, c > r, ⎩ 0, c ≤ 0, which is discontinuous at c = 0. The first moment of C is r (3.8) E(C) = r − e−λAγ (γ−c) dc, 0
and the second moment is (3.9)
2
2
E(C ) = r − 2
r
ce−λAγ (γ−c) dc.
0
3.1. Asymptotic analysis. The form of the area function (3.5) implies that it is not feasible to obtain analytic expressions for integrals such as those in the moment equations (3.8) and (3.9). As an alternative, we present an asymptotic result for these moment expressions. Theorem 3.1. For γ > r, under greedy routing the first hop moment E(C) ∼ r −
(3.10)
Γ(5/3) 2/3
,
(λa0 )
and the second hop moment (3.11)
E(C 2 ) ∼ r2 − 2r
Γ(5/3) 2/3
(λa0 )
+
Γ(7/3) 4/3
,
(λa0 )
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H. P. KEELER AND P. G. TAYLOR
as the node density λ → ∞, where Γ(·) is the gamma function, and
2r a0 = γ(γ − r)
1/2
4(γ − r) . 3
Proof. Expand the area expression at u = γ − r such that (3.12)
Aγ (u) ∼ a0 (u − γ + r)3/2 + a1 (u − γ + r)5/2 + O(u)7/2 as u → γ − r,
and introduce the change of variable t = u − γ + r = r − c, with a slight abuse of notation, which leads to the area function expansion (3.13)
Aγ (t) ∼ a0 t3/2 + a1 t5/2 + O(t7/2 ) as
t → 0,
where the first two terms a0 and a1 are, respectively,
(3.14) (3.15)
1/2 2r 4(γ − r) a0 = , γ(γ − r) 3 1/2 2r (−3γ 2 + 9γr + r2 ) a1 = . γ(γ − r) 15γr
Consider integrals of the form (3.16)
b
I(λ) =
tk e−λA(t) dt,
a
where A(t) > A(a) for all t ∈ (a, b), and asymptotically (3.17)
A(t) ∼ a0 (t − a)μ
as t → a.
Applying a generalized version of Laplace’s method (see Wong [18, page 58]) to this integral gives (3.18)
I(λ) ∼
Γ(τ ) μ(λa0 )τ
as λ → ∞,
where (3.19)
τ=
2(k + 1) . 3
With a = 0 and μ = 3/2, result (3.10) readily follows from (3.18), and, similarly, for the second moment expression r (3.20) E(C 2 ) = r2 − 2 (1 − t)e−Aγ (t) dt, 0
result (3.11) follows.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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1 Analytic Solution First Order Second Order
0.9 0.8 0.7
Fγ(u)
0.6 0.5 0.4 0.3 0.2 0.1 0
9
9.2
9.4
9.6
9.8
10
u
Fig. 3.2. Approximation of distribution of U (γ = 10 and λ = 3).
1 Numerical Asymptotic 0.9
0.8
E(C)
0.7
0.6
0.5
0.4
1
2
3
4
5
λ
6
7
8
9
10
Fig. 3.3. Approximation of E(C) by the generalized Laplace method ( = 10).
For all numerical results, unless stipulated otherwise, the length values will be rescaled by the transmission radius; thus each node has a unit transmission radius. The two-term area expansion (3.12) appears to give reasonably accurate results regardless of the value of γ. This accuracy carries over when the above expansion is substituted into (3.1) to yield an approximation to the hop advancement distribution, which is almost indistinguishable from the true distribution (see Figure 3.2) when γ = 10 and λ = 3. The closed-form moment approximations (3.10) and (3.11) agree well with numerical results for a practical range of λ (see Figures 3.3 and 3.4). The approximations start to break down for λ ∼ 1, particularly for the second moment (in Figure 3.4). However, the approximations give sufficiently accurate results for the first and second moments of C. 3.2. Effect of sink distance. The area function increases with respect to the parameter γ, which implies that hop lengths increase stochastically with γ. This area
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H. P. KEELER AND P. G. TAYLOR
0.9 Numerical Asymptotic 0.8
0.7
2
E(C )
0.6
0.5
0.4
0.3
0.2
1
2
3
4
5
λ
6
7
8
9
10
Fig. 3.4. Approximation of E(C 2 ) by the generalized Laplace method ( = 10).
behavior can be verified geometrically by observing that the arc of the intersecting sector of radius u flattens as γ increases; hence the total feasible region approaches a semicircle (refer to Figure 3.1). A simple geometrical argument can be used to obtain the area of this semicircle, which is the supremum of the area function. Alternatively, we consider the derivative of the area function (3.6) with respect to c = γ − u, that is, c2 − r 2 , (3.21) Aγ (γ − c) = 2 (γ − c) arccos 1 + 2γ (γ − c) and in the limit as γ approaches infinity we obtain
(3.22) A∞ (c) = −2 r2 − c2 , which is the derivative of the supremum of the area function; thus
0 ≤ c ≤ r. (3.23) A∞ (c) = r2 arccos(c/r) − c r2 − c2 , This is the feasible area function under the Most Forward within Radius routing model proposed by Takagi and Kleinrock [16] provided that messages travel only forward. Kleinrock and Silvester [11] stated earlier that the difference in the two area functions is negligible provided that γ is sufficiently large. The difference in the hop distribution parameterized by two different values, γ1 and γ2 , of γ can be expressed in terms of a goodness-of-fit measure. We choose the Kullback–Leibler divergence [12], which measures the difference between the “true” probability measure P and a proposed probability measure Q over some sample space Ω. In general, the Kullback–Leibler divergence, defined as dQ log (3.24) D(P, Q) = − dP, dP Ω is nonnegative, and zero when the two measures are identical. Consequently, the mixed discrete-continuous hop distribution leads to the expression ¯ ¯ r Fγ2 (0+ ) fγ2 (c) + ¯ ¯ dc + Fγ2 (0 ) log ¯ fγ2 (c) log ¯ , (3.25) D(γ1 , γ2 ) = Fγ1 (0+ ) fγ1 (c) 0
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
0.18
2221
=3 =6 =∞
0.16 0.14
D(, γ)
0.12 0.1 0.08 0.06 0.04 0.02 0
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Fig. 3.5. Kullback–Leibler divergence quantifying the change in the distribution (λ = 3).
where the probability of a routing void or zero advancement is (3.26)
F¯γ (0+ ) = lim F¯γ ( )
(3.27)
=e
→0 −λAγ (γ)
.
We evaluated the integral in (3.25) numerically to determine how the hop distribution changes when γ1 = and γ2 = γ are varied (refer to Figure 3.5). We observed that D(, γ) is virtually zero for the majority of γ, and then it increases markedly for γ ≤ 3. This implies that the algebraic dependence of F¯γ (c) on γ significantly affects the distribution of the remaining hop lengths only after this point. The negative consequence of this is that hop lengths cannot be assumed to be independent of the current sink distance for γ ≤ 3. Conversely, the hops up to this point can be modeled reasonably well with the first hop distribution. Moreover, for finite ≥ 3, the divergence D(, γ) is approximately equal to D(∞, γ), which suggests that the simpler area expression (3.23) may be used. 4. Multihop distribution. 4.1. Dependence between successive hops. We introduce some notation to describe the locations of the forwarding nodes. Specifically, we introduce indexing for each sink distance so that U0 = and U1 = U , and, after i hops, the remaining distance to the sink is Ui . Let the random variable Θi be the angle between the ith node and the previous node in relation to the sink. We note that Ui is a global coordinate (in relation to the sink) and Θi is a local coordinate (in relation to the previous node). We assign the point Xi = (Ui , Θi ) to the ith forwarding node. The source (or zeroth) node corresponds to the point X0 = (U0 , 0) = (, 0). A message travels i hops along
i = (X0 , X1 , . . . , Xi ). a path that corresponds to a sequence of random points X The decrease in the sink distance between ith and (i−1)th hops is Ci = Ui−1 −Ui . As we saw above, Ci depends on the forwarding node’s sink distance Ui−1 . We refer to this dependence of the distribution on the sink distance simply as the sink dependence. Zorzi and Rao [19] pointed out that each Ci+1 is stochastically dominated by Ci in the sense that, for i ≥ 0, there exists the stochastic ordering (4.1)
P(Ci+1 > c) ≤ P(Ci > c),
c ∈ (0, r).
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2222
H. P. KEELER AND P. G. TAYLOR
RA X1 X0
U1 Θ1 RB
Fig. 4.1. No awake nodes in the intersection region RB during the first message relay.
This follows directly from the observation in section 3.2 that the feasible area function increases with respect to the sink distance. The Kullback–Leibler analysis suggests that the sink dependence is negligible for the majority of hop advancements. However, after the first hop there is another source of dependence between the random variables Ci . To simplify our explanation of this phenomenon we assume that all nodes within the forwarding node’s feasible region are in their awake state. Then there can be no nodes closer to the sink in the feasible region of the original node other than the forwarding node. Hence, at the second hop there is a region where there can be no potential forwarding nodes (region RB in Figure 4.1). If this intersection region did contain a node, then this node would have been chosen as the new forwarding node at the previous hop instead of the one that was chosen. This implies that C2 is dependent on both U1 and Θ1 . We refer to this dependence on both the sink distance and the sink angle as path dependence and to the hop model that includes path dependence simply as the dependent model. Conversely, the independent model includes only the sink dependence. We have assumed for the purpose of our discussion of path dependence that all nodes within the forwarding node’s feasible region were in their awake state. However, the analysis can be extended to the situation when either a stochastic or deterministic power scheme is in operation. Then, there is a positive probability that asleep nodes found within the intersection of feasible regions when a message is received will be awake when the transmitting node is choosing the next forwarding node. Specifically, if α is the actual density of nodes and p is the probability of each node being awake, an event which is sampled independently at each time step, then the nodes in the intersection area will have a density α(1 − p)p, whereas the nodes in the nonintersecting area will have a density αp. Hence, the path dependence decreases as a new ensemble of possible forwarding nodes is sampled, implying the dependent model will more closely resemble the independent model. The path dependence will be reduced further as the number of nodes increases and the probability of them being awake decreases. In the limit as α → ∞ and p → 0 with λ = αp held constant, the dependence disappears entirely, and we have the independent model of Zorzi and Rao [19]. Future work lies in investigating the values of p and α for which the dependent model closely resembles the more tractable, independent model. We use F and G to denote the distributions of Ui under the independent and dependent models, respectively. The distribution of Ui+1 under the independent model is dependent only on the sink distance of the current forwarding node, and thus we can write (4.2)
Fi (ui+1 ) = PI (Ui+1 ≤ ui+1 |Ui = ui ),
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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while under the dependent model the distribution is dependent on the entire message path, and thus we write (4.3)
i = xi ), Gi (ui+1 ) = PD (Ui+1 ≤ ui+1 |X
where the subscripts I and D indicate probability measures under the two models.
i = xi , We stress that the distributions (4.2) and (4.3) depend on Ui = ui and X respectively, but for convenience, we have omitted this in the subscript notation. Further shorthand notation is used to denote the feasible area under the independent
i (ui+1 ). and dependent models, respectively, as Ai (ui+1 ) and A The area expression under the independent model is always given by the original area equation (3.5), and hence, the distribution and probability density of Ui+1 are obtained by setting = ui in (3.1) and (3.2). Under the dependent model, if the feasible area function is given after i hops, the conditional sink distribution immediately follows: ⎧ ⎨ 1 − e−λAi (ui+1 ) , ui − r ≤ u < ui , (4.4) Gi (ui+1 ) = 1, u ≥ ui , ⎩ 0, u < ui − r. and its probability density is defined on the region where it is absolutely continuous (4.5)
i (ui+1 )e−λAi (ui+1 ) . gi (ui+1 ) = λA
For Ci , we adopt notation similar to that used for the sink distance random ¯ denote the hop distributions under the two models. The variables; hence F¯ and G complement of the sink distribution yields the conditional hop distribution under both the independent and dependent models, the latter being ⎧ ⎨ e−λAi (ui −ci+1 ) , 0 < ci+1 ≤ r, ¯ (4.6) Gi (ci+1 ) = 1, ci+1 > r, ⎩ 0, ci+1 ≤ 0. and its probability density is defined on the region where it is absolutely continuous (4.7)
(ci+1 )e−λA i (ui −ci+1 ) . g¯i (ci+1 ) = λA i
Let Ii (ui+1 ) ⊂ R2 be the feasible region of the ith forwarding node as a function of ui+1 under the independent model. The set representing the feasible region under the dependent model follows by excluding the intersections of previous feasible regions; thus Di (ui+1 ) = Ii (ui+1 ) \ ∪j=i−1 j=0 Ij (uj+1 ). Methods of calculating the area of D1 (u2 ) and D2 (u3 ) are given in the appendix (sections C.1 and C.2). We note that analytic area expressions after two hops are intractable given the number of possible geometric configurations.
1 = (X0 , X1 ), the feasible region for the second hop For any two node path X under the dependent model is clearly a subset of the region under the independent model; hence (4.8)
1 (u1 − c2 ) ≤ A1 (u1 − c2 ), A
c2 ∈ [0, r].
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H. P. KEELER AND P. G. TAYLOR
This inequality allows us to compare the hop distributions (3.7) and (4.6) under the two models, which leads to the stochastic ordering (4.9)
1 = x1 ) ≤ PI (C2 > c|X
1 = x1 ), PD (C2 > c|X
c ∈ (0, r),
on the conditional distributions after one hop. Hence, for any two node path, C2 under the dependent model is stochastically dominated by C2 under the independent model. We observed that there is a stochastic ordering (4.1) on single hops stemming from the sink dependence. Furthermore, there is a conditional stochastic ordering on the second hop owing to the path dependence (4.9). We now investigate the possibility of a stochastic ordering for the multihop case. Subsequently, under the dependent model we introduce the distribution
i ). Gi (ui+1 , θi+1 ) = PD (Ui+1 ≤ ui+1 , Θi+1 ≤ θi+1 |X The respective angles −ψui (ui+1 ) and ψui (ui+1 ) denote the lower and upper values of the θi+1 domain. The distribution Gi (ui+1 , θi+1 ) is absolutely continuous on the domain Di (ui+1 ). However, it has a mass at the point Xi corresponding to the event when there are no nodes in the feasible region. In the dependent case, this means that there will never be a closer node in the feasible region, and hence, the sink is unreachable. We denote the density function of Gi (ui+1 , θi+1 ) by gi (ui+1 , θi+1 ), which is defined on the region where the distribution is absolutely continuous. Under the dependent model the joint probability density of Ui and Θi is (4.10)
gi (ui+1 , θi+1 ) = λDi (ui+1 , θi+1 )ui+1 e−λAi (ui+1 ) .
The spatially dependent density function is λDi (ui+1 , θi+1 ) = λIDi (ui+1 , θi+1 ) and the indicator function of the dependent feasible region is 1, (ui+1 , θ0i ) ∈ Di , IDi (ui+1 , θi+1 ) = 0 otherwise. The global angular coordinate θ0i is simply the angle between the source node and the ith forwarding node in relation to the sink. The indicator function gives a zero joint probability density in the regions where there is zero node density owing to the path dependence. Under the independent model we introduce a similar distribution:
i ). Fi (ui+1 , θi+1 ) = PI (Ui+1 ≤ ui+1 , Θi+1 ≤ θi+1 |X The distribution Fi (ui+1 , θi+1 ) is absolutely continuous on the domain Ii (ui+1 ), and, similarly to the dependent model, it has a mass at the point Xi . However, even if there is zero advancement, that is, (ui+1 , θi+1 ) = (ui , θi ), it is still possible that (ui+2 , θi+2 ) = (ui+1 , θi+1 ) since under the independent model we effectively resample node positions at every hop so there is a possibility that there will be a node in the feasible region at the next resampling. We also note that Fi (ui+1 , θi+1 ) depends
i−1 . Denote the only on the current point Xi and not on the previous points X corresponding density function by fi (ui+1 , θi+1 ), which is defined on the region where the distribution is absolutely continuous.
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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Write the distribution and the density for the remaining distance to the sink for the dependent model, respectively, as ψu (ui+1 ) i Gi (ui+1 ) = Gi (ui+1 , θi+1 )dθi+1 , −ψui (ui+1 )
gi (ui+1 ) =
ψui (ui+1 )
−ψui (ui+1 )
gi (ui+1 , θi+1 )dθi+1 ,
and similarly for the independent model: ψui (ui+1 ) Fi (ui+1 ) = Fi (ui+1 , θi+1 )dθi+1 , −ψui (ui+1 )
fi (ui+1 ) =
ψui (ui+1 )
−ψui (ui+1 )
fi (ui+1 , θi+1 )dθi+1 .
The stochastic dominance inequality after the first hop (4.9) is equivalent to (4.11)
G1 (u2 ) ≤ F1 (u2 ),
u1 − r ≤ u2 < u1 .
This result leads to a stochastic bound after two hops between the respective models. Lemma 4.1. For i = 2 hops, the sink distances Ui under the two models behave such that (4.12)
PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ).
Proof. This is equivalent to the inequality (4.13) PD (Ui ≤ ui ) = G0 (ui ) + (4.14) (4.15)
≤ F0 (ui ) +
u− 0
max(ui ,u0 −r) u0 max(ui ,u0 −r)
ψu0 (u1 ) −ψu0 (u1 )
g0 (u1 , θ1 )G1 (ui )dθ1 du1
F1 (ui )dF0 (u1 )
= PI (Ui ≤ ui ),
where we denote u− i as the left limit: u− i = lim (ui − ). →0
The zero-advancement probability (that is, the upper limit u0 ) is not included in the expression (4.13), as such an event under the dependent model implies that it is not possible for any further advancement toward the sink in future hops. The Riemann–Stieltjes-type integral (4.14) has the upper limit u0 instead of u− 0 , as the message may not advance during one hop and then advance during the next under the independent model. Under the independent model, the joint density and distribution of the first hop are identical to those under the dependent model. Thus the proof immediately follows; see Appendix B for details. It is tempting to conjecture that the two-hop stochastic order (4.12) can be extended to the multihop case, which would lead to the following conjecture.
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H. P. KEELER AND P. G. TAYLOR
Conjecture 1. For any number of hops i, the sink distances Ui under the two models behave such that PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ). The conjecture hinges upon the truth of the inequality (4.16)
PD (Ui ≤ ui ) = G0 (ui ) +
u− 0
+
u− 0
u− i−1
u0
max(ui ,u0 −r) ui
+
ψui−2 (ui−1 )
g0 (u1 , θ1 ) . . . Gi−1 (ui ) . . . dθ1 du1
f0 (u1 , θ1 )F1 (ui )dθ1 du1 + · · ·
max(ui ,ui−1 −r)
ψui−2 (ui−1 ) −ψui−2 (ui−1 )
f0 (u1 , θ1 ) . . . Fi−1 (ui ) . . . dθ1 du1
F1 (ui )dF0 (u1 ) + · · ·
... max(ui ,u0 −r)
g0 (u1 , θ1 )G1 (ui )dθ1 du1 + · · ·
−ψui−2 (ui−1 )
...
max(ui ,u0 −r) −ψu0 (u1 ) u0
≤ F0 (ui ) +
ψu0 (u1 )
ψu0 (u1 )
+
max(ui ,ui−1 −r)
−ψu0 (u1 )
−ψu0 (u1 )
u− i−1
max(ui ,u0 −r)
u− 0
ψu0 (u1 )
...
−ψu0 (u1 )
max(ui ,u0 −r)
≤ F0 (ui ) +
max(ui ,u0 −r)
ψu0 (u1 )
u− 0
max(ui ,ui −r)
Fi−1 (ui )dFi−2 (ui−1 ) . . . dF0 (u1 )
= PI (Ui ≤ ui ), where we observe that fi (ui+1 , θi+1 ) is independent of θi . For i ≥ 2, we observe that gi (ui+1 , θi+1 ) is greater than or equal to fi (ui+1 , θi+1 ) on the region where it is positive, but the range of θi over which fi (ui+1 , θi+1 ) is integrated into (4.16) is larger. The trade-off between these two competing influences is delicate.
2 (u3 ) can be calculated; see section C.1 for details. For The area function A three or fewer hops, the conjecture can be tested via regular numerical integration methods. The results depicted in Figure 4.2 provide support to the suggestion that PD (Ui ≤ ui ) ≤ PI (Ui ≤ ui ), but we have not been able to prove it. For higher hop numbers, deriving an analytic area expression is intractable, and regular numerical integration is too slow. Numerical investigation reveals that the difference between the two distributions is small for large λ values. This implies that the independent model is a good approximation for the dependent model at high node density. However, it is possible that even a small difference in the two models will accumulate over many hops. We stress that it is significantly easier to calculate the distributions using the independent model compared to the dependent model. If our conjecture is true, the independent model will always provide a stochastic lower bound for the hop advancement distributions irrespective of how close the two distributions are. 4.2. Distribution of the total number of hops. We give some motivation for calculating the distribution of Ci in general. Let the random variable N represent the total number of hops required for a message to reach the sink. We define the
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2227
1.2 Independent Dependent 1
P(U3≤ u3)
0.8
0.6
0.4
0.2
0
7
7.5
8
8.5 u
9
9.5
10
3
Fig. 4.2. Comparison of PD (Ui ≤ ui ) and PI (Ui ≤ ui ) (for = 10 and λ = 3).
random variable (4.17)
Zn =
n
Ci
i=1
to represent the distance advanced by a message in n hops assuming the parametric dependence discussed in section 3.2. We are interested in a means of quickly calculating the distribution of Zn owing to its inherent relation with N . This is encapsulated in the following simple result. Proposition 4.1. For a sink distance , the number of message hops N has the distribution (4.18) (4.19)
PD (N = 1) = 1 PD (N ≤ n) = 1 − PD (Zn−1 < − r)
∀n ≥ 1
if ≤ r, if > r.
Proof. Clearly, when ≤ r, only a single hop is needed to reach the sink. For > r, P(N > n) = PD (Un−1 > r). Observing that Un−1 = − Zn−1 , we see that P(N > n) = PD (Zn−1 < − r). The distribution of the random variable N arguably presents more critical information on the performance of the sensor network. For instance, if transmission errors are introduced at a given relay node with probability pE , then it is elementary to calculate the probability that errors are introduced along a path. However, the sink dependence of γ on F¯γ (c) implies that high dimensional integrals need to be evaluated in order to calculate the distribution of Zn ; more details will be given in the next section. It is possible, however, under the independent model, to obtain bounds for the average number of hops required to reach a sink given an initial sink distance (see Zorzi and Rao [19]). We develop this idea further by defining
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H. P. KEELER AND P. G. TAYLOR
n the general random variable Yn = i=1 Ci (γ), where Ci (γ) is a sequence of independent and identically distributed random variables which represent hop advancements
with a fixed sink distance γ. Let the random variable Sn = ni=1 Ci (r), noting that, for any γ > r, Ci (r) is stochastically smaller than Ci (γ). Thus, Sn is stochastically smaller than Yn , which provides a stochastic lower bound z ∈ (0, nr). P(Sn > z) ≤ PI (Zn > z),
n Let the random variable Tn = i=1 Ci (), which is the sum of n random variables that have the distribution of the first (that is, the stochastically largest) hop. The stochastic dominance inequality (4.20)
P(Sn > z) ≤ PI (Zn > z) ≤ P(Tn > z),
z ∈ (0, nr),
easily follows. Both Sn and Tn form renewal processes, and their distributions can be calculated as the n-fold convolutions of the hop distributions: P(Yn ≤ z) = F¯γ (c1 ) ∗ F¯γ (c2 ) ∗ · · · ∗ F¯γ (cn ).
(4.21)
The nature of the probability distributions motivates us to use numerical methods to perform the convolutions. We approximate hop distribution F¯γ (c) with a discrete distribution by partitioning the hop interval into nc subintervals of width ΔC = r/nc . Let the random variable J represent the jth subinterval in which a hop value C(γ) may lie. We approximate the probability mass function of J by the difference relation (4.22)
P(J = j) = pJ (j) ≈ F¯γ ((j + 1/2)ΔC) − F¯γ ((j − 1/2)ΔC),
where the integer j ∈ [0, nc ]. A fast Fourier transform is applied to the probability mass function (4.22), which is raised to the power of n, transformed back, and summed to give an approximation to the distribution of Yn . The final results (in Figure 4.3) show the difference in the distributions of multihop bounds S5 and T5 , the difference of which grows over multiple hops. 4.3. Integration approach. The Kullback–Leibler analysis (Figure 3.5) reveals that the single hop distribution varies markedly in the parameter range γ ≤ 3 and only slightly for large γ. This implies that the sink dependence needs to be included in our model for this parameter range. It follows that under the independent model the advancement after two hops has the distribution z F¯1 (z − c1 )dF¯0 (c1 ), z ∈ [0, 2r]. (4.23) PI (Z2 ≤ z) = 0
This integral is easily numerically evaluated and readily extends to the n-hop case: (4.24) PI (Zn ≤ z) = (4.25)
min(z,r)
min(z−c1 ,r)
dF¯1 (c2 ) . . . min(z−n−2 n−1 i=1 ci ,r) ¯ Fn−1 z − ci dF¯n−2 (cn−1 ), 0
0
dF¯0 (c1 )
0
z ∈ [0, nr].
i=1
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
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1.4 F (z) S
FT(z)
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
z
Fig. 4.3. Comparison of multihop distributions of S5 and T5 (λ = 3 and = 10).
1.4 Sink Indpendent γ = 10 Sink Dependent
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1 z
1.5
2
Fig. 4.4. Comparison of multihop distributions (λ = 3, = 2, and a fixed γ = 10).
Unfortunately, calculating the above distribution involves evaluating high dimensional integrals. Furthermore, the dependent model involves integrating over the θi domains, which further increases the integral dimensionality. However, we calculate the simple two-hop distribution (4.23) to offer some insight into the accuracy of the sink-independent renewal process model. We compare the renewal process model to our two-hop sink-dependent result for an initial sink distance = 2 to illustrate where the former approach fails. With this initial sink distance, the sink-dependent model differs markedly from a renewal model with fixed γ = 10 (refer to Figure 4.4). However, the distributions are relatively close when a fixed γ = 2 is used for the renewal model (see Figure 4.5). This confirms the observations about the Kullback divergence in Figure 3.5. Hence, in the range γ ≥ 3 we can use a renewal model with γ = . However, if γ ≤ 3, the sink-dependent model is not approximated well by the renewal model.
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2230
H. P. KEELER AND P. G. TAYLOR
1.4 Sink Indpendent γ = 10 Sink Dependent
Probability Distribution
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1 z
1.5
2
Fig. 4.5. Comparison of multihop distributions (λ = 3, = 2, and a fixed γ = 2).
5. Future work and conclusions. We presented and examined a multihop model that is based on homogeneous spatial Poisson processes. For the initial hop, we obtained closed-form asymptotic expressions of the first and second moments that gave accurate results. We examined the first hop distribution and the effects that the “sink” dependence has on message advancement via Kullback–Leibler analysis. After the first hop, we examined the emergence of “path” dependence with a focus on the effects of a sleep scheme. We examined in more detail the simplifying assumption of Zorzi and Rao by deriving intersection area expressions. The exclusion and inclusion of the path dependence gave rise, respectively, to the “independent” and “dependent” multihop forwarding models. We established a stochastic order that relates the distributions of two-hop advancements under the independent and dependent models. We conjectured that a similar order holds for the multihop case. We supported this conjecture with numerical calculation of the three-hop distributions, which further showed that the hop advancements under the dependent model are stochastically dominated by those under the independent model. Further investigation is needed for the n-hop case. However, the number of integration dimensions soon grows to unmanageable numbers. One approach to tackling such unwieldy dimensionality is to use quasi–Monte Carlo methods, which we plan to implement as a future research task. As it is, under the dependent model, we could calculate the area functions after two and three hops. Deriving expressions beyond this is intractable due to the large number of geometrical configurations. However, we believe that only a “two-hop memory” model is needed to accurately describe the n-hop case. Furthermore, for sufficiently large λ, perhaps only the location of the previous node is needed; hence further work, both simulation and model-based, is needed to shed light on these issues. Our node deployment model incorporates the fact that nodes are awake at each transmission with a simple probability p, independently of their awake state at other transmission times. The inclusion of more complex sleep schemes into the model would result in the awake probability becoming a function of a number of variables. The nature of the sleep scheme (stochastic, deterministic, or a combination of both)
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2231
will directly affect the complexity of the awake parameter. A simple example is a sleep scheme based on a continuous or discrete time Markov chain, which results in the awake probability becoming a function of time. Consequently, work lies in developing and studying models that include more complex awake probabilities due to sleep schemes, and quantifying their effect upon the sensor network’s performance. Future research lies in deriving local node density functions under a sleep scheme, which would have a direct effect on the Poisson model. It would be interesting to examine the range of the underlying node density α and the sleep parameter p that allows the dependent model to be closely approximated by the independent model. Under this setting, the multihop distribution can be quickly calculated with renewal models, as the Kullback–Leibler analysis showed that the sink distance has little effect on hop behavior for approximately γ ≥ 3. Although the final few hops are not represented well by the renewal process model with γ = , the difference may not be significant overall. Replacing the constant radius assumption so that the transmission radius of a node varies randomly is a more realistic model extension, which also leads to some intriguing research directions. For example, one might need to assume that nodes can forward data only when they are within mutual transmission radius of each other. It follows that not choosing or detecting potential forwarding nodes, owing to them having insufficiently small radii, induces another form of node thinning. This will have an effect on the path dependence and the performance of routing. Alternatively, a simple “one-way” communication model may be sufficient under the random radius assumption. Another meaningful model extension is to let the node density be a spatially dependent function. A model of this kind would reflect the need to examine more realistic node placement scenarios, which inspires various suggestions for node density functions. For example, the node density may increase as one nears the sink, which would allow nodes to better accommodate the convergence of messages approaching the sink. However, ideally suggestions for node density functions would still need to be amenable to analytic and asymptotic methods. Finally, future work lies in applying our stochastic analysis to other routing schemes. These schemes may already exist or are yet to be proposed, such as one that chooses to minimize the intersection regions between nodes during each hop, thus reducing the path dependence. Appendix A. Asymptotic expansions. Consider the angle function ψγ (u) = arccos
u2 + γ 2 − r 2 2uγ
= arccos W (u) which we wish to expand at u = γ − r. The function W (γ − r) = 1; thus, we expand the function arccos x at x = 1 by observing 2 −1/2
−(1 − x )
−1/2 1 = −2 (1 − x) 1 − (1 − x) 2 1 3 −1/2 −1/2 1/2 3/2 ≈ −2 + (1 − x) + (1 − x) (1 − x) , 4 32 −1/2
−1/2
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2232
H. P. KEELER AND P. G. TAYLOR
which, integrated, leads to arccos x ≈ 2
1/2
1/2
2(1 − x)
1 3 3/2 5/2 (1 − x) − (1 − x) + . 12 160
Expand the function W (u) at u = γ − r, so that W (u) ≈ 1 −
γ+r γ+r r 2 3 (u − γ + r)+ 2 (u − γ + r) + 3 (u − γ + r) . γ(γ − r) 2 (−γ + r) γ 2 (−γ + r) γ
After some work we arrive at the approximation ψγ (u) ≈ b0 (u − γ + r)1/2 + b0 (u − γ + r)3/2 + b2 (u − γ + r)5/2 , where the expansion terms are b0 =
2r γ(γ − r)
1/2 ,
1/2 2 2r r − 3rγ − 3γ 2 b1 = , γ(γ − r) 12(γ 2 r − γr2 ) 1/2 4 2r 3r + 25r2 γ 2 − 10r3 γ + 30γ 3 r − 5γ 4 . b2 = γ(γ − r) 160γ 2(γ − r)2 r2 The feasible area function is given by Aγ (u) = 2
u
γ−r
wψγ (w)dw;
hence the area expansion is Aγ (u) ≈ a0 (u − γ + r)3/2 + a1 (u − γ + r)5/2 + a2 (u − γ + r)7/2 , and the expansion terms are
1/2 2r 4(γ − r) a0 = , γ(γ − r) 3 1/2 2r (−3γ 2 + 9γr + r2 ) a1 = , γ(γ − r) 15γr 1/2 2r (−15γ 4 − 30γ 3 r − 45γ 2 r2 + 10γr3 + 9r4 ) . a2 = γ(γ − r) 840γ 2 (γ − r)r2
Appendix B. Integral inequality (4.14). Under the independent model, the joint density and distribution of the first hop are identical to those under the dependent model; that is, g0 (u1 , θ1 ) = f0 (u1 , θ1 ) and F0 (ui ) = G0 (ui ). This fact and
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
X1 X0
2233
u1 Xs u01 X01
Fig. C.1. The nature of the intersection region depends on the u2 interval.
inequality (4.11) immediately lead to
u− 0 max(ui ,u0 −r)
ψu0 (u1 )
−ψu0 (u1 )
u− 0
g0 (u1 , θ1 )G1 (ui )dθ1 du1 ψu0 (u1 )
= max(ui ,u0 −r)
≤ ≤
u− 0
max(ui ,u0 −r) u0 max(ui ,u0 −r)
−ψu0 (u1 )
ψu0 (u1 )
−ψu0 (u1 )
f0 (u1 , θ1 )G1 (ui )dθ1 du1 f0 (u1 , θ1 )F1 (ui )dθ1 du1
F1 (ui )dF0 (u1 ).
Appendix C. Feasible area expressions. C.1. After one hop. The source and the current forwarding nodes are located at the points X0 and X1 , respectively, and the sink is located at the point XS . Symmetry allows us to assume that X1 is located above the baseline which runs from X0 to XS (as shown in Figure C.1). The location of X1 is represented by the sink distance u1 and the sink angle θ1 . The point closest to the sink where the transmission circumferences of the source and the current forwarding nodes intersect is also of importance. Denote this point by X01 , and let u01 be the distance from this point to the sink (see Figure C.1). To calculate u01 we observe that an isosceles triangle with two r-sides is formed by the points X0 , X1 , and X01 (see Figure C.2). The third side of this triangle is the distance separating X0 and X1 ; hence 1/2 h01 = u21 + 2 − 2u1 cos θ1 . Let δ1 represent the angle X1 X0 X01 , given by h01 δ1 = arccos , 2r and let β1 be the angle X1 X0 XS , given by 2 h01 + 2 − u21 β1 = arccos . 2h01
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2234
H. P. KEELER AND P. G. TAYLOR
h01 δ1 r
u1 r u01
Fig. C.2. The intersection of transmission circumferences.
Represent the last angle, XS X0 X01 , by η1 = δ1 − β1 , thus giving the expression 1/2 u01 = 2 + r2 − 2r cos η1 . Consider the point of distance u from the sink that lies on the transmission circumference of a node that has a sink distance γ. Recall that the angle that is formed by connecting this point to the node via the sink is given by the expression 2 u + γ 2 − r2 ψγ (u) = arccos . 2uγ We place an emphasis on this angle function, as it appears in the kernels of the integral area expressions for both the independent and dependent cases, and we introduce the function Δψ(u2 ) to describe the angular width of the intersection of the source and current feasible regions. That is, Δψ(u2 ) represents the angle between the top and bottom edges of the intersection region at a distance u2 from the sink (refer to Figure C.3). Subsequently, the intersection area function u2 A1\0 (u2 ) = w2 Δψ(w2 )dw2 −r
allows us to calculate the feasible area under the dependent model. Hence, we need an analytic form of Δψ(u2 ) over the entire u2 domain. The intersection area is zero on the interval [u1 − r, − r], and hence, Δψ(u2 ) is zero on this interval (refer to Figure C.1). The boundary of the intersection region is formed by arcs from the two circles of radius r and a sector of radius u2 . The shape of this boundary naturally affects the value of Δψ(u2 ). Thus, we observe that the rest of the domain can be divided into two intervals, [ − r, u01 ] and [u01 , u1 ].
u2 Δψ(u2 )
Fig. C.3. The angle function Δψ(u2 ) describes the intersection region.
On the first interval, [−r, u01 ], the intersection area is zero when the point X01 is located above the baseline, which extends from X0 to XS . Conversely, the intersection
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2235
area on this interval is positive if δ1 > β1 . When this condition is met, we observe on the first interval that the upper and lower arcs, which form the feasible region boundary, coincide with the transmission circumference of the source node. Hence, on the first interval we have the intersection angle expression Δψ(u2 ) = 2ψ (u2 )I− 01 ,
− r ≤ u2 ≤ u01 ,
− where I− 01 is an indicator function for when X01 is below the baseline. That is, I01 = 1 when X01 is below the baseline or, equivalently, when δ1 > β1 . We observe on the second interval, [u01 , u1 ], that part of the feasible region boundary coincides with the transmission circumference of the current forwarding node. Hence, on the second interval we obtain the intersection angle expression
Δψ(u2 ) = ψ (u2 ) + ψu1 (u2 ) − θ1 ,
u01 ≤ u2 ≤ u1 .
To perform the integration step we note that our expression for the feasible area under the independent model gives us the general solution to the integral Aγ (u) = 2
u
γ−r
wψγ (w)dw
= r2 φγ (u) + u2 ψγ (u) − rγ sin φγ (u), where we recall the angle functions
r 2 + γ 2 − u2 , 2rγ 2 u + γ 2 − r2 ψγ (u) = arccos . 2uγ φγ (u) = arccos
Thus, on the first interval, [ − r, u01 ], we have the area expression A1\0 (u2 ) = 2
u2
−r
w2 ψ (w2 )dw2 I− 01
= A (u2 )I− 01 . On the second interval, [u01 , u1 ], we have the slightly more complicated area expression A1\0 (u2 ) = =
u2
−r
w2 [ψ (w2 ) + ψu1 (w2 ) − θ1 ] dw2 + A (u01 )I− 01
1 A (u2 ) + Au1 (u2 ) + θ1 u201 − u22 2 1 A (u01 )[2I− + 01 − 1] − Au1 (u01 ) . 2
Consequently, for i = 1 the feasible area function under the dependent model is given by
1 (u2 ) = Au1 (u2 ) − A1\0 (u2 ). A
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2236
H. P. KEELER AND P. G. TAYLOR
X2
X1
X01 X0
X02
Fig. C.4. The intersection region R0∩2\1 (u3 ) can consist of up to four arcs.
C.2. After two hops. After two hops, an area expression for the intersection region R0∩2\1 (u3 ) = I0 (u3 ) ∩ I2 (u3 ) \ I1 (u3 ) is needed to calculate the area function under the dependent model (see one example of the intersection region in Figure C.4). Again, the angle function Δψ(u3 ) describes the angular width of the area under consideration, thus giving the general integral expression A0∩2\1 (u3 ) =
u3
u0 −r
w3 Δψ(w3 )dw3 .
We assume again that the first forwarding node is above the baseline, which runs from the source node to the sink. We outline a more general approach to find an analytic expression for Δψ(u3 ). In relation to the baseline, we refer to the boundary of the feasible region that is above the baseline as simply the top path. We consider a point on the top path which is a distance u3 from the sink. Let ψT (u3 ) be the angle between the baseline and the line connecting the sink to this point. Likewise, we refer to the boundary below the baseline as the bottom path, and we let ψB (u3 ) be its corresponding angle in relation to the sink, thus leading to the angle expression Δψ(u3 ) = ψT (u3 ) − ψB (u3 ). The region R0∩2\1 (u3 ) is enclosed by the three transmission circles and a sector of radius u3 . This knowledge can be used to obtain general expressions for ψT (u3 ) and ψB (u3 ). We consider the positioning of the intersection points of the transmission circumferences. For the top path, if the intersection point X01 is above the baseline, then ψT (u3 ) = min[θ1 − ψu1 (u3 ), ψu0 (u3 )]. A similar approach applied to the bottom path gives
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A STOCHASTIC ANALYSIS OF SENSOR NETWORK ROUTING
2237
ψB (u3 ) = max[θ1 + θ2 − ψu2 (u3 ), −ψu0 (u3 )]. Combining these results gives the angular width function Δψ(u3 ) = min [θ1 − ψu1 (u3 ), ψu0 (u3 )] − max [θ1 + θ2 − ψu2 (u3 ), −ψu0 (u3 )] , which allows the calculation of the area of R0∩2\1 (u3 ). Thus, we have a method of calculating the feasible areas after one and two hops under the dependent model:
2 (u3 ) = Au2 (u3 ) − A2\1 (u3 ) − A0∩2\1 (u3 ). A Arguably, a simpler but slightly more computationally exhaustive method is to use a crude Monte Carlo method. Alternatively, all the different geometrical configurations could be laboriously listed and their intersection expressions derived. REFERENCES [1] I. F. Akyildiz, S. Weilian, Y. Sankarasubramaniam, and E. Cayirci, A survey on sensor networks, IEEE Communications Magazine, 40 (2002), pp. 102–114. [2] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, Volume II: Theory, NOW Publishers, Delft, The Netherlands, 2009. [3] P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia, Routing with guaranteed delivery in ad hoc wireless networks, in Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, ACM, New York, 1999, pp. 48–55. [4] S. Capkun, M. Hamdi, and J.-P. Hubaux, GPS-free positioning in mobile ad hoc networks, in Proceedings of the 34th Annual Hawaii International Conference on System Sciences, IEEE, Washington, DC, 2001, p. 10. [5] C. Chong and S. P. Kumar, Sensor networks: Evolution, opportunities and challenges, Proc. IEEE, 91 (2003), pp. 1274–1256. [6] K. Fabian, W. Roger, Z. Yan, and Z. Aaron, Geometric ad-hoc routing: Theory and practice, in Proceedings of the 22nd Annual Symposium on Principles of Distributed Computing, ACM, Boston, 2003, pp. 63–72. [7] H. Frey, Scalable geographic routing algorithms for wireless ad hoc networks, IEEE Trans. Networks, 18 (2004), pp. 18–22. [8] S. Giordano and J.-Y. Le Boudec, Self-organizing wide-area routing, in Proceedings of SCI 2000/ISAS 2000, Orlando, FL, 2000. [9] R. Iyengar and B. Sikdar, Scalable and distributed GPS free positioning for sensor networks, in Proceedings of the IEEE International Conference on Communications, Vol. 1, IEEE, Washington, DC, 2003, pp. 338–342. [10] B. Karp and H. T. Kung, Greedy perimeter stateless routing for wireless networks, in Proceedings of the 6th Annual ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom 2000), ACM, New York, IEEE, Washington, DC, 2000, pp. 243–254. [11] L. Kleinrock and J. Silvester, Optimum transmission radii for packet radio networks or why six is a magic number, in Proceedings of the IEEE National Telecommunications Conference, IEEE, Washington, DC, 1978, pp. 4.31–4.35. [12] S. Kullback, Information Theory and Statistics, Wiley, New York, 1959. [13] R. Madan and S. Lall, An energy-optimal algorithm for neighbor discovery in wireless sensor networks, Mobile Networks and Applications, 11 (2006), pp. 317–326. [14] A. Sinha and A. Chandrakasan, Dynamic power management in wireless sensor networks, IEEE Design and Test of Computers, 18 (2001), pp. 62–74. [15] I. Stojmenovic, Position-based routing in ad hoc networks, IEEE Communications Magazine, 40 (2002), pp. 128–134. [16] H. Takagi and L. Kleinrock, Optimal transmission ranges for randomly distributed packet radio terminals, IEEE Trans. Communications, 32 (1984), pp. 246–257.
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