A Geometric Interpretation of Fading in Wireless Networks: Theory and Applications
aa r X i v : . [ c s . I T ] N ov A Geometric Interpretation of Fading inWireless Networks: Theory and Applications
Martin Haenggi,
Senior Member, IEEE
Abstract
In wireless networks with random node distribution, the underlying point process model and thechannel fading process are usually considered separately. A unified framework is introduced that permitsthe geometric characterization of fading by incorporating the fading process into the point process model.Concretely, assuming nodes are distributed in a stationary Poisson point process in R d , the propertiesof the point processes that describe the path loss with fading are analyzed. The main applications areconnectivity and broadcasting. Index Terms
Wireless networks, geometry, point process, fading, connectivity, broadcasting.
I. I
NTRODUCTION AND S YSTEM M ODEL
A. Motivation
The path loss over a wireless link is well modeled by the product of a distance component (often calledlarge-scale path loss) and a fading component (called small-scale fading or shadowing). It is usuallyassumed that the distance part is deterministic while the fading part is modeled as a random process.This distinction, however, does not apply to many types of wireless networks, where the distance itselfis subject to uncertainty. In this case it may be beneficial to consider the distance and fading uncertaintyjointly, i.e. , to define a stochastic point process that incorporates both. Equivalently, one may regard thedistance uncertainty as a large-scale fading component and the multipath fading uncertainty as small-scalefading component.
This paper is an extension of preliminary work that has appeared at ISIT 2006, Seattle, WA, and ISIT 2007, Nice, France.M. Haenggi is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: [email protected]
November 4, 2018 DRAFT
We introduce a framework that offers such a geometrical interpretation of fading and some new insightinto its effect on the network. To obtain concrete analytical results, we will often use the Nakagami- m fading model, which is fairly general and offers the advantage of including the special cases of Rayleighfading and no fading for m = 1 and m → ∞ , respectively.The two main applications of the theoretical foundations laid in Section 2 are connectivity (Section 3)and broadcasting (Section 4). Connectivity.
We characterize the geometric properties of the set of nodes that are directly connectedto the origin for arbitrary fading models, generalizing the results in [1], [2]. We also show that if thepath loss exponent equals the number of network dimension, any fading model (with unit mean) isdistribution-preserving in a sense made precise later.
Broadcasting.
We are interested in the single-hop broadcast transport capacity , i.e. , the cumulateddistance-weighted rate summed over the set of nodes that can successfully decode a message sent from atransmitter at the origin. In particular, we prove that if the path loss exponent is smaller than the numberof network dimensions plus one, this transport capacity can be made arbitrarily large by letting the rateof transmission approach 0.In Section 5, we discuss several other applications, including the maximum transmission distance,probabilistic progress, the effect of retransmissions, and localization. B. Notation and symbols
For convenient reference, we provide a list of the symbols and variables used in the paper. Most ofthem are also explained in the text. Note that slanted sans-serif symbols such as x and f denote randomvariables, in contrast to x and f that are standard real numbers or “dummy” variables. Since we modelthe distribution of the network nodes as a stochastic point process, we use the terms points and nodesinterchangeably. November 4, 2018 DRAFT
Symbol Definition/explanation [ k ] the set { , , . . . , k } A ( x ) indicator function u ( x ) , { x > } ( x ) (unit step function) d number of dimensions of the network o origin in R d B a Borel subset of R or R d c d , π d/ / Γ(1 + d/ (volume of the d -dim. unit ball) α path loss exponent δ , d/α ∆ , ( d + 1) /αs minimum path gain for connection F, f fading distribution (cdf), fading r.v. F X distribution of random variable X (cdf) Φ = { x i } path loss process before fading (PLP) Ξ = { ξ i } path loss process with fading (PLPF) ˆΦ = { ˆ x i } points in Φ connected to origin ˆΞ = { ˆ ξ i } points in Ξ connected to origin Λ , λ counting measure and density for Φˆ N = ˆΞ( R + ) number of nodes connected to o A cardinality of A C. Poisson point process model
A well accepted model for the node distribution in wireless networks is the homogeneous Poissonpoint process (PPP) of intensity λ . Without loss of generality, we can assume λ = 1 (scale-invariance). Node distribution.
Let the set { y i } , i ∈ N consist of the points of a stationary Poisson point process in R d of intensity , ordered according to their Euclidean distance k y i − o k to the origin o . Define a new In particular, if nodes move around randomly and independently, or if sensor nodes are deployed from an airplane in largequantities.
November 4, 2018 DRAFT one-dimensional (generally inhomogeneous) PPP { r i , k y i − o k} such that < r < r < . . . a.s. Let α > be the path loss exponent of the network and Φ = { x i , r αi } be the path loss process (beforefading) (PLP). Let { f , f , f , . . . } be an iid stochastic process with f drawn from a distribution F , F f with unit mean, i.e. , E f = 1 , and supp f ⊂ R + . Finally, let Ξ = { ξ i , x i / f i } be the path loss processwith fading (PLPF). In order to treat the case of no fading in the same framework, we will allow thedegenerate case F ( x ) = u ( x − , resulting in Φ = Ξ . Note that the fading is static (unless mentionedotherwise), and that { ξ i } is no longer ordered in general. We will also interpret these point processes as random counting measures , e.g. , Φ( B ) = { Φ ∩ B } for any Borel subset B of R . Connectivity.
We are interested in connectivity to the origin. A node i is connected if its path loss issmaller than /s , i.e. , if ξ i < /s . The processes of connected nodes are denoted as ˆΦ = { x i : ξ i < /s } (PLP) and ˆΞ = { ξ i : ξ i < /s } = Ξ ∩ [0 , /s ) (PLPF). Counting measures.
Let Λ be the counting measure associated with Φ , i.e. , Λ( B ) = E Φ( B ) for Borel B . For Λ([0 , a )) = E Φ([0 , a )) , we will also use the shortcut Λ( a ) . Similarly, let ˆΛ be the countingmeasure for ˆΦ . All the point processes considered admit a density . Let λ ( x ) = d Λ( x ) / d x and and ˆ λ ( x ) = d ˆΛ( x ) / d x be the densities of Φ and ˆΦ , respectively. Fading model.
To obtain concrete results, we frequently use the Nakagami- m (power) fading model.The distribution and density are F ( x ) = 1 − Γ ic ( m, mx )Γ( m ) (1) f ( x ) = m m x m − exp( − mx )Γ( m ) , (2)where Γ ic denotes the upper incomplete gamma function. This distribution is a single-parameter versionof the gamma distribution where both parameters are the same such that the mean is always. D. The standard network
For ease of exposition, we often consider a standard network that has the following parameters: δ , d/α = 1 (path loss exponent equals the number of dimensions) and Rayleigh fading, i.e. , F ( x ) =(1 − e − x ) u ( x ) .Fig. 1 shows a PPP of intensity 1 in a × square, with the nodes marked that can be reachedfrom the center, assuming a path gain threshold of s = 0 . . The disk shows the maximum transmissiondistance in the non-fading case. The term “standard” here refers to the fact that in this case the analytical expressions are particularly simple. We do notclaim that these parameters are the ones most frequently observed in reality.
November 4, 2018 DRAFT −8 −6 −4 −2 0 2 4 6 8−8−6−4−202468
Fig. 1. A Poisson point process of intensity 1 in a × square. The reachable nodes by the center node are indicated by abold × for a path gain threshold of s = 0 . , a path loss exponent of α = 2 , and Rayleigh fading (standard network). The circleindicates the range of successful transmission in the non-fading case. Its radius is / √ s ≈ . , and there are about π/s ≈ nodes inside. II. P
ROPERTIES OF THE P OINT P ROCESSES
Proposition 1
The processes Φ , Ξ , and ˆΞ are Poisson.Proof: { y i } is Poisson by definition, so { r i } and Φ = { x i } are Poisson by the mapping theorem[3]. Ξ is Poisson since f i is iid, and ˆΞ( R ) = Ξ([0 , /s )) .The Poisson property of ˆΦ will be established in Prop. 6.Cor. 2 states some basic facts about these point processes that result from their Poisson property. Corollary 2 (Basic properties.) (a) Λ( x ) = E Φ([0 , x )) = c d x δ and λ ( x ) = c d δx δ − . In particular, for δ = 1 , Φ is stationary (on R + ).(b) r i is governed by the generalized gamma pdf f r i ( r ) = e − c d r d d ( c d r d ) i r Γ( i ) , (3) November 4, 2018 DRAFT and x i is distributed according to the cdf F x i ( x ) = 1 − Γ ic ( i, c d x δ )Γ( i ) , . (4) The expected path loss without fading is E x i = c − /δd Γ( i + 1 /δ )Γ( i ) . (5) In particular, for the standard network, the x i are Erlang with E x i = i/c d .(c) The distribution function of ξ i is F ξ i ( x ) = 1 − Z ∞ F ( r/x ) (cid:18) c id δr δi − exp( − c d r δ )Γ( i ) (cid:19) d r . (6) For δ = 1 and Nakagami- m fading, the pdf of ξ i is f ξ i ( x ) = m m +1 (cid:0) m + i − m (cid:1) c id x i − ( m + c d x ) m + i . (7) In particular, F ξ ( x ) = 1 − (cid:18) mc d x + m (cid:19) m (8) and E ξ i = mic d ( m − for m > (9) Var ξ i = m i ( m + i − c d ( m − ( m − for m > . (10) For the standard networks, F ξ i ( x ) = (cid:18) c d xc d x + 1 (cid:19) i . (11) Proof: (a) Since the original d -dimensional process { y i } is stationary, the expected number of points in a ballof radius x around the origin is c d x d . The one-dimensional process { r i } has the same number ofpoints in [0 , x ) , and x i = r αi , so E Φ([0 , x )) = c d x δ . For δ = 1 , λ ( x ) = c d is constant.(b) Follows directly from the fact that { y i } is stationary Poisson. ((3) has been established in [4].)(c) The cdf P [ ξ i < x ] is − E x i ( F ( x i /x )) with x i distributed according to (4). (7) is obtained bystraightforward (but tedious) calculation. Remarks: - For general (rational) values of m , d , and α , F ξ i can be expressed using hypergeometric functions. November 4, 2018 DRAFT - (8) approaches − exp( − c d x ) as m → ∞ , which is the distribution of x . Similarly, lim m →∞ E ξ i = i/c d = E x i and lim m →∞ Var ξ i = i/c d = Var x i .- Alternatively we could consider the path gain process ξ − i . Since F ξ − i ( x ) = 1 − F ξ i (1 /x ) , thedistribution functions look similar.- In the standard network, the expected path loss E ξ i does not exist for any i , and for i = 1 , theexpected path gain is infinite, too, since both x and f are exponentially distributed. For i > , E ( ξ − i ) = c d / ( i − , and for i > , Var( ξ − i ) = 2 c d / (( i − i − .- For the standard network, the differential entropy h ( ξ i ) , E [ − ln f ξ i ( ξ i )] is − log c d for i = 1 andgrows logarithmically with i . For Nakagami- m fading h ( ξ ) = 1 + 1 /m − log c d . For the path gainprocess in the standard network, the entropy has the simple expression h ( ξ − i ) = i + 1 i + log (cid:16) πi (cid:17) , (12)which is monotonically decreasing , reflecting the fact that the variance Var ξ − i is decreasing with i − .- The ξ i are not independent since the x i are ordered. For example, in the case of the standard network,the difference x i +1 − x i is exponentially distributed with mean /c d , thus the joint pdf is f x ... x n ( x , . . . , x n ) = c nd e − c d x n For δ = 1 and any fading distribution F with mean , Ξ( B ) d = Φ( B ) ∀ B ⊂ R + , i.e., fading is distribution-preserving.Proof: Since Ξ is Poisson, independence of Ξ( B ) and Ξ( B ) for B ∩ B = ∅ is guaranteed. Soit remains to be shown that the intensities (or, equivalently, the counting measures on Borel sets) are thesame. This is the case if for all a > , E ( { x i : x i > a, ξ i < a } ) = E ( { x i : x i < a, ξ i > a } ) , i.e. , the expected numbers of nodes crossing a from the left (leaving the interval [0 , a ) ) and the right(entering the same interval) are equal. This condition can be expressed as Z a λ ( x ) F ( x/a ) d x = Z ∞ a λ ( x )(1 − F ( x/a )) d x ∀ a > . November 4, 2018 DRAFT If δ = 1 , λ ( x ) = c d , and the condition reduces to Z F ( x ) d x = Z ∞ (1 − F ( x )) d x , which holds since Z (1 − F ( x )) d x | {z } − R F ( x ) d x + Z ∞ (1 − F ( x )) d x = E f = 1 . An immediate consequence is that a receiver cannot decide on the amount of fading present in thenetwork if δ = 1 and geographical distances are not known. Corollary 4 For Nakagami- m fading, δ = 1 , and any a > , the expected number of nodes with x i < a and ξ i > a , i.e. , nodes that leave the interval [0 , a ) due to fading, is E ( { x i : x i < a, ξ i > a } ) = c d a m m − Γ( m ) e − m . (14) The same number of nodes is expected to enter this interval. For Rayleigh fading ( m = 1 ), the fractionof nodes leaving any interval [0 , a ) is /e .Proof: E ( { x i : x i < a } ) = Λ( a ) = c d a , and for Nakagami- m , the fraction of nodes leaving theinterval is Z F ( x ) d x = m m − Γ( m ) e − m . Clearly, fading can be interpreted as a stochastic mapping from x i to ξ i . So, { x i } are the points inthe geographical domain (they indicate distance), whereas { ξ i } are the points in the path loss domain,since ξ i is the actual path loss including fading. This mapping results in a partial reordering of thenodes, as visualized in Fig. 2. In the path loss domain, the connected nodes are simply given by { ˆ ξ i } = { ξ i } ∩ [0 , /s ] .Fig. 3 illustrates the situation for 200 nodes randomly chosen from [0 , with a threshold s = 1 . Beforefading, we expect 40 nodes inside. From these, a fraction e − is moving out (right triangles), the reststays in (marked by × ). From the ones outside, a fraction (1 − e − )( ae ) ≈ moves in (left triangles),the rest stays out (circles).For the standard network, the probability of point reordering due to fading can be calculated explicitly.Let P i,j , P [ ξ i > ξ i + j ] . By this definition, P i,j = P [ x i / f i > x i + j / f i + j ] = P (cid:20) x i x i + y j > f i f i + j (cid:21) . (15) November 4, 2018 DRAFT x ξ Fig. 2. The points of a Poisson point process x i are mapped and reordered according to ξ i := x i / f i , where f i is iid exponentialwith unit mean. In the lower axis, the nodes to the left of the threshold /s are connected to the origin (path loss smaller than /s ). good f ad i ngbad f ad i ng Geographical domain P a t h l o ss do m a i n Fig. 3. Illustration of the Rayleigh mapping. 200 points x i are chosen uniformly randomly in [0 , . Plotted are the points ( x i , x i /f i ) , where the f i are drawn iid exponential with mean 1. Consider the interval [0 , ( i.e. , assume a threshold s = 1 ).Points marked by × are points that remain inside [0 , , those marked by o remain outside, the ones marked with left- andright-pointing triangles are the ones that moved in and out, respectively. The node marked with a double triangle is the furthestreachable node. On average the same number of nodes move in and out. Note that not all points are shown, since a fraction e − is mapped outside of [0 , . November 4, 2018 DRAFT0 x i is Erlang with parameters i and c d , y j is the distance from x i to x i + j and thus Erlang with parameters j and c d , and the cdf of z := f n / f n + m is F z ( x ) = x/ ( x + 1) . Hence P i,j = E x , y (cid:18) x i x i + y j (cid:19) = Z ∞ Z ∞ x x + y c i + jd x i − y j − Γ( i )Γ( j ) e − c d ( x + y ) d x d y .P i,j does not depend on c d . Closed-form expressions include P , = 1 − ln 2 ≈ . , and P , =3 − ≈ . . Generally P k,k can be determined analytically. For k = 1 , , , , we obtain − ln 2 , 12 ln 2 − , / − 120 ln 2 , − . Further, lim k →∞ P k,k = 1 / , which is the probabilitythat an exponential random variable is larger than another one that has twice the mean.In the limit, as i → ∞ , P i,j = 1 / ( j + 1) , which is the probability that a node has the largest fadingcoefficient among j + 1 nodes that are at the same distance. Indeed, as i → ∞ , x i + j < x i (1 + ǫ ) a.s. forany ǫ > and finite j .While the ξ i are dependent, it is often useful to consider a set of independent random variables, obtainedby conditioning the process on having a certain number of nodes n in an interval [0 , a ) (or, equivalently,conditioning on x n +1 = a ) and randomly permuting the n nodes. In doing so, the n points { x i } and { ξ i } , i = 1 , , . . . , n are iid distributed as follows. Corollary 5 Conditioned on x n +1 = a :(a) The nodes { x i } ni =1 are iid distributed with f a x i ( x ) = λ ( x )Λ( x ) = δ (cid:16) xa (cid:17) δ x , x < a (16) and cdf F a x i ( x ) = ( x/a ) δ .(b) The path loss with fading { ξ i } ni =1 is distributed as F aξ i ( x ) = 1 − Z a F ( y/x ) δ (cid:16) ya (cid:17) δ y d y . (17) (c) For the standard network, F aξ i ( x ) = xa (cid:16) − e − a/x (cid:17) (18) (d) For Rayleigh fading and δ = 1 / , F aξ i ( x ) = √ π r xa erf (cid:18)r ax (cid:19) . (19) Proof: As in (6), the cdf is given by − E ( F ( y /x )) with y distributed as (16). November 4, 2018 DRAFT1 III. C ONNECTIVITY Here we investigate the processes ˆΦ and ˆΞ = Ξ ∩ [0 , /s ) of connected nodes. A. Single-transmission connectivity and fading gain Proposition 6 (Connectivity) Let a transmitter situated at the origin transmit a single message, andassume that nodes with path loss smaller than /s can decode, i.e. , are connected. We have:(a) ˆΦ is Poisson with ˆ λ ( x ) = λ ( x )(1 − F ( sx )) .(b) With Nakagami- m fading, the number ˆ N = ˆΦ( R + ) of connected nodes is Poisson with mean E ˆ N m = c d ( ms ) δ Γ( δ + m )Γ( m ) (20) and the connectivity fading gain , defined as the ratio of the expected numbers of connected nodeswith and without fading, is E ˆ N m E ˆ N ∞ = 1 m δ Γ( δ + m )Γ( m ) = E ( f δ ) . (21) Proof: (a) The effect of fading on the connectivity is independent (non-homogeneous) thinning by − F ( sx ) = P [ x/ f < /s ] .(b) Using (a), the expected number of connected nodes is Z ∞ ˆ λ ( x ) d x = Z ∞ c d δx δ − Γ ic ( m, msx )Γ( m ) d x which equals E ˆ N m in the assertion. Without fading, E ˆ N ∞ = lim m →∞ = Λ(1 /s ) = c d s − δ , whichresults in the ratio (21). Remarks: 1) (20) is a generalization of a result in [1] where the connectivity of a node in a two-dimensionalnetwork with Rayleigh fading was studied.2) E ˆ N can also be expressed as E ˆ N = ∞ X i =1 P [ ξ i < /s ] . (22)The relationship with part (b) can be viewed as a simple instance of Campbell’s theorem [5]. Since ˆ N is Poisson, the probability of isolation is P ( ˆ N = 0) = exp( − E ˆ N ) .3) E ˆ N = c d s − δ Γ( δ + 1) , and E ˆ N ∞ = c d s − δ . For δ = 1 , ˆ N does not depend on the type (or presence)of fading. November 4, 2018 DRAFT2 δ f ad i ng ga i n Fig. 4. Connectivity fading gain for Nakagami- m fading as a function of δ ∈ [0 , / and m ∈ [1 , . For δ = 1 , the gain is independent of m (thick line). 4) The connectivity fading gain equals the δ -th moment of the fading distribution, which, by definition,approaches one as the fading vanishes, i.e. , as m → ∞ . For a fixed δ , it is decreasing in m if δ > ,increasing if δ < , and equal to for all m if δ = 1 . It also equals if δ = 0 . For a fixed m ,it is not monotonic with δ , but exhibits a minimum at some δ min ∈ (0 , . The fading gain as afunction of δ and m is plotted in Fig. 4. For Rayleigh fading and δ = 1 / , the fading gain is π/ ,and the minimum is assumed at δ min ≈ . , corresponding to α ≈ . for d = 2 . So, dependingon the type of fading and the ratio of the number of network dimensions to the path loss exponent α , fading can increase or decrease the number of connected nodes.5) For the standard network, E ˆ N = c d /s and the probability of isolation is e − c d /s .6) The expected number of connected nodes ˆ N a with x i < a is E ˆ N a = c d a δ F aξ i (1 /s ) . (23)where F aξ i is given in (17). Corollary 7 Under Nagakami- m fading, a uniformly randomly chosen connected node ˆ x ∈ ˆΦ has mean E ˆ x = δ ( δ + m ) ms ( δ + 1) , (24) which is δ/m times the value without fading. November 4, 2018 DRAFT3 Proof: A random connected node ˆ x is distributed according to f ˆ x ( x ) = ˆ λ ( x ) E ˆ N . (25)Without fading, the distribution is s δ δx δ − , x /s , resulting in an expectation of δ/ ( s ( δ + 1)) .For Rayleigh fading, for example, the density f ˆ x is a gamma density with mean δ/s , so the averageconnected node is δ times further away than without fading. B. Connectivity with retransmissions Assuming a block fading network and n transmissions of the same packet, what is the process of nodesthat receive the packet at least once? Corollary 8 In a network with iid block fading, the density of the process of nodes ˆ λ n that receive atleast one of n transmissions is ˆ λ n ( x ) = (1 − F ( sx ) n ) c d δx δ − . (26) Proof: This is a straightforward generalization of Prop. 6(a).So, in a standard network, the number of connected nodes with n transmissions E ˆ N n = Z ∞ ˆ λ n ( x ) d x = c d s (Ψ( n + 1) + γ ) , (27)where Ψ is the digamma function (the logarithmic derivative of the gamma function), which grows with log n . Alternatively if the threshold s k for the k -th transmission is chosen as s k , s /k , k ∈ [ n ] , theexpected number of nodes reached increases linearly with the number of transmissions.IV. B ROADCASTING A. Broadcasting reliability Proposition 9 For δ = 1 and Nakagami- m fading, m ∈ N , the probability that a randomly chosen node x ∈ [0 , a ) can be reached is p m (˜ s ) = 1˜ s − exp( − m ˜ s ) m − X k =0 m k (1 − k/m ) k ! ˜ s k ! , (28) where ˜ s , as . p m is increasing in m for all ˜ s > and converges uniformly to lim m →∞ p m (˜ s ) = min { , ˜ s − } . (29) November 4, 2018 DRAFT4 Proof: p m (˜ s ) is given by p m (˜ s ) = Z (1 − F (˜ sx )) d x = Z Γ( m, m ˜ sx )Γ( m ) d x . (30)For m ∈ N , this is p m (˜ s ) = m − X k =0 Z exp( − m ˜ sx ) ( m ˜ sx ) k k ! d x , (31)which, after some manipulations, yields p m (˜ s ) = 1˜ s − m exp( − m ˜ s ) m − X k =0 k X j =0 ( m ˜ s ) j j ! (32) = 1˜ s − exp( − m ˜ s ) m − X k =0 m k (1 − k/m ) k ! ˜ s k | {z } P m − (˜ s ) . (33)The polynomial P m − is the Taylor expansion of order m of (1 − ˜ s ) exp( m ˜ s ) at ˜ s = 0 (the coefficientfor ˜ s m is zero). So exp( − m ˜ s ) P m − (˜ s ) = 1 − s + O ( s m +1 ) from which the limit for ˜ s < follows.For ˜ s > , the exponential dominates the polynomial so that their product tends to zero and / ˜ s remainsas the limit.The convergence to min { , ˜ s − } is the expected behavior, since without fading a node is connected ifit is positioned within [0 , /s ] ( ˜ s < ) and for a randomly chosen node in [0 , a ] for a > /s or ˜ s > , thishas probability /as . So with increasing m , derivatives of higher and higher order become 0 at ˜ s = 0 .From the previous discussion we know that p m (˜ s ) = 1 + O (˜ s m ) . Calculating the coefficient for ˜ s m yields p m (˜ s ) = 1 − m m Γ( m + 2) ˜ s m + O (˜ s m +1 ) . (34)The m -th order Taylor expansion at ˜ s = 0 is a lower bound. Upper bounds are obtained by truncatingthe polynomial; a natural choice is the first-order version m − s to obtain (cid:18) − m m Γ( m + 2) ˜ s m (cid:19) + < p m (˜ s ) min (cid:26) , s (1 − exp( − m ˜ s )(1 + ( m − s )) (cid:27) . (35)Using the lower bound, we can establish the following Corollary. Corollary 10 ( ǫ -reachability.) If as < (Γ( m + 2) · ǫ ) /m m . (36) at least a fraction − ǫ of the nodes x i ∈ [0 , a ) are connected. In the standard network (specializing to m = 1 ), the sufficient condition is as < ǫ , (37) November 4, 2018 DRAFT5 This follows directly from the lower bound in (35). Remarks: - For m → ∞ , the bound (36) is not tight since the RHS converges to /e for all positive ǫ (byStirling’s approximation), while the exact condition is as < / (1 − ǫ ) .- The sufficient condition (37) is tight (within 7%) for ǫ < . . With p ( as ) = (1 − e − as ) /as , thecondition p ( as ) > − ǫ can be solved exactly using the Lambert W function: as < W ( − qe − q ) + q , where q , − ǫ . (38)A linear approximation yields the same bound as before, while a quadratic expansion yields thesufficient condition as < ǫ + 4 / ǫ which is within . for ǫ < . . B. Broadcast transport sum-distance and capacity Assuming the origin o transmits, the set of nodes that receive the message is { ˆ x i } . We shall determinethe broadcast transport sum-distance D , i.e. , the expected sum over the all the distances ˆ x /αi from theorigin: D , E X x ∈ ˆΦ x /α (39) Proposition 11 The broadcast transport sum-distance for Nakagami- m fading is D m = c d δ ∆ 1( ms ) ∆ Γ( m + ∆)Γ( m ) , (40) and the (broadcast) fading gain D m /D ∞ is D m D ∞ = 1 m ∆ Γ( m + ∆)Γ( m ) = E ( f ∆ ) . (41) Proof: From Campbell’s theorem E X x ∈ ˆΦ x /α = Z ∞ x /α ˆ λ ( x ) d x = c d δ Z ∞ x /α + δ − (1 − F ( sx )) d x , which equals (40) for Nakagami- m fading. November 4, 2018 DRAFT6 Without fading, a node x i is connected if x i < /s , therefore D ∞ = Z /s x /α λ ( x ) d x (42) = c d δ ∆ s − ∆ = c d dd + 1 s − ∆ . (43)So the fading gain D m /D ∞ is the ∆ -th moment of f as given in (41). Remarks: 1) The fading gain is independent of the threshold s . D m ∝ s − ∆ for all m . It strongly resembles theconnectivity gain (Prop. 6), the only difference being the parameter ∆ instead of δ . In particular, D m is independent of m if ∆ = 1 . See Remark 3 to Prop. 6 and Fig. 4 for a discussion and visualizationof the behavior of the gain as a function of m and ∆ .2) For Rayleigh fading ( m = 1 ), D = c d δs − ∆ , and the fading gain is Γ(1 + ∆) . For d = α = 2 , D ∞ = π s / .3) The formula for the broadcast transport sum-distance reminds of an interference expression. Indeed,by simply replacing x /α by x − , a well-known result on the mean interference is reproduced:Assuming each node transmits at unit power, the total interference at the origin is E X x ∈ Φ x − ! = Z ∞ x − λ ( x ) d x = c d δδ − x δ − (cid:12)(cid:12)(cid:12) ∞ which for δ < diverges due to the lower bound integration bound ( i.e. , the one or two closestnodes) and for δ > diverges due to the upper bound ( i.e. , the large number of nodes that are faraway).So far, we have ignored the actual rate of transmission R and just used the threshold s for the sum-distance. To get to the single-hop broadcast transport capacity C (in bit-meters/s/Hz), we relate the(bandwidth-normalized) rate of transmission R and the threshold s by R = log (1 + s ) and define C , max R> { R · D (2 R − } = max s> { log (1 + s ) D ( s ) } . (44)Let D m be the broadcast transport sum-distance for s = 1 (see Prop. 11) such that D m = D m s − ∆ . Proposition 12 For Nakagami- m fading:(a) For ∆ ∈ (0 , , the broadcast transport capacity is achieved for R opt = W (cid:16) − e − / ∆ ∆ (cid:17) + ∆ − log 2 , ∆ ∈ (0 , . (45) November 4, 2018 DRAFT7 The resulting broadcast transport capacity is tightly (within at most 0.13%) lower bounded by C m > D m (∆)log 2 (∆ − − ∆) (cid:16) e ∆ − − ∆ − (cid:17) − ∆ . (46) (b) For ∆ = 1 , C m = c d δ log 2 (47) independent of m , and R opt = 0 .(c) For ∆ > , the broadcast transport capacity increases without bounds as R → , independent of thetransmit power.Proof: (a) D m ∝ s − ∆ , so C m ∝ R (2 R − − ∆ which, for ∆ , has a maximum at R opt given in (45). Thelower bound stems from an approximation of R opt using W ( − exp( − / ∆) / ∆) ' − ∆ which holdssince for ∆ = 1 , the two expressions are identical, and the derivative of the Lambert W expressionis smaller than -1 for ∆ < .(b) For ∆ = 1 , C m increases as the rate is lowered but remains bounded as R → . The limit is c d δ/ log 2 .(c) For ∆ > , R (2 R − − ∆ is decreasing with R , and lim R → R (2 R − − ∆ = lim R → (log 2) − ∆ R − ∆ = ∞ . Remarks: - The optima for R , s are independent of the type of fading (parameter m ).- For ∆ < , the optimum s is tightly lower bounded by s opt > exp(∆ − − ∆) − . (48)This is the expression appearing in the bound (46).- (c) is also apparent from the expression D ( s ) log (1 + s ) , which, for s → , is approximately D m s − ∆ / log 2 . So, the intuition is that in this regime, the gain from reaching additional nodesmore than offsets the loss in rate.- For ∆ = 1 / (2 log 2) , s opt = R opt = 1 and C m = D m . This is, however, not the minimum. Thecapacity is minimum around ∆ ≈ . , depending slightly on m .Fig. 5 depicts the optimum rate as a function of ∆ , together with the lower bound (∆ − − ∆) / log 2 ,and Fig. 6 plots the broadcast transport capacity for Rayleigh fading and no fading for a two-dimensional November 4, 2018 DRAFT8 ∆ R op t Optimum rate vs. ∆ exactlower bound Fig. 5. Optimum transmission rates for ∆ ∈ [0 . , . The optimum rate is for ∆ = 1 / (2 log 2) ≈ . . network. The range ∆ ∈ [0 . , . corresponds to a path loss exponent range α ∈ [3 , . It can be seenthat Nakagami fading is harmful. For small values of ∆ , the capacity for Rayleigh fading is about 10%smaller. C. Optimum broadcasting (superposition coding) Assuming that nodes can decode at a rate corresponding to their SNR, the broadcast transport capacity(without fading) is ˜ C = E "X x ∈ Φ x /α log (1 + x − ) (49)To avoid problems with the singularity of the path loss law at the origin, we replace the log by 1 for x < . For x > , we use the lower bound log (1 + x − ) > /x . Proceeding as in the proof of Prop. 11,we obtain ˜ C > c d δ (cid:18) 1∆ + Z ∞ x ∆ − d x (cid:19) , (50)which is significantly larger than in the case with single-rate decoding. For ∆ < , ˜ C > c d δ ∆(1 − ∆) . (51) November 4, 2018 DRAFT9 ∆ C Broadcast transport capacity vs. ∆ m=1m= ∞ Fig. 6. Broadcast transport capacity for d = 2 , ∆ ∈ [0 . , . and m = 1 and m = ∞ . For ∆ = 1 , the capacity is π/ (3 log 2) ≈ . irrespective of m . For the no fading case, the minimum occurs at ∆ = 1 / (2 log 2) , where C = 2 π/ . For ∆ > , this lower bound and thus ˜ C is unbounded, in agreement with the previous result. The onlydifference is that for ∆ = 1 , ˜ C diverges whereas C is finite. Note that since log (1 + x − ) < / ( x log 2) for x > , the lower bound is within a factor log 2 of the correct value.If the actual Shannon capacity were considered for nodes that are very close, ˜ C would diverge morequickly as ∆ → ( α → ∞ ) since the contribution from the nodes within distance one would be: ˜ C [0 , > c d δ Z − x ∆ − log x d x = 1log(2)∆ . (52)V. O THER A PPLICATIONS A. Maximum transmission distance How far can we expect to transmit, i.e. , what is the (average) maximum transmission distance M , E (cid:0) max x ∈ ˆΦ { x /α } (cid:1) ?Let ˆ x be a uniformly randomly chosen connected node. The pdf f ˆ x is given by (25). The distribution November 4, 2018 DRAFT0 of the maximum x M of a Poisson number of RVs is given by the Gumbel distribution F ˆ x M ( x ) = exp (cid:16) − E ˆ N (1 − F ˆ x ( x ) (cid:17) . (53)So, in principle, M = E (ˆ x /αM ) can be calculated. However, even for the standard network, where F ˆ x M ( x ) = exp( − c d s exp( − sx )) , there does not seem to exist a closed-form expression. If the number ofconnected nodes was fixed to c d /s (instead of being Poisson distributed with this mean), we would have F ˆ x M ( x ) = (1 − e − xs ) c d /s with mean E ˆ x M = 1 s (cid:16) Ψ (cid:16) c d s + 1 (cid:17) + γ (cid:17) . (54)Since Ψ is concave, this upperbounds the true mean by Jensen’s inequality. Finally, we invoke Jensenagain by replacing E (ˆ x /αM ) by E (ˆ x M ) /α to obtain M < (cid:18) s (cid:16) Ψ (cid:16) c d s + 1 (cid:17) + γ (cid:17)(cid:19) /α . (55)Without much harm, Ψ( x ) could be replaced by (the slightly larger) log( x ) . Even replacing Ψ( x + 1) by log( x ) still appears to be an upper bound. The bound is quite tight, see Fig. 7. Also compare with Fig. 1,where the most distant node is quite exactly 6 units away ( s = 0 . ). The factor s − /α is the bound inthe non-fading case, so the Rayleigh fading (diversity) gain for the maximum transmission distance isroughly log(1 /s ) /α which grows without bounds as s → . B. Probabilistic progress In addition to the maximum transmission distance or the distance-rate product, the product distancestimes probability of success may be of interest. Without considering the actual node positions, one maywant to maximize the continuous probabilistic progress G ( x ) , max { x /α P [ f > sx ] } . For the standardnetwork with α = 2 , this is maximized at x = 1 / s . If there was no fading, the optimum would be x = p /s . Of course there is no guarantee that there is a node very close to this optimum location.Alternatively, define the (discrete) probabilistic progress when transmitting to node i by G i , E (cid:16) x /αi · P [ f > s x i | x i ] (cid:17) (56)We would like to find i opt = arg max i G i . For the standard network, G i = E (cid:16) x /αi exp( − s x i ) (cid:17) = c id ( s + c d ) i +1 /α Γ( i + 1 /α )Γ( i ) . (57) Note that the Gumbel cdf is not zero at + . This reflects the fact that the number of connected nodes may be zero, in whichcase the maximum transmission distance would be zero. Accordingly the pdf includes a pulse at , the term exp( − E ˆ N ) δ ( x ) . November 4, 2018 DRAFT1 M Expected max. transmission distance vs. threshold s simulationupper boundno fading Fig. 7. Expected maximum transmission distances for the standard two-dimensional network and s ∈ [0 . , . . Forcomparison, the curve s − / for the non-fading case is also displayed. The maximum of G i cannot be found directly, but since Γ( i + 1 /α ) / Γ( i ) is very tightly lower boundedby i /α we have G i / c id i /α ( s + c d ) i +1 /α (58)which, assuming a continuous parameter ˜ i , is maximized at ˜ i opt = 1 α log(1 + sc d ) . (59)Note that the same expression for i opt would be obtained if G i was approximated by the factorization G ′ i = E ( x /αi ) P [ ξ i < /s ] . For the standard network, E ( x /αi ) = Γ( i +1 /α )Γ( i ) c /αd , and P [ ξ i < /s ] = ( π/ ( π + s )) i .So G ′ i differs from G i only by the factor (1 + s/c d ) /α which is independent of i and quite small fortypical s .Now, the question is how to round ˜ i opt to i opt . For large s , i opt = 1 . For small s , ˜ i opt ≈ c d / ( αs ) so i opt = ⌈ c d αs ⌉ (60)is a good choice. It can be verified that this is indeed the optimum. The expected distance to this i opt -thnode is quite exactly / ( αs ) /α . So in this non-opportunistic setting when reliability matters, Rayleighfading is harmful; it reduces the range of transmissions by a factor α − /α . November 4, 2018 DRAFT2 C. Retransmissions and localization Proposition 13 (Retransmissions) Consider a network with block Rayleigh fading. The expected numberof nodes that receive k out of n transmitted packets E N nk is E N nk = c d Γ(1 + δ )( ks ) δ , k ∈ { , , . . . , n } . (61) Proof: Let p ( x ) , − F ( sx ) . The density of nodes that receive k packets out of n transmissions isgiven by λ nk ( x ) = λ ( x ) (cid:18) nk (cid:19) p ( x ) k (1 − p ( x )) n − k . (62)Plugging in p ( x ) = exp( − sx ) for Rayleigh fading and integrating (62) yields E N nk = Λ nk ( R + ) . Remarks: - Interestingly, (61) is independent of n . So, the mean number of nodes that receive k packets does notdepend on how often the packet was transmitted.- Summing λ nk over k ∈ [ n ] reproduces Cor. 8.- (61) is valid even for k = 0 since E N n = ∞ .- For the standard networks, the expression simplifies to E N nk = c d ks , which, when summed over k ∈ [ n ] ,yields (27).Let x nk be the position of a randomly chosen node from the nodes that received k out of n packets.From Prop. 13, the pdf (normalized density) is f x nk ( x ) = λ nk ( x ) ( ks ) δ c d Γ(1 + δ ) , k ∈ [ n ] . (63)For the standard network, we have E x nn = ( ns ) − , Var x nn = ( ns ) − , and E x n = s (Ψ( n + 1) + γ ) ,which is again related to (27) (division by the constant density c d ).The densities of the nodes receiving exactly k of messages is plotted in Fig. 8 for the standardnetwork with α = 2 .This expression permits the evaluation of the contribution that each additional transmission makes tothe broadcast transport sum-distance and capacity.These results can also be applied in localization. If a node receives k out of n transmissions, E x nk isan obvious estimate for its position, and Var x nk for the uncertainty. Alternatively, if the path loss x canbe measured, then the corresponding node index ˆ i ( x ) can be determined by the ML estimate ˆ i ( x ) = arg max i f ξ i ( x ) , (64) November 4, 2018 DRAFT3 D en s i t y λ k ( x ) k=6 k=1 k=0sum λ k6 (x), k ∈ [6] Fig. 8. Densities λ k ( x ) for the standard network with α = 2 ( c d = π ) and s = 1 . The maximum of the density for k = n = 6 is λ (0) = π . The dashed curve is the density of the nodes that receive at least 1 packet. Normalized by E N k these densitiesare the pdfs of x k . with the pdf f ξ i given in Cor. 2. For the standard networks, for example, the ML decision is ˆ i ( x ) = ⌈ c d /x ⌉ since ˆ i ( x ) = i ⇐⇒ c d i x < c d i − . (65)This is of course related to the fact E x i = i/c d .VI. C ONCLUDING R EMARKS We have offered a geometric interpretation of fading in wireless networks which is based on apoint process model that incorporates both geometry and fading. The framework enables analyticalinvestigations of the properties of wireless networks and the impact of fading, leading to closed-formresults that are obtained in a rather convenient manner.For Nakagami- m fading, it turns out that the connectivity fading gain is the δ -th moment of the fadingdistribution, while the fading gain in the broadcast transport sum-distance is its ∆ -th moment. A pathloss exponent larger than the number of dimensions d ( d + 1 for broadcasting) leads to a negative impactof fading. Interestingly, the broadcast transport capacity turns out to be unbounded if ∆ > , i.e. , if thepath loss exponent is smaller than d + 1 . While this result may be of interest for the design of efficientbroadcasting protocols, it also raises doubts on the validity of transport capacity as a performance metric. November 4, 2018 DRAFT4 Generally, it can be observed that the parameters δ and/or ∆ appear ubiquitously in the expressions.So the network behavior critically depends on the ratio of the number of dimensions to the path lossexponent.Other applications considered include the maximum transmission distance, probabilistic progress, andthe effect of retransmissions. We are convinced that there are many more that will benefit from thetheoretical foundations laid in this paper.A CKNOWLEDGMENTS The support of NSF (Grants CNS 04-47869, DMS 505624) and the DARPA IT-MANET program(Grant W911NF-07-1-0028) is gratefully acknowledged.R EFERENCES [1] D. Miorandi and E. Altman, “Coverage and Connectivity of Ad Hoc Networks in Presence of Channel Randomness,” in IEEE INFOCOM’05 , (Miami, FL), Mar. 2005.[2] M. Haenggi, “A Geometry-Inclusive Fading Model for Random Wireless Networks,” in ∼ mhaenggi/pubs/isit06.pdf.[3] J. F. C. Kingman, Poisson Processes . Oxford Science Publications, 1993.[4] M. Haenggi, “On Distances in Uniformly Random Networks,” IEEE Trans. on Information Theory ∼ mhaenggi/pubs/tit05.pdf.[5] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and its Applications . John Wiley & Sons, 1995. 2nd Ed.. John Wiley & Sons, 1995. 2nd Ed.