The Spatial Outage Capacity of Wireless Networks
aa r X i v : . [ c s . I T ] J a n The Spatial Outage Capacityof Wireless Networks
Sanket S. Kalamkar,
Member, IEEE, and Martin Haenggi,
Fellow, IEEE
Abstract
We address a fundamental question in wireless networks that, surprisingly, has not been studiedbefore: what is the maximum density of concurrently active links that satisfy a certain outage constraint?We call this quantity the spatial outage capacity (SOC), give a rigorous definition, and analyze itfor Poisson bipolar networks with ALOHA. Specifically, we provide exact analytical and approximateexpressions for the density of links satisfying an outage constraint and give simple upper and lowerbounds on the SOC. In the high-reliability regime where the target outage probability is close to zero, weobtain an exact closed-form expression of the SOC, which reveals the interesting and perhaps counter-intuitive result that all transmitters need to be always active to achieve the SOC, i.e. , the transmitprobability needs to be set to to achieve the SOC. Index Terms
Interference, outage probability, Poisson point process, spatial outage capacity, stochastic geometry,wireless networks.
I. I
NTRODUCTION
A. Motivation
In a wireless network, the outage probability of a link is a key performance metric thatindicates the quality-of-service. To ensure a certain reliability, it is desirable to impose a limiton the outage probability, which depends on path loss, fading, and interferer locations. Forexample, in an interference-limited network, the outage probability of a link is the probability
S. S. Kalamkar and M. Haenggi are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame,IN, 46556 USA. (e-mail: { skalamka, mhaenggi } @nd.edu).This work is supported by the US National Science Foundation (grant CCF 1525904).Part of this work was presented at the 2017 IEEE International Conference on Communications (ICC’17) [1]. that the signal-to-interference ratio (SIR) at the receiver of that link is below a certain threshold.The interference originates from concurrently active transmitters as governed by a medium accesscontrol (MAC) scheme. Clearly, if more transmitters are active, then the interference at a receiveris higher, which increases the outage probability. Hence, given the outage constraint, a natural anda fundamental question, which has surprisingly remained unanswered, is “What is the maximumdensity of concurrently active links that meet the outage constraint?” To rigorously formulatethis question, we introduce a quantity termed the spatial outage capacity (SOC). The SOC hasapplications in a wide range of wireless networks, including cellular, ad hoc, device-to-device(D2D), machine-to-machine (M2M), and vehicular networks. In this paper we focus on thePoisson bipolar model, which is applicable to infrastructureless networks such as ad hoc, D2D,and M2M networks. B. Definition and Connection to SIR Meta Distribution
Modeling the random node locations as a point process, formally, the SOC is defined asfollows.
Definition 1 ( Spatial outage capacity ) . For a stationary and ergodic point process model where λ is the density of potential transmitters, p is the fraction of links that are active at a time, and η ( θ, ǫ ) is the fraction of links in each realization of the point process that have an SIR greaterthan θ with probability at least − ǫ , the SOC is S ( θ, ǫ ) , sup λ,p λpη ( θ, ǫ ) , (1) where θ ∈ R + , ǫ ∈ (0 , , and the supremum is taken over λ > and p ∈ (0 , . The SOC formulation applies to all MAC schemes where the fraction of active links in eachtime slot is p and each link is active for a fraction p of the time. This includes MAC schemeswhere the events that nodes are transmitting are dependent on each other, such as carrier-sensemultiple access (CSMA). In Def. 1 ǫ represents an outage constraint. Thus the SOC yieldsthe maximum density of links that satisfy an outage constraint. Alternatively, the SOC is themaximum density of concurrently active links that have a success probability (reliability) greaterthan − ǫ . Hence the SOC represents the maximum density of reliable links, where ǫ denotesa reliability threshold. We call the pair of λ and p that achieves the SOC as the SOC point . We denote the density of concurrently active links that have an outage probability less than ǫ (alternatively, a reliability of − ǫ or higher) as λ ǫ , λpη ( θ, ǫ ) , (2)which results in S ( θ, ǫ ) = sup λ,p λ ǫ . Due to the ergodicity of the point process Φ , we can express λ ǫ as the limit λ ǫ = lim r →∞ πr X y ∈ Φ k y k
Fig. 1. The histogram of the empirical probability density function of the link success probability in a Poisson bipolar networkwith ALOHA channel access scheme for transmit probabilities p = 1 / and p = 1 . Both cases have the same mean successprobability of p s ( θ ) = 0 . , but we see a different distribution of link success probabilities for different values of the pairdensity λ and transmit probability p . For p = 1 / , the link success probabilities mostly lie between . and . (concentratedaround their mean), while for p = 1 , they are spread much more widely. The SIR threshold θ = −
10 dB , distance between atransmitter and its receiver R = 1 , path loss exponent α = 4 , and λp = 1 / . The conditional link success probability P s ( θ ) (and thus the meta distribution η ( θ, ǫ ) ) allows usto directly calculate the standard (mean) success probability, which is a key quantity of interestin wireless networks. In particular, we can express the mean success probability as p s ( θ ) , P ( SIR > θ ) = E !t ( P s ( θ )) = Z η ( θ, x )d x, where the SIR is calculated at the typical receiver and E !t ( · ) denotes the expectation with respectto the reduced Palm distribution. The standard success probability can be easily calculated bytaking the average of the link success probabilities. Hence, in a realization of the network, p s ( θ ) can be interpreted as a spatial average which provides limited information about the outageperformance of an individual link. As Fig. 1 shows, for a Poisson bipolar network with ALOHAwhere each transmitter has an associated receiver at a distance R , depending on the networkparameters, the distribution of P s ( θ ) varies greatly for the same p s ( θ ) . Hence the link successprobability distribution is a much more comprehensive metric than the mean success probabilitythat is usually considered. Since the SOC can be evaluated using the distribution of link successprobabilities, it provides fine-grained information about the network. C. Contributions
This paper makes the following contributions: • We introduce a new notion of capacity—the spatial outage capacity. • For the Poisson bipolar network with Rayleigh fading and ALOHA, we give exact andapproximate expressions of the density of reliable links. We also derive simple upper andlower bounds on the SOC. • We show the trade-off between the density of active links and the fraction of reliable links. • In the high-reliability regime where the target outage probability is close to , we givea closed-form expression of the SOC and prove that the SOC is achieved at p = 1 . ForRayleigh distributed link distances, we show that the density of reliable links is asymptot-ically independent of the density of (potential) transmitters λ as ǫ → . D. Related Work
For Poisson bipolar networks, the mean success probability p s ( θ ) is calculated in [3] and [4].For ad hoc networks modeled by the Poisson point process (PPP), the link success probability P s ( θ ) is studied in [5], where the focus is on the mean local delay, i.e. , the − st moment of P s ( θ ) in our notation. The notion of the transmission capacity (TC) is introduced in [6], whichis defined as the maximum density of successful transmissions provided the outage probability of the typical user stays below a predefined threshold ǫ . While the results obtained in [6] arecertainly important, the TC does not represent the maximum density of successful transmissionsfor the target outage probability, as claimed in [6], since the metric implicitly assumes that eachlink in a realization of the network is typical.A version of the TC based on the link success probability distribution is introduced in [7],but it does not consider a MAC scheme, i.e. , all nodes always transmit ( p = 1 ). The choice of p is important as it greatly affects the link success probability distribution as shown in Fig. 1.In this paper, we consider the general case with the transmit probability p ∈ (0 , .The meta distribution η ( θ, ǫ ) for Poisson bipolar networks with ALOHA and cellular networksis calculated in [2], where a closed-form expression for the moments of P s ( θ ) is obtained, andan exact integral expression and simple bounds on η ( θ, ǫ ) are provided. A key result in [2] isthat, for constant transmitter density λp , as the Poisson bipolar network becomes very dense( λ → ∞ ) with a very small transmit probability ( p → ), the disparity among link successprobabilities vanishes and all links have the same success probability, which is the mean success probability p s ( θ ) . For the Poisson cellular network, the meta distribution of the SIR is calculatedfor the downlink and uplink scenarios with fractional power control in [8], with base stationcooperation in [9], and for D2D networks underlaying the cellular network (downlink) in [10].Furthermore, the meta distribution of the SIR is calculated for millimeter-wave D2D networksin [11] and for D2D networks with interference cancellation in [12]. E. Comparison of the SOC with the TC
The TC defined in [6] can be written as c ( θ, ǫ ) , (1 − ǫ ) sup { λp : E !t ( P s ( θ )) > − ǫ } , while the SOC can be expressed as S ( θ, ǫ ) , sup λ,p { λp P ( P s ( θ ) > − ǫ ) } . The mean success probability p s ( θ ) , E !t ( P s ( θ )) depends only on the product λp and ismonotonic. Hence the TC can be written as c ( θ, ǫ ) , (1 − ǫ ) p − (1 − ǫ ) . The TC yields themaximum density of links such that the typical link satisfies the outage constraint. In otherwords, in the TC framework, the outage constraint is applied at the typical link, i.e. , afteraveraging over the point process. This means that the outage constraint is not applied at theactual links, but at a fictive link whose SIR statistics correspond to the average over all links.The supremum is taken over only one parameter, namely λp . On the other hand, in the SOCframework, the outage constraint is applied at each individual link . It accurately yields themaximum density of links that satisfy an outage constraint. This means that λ and p need to beconsidered separately. We further illustrate the difference between the SOC and the TC throughthe following example. Example 1 ( Difference between the SOC and the TC ) . For Poisson bipolar networks withALOHA and SIR threshold θ = 1 / , link distance R = 1 , path loss exponent α = 4 , andtarget outage probability ǫ = 1 / , c (1 / , /
10) = 0 . (see [13, (4.15)]), which is achievedat λp = 0 . . At this value of the TC, p s ( θ ) = 0 . . But at p = 1 , actually only ofthe active links satisfy the outage. Hence the density of links that achieve outage isonly . . On the other hand, S (1 / , /
10) = 0 . which is the actual maximum density Hence the TC can be interpreted as a mean-field approximation of the SOC. of concurrently active links that have an outage probability smaller than . The SOC pointcorresponds to λ = 0 . and p = 1 , resulting in p s ( θ ) = 0 . . Thus the maximum density oflinks given the outage constraint is more than larger than the TC. The version of the TC proposed in [7] applies an outage constraint at each link, similar to theSOC, but assumes that each link is always active ( i.e. , there is no MAC scheme) and calculatesthe maximum density of concurrently active links subject to the constraint that a certain fractionof active links satisfy the outage constraint. Such a constraint is not required by our definitionof the SOC, and the SOC corresponds to the actual density of active links that satisfy the outageconstraint.
F. Organization of the Paper
The rest of the paper is organized as follows. In Sec. II, we provide the network model,formulate the SOC, give upper and lower bounds on the SOC, and obtain an exact closed-formexpression of the SOC in the high-reliability regime. In Sec. III, we consider the random linkdistance case where the link distances are Rayleigh distributed. We draw conclusions in Sec. IV.II. P
OISSON B IPOLAR N ETWORKS WITH D ETERMINISTIC L INK D ISTANCE
As seen from Def. 1, the notion of the SOC is applicable to a wide variety of wirelessnetworks. To gain crisp insights into the design of wireless networks, in this paper, we studythe SOC for Poisson bipolar networks where we consider deterministic as well as random linkdistances and obtain analytical results for both cases. Table I provides the key notation used inthe paper.
A. Network Model
We consider the Poisson bipolar network model in which the locations of transmitters form ahomogeneous Poisson point process (PPP) Φ ⊂ R with density λ [14, Def. 5.8]. Each transmitterhas a dedicated receiver at a distance R in a uniformly random direction. In a time slot, each nodein Φ independently transmits at unit power with probability p and stays silent with probability − p . Thus the active transmitters form a homogeneous PPP with density λp . We consider astandard power law path loss model with path loss exponent α . We assume that a channel issubject to independent Rayleigh fading with channel power gains as i.i.d. exponential randomvariables with mean . TABLE IS
UMMARY OF N OTATION
Notation Definition/Meaning Φ , Φ t Point process of transmitters θ SIR threshold ǫ Target outage probability S ( θ, ǫ ) Spatial outage capacity (SOC) η ( θ, ǫ ) Fraction of reliable transmissions λ Density of potential transmitters µ Density of receivers for the random link distances case p Fraction of links that are active at a time λ ǫ Density of reliable transmissions P s ( θ ) Conditional link success probability p s ( θ ) Mean success probability α Path loss exponent δ /αM b ( θ ) b th moment of the conditional link success probability R Link distance in a bipolar network
We focus on the interference-limited case, where the received SIR is a key quantity of interest.To the PPP, we add a (desired) transmitter at location ( R, ) and a receiver at the origin o . Underthe expectation over the PPP, this link is the typical link. The success probability p s ( θ ) of thetypical link is the ccdf of the SIR calculated at the origin. For Rayleigh fading, from [4], [14],it is known that p s ( θ ) = exp (cid:0) − λpCθ δ (cid:1) , (4)where C , πR Γ(1 + δ )Γ(1 − δ ) with δ , /α . The model is scale-invariant in the followingsense: The SIR of all links in any realization of the bipolar network with transmitter locations ϕ remains unchanged if the plane is scaled by an arbitrary factor a > . Such scaling resultsin transmitter locations aϕ and link distances aR . The density of the scaled network is λ/a .By setting a = 1 /R to obtain unit distance links, the resulting density is λR . Hence withoutloss of generality, we can set R = 1 . Applied to the meta distribution and the SOC, this meansthat the model with parameters ( R, λ ) behaves exactly the same as the model with parameters (1 , λR ) . B. Exact Formulation
Observe from Def. 1 that the SOC depends on η ( θ, ǫ ) = P ( P s ( θ ) > − ǫ ) whose directcalculation seems infeasible. But the moments of P s ( θ ) are available in closed-form [2], fromwhich we can derive exact and approximate expressions of λ ǫ and obtain simple upper and lowerbounds on the SOC. Let M b ( θ ) denote the b th moment of P s ( θ ) , i.e., M b ( θ ) , E (cid:0) P s ( θ ) b (cid:1) . (5)The mean success probability is p s ( θ ) ≡ M ( θ ) .From [2, Thm. 1], we can express M b ( θ ) as M b ( θ ) = exp (cid:0) − λCθ δ D b ( p, δ ) (cid:1) , b ∈ C , (6)where D b ( p, δ ) , ∞ X k =1 (cid:18) bk (cid:19)(cid:18) δ − k − (cid:19) p k , p, δ ∈ (0 , . (7)For b ∈ N , the sum is finite and D b ( p, δ ) becomes a polynomial which is termed diversitypolynomial in [15]. The series in (7) converges for p < , and at p = 1 it is defined if b / ∈ Z − or b + δ / ∈ Z − and converges if ℜ ( b + δ ) > . Here ℜ ( z ) is the real part of the complex number z . For b = 1 (the first moment), D ( p, δ ) = p , and we get the expression of p s ( θ ) as in (4). Wecan also express D b ( p, δ ) using the Gaussian hypergeometric function F as D b ( p, δ ) = pb F (1 − b, − δ ; 2; p ) . (8)Using the Gil-Pelaez theorem [16], the exact expression of λ ǫ = λpη ( θ, ǫ ) can be obtained inan integral form from that of η ( θ, ǫ ) given in [2, Cor. 3] as λ ǫ = λp − λpπ ∞ Z sin( u ln(1 − ǫ ) + λCθ δ ℑ ( D ju )) ue λCθ δ ℜ ( D ju ) d u, (9)where j , √− , D ju = D ju ( p, δ ) , and ℑ ( z ) is the imaginary part of the complex number z .The SOC is then obtained by taking the supremum of λ ǫ over λ > and p ∈ (0 , . C. Approximation with Beta Distribution
We can accurately approximate λ ǫ in a semi-closed form using the beta distribution, asshown in [2]. The rationale behind such approximation is that the support of the link success probability P s ( θ ) is [0 , , making the beta distribution a natural choice. With the beta distributionapproximation, λ ǫ can be approximated as λ ǫ ≈ λp (cid:18) − I ǫ (cid:18) µβ − µ , β (cid:19)(cid:19) , (10)where I ǫ ( y, z ) , R − ǫ t y − (1 − t ) z − d t/B ( y, z ) is the regularized incomplete beta function with B ( · , · ) denoting beta function, µ = M , and β = ( M − M )(1 − M ) / ( M − M ) .The advantage of the beta approximation is the faster computation of λ ǫ compared to the exactexpression without losing much accuracy [2, Tab. I, Fig. 4] (also see Fig. 7 of this paper). Ingeneral, it is difficult to obtain the SOC analytically due to the forms of λ ǫ given in (9) and (10).But we can obtain the SOC numerically with ease. We can also gain useful insights consideringsome specific scenarios, on which we focus in the following subsection. D. Constrained SOC1) Constant λp in dense networks: For constant λp (or, equivalently, a fixed p s ( θ ) ), we nowstudy how the density of reliable links λ ǫ behaves in an ultra-dense network. Given θ , α , and ǫ , this case is equivalent to asking how λ ǫ varies as λ → ∞ while letting p → for constanttransmitter density λp (constant p s ( θ ) ). We denote the constrained SOC by ˜ S ( θ, ǫ ) . Lemma 1 ( p → for constant λp ) . Let ν = λp . Then, for constant ν while letting p → , theSOC constrained on the density of concurrent transmissions is ˜ S ( θ, ǫ ) = λp, if 1 − ǫ < p s ( θ )0 , if 1 − ǫ > p s ( θ ) . (11) Proof:
Applying Chebyshev’s inequality to (3), for − ǫ < p s ( θ ) = M , we have η ( θ, ǫ ) > − var( P s ( θ ))((1 − ǫ ) − M ) , (12)where var( P s ( θ )) = M − M is the variance of P s ( θ ) . From [2, Cor. 1], for constant ν , weknow that lim p → λp = ν var( P s ( θ )) = 0 . Hence the lower bound in (12) approaches , which leads to η ( θ, ǫ ) → . This results in the SOC constrained on the density of concurrent transmissionsequal to λp .On the other hand, for − ǫ > M , η ( θ, ǫ ) ≤ var( P s ( θ ))((1 − ǫ ) − M ) . (13) As we let p → for constant ν , the upper bound in (13) approaches , which leads to η ( θ, ǫ ) → .This results in the SOC constrained on the density of concurrent transmissions equal to .In fact, as var( P s ( θ )) → , the ccdf of P s ( θ ) approaches a step function that drops from to at the mean of P s ( θ ) , i.e. , at − ǫ = p s ( θ ) . This behavior is in agreement with (11). Remark : Lemma 1 shows that, if p → while λp is fixed to the value ν at which p s ( θ ) equals the target reliability − ǫ , the maximum value of the constrained SOC is the value ofthe TC times / (1 − ǫ ) , and that value of the TC is ν (1 − ǫ ) .This observation can be explained as follows: As p → while keeping λp = ν , all links ina realization of the network have the same success probability, and that value of the successprobability equals p s ( θ ) ( i.e. , the success probability of transmissions over the typical link) [2].This implies that, from the outage perspective, each link in the network can now be treated asif it were the typical link, as in the TC framework. If ν is initially set to a value that resultsin p s ( θ ) > − ǫ , we can always increase it till p s ( θ ) = 1 − ǫ while all active links satisfyingthe outage constraint (or, equivalently, the typical link satisfying the outage constraint withprobability one). Accordingly, the value of the TC equals − ǫ times the value of ν at which p s ( θ ) = 1 − ǫ .Fig. 2 shows that at small values of the target outage probability ǫ , the density of reliabletransmissions monotonically increases with p . On the other hand, at larger values of ǫ , it firstdecreases with p . λp → : For λp → , λ ǫ depends linearly on λp , which we prove in the next lemma. Lemma 2 ( λ ǫ as λp → ) . As λp → , λ ǫ ∼ λp. Proof: As λp → , M approaches and thus var( P s ( θ )) = M ( M p ( δ − − approaches .Since ǫ ∈ (0 , , we have − ǫ < M as λp → . Using Chebyshev’s inequality for − ǫ < M asin (12) and letting var( P s ( θ )) → , the lower bound in (12) approaches , leading to η ( θ, ǫ ) → .Lemma 2 can be understood as follows. As λp → , the density of active transmitters is verysmall. Thus each transmission succeeds with high probability and η ( θ, ǫ ) → . In this regime,the density of reliable links λ ǫ is directly given by λp .The case λp → can be interpreted in two ways: 1) λ → for constant p and 2) p → forconstant λ . Lemma 2 is valid for both cases, or any combination thereof. The case of constant Fig. 2. The density of reliable links λ ǫ against the transmit probability p for λp = 1 / , θ = −
10 dB , α = 4 , and R = 1 .The values at the curves are ǫ = 0 . , . , . , . , . , . (bottom to top). The mean success probability is p s ( θ ) = 0 . . p is relevant since it can be interpreted as a delay constraint: As p gets smaller, the probabilitythat a node makes a transmission attempt in a slot is reduced, increasing the delay. Since themean delay until successful reception is larger than the mean channel access delay /p , it getslarge for small values of p . Thus, a delay constraint prohibits p from getting too small.Fig. 3 illustrates Lemma 2. Also, observe that, as p → ( p = 10 − in Fig. 3), λ ǫ increaseslinearly with λp until λp reaches the value . which corresponds to p s ( θ ) = 1 − ǫ = 0 . and then drops to . This behavior is in accordance with Lemma 1. In general, as λp increases, λ ǫ increases first and then decreases after a tipping point. This is due to the two opposite effectsof λp on λ ǫ : The density λp of active transmitters increases, but at the same time, more activetransmissions cause higher interference, which in turn, reduces the fraction η ( θ, ǫ ) of links thathave a reliability at least − ǫ .The contour plot in Fig. 4(a) visualizes the trade-off between λp and η ( θ, ǫ ) . The contourcurves for small values of λp run nearly parallel to those for λ ǫ , indicating that η ( θ, ǫ ) is closeto . Specifically, the contour curves for λp = 0 . and λp = 0 . match those for λ ǫ = 0 . and λ ǫ = 0 . almost exactly. This behavior is in accordance with Lemma 2. In contrast, forlarge values of λp , the decrease in η ( θ, ǫ ) dominates λ ǫ . Also, for larger values of λ ( λ > . forFig. 4(a)), λ ǫ first increases and then decreases with the increase in p . This behavior is due to the Fig. 3. The density of reliable links λ ǫ given in (2) for different values of the transmit probability p for θ = −
10 dB , α = 4 ,and ǫ = 1 / . Observe that the slope of λ ǫ is one for small λp . The dashed arrow points to the value of λp = 0 . , whichcorresponds to − ǫ = p s ( θ ) = 0 . . following trade-off in p . For a small p , there are few active transmitters in the network per unitarea, but a higher fraction of links are reliable. On the other hand, a large p means more activetransmitters per unit area, but also a higher interference which reduces the fraction of reliablelinks. For λ < . , the increase in the density of active transmitters dominates the decrease in η ( θ, ǫ ) , and λ ǫ increases monotonically with p . The three-dimensional plot corresponding to thecontour plot in Fig. 4(a) is shown in Fig. 4(b). E. Bounds on the SOC
In this subsection, we obtain simple upper and lower bounds on the SOC.
Theorem 1 ( Upper bound on the SOC ) . For any b > , the SOC is upper bounded as S ( θ, ǫ ) ≤ eπθ δ Γ(1 − δ )Γ(1+ δ ) 1 b (1 − ǫ ) b , < b ≤ , eπθ δ Γ(1 − δ ) Γ( b )Γ( b + δ )(1 − ǫ ) b , b > . (14) Proof:
Using Markov’s inequality, η ( θ, ǫ ) can be upper bounded as η ( θ, ǫ ) ≤ M b ( θ )(1 − ǫ ) b , b > , (15) SOC point (a)
01 00.05 10.5 20.1 0 3
X: 1Y: 0.23Z: 0.09227
SOC point (b)Fig. 4. (a) Contour plots of λ ǫ and the product λp for θ = −
10 dB , α = 4 , and ǫ = 1 / . The solid lines represent the contourcurves for λ ǫ and the dashed lines represent the contour curves for λp . The numbers in “black” and “red” indicate the contourlevels for λ ǫ and λp , respectively. The SOC is S ( θ, ǫ ) = 0 . . The values of λ and p at the SOC point are . and ,respectively, and the corresponding mean success probability is p s ( θ ) = 0 . . The arrow corresponding to the “SOC point”points to the pair of λ and p for which the SOC is achieved. (b) Three-dimensional plot of λ ǫ corresponding to the contourplot. where M b ( θ ) = e − λCθ δ D b ( p,δ ) . Hence we can upper bound the SOC as S ( θ, ǫ ) ≤ S u , where S u = sup λ,p λp e − λCθ δ D b ( p,δ ) (1 − ǫ ) b , (16)with C = π Γ(1 − δ )Γ(1 + δ ) and D b ( p, δ ) = pb F (1 − b, − δ ; 2; p ) . Let us denote f u ( λ, p ) = λpe − λCθ δ D b ( p,δ ) . We can then write ∂f u ( λ, p ) ∂λ = pe − λCθ δ D b ( p,δ ) | {z } > (1 − λCθ δ D b ( p, δ )) . Setting ∂f u ( λ,p ) ∂λ = 0 , we obtain the critical point as λ ( p ) = 1 / ( Cθ δ D b ( p, δ )) . For any given p ,the objective function is quasiconcave. Thus λ ( p ) is the global optimum for each p . As a result,the optimization problem in (16) reduces to S u = 1(1 − ǫ ) b sup p f u ( λ ( p ) , p )= 1 eCθ δ b (1 − ǫ ) b sup p F (1 − b, − δ ; 2; p ) . (17) For < b < , F (1 − b, − δ ; 2; p ) monotonically increases with p . In this case, p → solves(17). On the other hand, for b > , F (1 − b, − δ ; 2; p ) monotonically decreases with p . Thus p = 1 solves (17). Overall the value of p that solves (17) is p → , < b < , = 1 , b > . (18)Note that the objective function in (17) is monotonic in p . Hence p in (18) is again the globaloptimum.Finally, for < b < , the upper bound on the SOC is obtained by substituting p = 0 in theobjective of (17). Since F (1 − b, − δ ; 2; 0) = 1 , S u = 1 eπθ δ Γ(1 − δ )Γ(1 + δ ) 1 b (1 − ǫ ) b , < b < . (19)Similarly since b F (1 − b, − δ ; 2; 1) = Γ( b + δ )Γ( b )Γ(1+ δ ) , S u = 1 eπθ δ Γ(1 − δ ) Γ( b )Γ( b + δ )(1 − ǫ ) b , b > . (20)For b = 1 , the hypergeometric function returns irrespective of the other parameters, and thus(19) and (20) are identical.The tightest Markov upper bound can be obtained by minimizing S u in (19) and (20) over b .Now, the value of b that minimizes S u in (19) is b m = − − ǫ ) . (21)Since S u takes two different values depending on whether < b ≤ or b > , to obtainthe tightest Markov upper bound, we need to consider following two cases based on whether b m ∈ (0 , or b m > . b m ∈ (0 , : If b m ∈ (0 , , it is the optimum value of b that minimizes S u since S u in (19)is smaller than S u in (20) for < b < , greater for b > , and equal to for b = 1 . From (21),it is apparent that the case b m ∈ (0 , is equivalent to ǫ ∈ [0 . , . Hence, if ǫ ∈ [0 . , ,the optimum b that gives the tightest Markov upper bound is given by (21). Substituting b = b m in (19), we get the exact closed-form expression of the tightest Markov upper bound as S tu = − ln(1 − ǫ ) πθ δ Γ(1 − δ )Γ(1 + δ ) , if ǫ ∈ [0 . , , (22)where ‘ t ’ in the superscript of S tu indicates the tightest bound.2) b m > : If b m > , i.e. , ǫ ∈ (0 , . , the optimum b is the value of b that minimizes S u in (20). However, due to the form of S u in (20), the optimum b cannot be expressed in a closed-form. Hence the tightest Markov upper bound also cannot be expressed in a closed-form,but it can be easily evaluated numerically. Furthermore, for b > , we can get a closed-formexpression of the approximate tightest Markov bound by using the approximation Γ( b + δ )Γ( b ) ≈ b δ (23)in (20). Then, for b > , we can express (20) as S u ≈ eπθ δ Γ(1 − δ ) 1 b δ (1 − ǫ ) b . (24)The value of b that minimizes (24) is given as ¯ b m = − δ ln(1 − ǫ ) . The corresponding closed-form expression of the tightest approximate Markov upper bound isobtained by substituting ¯ b m in (24) which is given as S tu ≈ (cid:18) − ln(1 − ǫ ) θ (cid:19) δ e − (1 − δ ) πδ δ Γ(1 − δ ) , if ǫ ∈ (0 , . . (25)Fig. 5 illustrates upper bounds on the SOC.Letting ǫ → , from (25), we observe that the lower tail of the SOC decreases exponentially, i.e. , S ( θ, ǫ ) / (cid:16) ǫθ (cid:17) δ e − (1 − δ ) πδ δ Γ(1 − δ ) , ǫ → , (26)where ‘ / ’ denotes an upper bound which gets tighter asymptotically (here as ǫ → ). In thenext subsection, we shall show that the bound in (26) is in fact asymptotically tight, i.e. , (26)matches the exact expression of the SOC as ǫ → .We now obtain lower bounds on the SOC. Theorem 2 ( Lower bound on the SOC ) . The SOC is lower bounded as S ( θ, ǫ ) > (cid:18) − W ((1 − ǫ ) e ) πθ δ Γ(1 + δ )Γ(1 − δ ) (cid:19) (cid:18) e − (1 − W ((1 − ǫ ) e )) − (1 − ǫ ) ǫ (cid:19) , (27) where W ( · ) denotes the Lambert W function.Proof: By the reverse Markov’s inequality, we have − E !t ((1 − P s ( θ )) b ) ǫ b < η ( θ, ǫ ) , b > . For b ∈ N we can lower bound the SOC as S ( θ, ǫ ) > S l , -2 -1 Fig. 5. Analytical and numerical results for the SOC. The tightest Markov upper bound on the SOC obtained numericallyuses (19) and (20), which are optimized over b . The tightest Markov upper bound obtained analytically uses (22) and (25). TheSOC upper bound obtained analytically is quite close to that obtained numerically for the almost complete range of reliabilitythreshold − ǫ , except near − ǫ = 0 . (which is due to the approximation in (23)). The classical Markov bounds areplotted using (19) and (20) for b = 1 , b = 2 , and b = 4 . The lower bound for b = 1 is plotted using (27), while the lowerbounds for b = 2 and b = 4 are plotted numerically using (28). θ = − dB and α = 4 . where S l = sup λ,p λp − P bk =0 (cid:0) bk (cid:1) ( − k M k ( θ ) ǫ b ! , (28)with M k ( θ ) = e − λCθ δ D k ( p,δ ) . For b = 1 , (28) reduces to S l = sup λ,p λp − − e − λpCθ δ ǫ ! . (29)Since λ and p appear together as their product λp , S l can be obtained by taking the supremumover t = λp , i.e. , S l = sup t t − − e − tCθ δ ǫ !| {z } f ( t ) . (30)Substituting the value of t that results in ∂f ( t ) ∂t = 0 in f ( t ) , we get the desired expression in (27). Fig. 6. The solid lines represent the exact D b ( p, δ ) as in (7), while the dashed lines represent the asymptotic form of D b ( p, δ ) as in (31). For the values of b ∈ R + \ { } , an analytical expression for S l is difficult to obtain due to theform of (28), but we can easily obtain corresponding lower bounds numerically. Fig. 5 showsMarkov lower bounds on the SOC for b = 1 , b = 2 , and b = 4 . F. High-reliability Regime
In this section, we investigate the behavior of λ ǫ and the SOC in the high-reliability regime, i.e. , as ǫ → . To this end, we first provide an asymptote of D b ( p, δ ) as b → ∞ , which willbe used to obtain a closed-form expression of the SOC in the high-reliability regime. Then westate a simplified version of de Bruijn’s Tauberian theorem (see [17, Thm. 4.12.9]) which allowsa convenient formulation of η ( θ, ǫ ) = P ( P s ( θ ) > − ǫ ) in terms of the Laplace transform as ǫ → . ‘ . ’ denotes an upper bound with asymptotic equality (here as b → ∞ ). Lemma 3 ( Asymptote of D b ( p, δ ) as b → ∞ ) . For b ∈ R , we have D b ( p, δ ) . p δ b δ Γ(1 + δ ) , b → ∞ , (31) Proof:
See Appendix A.Fig. 6 illustrates how quickly D b ( p, δ ) approaches the asymptote. Theorem 3 ( de Bruijn’s Tauberian theorem [18, Thm. 1] ) . For a non-negative random variable Y , the Laplace transform E [exp( − sY )] ∼ exp( rs u ) for s → ∞ is equivalent to P ( Y ≤ ǫ ) ∼ exp( q/ǫ v ) for ǫ → , when /u = 1 /v + 1 (for u ∈ (0 , and v > ), and the constants r and q are related as | ur | /u = | vq | /v . Theorem 4 ( λ ǫ in the high-reliability regime ) . For ǫ → , the density of reliable links λ ǫ satisfies λ ǫ ∼ λp exp − (cid:18) θpǫ (cid:19) κ ( δλC ′ ) κ/δ κ ! , ǫ → , (32) where κ = δ − δ = α − and C ′ = π Γ(1 − δ ) .Proof: Let Y = − ln( P s ( θ )) . The Laplace transform of Y is E (exp( − sY )) = E ( P s ( θ ) s ) = M s ( θ ) . Using (6) and Lemma 3, we have M s ( θ ) ∼ exp (cid:18) − λC ( θp ) δ s δ Γ(1 + δ ) (cid:19) , | s | → ∞ . Comparing this expression with that in Thm. 3, we have r = − λC ( θp ) δ Γ(1+ δ ) , u = δ , v = δ/ (1 − δ ) = κ ,and thus q = κ ( δλC ′ ) κ/δ ( θp ) κ , where C ′ = π Γ(1 − δ ) . Using Thm. 3, we can now write P ( Y ≤ ǫ ) = P ( P s ( θ ) ≥ exp( − ǫ )) (a) ∼ P ( P s ( θ ) ≥ − ǫ ) , ǫ →
0= exp − ( θp ) κ ( δλC ′ ) κ/δ κǫ κ ! , (33)where (a) follows from exp( − ǫ ) ∼ − ǫ as ǫ → . Since we have λ ǫ = λp P ( P s ( θ ) > − ǫ ) , (34)the desired result in (32) follows from substituting (33) in (34).For the special case of p = 1 (all transmitters are active), P ( P s ( θ ) ≥ − ǫ ) in (33) simplifiesto P ( P s ( θ ) ≥ − ǫ ) ∼ exp − (cid:0) δλC ′ θ δ (cid:1) κ/δ κǫ κ ! , ǫ → , in agreement with [7, Thm. 2] where the result for this special case was derived in a less directway than Thm. 4. Fig. 7 shows the behavior of (32) in the non-asymptotic regime and also theaccuracy of the beta approximation given by (10).We now investigate the scaling of S ( θ, ǫ ) in the high-reliability regime. Fig. 7. The solid line with marker ‘o’ represents the exact expression of λ ǫ as in (9), the dotted line represents the asymptoticexpression of λ ǫ given by (32) as ǫ → , and the dashed line represents the approximation by the beta distribution given by(10). Observe that the beta approximation is quite accurate. θ = 0 dB , α = 4 , λ = 1 / , and p = 1 / . Corollary 1 ( SOC in high-reliability regime ) . For ǫ → , S ( θ, ǫ ) ∼ (cid:16) ǫθ (cid:17) δ e − (1 − δ ) πδ δ Γ(1 − δ ) , (35) and the SOC is achieved at p = 1 .Proof: For notational simplicity, let us define the rate-reliability ratio as ρ , ǫ/θ anddenote ξ ρ , ρ − κ ( δC ′ ) κ/δ κ and f ρ ( λ, p ) , λp exp( − λ κ/δ p κ ξ ρ ) . From (32), we can then write λ ǫ ∼ f ρ ( λ, p ) , ǫ → , and the SOC is S ( θ, ǫ ) ∼ sup λ,p f ρ ( λ, p ) , ǫ → . (36)We can then write ∂f ρ ( λ, p ) ∂λ = p exp (cid:0) − λ κ/δ p κ ξ ρ (cid:1)| {z } > (cid:20) − κξ ρ δ λ κ/δ p κ (cid:21) . Setting ∂f ρ ( λ,p ) ∂λ = 0 , we obtain the critical point as λ ( p ) = (cid:18) δξ ρ κp κ (cid:19) δ/κ . (37) For any given p , the objective function is quasiconcave. Hence the optimization problem in (36)reduces to S ( θ, ǫ ) ∼ sup p f ρ ( λ ( p ) , p ) , ǫ → , = (cid:18) δeκξ ρ (cid:19) δ/κ sup p p − δ , ǫ → . Observe that f ρ ( λ ( p ) , p ) monotonically increases with p and thus attains the maximum at p = 1 .Hence the SOC is achieved at p = 1 and λ = (cid:16) δξ ρ κ (cid:17) δ/κ and is given by (35) after simplification.The equation (35) confirms the asymptotic bound on the SOC given in (26). Corollary 2 ( The meta distribution at the SOC point ) . As ǫ → , the value of the metadistribution at the SOC point can be simply expressed as η ( θ, ǫ ) ∼ e − (1 − δ ) . (38) Proof:
From Cor. 1, as ǫ → , the SOC can be expressed as S ( θ, ǫ ) ∼ λ opt p opt η ( θ, ǫ ) , (39)where λ opt = λ (given by (37)) and p opt = 1 correspond to the SOC point as ǫ → . Then,comparing (39) with (35), we get the desired expression of η ( θ, ǫ ) as in (38). Corollary 3 ( The mean success probability at the SOC point ) . As ǫ → , the mean successprobability at the SOC point can be expressed as p s , opt ∼ − (cid:16) ǫδ (cid:17) δ Γ(1 + δ ) . (40) Proof:
Substituting λ = λ (given by (37)) and p = 1 in (4) and using e − x ∼ − x as x → yield the desired expression.We now provide few remarks pertaining to the high-reliability regime. Remarks : • Letting C δ = (cid:0) δ (cid:1) δ e − (1 − δ ) Γ(1 − δ ) , the density of transmitters λ ∗ , S ( θ, ǫ ) that maximizes thedensity of active links that achieve a reliability at least − ǫ behaves as λ ∗ π ∼ C δ (cid:16) ǫθ (cid:17) δ , ǫ → . (41) The coefficient C δ depends only on δ . In the practically relevant regime / ≤ δ < , i.e. , < α ≤ , C δ ≈ − δ . In (41), the left side λ ∗ π is the mean number of reliable receivers ina disk of unit radius in the network. Equation (41) reveals an interesting trade-off betweenthe spectral efficiency (captured by θ ) and the reliability (captured by ǫ ), where only theirratio matters. For example, at low rates, ln(1 + θ ) ∼ θ ; thus, a × higher reliability canbe achieved by lowering the rate by a factor of . • The (potential) transmitter density λ opt that achieves the SOC is λ opt ∼ (cid:16) ǫθ (cid:17) δ πδ δ Γ(1 − δ ) , ǫ → . (42)Here λ opt π is the mean number of (potential) transmitters in a disk of unit radius in thenetwork that achieves the SOC. • From (38), it is apparent that, at the SOC point, the fraction of links that satisfy the outageconstraint depends only on the path loss exponent α , as δ , /α . • The mean success probability p s , opt at the SOC point (given by (40)) allows us to relate theSOC and the TC. Substituting q ∗ = 1 − p s , opt in [13, (4.29)], we can express the TC as c ( θ, ǫ ) ∼ (cid:16) ǫθ (cid:17) δ πδ δ Γ(1 − δ ) , ǫ → , which is the same as the expression of the optimum λ that achieves the SOC (given by(42)). Hence S ( θ, ǫ ) = c ( θ, ǫ ) e − (1 − δ ) if the TC framework used p s ( θ ) = p s , opt instead of p s ( θ ) = 1 − ǫ (given that p = 1 is optimum). • From Cor. 1, observe that the exponents of θ and ǫ are the same. The SOC scales in ǫ similar to the TC defined in [7], i.e. , as Θ( ǫ δ ) , while the original TC defined in [6] scaleslinearly in ǫ . • For α = 4 , the expression of SOC in (35) simplifies to S ( θ, ǫ ) ∼ . (cid:16) ǫθ (cid:17) / , ǫ → , and the meta distribution gives η ≈ . . In other words, approximately of active linkssatisfy the outage constraint if α = 4 . Also, for α = 4 , the mean success probability at theSOC point is simply given by p s , opt ∼ − . √ ǫ as ǫ → . Fig. 8 plots λ ǫ versus λ and p for ǫ = 0 . and α = 4 . In this case, the SOC is achieved at p = 1 .III. P OISSON B IPOLAR N ETWORKS WITH R ANDOM L INK D ISTANCES
We now consider the case of random link distance, where the link distances are i.i.d. randomvariables (which are constant over time). X: 1Y: 0.0701Z: 0.04068
SOC point
Fig. 8. Three-dimensional plot of λ ǫ for ǫ = 0 . , θ = −
10 dB , and α = 4 . Observe that p = 1 achieves the SOC. The meansuccess probability p s ( θ ) at the SOC point is . . A. Network Model
Let R i denote the random link distance between a transmitter i and its associated receiver in aPoisson bipolar network. We assume that R i is Rayleigh distributed with mean / (2 √ µ ) as it isthe distribution of the nearest-neighbor distance in a PPP of density µ [19]. This scenario canbe interpreted as the one where an active transmitter tries to communicate to its nearest receiverin a network where the potential transmitters form a PPP Φ t of density λ and the receivers forman another PPP (independent of Φ t ) of density µ . Similar to the deterministic link distance case,we add a receiver at the origin o to the receiver PPP and an always active transmitter at location( R o , ), where R o is the Rayleigh distributed link distance. Under the expectation over the pointprocess, this link is the typical link. B. Exact Formulation of the SOC
Lemma 4 ( b th moment of the link success probability ) . For Rayleigh distributed link distanceswith mean / (2 √ µ ) , the b th moment of P s ( θ ) is M b ( θ ) = µµ + λθ δ Γ(1 + δ )Γ(1 − δ ) D b ( p, δ ) . (43) Generalizations to other link distance distributions are beyond the scope of this paper. This is because some new techniquesmay need to be developed, and it is unclear what other distribution to assume. Proof:
See Appendix B.For b = 1 , using D ( p, δ ) = p , we get the expression of the mean success probability p s ( θ ) .Moreover, for b ∈ N , (43) represents the joint success probability of b transmissions with randomlink distance, as obtained in [15, (23)].As in the deterministic link distance case, using the Gil-Pelaez theorem, we can calculate thedensity of reliable links from (9), and the SOC is obtained by taking the supremum of λ ǫ over λ and p . Like the deterministic link distance case, the beta approximation is quite accurate.In the rest of the paper, we assume µ = 1 without loss of generality. C. Bounds on the SOC
Theorem 5 ( Upper bound on the SOC ) . For any b > , the SOC for Rayleigh distributed linkdistances is upper bounded as S ( θ, ǫ ) ≤ θ δ Γ(1 − δ )Γ(1+ δ ) 1 b (1 − ǫ ) b , < b ≤ , θ δ Γ(1 − δ ) Γ( b )Γ( b + δ )(1 − ǫ ) b , b > . (44) Proof:
Again using Markov’s inequality, η ( θ, ǫ ) can be upper bounded as η ( θ, ǫ ) ≤ M b ( θ )(1 − ǫ ) b , b > , where M b ( θ ) = λθ δ Γ(1+ δ )Γ(1 − δ ) D b ( p,δ ) . Hence for any b > , we have S ≤ S u , where S u = 1(1 − ǫ ) b sup λ,p λp λθ δ Γ(1 + δ )Γ(1 − δ ) D b ( p, δ ) | {z } A λ,p . (45) A λ,p is maximized at λ = ∞ , and it follows that S u = 1 θ δ Γ(1 + δ )Γ(1 − δ ) b (1 − ǫ ) b sup p F (1 − b, − δ ; 2; p ) , (46)where we have used D b ( p, δ ) = pb F (1 − b, − δ ; 2; p ) as in (8).Notice that the optimization problem in (46) is similar to that in (17). Thus, following thesteps after (17) in the proof of Thm. 1, we get (44).Similar to the deterministic link distance case (as discussed in Sec. II-E after Thm. 1), from(44), for ǫ ∈ [0 . , , we can obtain the exact closed-form expression of the tightest Markovbound as S tu = − e ln(1 − ǫ ) θ δ Γ(1 − δ )Γ(1 + δ ) . (47) -1 Fig. 9. Analytical and numerical results for the SOC. The tightest Markov upper bound on the SOC obtained numerically uses(44), which is optimized over b . The tightest Markov upper bound obtained analytically uses (47) when ǫ ∈ [0 . , and (48)when ǫ ∈ (0 , . . Observe that the analytical approximation of the SOC upper bound provides a tight upper bound for thecomplete range of reliability threshold − ǫ . The curve corresponding to the tightest Markov upper bound obtained analyticallydeviates from that obtained numerically at − ǫ = 0 . due to the approximation as in (23). The classical Markov boundsare plotted using (44) for b = 1 , b = 2 , and b = 4 . θ = − dB and α = 4 . For ǫ ∈ (0 , . , we can obtain the exact tightest Markov bound numerically. Alternatively,using the approximation in (23), we get a closed-form expression of the approximate tightestMarkov bound as S tu ≈ (cid:18) − ln(1 − ǫ ) θ (cid:19) δ e δ δ δ Γ(1 − δ ) . (48)As Fig. 9 shows, the tightest Markov upper bound on the SOC obtained analytically using (48)deviates slightly from that obtained numerically at ǫ = 0 . due to the approximation in (23).As ǫ becomes smaller, i.e. , − ǫ becomes closer to , the approximation (48) becomes better.For ǫ < . , the gap between the approximation of the upper bound and the beta approximationis less than . dB. Theorem 6 ( Lower bound on the SOC ) . The SOC is lower bounded as S ( θ, ǫ ) > (1 − √ − ǫ ) ǫθ δ Γ(1 + δ )Γ(1 − δ ) . (49) Proof:
The proof follows the proof of Thm. 2 with M ( θ ) = λpθ δ Γ(1+ δ )Γ(1 − δ ) . Fig. 9 shows lower bounds on the SOC. Similar to the deterministic link distance case, theMarkov lower bounds for b ∈ R + \ { } are analytically intractable. D. High-reliability Regime
Theorem 7 ( SOC in the high-reliability regime ) . For Rayleigh distributed link distances, S ( θ, ǫ ) ∼ (cid:16) ǫθ (cid:17) δ δ )Γ(1 − δ ) , ǫ → , and the SOC is achieved at p = 1 .Proof: As in the proof of Thm. 4, let Y = − ln( P s ( θ )) with its Laplace transform as L Y ( s ) = M s ( θ ) . Asymptotically, L Y ( s ) (a) ∼ As δ , | s | → ∞ , (50)where (a) follows from using D b ( p, δ ) ∼ p δ b δ Γ(1+ δ ) as | b | → ∞ in (43) and thus A = 1 / ( λθ δ p δ Γ(1 − δ )) . We claim that the expression in (50) is equivalent to F Y ( ǫ ) ∼ Aǫ δ Γ(1 + δ ) , ǫ → . (51)The proof that (50) and (51) are equivalent is given in Appendix C.As ǫ → , since F Y ( ǫ ) = P ( − ln( P s ( θ )) < ǫ ) ∼ P ( P s ( θ ) > − ǫ )= η ( θ, ǫ ) , the density of reliable links in the high-reliability regime can be expressed as λ ǫ ∼ ǫ δ p − δ θ δ Γ(1 + δ )Γ(1 − δ ) , ǫ → . (52)Here, λ ǫ is independent of the density λ of (potential) transmitters. As a result, the SOC is S ( θ, ǫ ) ∼ ǫ δ θ δ Γ(1 + δ )Γ(1 − δ ) sup p p − δ , ǫ → . Setting p = 1 achieves the SOC, which is given as S ( θ, ǫ ) ∼ (cid:16) ǫθ (cid:17) δ δ )Γ(1 − δ ) , ǫ → . Similar to the deterministic link distance case, only the ratio of the spectral efficiency and thereliability matters. As we observe from (52), λ ǫ does not depend on λ . This is due to the factthat, in the high-reliability regime, the increase in λ decreases linearly the fraction of reliablelinks η ( θ, ǫ ) . For example, a 2 × increase in λ decreases η ( θ, ǫ ) by a factor of . Also, the SOC isa function of just two parameters, the reliability-to-target-SIR ratio and a constant that dependsonly on δ , i.e. , on the path loss exponent α .IV. C ONCLUSIONS
This paper introduces a new notion of capacity, termed spatial outage capacity (SOC), whichis the maximum density of concurrently active links that meet a certain constraint on thesuccess probability. Hence the SOC provides a mathematical foundation for questions of networkdensification under strict reliability constraints. Since the definition of the SOC is very general, i.e. , it is not restricted to a specific point process model, link distance distribution, MAC scheme,transmitter-receiver association schemes, fading distribution, power control scheme, etc., it isapplicable to a wide range of wireless networks.For Poisson bipolar networks with ALOHA and Rayleigh fading, we provide an exact ana-lytical expression and a simple approximation for the density of reliable links λ ǫ . The SOC canbe easily calculated numerically as the supremum of λ ǫ obtained by optimizing over the densityof (potential) transmitters λ and the transmit probability p .In the high-reliability regime where the target outage probability ǫ of a link goes to , wegive a closed-form expression of the SOC which reveals • the trade-off between the spectral efficiency and the reliability where only their ratio matterswhile calculating the SOC. • insights on the scaling behavior of the SOC where, for both deterministic and Rayleighdistributed link distance cases, we show that the SOC scales in ǫ as Θ( ǫ δ ) .Interestingly, p = 1 achieves the SOC in the high-reliability regime. This means that withALOHA, all transmitters should be active in order to maximize the number of reliable transmis-sions in a unit area that succeed with a probability close to one. Hence, in the high-reliabilityregime, backing off is not SOC-achieving. This happens because the reduction in the density ofactive links with p cannot be overcome by the increase in the fraction of reliable links.For Rayleigh distributed link distances, in the high-reliability regime, we have shown thatthe density of reliable links does not depend on λ as the increase in λ is exactly offset by the fraction of reliable links. To be precise, a t -fold increase in λ decreases the density of reliablelinks by a factor of t .As a future work, it is important to generalize the results obtained for Rayleigh fading toother fading distributions. However, since the current bounds on the SOC and the high-reliabilityregime results exploit a structure induced by Rayleigh fading assumption, one might need todevelop new techniques depending on the fading distribution considered.A PPENDIX AP ROOF OF L EMMA D b ( p, δ ) = ∞ X k =1 (cid:18) bk (cid:19)(cid:18) δ − k − (cid:19) p k = p ∞ X k =1 (cid:18) bk (cid:19)(cid:18) δ − k − (cid:19) p k − | {z } A k ( p ) . (53)By Taylor’s theorem, A k ( p ) = ∞ X j =0 A ( j ) k (1) j ! ( p − j , (54)where A ( j ) k (1) is the j th derivative of A k ( p ) at p = 1 . Let ( k ) j , k ( k − k − · · · ( k − j + 1) denote the falling factorial. Then A ( j ) k (1) can be written as A ( j ) k (1) = ∞ X k =1 (cid:18) bk (cid:19)(cid:18) δ − k − (cid:19) ( k − j = Γ( b + δ − j )Γ( b − j )Γ(1 + δ ) ( δ − j (a) . b δ ( δ − j Γ(1 + δ ) , (55)where (a) follows from Γ( b + δ − j )Γ( b − j ) . b δ as b → ∞ . From (54) and (55), A k ( p ) . b δ Γ(1 + δ ) ∞ X j =0 ( δ − j j ! ( p − j | {z } p δ − . (56)From (53) and (56), we get the desired result. A PPENDIX BP ROOF OF L EMMA R . Then the probability densityfunction of R is f R ( a ) = 2 πµa exp( − πµa ) . Let k z k denote the distance between a receiverand a potential interferer z ∈ Φ t . Given Φ t , the conditional link success probability P s ( θ ) is P s ( θ ) = P (cid:18) hR − α I > θ | Φ t (cid:19) = E ( ( h > θR α I ) | Φ t ) , where I = X z ∈ Φ t \{ z o } h z k z k − α ( z ∈ Φ t ) , where z o ∈ Φ t denotes the desired transmitter. Conditioning on R and then averaging over fadingand ALOHA results in P s ( θ ) | R = Y z ∈ Φ t \{ z o } p θ (cid:16) R k z k (cid:17) α + 1 − p . Let f ( r ) = (cid:0) p θr α + 1 − p (cid:1) b . Then the b th moment of P s ( θ ) is M b ( θ ) = E Y z ∈ Φ t \{ z o } f (cid:18) R k z k (cid:19) (a) = E R (cid:20) exp (cid:18) − πλ Z ∞ t (cid:18) − f (cid:18) Rt (cid:19)(cid:19) d t (cid:19)(cid:21) (b) = 2 πµ ∞ Z a exp − πλ ∞ Z t (cid:16) − f (cid:16) at (cid:17)(cid:17) d t e − µπa d a (c) = 2 πµ ∞ Z a exp − πλa ∞ Z y (cid:18) − f (cid:18) y (cid:19)(cid:19) d y e − µπa d a = µµ + 2 λ R ∞ y (1 − f (1 /y )) d y (d) = µµ + 2 λ ∞ Z − (cid:18) − pθr α θr α (cid:19) b ! r − d r | {z } F b , (57)where (a) follows from the probability generating functional of the PPP [14, Chapter 4], (b) follows from the de-conditioning on R , (c) follows from the substitution y = t/a , and (d) follows from the substitution y = 1 /r and plugging f ( r ) back. With as the upper limit of theintegral and µ = λ , (57) reduces to the expression of the b th moment of the success probabilityin a Poisson cellular network as in [2, (27)].With r α = x , the integral in (57) can be expressed as F b = 1 α ∞ Z − (cid:18) − pθx θx (cid:19) b ! x − δ − d x (e) = ∞ X k =1 (cid:18) bk (cid:19) ( − k +1 ( pθ ) k α ∞ Z x k − δ − (1 + θx ) k d x (f) = θ δ πδ sin( πδ ) D b ( p, δ ) , (58)where (e) follows from the binomial expansion of (cid:0) − pθx θx (cid:1) b and Fubini’s theorem, and (f) follows from ∞ Z x k − δ − (1 + θx ) k d x = θ δ − k (cid:20) π sin( πδ ) Γ( k − δ )Γ( k )Γ(1 − δ ) (cid:21) and ( − k +1 (cid:0) k − δ − k − (cid:1) = (cid:0) δ − k − (cid:1) . Finally, substituting (58) in (57) and using πδ sin( πδ ) ≡ Γ(1 + δ )Γ(1 − δ ) , (43) follows. A PPENDIX CP ROOF THAT (50)
AND (51)
ARE EQUIVALENT
The proof uses the Weierstrass approximation theorem that any continuous function f :[ t , t ] → R can be approximated by a sequence of polynomials from above and below.In our case, t = 0 and t = 1 . Thus, for any given t > , if f ( y ) is a continuous real-valuedfunction on [0 , , for n ≥ , there exists a sequence of polynomials P n ( y ) and Q n ( y ) such that P n ( y ) ≤ f ( y ) ≤ Q n ( y ) ∀ y ∈ [0 , , (59) Z ( Q n ( y ) − f ( y ))d y ≤ t, (60)and Z ( f ( y ) − P n ( y ))d y ≤ t. (61)Even if f ( y ) has a discontinuity of the first kind, we can still construct polynomials P n ( y ) and Q n ( y ) that satisfy (59)-(61). See [20, Sec. 7.53] for the details of the construction of such polynomials. To prove the desired result, we first show that lim s →∞ s δ Z ∞ e − sy f ( e − sy )d F Y ( y ) = A Γ( δ ) Z ∞ y δ − f ( e − y ) e − y d y. (62)Let Q n ( y ) = P nk =0 a k y k with a k ∈ R for k = 0 , , . . . , n . We then have lim sup s →∞ s δ Z ∞ e − sy f ( e − sy ) dF Y ( y ) ≤ lim s →∞ s δ Z ∞ e − sy Q n ( e − sy ) dF Y ( y )= lim s →∞ n X k =0 a k s δ Z ∞ e − ( k +1) sy dF Y ( y ) (a) = A n X k =0 a k ( k + 1) δ (b) = A Γ( δ ) Z ∞ y δ − e − y Q n ( e − y ) d y (c) = A Γ( δ ) Z ∞ y δ − e − y f ( e − y ) d y, where (a) follows from lim s →∞ s δ R ∞ e − sy dF Y ( y ) = A , (b) follows from the definition of thegamma function as Γ( δ ) , R ∞ y δ − e − y d y , and (c) follows from the dominated convergencetheorem as n → ∞ .By a similar argument for P n ( y ) , we have lim inf s →∞ s δ Z ∞ e − sy f ( e − sy ) dF Y ( y ) ≥ A Γ( δ ) Z ∞ y δ − e − y f ( e − y ) d y, and (62) follows.Now let f ( y ) = y , e ≤ y ≤ , ≤ y < e . (63)Letting s = 1 /ǫ in (62) and using (63), we have lim ǫ → ǫ − δ F Y ( ǫ ) = lim s →∞ s δ Z ∞ e − sy f ( e − sy ) dF Y ( y ) (d) = A Γ( δ ) Z y δ − d y = A Γ(1 + δ ) . where (d) follows from (62) and (63).A CKNOWLEDGMENT
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