Spatial Outage Capacity of Poisson Bipolar Networks

Transcription

1 Satial Outage Caacity of Poisson Biolar Networks Sanket S. Kalamkar and Martin Haenggi Deartment of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Abstract We introduce a new notion of caacity, termed satial outage caacity (SOC), which is defined as the maximum density of concurrently active links that have a success robability greater than a redefined threshold. For Poisson biolar networks, we rovide exact analytical and aroximate exressions for the density of concurrently active links satisfying an outage constraint. In the high-reliability regime, we obtain an exact closed-form exression of the SOC, which gives its asymtotic scaling behavior. Index Terms Biolar network, interference, Poisson oint rocess, SIR, satial outage caacity, stochastic geometry A. Motivation I. INTRODUCTION Stochastic geometry rovides the mathematical tools to study wireless networks where node locations are modeled by a random oint rocess. By satial averaging, stochastic geometry allows us to evaluate the statistics of the wireless network such as interference distribution and average success robability. In this aroach, the erformance evaluation is usually done with resect to the tyical user or tyical link. Such satial averaging often leads to tractable erformance metrics for the given network arameters, which allows us to choose network arameters that otimize the network erformance. While such a macroscoic view based on satial averaging is imortant, it does not give fine-grained information about the network, such as the link-wise erformance characterized by the link success robability. Due to the random node locations, the success robability of each link is a random variable that deends on ath loss, fading, and interferer locations. In fact, as Fig. shows, for the same average success robability, deending on the network arameters, the distribution of link success robabilities in a Poisson biolar network varies significantly. Thus the link success robability distribution is a much more comrehensive metric than the average success robability that is usually considered. In this regard, we introduce a new notion of caacity, termed satial outage caacity (SOC). The SOC is defined as the maximum density of concurrently active links that have a success robability greater than a certain threshold. In other words, the SOC is the maximum density of concurrently active links that achieve a certain reliability and thus reresents the maximum density of reliable links. The definition of the SOC is based on the distribution of link success robabilities across the network. Fig.. The emirical robability density function of the link success robability in a Poisson biolar network for transmit robabilities = / and =. Both cases have the same average success robability s =.5944, but we see a different distribution of link success robabilities for different values of the air of density λ and transmit robability. For = /, the link success robabilities lie between.3 and.9 (concentrated around their average), while for =, they are sread more widely. The SIR threshold θ = db, distance between a transmitter and its associated receiver R =, ath loss exonent α = 4, and λ = /3. Definition (Satial outage caacity). For a stationary and ergodic oint rocess model where λ is the density of otential transmitters, is the fraction of nodes that are active at a time, and η(θ,x) is the fraction of links in each realization of the oint rocess that have a signal-to-interference ratio (SIR) greater than θ with robability at least x, the SOC is S(θ,x) su λ, λη(θ,x), () where θ R +, x (,), λ >, and (,]. We denote the density of concurrently active links that have a success robability greater than x (alternatively, a reliability of x or higher) as which results in τ(θ,x) λη(θ,x), (2) S(θ,x) = su λ, τ(θ,x). (3) Hence τ(θ,x) is the density of reliable links. The SOC rovides a useful ractical measure which tells us the maximum

2 number of active links er unit area a wireless network can handle at a time while guaranteeing a certain reliability. The SOC has alications in a wide range of wireless networks, such as ad hoc, D2D, M2M, and vehicular networks. B. Background Given the oint rocess Φ, the link success robability is a random variable that is given as P s (θ) P(SIR > θ Φ), (4) where the conditional robability is calculated by averaging over the fading and the medium access scheme (if random) of the interferers. The robability η(θ, x) in (), termed meta distribution of the SIR in [], is the comlementary cumulative distribution function (ccdf) of the link success robability. Thus the meta distribution is given as η(θ,x) P!t (P s (θ) > x), (5) where P!t ( ) denotes the reduced Palm robability, given that an active transmitter is resent at the rescribed location, and the SIR is calculated at its associated receiver. Due to the ergodicity of the oint rocess, given the random locations of nodes, η(θ,x) is also the robability that the tyical link has a success robability at least x. The average success robability follows as s (θ) = P(SIR > θ) = E!t (P s (θ)) = η(θ, x)dx, (6) where E!t ( ) denotes the exectation with resect to the reduced Palm distribution. C. Contributions The contributions of the aer are as follows: We introduce a new notion of caacity satial outage caacity based on the link success robability distribution. For the Poisson biolar network with ALOHA, we evaluate the density of concurrently active links satisfying an outage constraint. 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 robability is close to, we give a closed-form exression of the SOC and rove that the SOC is achieved at =. D. Related Work For Poisson biolar networks, the success robability of the tyical link s (θ) is calculated in [2] and [3]. The notion of the transmission caacity (TC) is introduced in [4], which is defined as the maximum density of successful transmissions rovided the outage robability of the tyical user stays below a certain threshold ǫ. While the results obtained in [4] are certainly imortant, the TC does not reresent the actual maximum density of successful transmissions for the target outage robability, as claimed in [4], since the metric imlicitly assumes that each link is tyical. We illustrate the difference between the SOC and the TC through the following examle. Examle. For Poisson biolar networks with ALOHA where SIR threshold θ = /, link distance R =, ath loss exonent α = 4, and target outage robability ǫ = /, the TC is.68 (see [5, (4.5)]), which is achieved at λ =.675. At this value of the TC, s (θ) =.9. But at =, actually only 82% active links satisfy the % outage. Hence the density of links that achieve % outage is only.55. On the other hand, the SOC is.9227 which is the actual maximum density of concurrently active links that have an outage robability smaller than %, and is achieved at λ =.23 and =, resulting in s (θ) = Hence the maximum density of links given the % outage constraint is more than 5% larger than the the TC. A version of the TC based on the link success robability distribution is introduced in [6], but it does not consider a medium access control (MAC) scheme, i.e., all nodes always transmit ( = ). Here, we consider the general case with (,]. The choice of is imortant as it greatly affects the link success robability distribution as shown in Fig.. Also, the TC defined in [6] calculates the maximum density of concurrently active links subject to the constraint that a certain fraction of active links satisfy the outage constraint. Such constraint of certain fraction of active links satisfying the outage constraint is not required by our definition of the SOC, and the SOC corresonds to the actual density of active links that satisfy an outage constraint. The meta distribution η(θ, x) for Poisson biolar networks and cellular networks is studied in [], where a closed-form exression for the moments of P s (θ) is obtained, and an exact integral exression and simle bounds for η(θ, x) are rovided. A key result in [] is that, for constant transmitter density λ, as the Poisson biolar network becomes very dense (λ ) with a very small transmit robability ( ), all links have the same success robability, which is the average success robability s (θ). II. NETWORK MODEL We consider the Poisson biolar network model in which the locations of transmitters form a homogeneous Poisson oint rocess (PPP) Φ R 2 with density λ [7, Def. 5.8]. Each transmitter has a dedicated receiver at a distance R in a uniformly random direction. In a time slot, each node in Φ indeendently transmits at unit ower with robability and stays silent with robability. Thus the active transmitters form a homogeneous PPP with density λ. We consider a standard ower law ath loss model with ath loss exonentα. We assume that a channel is subject to indeendent Rayleigh fading with channel ower gains as i.i.d. exonential random variables with mean. We focus on the interference-limited case, where the received SIR is a key quantity of interest. The success roba-

3 bility s (θ) of the tyical link deends on the SIR. From [3], [7], [8], it is known that s (θ) = ex ( λcθ δ), (7) where C πr 2 Γ(+δ)Γ( δ) with δ 2/α. A. Exact Formulation III. SPATIAL OUTAGE CAPACITY Observe from Def. that, the SOC deends on η(θ,x) = P(P s (θ) > x). Let M b (θ) denote the bth moment of P s (θ). Then M b (θ) E ( P s (θ) b). (8) The average success robability is s (θ) M (θ). From [, Thm. ], we can exress M b (θ) as where M b (θ) = ex ( λcθ δ D b (,δ) ), b C, (9) D b (,δ) k= ( )( ) b δ k,,δ (,]. () k k With the finite uer limit of the sum, D b (,δ) in () becomes a olynomial which is termed diversity olynomial in [9]. For b = (the first moment), D (,δ) =, and we get the exression of s (θ) as in (7). We can also exressd b (,δ) using the Gaussian hyergeometric function 2 F as D b (,δ) = b 2 F ( b, δ;2;). () Using the Gil-Pelaez theorem [], the exact exression of τ(θ,x) = λη(θ,x) can be obtained in integral form from that of η(θ,x) given in [, Cor. 3] as τ(θ,x) = λ 2 λ π sin(ulnx+λcθ δ I(D ju )) du, (2) ue λcθδ R(D ju) where j, D ju = D ju (,δ) is given by (), while R(z) and I(z) are the real and imaginary arts of the comlex number z, resectively. Note that the SOC is obtained by taking the suremum of τ(θ,x) over λ and. B. Aroximation with Beta Distribution We can accurately aroximate (2) in a semi-closed form using the beta distribution, which is a good aroximation (and a simle one) as shown in []. The rationale behind such aroximation is that the suort of the link success robability P s (θ) is [,], making the beta distribution a natural choice. With the beta distribution aroximation, from [, Sec. II.F], τ(θ,x) is aroximated as ( ( )) µβ τ(θ,x) λ I x µ,β, (3) where I x (y,z) x ty ( t) z dt/b(y,z) is the regularized incomlete beta function with B(, ) denoting beta function, µ = M, and β = (M M 2 )( M )/(M 2 M 2 ). The advantage of the beta aroximation is the faster comutation of τ(θ, x) comared to the exact exression without losing much accuracy [, Tab. I, Fig. 4] (also see Fig. 6 of this aer). In general, it is difficult to obtain the SOC analytically due to the forms of τ(θ,x) given in (2) and (3). But we can obtain the SOC numerically with ease. We can also gain useful insights considering some secific scenarios, on which we focus in the following two subsections of the aer. C. Constrained SOC ) Constant λ: For constant λ (or, equivalently, a fixed s (θ)), we now study how the density of reliable links τ(θ,x) behaves in an ultra-dense network. Given θ, R, α, and x, this case is equivalent to asking how τ(θ,x) varies as λ while letting for constant transmitter density λ (constant s (θ)). Lemma ( for constant λ). Let ν = λ. Then, for constant ν while letting, the SOC constrained on the density of concurrent transmissions is S(θ,x) = { λ, if x < s (θ), if x > s (θ). (4) Proof: Alying Chebyshev s inequality to (5), for x < s (θ) = M, we have η(θ,x) > var(p s(θ)) (x M ) 2, (5) where var(p s (θ)) = M 2 M 2. From [, Cor. ], for constant ν, we know that lim var(p s (θ)) =. λ=ν Thus the lower bound in (5) aroaches, leading to η(θ, x). This results in the SOC constrained on the density of concurrent transmissions equal to λ. On the other hand, for x > M, η(θ,x) var(p s(θ)) (x M ) 2. (6) As we let for constant ν, the uer bound in (6) aroaches, leading to η(θ,x). 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) aroaches a ste function that leas from to at the mean of P s (θ), i.e., at x = s (θ). This behavior justifies (4). 2) λ : For λ, τ(θ,x) deends linearly on λ, which we rove in the next lemma. Lemma 2 (τ(θ,x) as λ ). As λ, τ(θ,x) λ. (7) Proof: As λ, M aroaches and thus var(p s (θ)) = M(M 2 (δ ) ) aroaches. Since x (,), we have x < M as λ. Using Chebyshev s inequality for x < M as in (5) and letting var(p s (θ)), the lower bound in (5) aroaches, leading to η(θ,x).

4 = -5 =.3 corresonding to x = s =.9 =.6 = Fig. 2. The density of reliable links τ(θ,x) given in (2) for different values of the transmit robability for θ = db, R =, α = 4, and x =.9. Observe that the sloe of τ(θ,x) is one for small values of λ. Lemma 2 can be understood as follows. As λ, the density of active transmitters is very small. Thus each transmission succeeds with high robability, and η(θ, x) in this regime, increasing the density of reliable links linearly with λ. The case λ can be interreted in two ways: ) λ for constant and 2) for constant λ. Lemma 2 is valid for both cases, or any combination thereof. The case of constant is relevant since it can be interreted as a delay constraint: Asgets smaller, the robability that a node makes a transmission attemt in a slot reduces, increasing the delay. Since the mean delay until successful recetion is larger than the mean channel access delay /, it gets large for small values of. Thus, a delay constraint rohibits from getting too small. Fig. 2 illustrates Lemma 2. Also, observe that, as ( = 5 in Fig. 2), τ(θ,x) increases linearly with λ till the roduct λ reaches to the value that corresonds to s (θ) = x =.9 and then leas to. This behavior is in accordance with Lemma. In general, as λ increases, τ(θ,x) increases first and then decreases after a tiing oint. This is due to the two oosite effects of λ on τ(θ,x). Because of the term λ in the exression of τ(θ,x) (see (2)), the increase in λ increases τ(θ, x) first, but contributes to more interference at the same time, which reduces the fraction η(θ,x) of links that have a reliability at least x, thereby reducing τ(θ,x). The contour lot in Fig. 3 visualizes the trade-off between λ and η(θ,x). The contour curves for small values of λ run nearly arallel to those for τ(θ,x), indicating that η(θ,x) is close to. Secifically the contour curves for λ =. and λ =.2 match almost exactly with those for τ(θ,x) =. and τ(θ,x) =.2, resectively. This behavior is in accordance with Lemma 2. Conversely, for large values of λ, the decrease in η(θ,x) dominates τ(θ,x). Also, notice that, for larger values of λ (λ >.4 for Fig. 3), τ(θ,x) first increases and then decreases with the increase in. This behavior is due to the following trade-off in. For a small, there are few SOC oint Fig. 3. Contour lots of τ(θ,x) and the roduct λ for θ = db,r =, α = 4, and x =.9. The solid lines reresent the contour curves for τ(θ,x) and the dashed lines reresent the contour curves for λ. The numbers in black and red indicate the contour levels for τ(θ,x) and λ, resectively. The SOC oint corresonds to the suremum of τ(θ, x), and gives the SOC equal to S(θ,x) = The values of λ and at the SOC oint are.23 and, resectively, and the corresonding average success robability is s(θ) = SOC oint 3 X: Y:.23 Z:.9227 Fig. 4. Three-dimensional lot of τ(θ, x) corresonding to the contour lot in Fig. 3. active transmitters in the network er unit area, but a higher fraction of links are reliable. On the other hand, a large means more active transmitters er unit area, but also a higher interference which reduces the fraction of reliable links. For λ <.4, the increase in the density of active transmitters dominates the increase in interference, and τ(θ, x) increases monotonically with. An observation for Fig. 3 is that the average success robability s (θ) at the SOC oint is.6984 for 9% reliability. The three-dimensional lot corresonding to the contour lot in Fig. 3 is shown in Fig. 4. D. High-reliability Regime In the high-reliability regime, the reliability threshold is close to, i.e., x. Alternatively, the outage robability threshold ǫ = x of a link is close to, i.e., ǫ. 2

5 5 D b (Exact) D b (Asymtotic).7.6 Exact (2) Asymtotic (8) Beta aroximation (3) = =.9 = b Fig. 5. The solid lines reresent the exact D b (,δ) as in (), while the dashed lines reresent asymtotic form of D b (,δ) as in (9). Observe that (9) is a good aroximation of () and is asymtotically tight. In this section, we investigate the behavior of τ(θ,x) and the SOC in the high-reliability regime. To this end, we first state a simlified version of de Bruijn s Tauberian theorem (see [, Thm ]) which allows a convenient formulation of η(θ, ǫ) = P(P s (θ) > ǫ) as ǫ in terms of the Lalace transform. The following simlified version of de Bruijn s Tauberian theorem suffices for our uroses. Theorem (de Bruijn s Tauberian theorem [2, Thm. ]). For a non-negative random variable Y, the Lalace transform E[ex( sy)] ex(rs u ) for s is equivalent to P(Y ǫ) ex(q/ǫ v ) forǫ, when /u = /v+ (for u (,) andv > ), and the constantsr andq are related as ur /u = vq /v. Theorem 2. For ǫ, the density of reliable links τ(θ, ǫ) satisfies ( ( ) ) κ θ (δλc ) κ δ τ(θ, ǫ) λex, ǫ, (8) ǫ κ where κ = δ δ = 2 α 2 and C = πr 2 Γ( δ). = 4 Proof: First, note that for b C, D b (,δ) δ b δ /Γ(+δ), b, (9) where D b (,δ) is given by (). In Fig. 5, we illustrate how quickly D b aroaches the asymtote. Let Y = ln(p s (θ)). The Lalace transform of Y is E(ex( sy)) = E(P s (θ) s ) = M s (θ). Using (9) and (9), we have M s (θ) ex ( λc(θ)δ s δ ), s. Γ(+δ) From Thm., we have r = λc(θ)δ Γ(+δ), u = δ, v = δ/( δ) = κ, and thus = 4 q = κ (δλc ) κ δ (θ) κ, = x Fig. 6. The solid line with marker o reresents the exact exression of τ(θ, x) as in (2), the dotted line reresents the asymtotic exression of τ(θ, x) given by (8) as ǫ, and the dashed line reresents the aroximation by the beta distribution given by (3). Observe that the beta aroximation is good. θ = db,r =,α = 4,λ = /2, and = /3. where C = πγ( δ). Using Thm., we can now write P(Y ǫ) = P(P s (θ) ex( ǫ)) (a) P(P s (θ) ǫ), ǫ ( ) = ex (θ)κ (δλc ) κ δ κǫ κ, (2) where (a) follows from ex( ǫ) ǫ as ǫ. Since we have τ(θ, ǫ) = λp(p s (θ) > ǫ). (2) the desired result in (8) follows from substituting (2) in (2). For the secial case of = (all transmitters are active), P(P s (θ) ǫ) as in (2) simlifies to ( ( ) δλc θ δ)κ δ P(P s (θ) ǫ) ex κǫ κ, ǫ, which is in agreement with [6, Thm. 2] where it was derived in a less direct way than Thm. 2. Fig. 6 shows the tightness of (8) in the high-reliability regime and also the accuracy of the beta aroximation given by (3). We now investigate the scaling of S(θ, ǫ) in the highreliability regime. Corollary (SOC in high-reliability regime). For ǫ, the SOC is asymtotically equal to ( ǫ δ e ( δ) S(θ, ǫ) θ) πr 2 δ δ Γ( δ), (22) and it is achieved at =. Proof: Let us denote ξ(θ,ǫ) = ( ) κ θ (δc ) κ/δ. ǫ κ

6 From (8), we can then write ( ) τ(θ, ǫ) λex λ κ/δ κ ξ(θ,ǫ), ǫ..5 SOC oint X: Y:.7 Z:.468 Thus we have.4 S(θ, ǫ) su λ, f(λ,), ǫ,.3 where f(λ,) = λex( λ κ/δ κ ξ(θ,ǫ)). First, fix (,]. As ǫ, we can then write f ( )[ λ = ex λ κ/δ κ ξ(θ,ǫ) κξ(θ,ǫ) ] λ κ/δ κ. }{{} δ > Setting f λ =, we obtain the critical oint as ( ) δ/κ δ λ = ξ(θ,ǫ)κ κ. Note that, for a fixed, f is strictly increasing for λ (,λ ] and strictly decreasing for λ > λ. Hence we have S(θ, ǫ) su ( = f(λ,), ǫ, δ eκξ(θ, ǫ) ) δ/κ su δ. Observe thatf(λ,) monotonically increases with, and thus attains the maximum at =. Thus the SOC is achieved at = and is given by (22) after simlification. Remark. From Cor., we observe that, as ǫ, the exonents of θ and ǫ are the same. In the high-reliability regime, the SOC scales in ǫ similar to the TC defined in [6], while the TC defined in [4] scales linearly inǫ(see [5, (4.29)]). For α = 4, the exression of SOC in (22) simlifies to ( ) 2ǫ 2 S(θ, ǫ) ǫ. θe π 3 2R2, For α = 4, Fig. 7 lots τ(θ,x) versus λ and for x =.993. In this case, the SOC is achieved at =. IV. CONCLUSIONS The first main contribution is a new notion of caacity, termed satial outage caacity (SOC), which is the maximum density of concurrently active intermissions while ensuring a certain reliability. The SOC gives fine-grained information about the network comared to the TC whose framework is based on the average success robability. The SOC has alications in wireless networks with strict reliability constraints. Secondly, for Poisson biolar networks with ALOHA, we have obtained an exact analytical exression and a simle aroximation for the density τ of concurrently active links satisfying an outage constraint. The SOC can be easily calculated numerically as the suremum of τ obtained by otimizing over the density λ and the transmit robability. When constrained on the density of concurrent transmissions, i.e., for constant λ, while letting, the suremum of τ Fig. 7. Three-dimensional lot of τ(θ,x) for x =.993,θ = db, R =, and α = 4. Observe that = achieves the SOC. The average success robability s(θ) at the SOC oint is is equal to the roduct λ if the reliability threshold is smaller than the average success robability and zero if the reliability threshold is larger than the average success robability. In the high-reliability regime where the target outage robability of a link goes to, we give a closed-form exression of the SOC and show that = achieves the SOC. ACKNOWLEDGMENT The artial suort of the U.S. National Science Foundation through grant CCF is gratefully acknowledged. REFERENCES [] M. Haenggi, The meta distribution of the SIR in Poisson biolar and cellular networks, IEEE Trans. Wireless Commun., vol. 5, no. 4, , Ar. 26. [2] M. Zorzi and S. Puolin, Otimum transmission ranges in multiho acket radio networks in the resence of fading, IEEE Trans. Commun., vol. 43, no. 7, , Jul [3] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, An ALOHA rotocol for multiho mobile wireless networks, IEEE Trans. Inf. Theory, vol. 52, no. 2, , Feb. 26. [4] S. P. Weber, X. Yang, J. G. Andrews, and G. de Veciana, Transmission caacity of wireless ad hoc networks with outage constraints, IEEE Trans. Inf. Theory, vol. 5, no. 2, , Dec. 25. [5] S. P. Weber and J. G. Andrews, Transmission caacity of wireless networks, Found. Trends Netw., vol. 5, no. 2-3,. 9 28, 22. [6] R. K. Ganti and J. G. Andrews, Correlation of link outages in lowmobility satial wireless networks, in Proc. Asilomar Conf. Signals Syst. Comut. (Asilomar ), , Nov. 2. [7] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge, U.K.: Cambridge Univ. Press, 22. [8] M. Haenggi and R. K. Ganti, Interference in large wireless networks, Found. Trends Netw., vol. 3, no. 2, , 29. [9] M. Haenggi and R. Smarandache, Diversity olynomials for the analysis of temoral correlations in wireless networks, IEEE Trans. Wireless Commun., vol. 2, no., , Nov. 23. [] J. Gil-Pelaez, Note on the inversion theorem, Biometrika, vol. 38, no. 3-4, , Dec. 95. [] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation. Cambridge, U.K.: Cambridge Univ. Press, 987. [2] J. Voss, Uer and lower bounds in exonential Tauberian theorems, Tbilisi Mathematical Journal, vol. 2,. 4 5, 29.