1

The Spatial Outage Capacity
of Wireless Networks

arXiv:1708.05870v2 [] 23 Jan 2018

Sanket S. Kalamkar, Member, IEEE, and Martin Haenggi, Fellow, IEEE

Abstract
We address a fundamental question in wireless networks that, surprisingly, has not been studied
before: 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 it
for Poisson bipolar networks with ALOHA. Specifically, we provide exact analytical and approximate
expressions for the density of links satisfying an outage constraint and give simple upper and lower
bounds on the SOC. In the high-reliability regime where the target outage probability is close to zero, we
obtain an exact closed-form expression of the SOC, which reveals the interesting and perhaps counterintuitive result that all transmitters need to be always active to achieve the SOC, i.e., the transmit
probability needs to be set to 1 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 that
indicates the quality-of-service. To ensure a certain reliability, it is desirable to impose a limit
on the outage probability, which depends on path loss, fading, and interferer locations. For
example, 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].

2

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 access
control (MAC) scheme. Clearly, if more transmitters are active, then the interference at a receiver
is higher, which increases the outage probability. Hence, given the outage constraint, a natural and
a fundamental question, which has surprisingly remained unanswered, is “What is the maximum
density of concurrently active links that meet the outage constraint?” To rigorously formulate
this question, we introduce a quantity termed the spatial outage capacity (SOC). The SOC has
applications 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 the
Poisson 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 as
follows.
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 greater
than θ with probability at least 1 − ǫ, the SOC is
S(θ, ǫ) , sup λpη(θ, ǫ),

(1)

λ,p

where θ ∈ R+ , ǫ ∈ (0, 1), and the supremum is taken over λ > 0 and p ∈ (0, 1].
The SOC formulation applies to all MAC schemes where the fraction of active links in each
time slot is p and each link is active for a fraction p of the time. This includes MAC schemes
where the events that nodes are transmitting are dependent on each other, such as carrier-sense
multiple access (CSMA). In Def. 1 ǫ represents an outage constraint. Thus the SOC yields
the maximum density of links that satisfy an outage constraint. Alternatively, the SOC is the
maximum density of concurrently active links that have a success probability (reliability) greater
than 1 − ǫ. Hence the SOC represents the maximum density of reliable links, where ǫ denotes
a reliability threshold. We call the pair of λ and p that achieves the SOC as the SOC point.

3

We denote the density of concurrently active links that have an outage probability less than ǫ
(alternatively, a reliability of 1 − ǫ or higher) as
λǫ , λpη(θ, ǫ),

(2)

which results in S(θ, ǫ) = sup λǫ . Due to the ergodicity of the point process Φ, we can express
λ,p

λǫ as the limit

1 X
1 (P(SIRỹ > θ | Φ) > 1 − ǫ) ,
r→∞ πr 2
y∈Φ

λǫ = lim

kyk<r

where ỹ is the receiver paired with transmitter y and 1(·) is the indicator function. From this
formulation, it is apparent that the outage constraint results in a static dependent thinning of Φ
to a point process of density λǫ .
The probability η(θ, ǫ) in (1), termed meta distribution of the SIR in [2], is the complementary
cumulative distribution function (ccdf) of the conditional link success probability which is given
as
Ps (θ) , P(SIR > θ | Φ),
where the conditional probability is calculated by averaging over the fading and the medium
access scheme (if random) of the interferers, and the SIR is calculated at the receiver of the link
under consideration. Accordingly, the meta distribution is given as
η(θ, ǫ) , P!t (Ps (θ) > 1 − ǫ),

(3)

where P!t (·) denotes the reduced Palm probability, given that an active transmitter is present at
the prescribed location, and the SIR is calculated at its associated receiver. Under the expectation
over the point process, it is the typical receiver. The meta distribution is the distribution of the
conditional link success probability, which is obtained by taking an expectation over the point
process. In other words, the meta distribution is the probability that the success probability of
the transmission over the typical link is at least 1 − ǫ. As a result, as is standard in stochastic
geometry, the calculation of the SOC is done at the typical user and involves averaging over the
point process. Due to the ergodicity of the point process, η(θ, ǫ) corresponds to the fraction of
reliable links in each realization of the point process. Hence we can calculate the SOC using
the meta distribution framework.1
1

Note that the meta distribution provides the tool to analyze the SOC, but the meta distribution is not needed to define the

SOC.

4

Fig. 1. The histogram of the empirical probability density function of the link success probability in a Poisson bipolar network
with ALOHA channel access scheme for transmit probabilities p = 1/10 and p = 1. Both cases have the same mean success
probability of ps (θ) = 0.5944, but we see a different distribution of link success probabilities for different values of the pair
density λ and transmit probability p. For p = 1/10, the link success probabilities mostly lie between 0.4 and 0.8 (concentrated
around their mean), while for p = 1, they are spread much more widely. The SIR threshold θ = −10 dB, distance between a
transmitter and its receiver R = 1, path loss exponent α = 4, and λp = 1/3.

The conditional link success probability Ps (θ) (and thus the meta distribution η(θ, ǫ)) allows us
to directly calculate the standard (mean) success probability, which is a key quantity of interest
in wireless networks. In particular, we can express the mean success probability as
Z 1
!t
ps (θ) , P(SIR > θ) = E (Ps (θ)) =
η(θ, x)dx,
0

!t

where the SIR is calculated at the typical receiver and E (·) denotes the expectation with respect
to the reduced Palm distribution. The standard success probability can be easily calculated by
taking the average of the link success probabilities. Hence, in a realization of the network, ps (θ)
can be interpreted as a spatial average which provides limited information about the outage
performance of an individual link. As Fig. 1 shows, for a Poisson bipolar network with ALOHA
where each transmitter has an associated receiver at a distance R, depending on the network
parameters, the distribution of Ps (θ) varies greatly for the same ps (θ). Hence the link success
probability distribution is a much more comprehensive metric than the mean success probability
that is usually considered. Since the SOC can be evaluated using the distribution of link success
probabilities, it provides fine-grained information about the network.

5

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 and
approximate expressions of the density of reliable links. We also derive simple upper and
lower 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 0, we give
a closed-form expression of the SOC and prove that the SOC is achieved at p = 1. For
Rayleigh distributed link distances, we show that the density of reliable links is asymptotically independent of the density of (potential) transmitters λ as ǫ → 0.

D. Related Work
For Poisson bipolar networks, the mean success probability ps (θ) is calculated in [3] and [4].
For ad hoc networks modeled by the Poisson point process (PPP), the link success probability
Ps (θ) is studied in [5], where the focus is on the mean local delay, i.e., the −1st moment of
Ps (θ) in our notation. The notion of the transmission capacity (TC) is introduced in [6], which
is 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] are
certainly important, the TC does not represent the maximum density of successful transmissions
for the target outage probability, as claimed in [6], since the metric implicitly assumes that each
link 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, 1].
The meta distribution η(θ, ǫ) for Poisson bipolar networks with ALOHA and cellular networks
is calculated in [2], where a closed-form expression for the moments of Ps (θ) is obtained, and
an exact integral expression and simple bounds on η(θ, ǫ) are provided. A key result in [2] is
that, for constant transmitter density λp, as the Poisson bipolar network becomes very dense
(λ → ∞) with a very small transmit probability (p → 0), the disparity among link success
probabilities vanishes and all links have the same success probability, which is the mean success

6

probability ps (θ). For the Poisson cellular network, the meta distribution of the SIR is calculated
for the downlink and uplink scenarios with fractional power control in [8], with base station
cooperation 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 networks
in [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 (Ps (θ)) > 1 − ǫ},
while the SOC can be expressed as
S(θ, ǫ) , sup{λpP(Ps (θ) > 1 − ǫ)}.
λ,p

The mean success probability ps (θ) , E!t (Ps (θ)) depends only on the product λp and is
monotonic. Hence the TC can be written as c(θ, ǫ) , (1 − ǫ)p−1
s (1 − ǫ). The TC yields the
maximum density of links such that the typical link satisfies the outage constraint. In other
words, in the TC framework, the outage constraint is applied at the typical link, i.e., after
averaging over the point process. This means that the outage constraint is not applied at the
actual 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 SOC
framework, the outage constraint is applied at each individual link.2 It accurately yields the
maximum density of links that satisfy an outage constraint. This means that λ and p need to be
considered separately. We further illustrate the difference between the SOC and the TC through
the following example.
Example 1 (Difference between the SOC and the TC). For Poisson bipolar networks with
ALOHA and SIR threshold θ = 1/10, link distance R = 1, path loss exponent α = 4, and
target outage probability ǫ = 1/10, c(1/10, 1/10) = 0.0608 (see [13, (4.15)]), which is achieved
at λp = 0.0675. At this value of the TC, ps (θ) = 0.9. But at p = 1, actually only 82% of
the active links satisfy the 10% outage. Hence the density of links that achieve 10% outage is
only 0.055. On the other hand, S(1/10, 1/10) = 0.09227 which is the actual maximum density
2

Hence the TC can be interpreted as a mean-field approximation of the SOC.

7

of concurrently active links that have an outage probability smaller than 10%. The SOC point
corresponds to λ = 0.23 and p = 1, resulting in ps (θ) = 0.6984. Thus the maximum density of
links given the 10% outage constraint is more than 50% larger than the TC.
The version of the TC proposed in [7] applies an outage constraint at each link, similar to the
SOC, but assumes that each link is always active (i.e., there is no MAC scheme) and calculates
the maximum density of concurrently active links subject to the constraint that a certain fraction
of active links satisfy the outage constraint. Such a constraint is not required by our definition
of the SOC, and the SOC corresponds to the actual density of active links that satisfy the outage
constraint.
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-form
expression of the SOC in the high-reliability regime. In Sec. III, we consider the random link
distance 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 wireless
networks. To gain crisp insights into the design of wireless networks, in this paper, we study
the SOC for Poisson bipolar networks where we consider deterministic as well as random link
distances and obtain analytical results for both cases. Table I provides the key notation used in
the paper.
A. Network Model
We consider the Poisson bipolar network model in which the locations of transmitters form a
homogeneous Poisson point process (PPP) Φ ⊂ R2 with density λ [14, 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 independently transmits at unit power with probability p and stays silent with probability
1 − p. Thus the active transmitters form a homogeneous PPP with density λp. We consider a
standard power law path loss model with path loss exponent α. We assume that a channel is
subject to independent Rayleigh fading with channel power gains as i.i.d. exponential random
variables with mean 1.

8

TABLE I
S UMMARY OF N OTATION

Notation
Φ, Φt

Definition/Meaning
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

λǫ

Density of reliable transmissions

Ps (θ)

Conditional link success probability

ps (θ)

Mean success probability

α

Path loss exponent

δ

2/α

Mb (θ)
R

bth moment of the conditional link success probability
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, 0) and a receiver at the origin o. Under
the expectation over the PPP, this link is the typical link. The success probability ps (θ) of the
typical link is the ccdf of the SIR calculated at the origin. For Rayleigh fading, from [4], [14],
it is known that

ps (θ) = exp −λpCθδ ,

(4)

where C , πR2 Γ(1 + δ)Γ(1 − δ) with δ , 2/α. The model is scale-invariant in the following
sense: 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 > 0. Such scaling results
in transmitter locations aϕ and link distances aR. The density of the scaled network is λ/a2 .
By setting a = 1/R to obtain unit distance links, the resulting density is λR2 . Hence without
loss of generality, we can set R = 1. Applied to the meta distribution and the SOC, this means
that the model with parameters (R, λ) behaves exactly the same as the model with parameters
(1, λR2 ).

9

B. Exact Formulation
Observe from Def. 1 that the SOC depends on η(θ, ǫ) = P(Ps (θ) > 1 − ǫ) whose direct
calculation seems infeasible. But the moments of Ps (θ) are available in closed-form [2], from
which we can derive exact and approximate expressions of λǫ and obtain simple upper and lower
bounds on the SOC. Let Mb (θ) denote the bth moment of Ps (θ), i.e.,

Mb (θ) , E Ps (θ)b .

(5)

The mean success probability is ps (θ) ≡ M1 (θ).
From [2, Thm. 1], we can express Mb (θ) as


Mb (θ) = exp −λCθδ Db (p, δ) ,

where


∞  
X
δ−1 k
b
p ,
Db (p, δ) ,
k k−1
k=1

b ∈ C,

(6)

p, δ ∈ (0, 1].

(7)

For b ∈ N, the sum is finite and Db (p, δ) becomes a polynomial which is termed diversity

polynomial in [15]. The series in (7) converges for p < 1, and at p = 1 it is defined if b ∈
/ Z−

or b + δ ∈
/ Z− and converges if ℜ(b + δ) > 0. Here ℜ(z) is the real part of the complex number
z. For b = 1 (the first moment), D1 (p, δ) = p, and we get the expression of ps (θ) as in (4). We
can also express Db (p, δ) using the Gaussian hypergeometric function 2 F1 as
Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p).

(8)

Using the Gil-Pelaez theorem [16], the exact expression of λǫ = λpη(θ, ǫ) can be obtained in
an integral form from that of η(θ, ǫ) given in [2, Cor. 3] as
λp λp
−
λǫ =
2
π
where j ,

√

Z∞
0

sin(u ln(1 − ǫ) + λCθδ ℑ(Dju ))
du,
ueλCθδ ℜ(Dju )

(9)

−1, Dju = Dju (p, δ), and ℑ(z) is the imaginary part of the complex number z.

The SOC is then obtained by taking the supremum of λǫ over λ > 0 and p ∈ (0, 1].
C. Approximation with Beta Distribution
We can accurately approximate λǫ in a semi-closed form using the beta distribution, as
shown in [2]. The rationale behind such approximation is that the support of the link success

10

probability Ps (θ) is [0, 1], making the beta distribution a natural choice. With the beta distribution
approximation, λǫ can be approximated as



µβ
,β
,
(10)
λǫ ≈ λp 1 − Iǫ
1−µ
R 1−ǫ
where Iǫ (y, z) , 0 ty−1 (1 − t)z−1 dt/B(y, z) is the regularized incomplete beta function with
B(·, ·) denoting beta function, µ = M1 , and β = (M1 − M2 )(1 − M1 )/(M2 − M12 ).

The advantage of the beta approximation is the faster computation of λǫ compared to the exact
expression without losing much accuracy [2, Tab. I, Fig. 4] (also see Fig. 7 of this paper). In
general, it is difficult to obtain the SOC analytically due to the forms of λǫ given in (9) and (10).
But we can obtain the SOC numerically with ease. We can also gain useful insights considering
some specific scenarios, on which we focus in the following subsection.
D. Constrained SOC
1) Constant λp in dense networks: For constant λp (or, equivalently, a fixed ps (θ)), we now
study how the density of reliable links λǫ behaves in an ultra-dense network. Given θ, α, and
ǫ, this case is equivalent to asking how λǫ varies as λ → ∞ while letting p → 0 for constant
transmitter density λp (constant ps (θ)). We denote the constrained SOC by S̃(θ, ǫ).
Lemma 1 (p → 0 for constant λp). Let ν = λp. Then, for constant ν while letting p → 0, the
SOC constrained on the density of concurrent transmissions is

 λp,
if 1 − ǫ < ps (θ)
S̃(θ, ǫ) =
 0,
if 1 − ǫ > ps (θ).

(11)

Proof: Applying Chebyshev’s inequality to (3), for 1 − ǫ < ps (θ) = M1 , we have
η(θ, ǫ) > 1 −

var(Ps (θ))
,
((1 − ǫ) − M1 )2

(12)

where var(Ps (θ)) = M2 − M12 is the variance of Ps (θ). From [2, Cor. 1], for constant ν, we
know that lim var(Ps (θ)) = 0. Hence the lower bound in (12) approaches 1, which leads to
p→0

λp=ν

η(θ, ǫ) → 1. This results in the SOC constrained on the density of concurrent transmissions
equal to λp.
On the other hand, for 1 − ǫ > M1 ,
η(θ, ǫ) ≤

var(Ps (θ))
.
((1 − ǫ) − M1 )2

(13)

11

As we let p → 0 for constant ν, the upper bound in (13) approaches 0, which leads to η(θ, ǫ) → 0.
This results in the SOC constrained on the density of concurrent transmissions equal to 0.
In fact, as var(Ps (θ)) → 0, the ccdf of Ps (θ) approaches a step function that drops from 1 to
0 at the mean of Ps (θ), i.e., at 1 − ǫ = ps (θ). This behavior is in agreement with (11).

Remark: Lemma 1 shows that, if p → 0 while λp is fixed to the value ν at which ps (θ)

equals the target reliability 1 − ǫ, the maximum value of the constrained SOC is the value of
the TC times 1/(1 − ǫ), and that value of the TC is ν(1 − ǫ).

This observation can be explained as follows: As p → 0 while keeping λp = ν, all links in

a realization of the network have the same success probability, and that value of the success
probability equals ps (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 as
if it were the typical link, as in the TC framework. If ν is initially set to a value that results
in ps (θ) > 1 − ǫ, we can always increase it till ps (θ) = 1 − ǫ while all active links satisfying
the outage constraint (or, equivalently, the typical link satisfying the outage constraint with
probability one). Accordingly, the value of the TC equals 1 − ǫ times the value of ν at which
ps (θ) = 1 − ǫ.

Fig. 2 shows that at small values of the target outage probability ǫ, the density of reliable

transmissions monotonically increases with p. On the other hand, at larger values of ǫ, it first
decreases with p.
2) λp → 0: For λp → 0, λǫ depends linearly on λp, which we prove in the next lemma.
Lemma 2 (λǫ as λp → 0). As λp → 0,
λǫ ∼ λp.
p(δ−1)

Proof: As λp → 0, M1 approaches 1 and thus var(Ps (θ)) = M12 (M1

− 1) approaches 0.

Since ǫ ∈ (0, 1), we have 1−ǫ < M1 as λp → 0. Using Chebyshev’s inequality for 1−ǫ < M1 as
in (12) and letting var(Ps (θ)) → 0, the lower bound in (12) approaches 1, leading to η(θ, ǫ) → 1.
Lemma 2 can be understood as follows. As λp → 0, the density of active transmitters is very
small. Thus each transmission succeeds with high probability and η(θ, ǫ) → 1. In this regime,
the density of reliable links λǫ is directly given by λp.
The case λp → 0 can be interpreted in two ways: 1) λ → 0 for constant p and 2) p → 0 for
constant λ. Lemma 2 is valid for both cases, or any combination thereof. The case of constant

12

0.1

0.08

0.06

0.04

0.02

0
0

0.2

0.4

0.6

0.8

1

Fig. 2. The density of reliable links λǫ against the transmit probability p for λp = 1/10, θ = −10 dB, α = 4, and R = 1.
The values at the curves are ǫ = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 (bottom to top). The mean success probability is ps (θ) = 0.855.

p is relevant since it can be interpreted as a delay constraint: As p gets smaller, the probability
that a node makes a transmission attempt in a slot is reduced, increasing the delay. Since the
mean delay until successful reception is larger than the mean channel access delay 1/p, it gets
large 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 → 0 (p = 10−5 in Fig. 3), λǫ increases
linearly with λp until λp reaches the value 0.0675 which corresponds to ps (θ) = 1 − ǫ = 0.9
and then drops to 0. This behavior is in accordance with Lemma 1. In general, as λp increases,
λǫ increases first and then decreases after a tipping point. This is due to the two opposite effects
of λp on λǫ : The density λp of active transmitters increases, but at the same time, more active
transmissions cause higher interference, which in turn, reduces the fraction η(θ, ǫ) of links that
have a reliability at least 1 − ǫ.
The contour plot in Fig. 4(a) visualizes the trade-off between λp and η(θ, ǫ). The contour
curves for small values of λp run nearly parallel to those for λǫ , indicating that η(θ, ǫ) is close
to 1. Specifically, the contour curves for λp = 0.01 and λp = 0.02 match those for λǫ = 0.01
and λǫ = 0.02 almost exactly. This behavior is in accordance with Lemma 2. In contrast, for
large values of λp, the decrease in η(θ, ǫ) dominates λǫ . Also, for larger values of λ (λ > 0.4 for
Fig. 4(a)), λǫ first increases and then decreases with the increase in p. This behavior is due to the

13

0.1

0.08

0.06

0.04

0.02

0
0

0.2

0.4

0.6

0.8

1

Fig. 3. The density of reliable links λǫ given in (2) for different values of the transmit probability p for θ = −10 dB, α = 4,
and ǫ = 1/10. Observe that the slope of λǫ is one for small λp. The dashed arrow points to the value of λp = 0.0675, which
corresponds to 1 − ǫ = ps (θ) = 0.9.

following trade-off in p. For a small p, there are few active transmitters in the network per unit
area, but a higher fraction of links are reliable. On the other hand, a large p means more active
transmitters per unit area, but also a higher interference which reduces the fraction of reliable
links. For λ < 0.4, the increase in the density of active transmitters dominates the decrease in
η(θ, ǫ), and λǫ increases monotonically with p. The three-dimensional plot corresponding to the
contour 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 > 0, the SOC is upper bounded as

1
1

, 0 < b ≤ 1,
eπθ δ Γ(1−δ)Γ(1+δ) b(1−ǫ)b
S(θ, ǫ) ≤
Γ(b)
1

, b > 1.
eπθ δ Γ(1−δ) Γ(b+δ)(1−ǫ)b

(14)

Proof: Using Markov’s inequality, η(θ, ǫ) can be upper bounded as
η(θ, ǫ) ≤

Mb (θ)
,
(1 − ǫ)b

b > 0,

(15)

14

1.8
1.6

1.8
0.04

X: 1
Y: 0.23
Z: 0.09227

1.4

1.6

1.2

0.1

SOC point

1

1.4
0.8

0.03

1.2

0.4

0.2

0.03

1

0.6
SOC point

0.8

0.02

0.05
0.04

0.6

0.1

0.4
0.01

0.2

0.02

0.05

0.06

0.07

0.02

0.2

0
1

0.08
0.09

0.01

0

0.05

0.01

0.4

0.6

0.8

0
1

0.5
2

1

0

(a)

3

(b)

Fig. 4. (a) Contour plots of λǫ and the product λp for θ = −10 dB, α = 4, and ǫ = 1/10. The solid lines represent the contour
curves for λǫ and the dashed lines represent the contour curves for λp. The numbers in “black” and “red” indicate the contour
levels for λǫ and λp, respectively. The SOC is S(θ, ǫ) = 0.09227. The values of λ and p at the SOC point are 0.23 and 1,
respectively, and the corresponding mean success probability is ps (θ) = 0.6984. 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 λǫ corresponding to the contour
plot.

where Mb (θ) = e−λCθ

δD

b (p,δ)

. Hence we can upper bound the SOC as
S(θ, ǫ) ≤ Su ,

where
δ

e−λCθ Db (p,δ)
Su = sup λp
,
(1 − ǫ)b
λ,p

(16)

with C = πΓ(1 − δ)Γ(1 + δ) and Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p). Let us denote fu (λ, p) =
λpe−λCθ

δD

b (p,δ)

. We can then write
∂fu (λ, p)
δ
= pe−λCθ Db (p,δ) (1 − λCθδ Db (p, δ)).
|
{z
}
∂λ
>0

Setting

∂fu (λ,p)
∂λ

= 0, we obtain the critical point as λ0 (p) = 1/(Cθδ Db (p, δ)). For any given p,

the objective function is quasiconcave. Thus λ0 (p) is the global optimum for each p. As a result,
the optimization problem in (16) reduces to
1
sup fu (λ0 (p), p)
(1 − ǫ)b p
1
1
sup
.
=
δ
b
eCθ b(1 − ǫ)
p
2 F1 (1 − b, 1 − δ; 2; p)

Su =

(17)

15

For 0 < b < 1, 2 F1 (1 − b, 1 − δ; 2; p) monotonically increases with p. In this case, p → 0 solves
(17). On the other hand, for b > 1, 2 F1 (1 − b, 1 − δ; 2; p) monotonically decreases with p. Thus
p = 1 solves (17). Overall the value of p that solves (17) is

 → 0, 0 < b < 1,
p0
 = 1, b > 1.

(18)

Note that the objective function in (17) is monotonic in p. Hence p0 in (18) is again the global
optimum.
Finally, for 0 < b < 1, the upper bound on the SOC is obtained by substituting p = 0 in the
objective of (17). Since 2 F1 (1 − b, 1 − δ; 2; 0) = 1,
Su =

eπθδ Γ(1

1
1
,
− δ)Γ(1 + δ) b(1 − ǫ)b

Similarly since b 2 F1 (1 − b, 1 − δ; 2; 1) =
Su =

1
eπθδ Γ(1

0 < b < 1.

(19)

b > 1.

(20)

Γ(b+δ)
,
Γ(b)Γ(1+δ)

Γ(b)
,
− δ) Γ(b + δ)(1 − ǫ)b

For b = 1, the hypergeometric function returns 1 irrespective of the other parameters, and thus
(19) and (20) are identical.
The tightest Markov upper bound can be obtained by minimizing Su in (19) and (20) over b.
Now, the value of b that minimizes Su in (19) is
bm = −

1
.
ln(1 − ǫ)

(21)

Since Su takes two different values depending on whether 0 < b ≤ 1 or b > 1, to obtain
the tightest Markov upper bound, we need to consider following two cases based on whether
bm ∈ (0, 1] or bm > 1.
1) bm ∈ (0, 1]: If bm ∈ (0, 1], it is the optimum value of b that minimizes Su since Su in (19)
is smaller than Su in (20) for 0 < b < 1, greater for b > 1, and equal to for b = 1. From (21),
it is apparent that the case bm ∈ (0, 1] is equivalent to ǫ ∈ [0.6321, 1). Hence, if ǫ ∈ [0.6321, 1),
the optimum b that gives the tightest Markov upper bound is given by (21). Substituting b = bm
in (19), we get the exact closed-form expression of the tightest Markov upper bound as
Sut =

− ln(1 − ǫ)
,
− δ)Γ(1 + δ)

πθδ Γ(1

if ǫ ∈ [0.6321, 1),

(22)

where ‘t’ in the superscript of Sut indicates the tightest bound.
2) bm > 1: If bm > 1, i.e., ǫ ∈ (0, 0.6321), the optimum b is the value of b that minimizes
Su in (20). However, due to the form of Su in (20), the optimum b cannot be expressed in a

16

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 > 1, we can get a closed-form
expression of the approximate tightest Markov bound by using the approximation
Γ(b + δ)
≈ bδ
Γ(b)

(23)

in (20). Then, for b > 1, we can express (20) as
Su ≈

1
eπθδ Γ(1

The value of b that minimizes (24) is given as
b̄m = −

− δ)

bδ (1

1
.
− ǫ)b

(24)

δ
.
ln(1 − ǫ)

The corresponding closed-form expression of the tightest approximate Markov upper bound is
obtained by substituting b̄m in (24) which is given as

δ
− ln(1 − ǫ)
e−(1−δ)
t
Su ≈
,
θ
πδ δ Γ(1 − δ)

if ǫ ∈ (0, 0.6321).

(25)

Fig. 5 illustrates upper bounds on the SOC.

Letting ǫ → 0, from (25), we observe that the lower tail of the SOC decreases exponentially,
i.e.,
S(θ, ǫ) /

 ǫ δ
θ

e−(1−δ)
,
πδ δ Γ(1 − δ)

ǫ → 0,

(26)

where ‘/’ denotes an upper bound which gets tighter asymptotically (here as ǫ → 0). In the
next 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 → 0.
We now obtain lower bounds on the SOC.
Theorem 2 (Lower bound on the SOC). The SOC is lower bounded as

  −(1−W ((1−ǫ)e))

1 − W ((1 − ǫ)e)
e
− (1 − ǫ)
S(θ, ǫ) >
,
πθδ Γ(1 + δ)Γ(1 − δ)
ǫ

where W (·) denotes the Lambert W function.

Proof: By the reverse Markov’s inequality, we have
1−

E!t ((1 − Ps (θ))b )
< η(θ, ǫ),
ǫb

For b ∈ N we can lower bound the SOC as
S(θ, ǫ) > Sl ,

b > 0.

(27)

17

101

100

10-1

10

-2

0

0.2

0.4

0.6

0.8

1

Fig. 5. Analytical and numerical results for the SOC. The tightest Markov upper bound on the SOC obtained numerically
uses (19) and (20), which are optimized over b. The tightest Markov upper bound obtained analytically uses (22) and (25). The
SOC upper bound obtained analytically is quite close to that obtained numerically for the almost complete range of reliability
threshold 1 − ǫ, except near 1 − ǫ = 0.3679 (which is due to the approximation in (23)). The classical Markov bounds are
plotted using (19) and (20) for b = 1, b = 2, and b = 4. The lower bound for b = 1 is plotted using (27), while the lower
bounds for b = 2 and b = 4 are plotted numerically using (28). θ = −10 dB and α = 4.

where
Sl = sup λp 1 −
λ,p

with Mk (θ) = e−λCθ

δD

k (p,δ)

Pb

b
k=0 k



(−1)k Mk (θ)
ǫb

!

,

(28)

. For b = 1, (28) reduces to
1 − e−λpCθ
Sl = sup λp 1 −
ǫ
λ,p

δ

!

.

(29)

Since λ and p appear together as their product λp, Sl can be obtained by taking the supremum
over t = λp, i.e.,
!
δ
1 − e−tCθ
.
Sl = sup t 1 −
ǫ
t
{z
}
|

(30)

f (t)

Substituting the value of t that results in

∂f (t)
∂t

= 0 in f (t), we get the desired expression in (27).

18

15

10

5

0
0

10

20

30

40

50

Fig. 6. The solid lines represent the exact Db (p, δ) as in (7), while the dashed lines represent the asymptotic form of Db (p, δ)
as in (31).

For the values of b ∈ R+ \ {1}, an analytical expression for Sl is difficult to obtain due to the
form of (28), but we can easily obtain corresponding lower bounds numerically. Fig. 5 shows
Markov 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 λǫ and the SOC in the high-reliability regime,
i.e., as ǫ → 0. To this end, we first provide an asymptote of Db (p, δ) as b → ∞, which will
be used to obtain a closed-form expression of the SOC in the high-reliability regime. Then we
state a simplified version of de Bruijn’s Tauberian theorem (see [17, Thm. 4.12.9]) which allows
a convenient formulation of η(θ, ǫ) = P(Ps (θ) > 1 − ǫ) in terms of the Laplace transform as
ǫ → 0. ‘.’ denotes an upper bound with asymptotic equality (here as b → ∞).
Lemma 3 (Asymptote of Db (p, δ) as b → ∞). For b ∈ R, we have
Db (p, δ) .

pδ bδ
,
Γ(1 + δ)

b → ∞,

Proof: See Appendix A.
Fig. 6 illustrates how quickly Db (p, δ) approaches the asymptote.

(31)

19

Theorem 3 (de Bruijn’s Tauberian theorem [18, Thm. 1]). For a non-negative random variable
Y , the Laplace transform E[exp(−sY )] ∼ exp(rsu ) for s → ∞ is equivalent to P(Y ≤ ǫ) ∼

exp(q/ǫv ) for ǫ → 0, when 1/u = 1/v + 1 (for u ∈ (0, 1) and v > 0), and the constants r and
q are related as |ur|1/u = |vq|1/v .

Theorem 4 (λǫ in the high-reliability regime). For ǫ → 0, the density of reliable links λǫ
satisfies



θp
λǫ ∼ λp exp −
ǫ
where κ =

δ
1−δ

=

2
α−2

κ

(δλC ′ )κ/δ
κ

!

,

ǫ → 0,

(32)

and C ′ = πΓ(1 − δ).

Proof: Let Y = − ln(Ps (θ)). The Laplace transform of Y is E(exp(−sY )) = E(Ps (θ)s ) =
Ms (θ). Using (6) and Lemma 3, we have


λC(θp)δ sδ
,
Ms (θ) ∼ exp −
Γ(1 + δ)

|s| → ∞.
δ

Comparing this expression with that in Thm. 3, we have r = − λC(θp)
, u = δ, v = δ/(1−δ) = κ,
Γ(1+δ)
and thus q =

1
κ

(δλC ′ )κ/δ (θp)κ , where C ′ = πΓ(1 − δ). Using Thm. 3, we can now write
P(Y ≤ ǫ) = P(Ps (θ) ≥ exp(−ǫ))
(a)

∼ P(Ps (θ) ≥ 1 − ǫ),

= exp −

ǫ→0
!

(θp)κ (δλC ′ )κ/δ
κǫκ

,

(33)

where (a) follows from exp(−ǫ) ∼ 1 − ǫ as ǫ → 0. Since we have
λǫ = λpP(Ps (θ) > 1 − 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(Ps (θ) ≥ 1 − ǫ) in (33) simplifies
to
δλC ′ θδ
P(Ps (θ) ≥ 1 − ǫ) ∼ exp −
κǫκ

κ/δ !

,

ǫ → 0,

in agreement with [7, Thm. 2] where the result for this special case was derived in a less direct
way than Thm. 4. Fig. 7 shows the behavior of (32) in the non-asymptotic regime and also the
accuracy of the beta approximation given by (10).
We now investigate the scaling of S(θ, ǫ) in the high-reliability regime.

20

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0.5

0.6

0.7

0.8

0.9

1

Fig. 7. The solid line with marker ‘o’ represents the exact expression of λǫ as in (9), the dotted line represents the asymptotic
expression of λǫ given by (32) as ǫ → 0, 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 = 1/2, and p = 1/3.

Corollary 1 (SOC in high-reliability regime). For ǫ → 0,
 ǫ δ e−(1−δ)
,
S(θ, ǫ) ∼
θ 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 and
′ κ/δ

denote ξρ , ρ−κ (δC κ)
λǫ ∼ fρ (λ, p),

and fρ (λ, p) , λp exp(−λκ/δ pκ ξρ ). From (32), we can then write

ǫ → 0, and the SOC is
S(θ, ǫ) ∼ sup fρ (λ, p),
λ,p

ǫ → 0.

(36)

We can then write



∂fρ (λ, p)
κξρ κ/δ κ
κ/δ κ
= p exp −λ p ξρ 1 −
λ p .
∂λ
δ
{z
}
|
>0

Setting

∂fρ (λ,p)
∂λ

= 0, we obtain the critical point as
δ/κ

δ
.
λ0 (p) =
ξρ κpκ

(37)

21

For any given p, the objective function is quasiconcave. Hence the optimization problem in (36)
reduces to
S(θ, ǫ) ∼ sup fρ (λ0 (p), p),
p

=



δ
eκξρ

δ/κ

ǫ → 0,

sup p1−δ ,
p

ǫ → 0.

Observe that fρ (λ0 (p), p) monotonically increases with p and thus attains the maximum at p = 1.
 δ/κ
and is given by (35) after simplification.
Hence the SOC is achieved at p = 1 and λ = ξρδκ
The equation (35) confirms the asymptotic bound on the SOC given in (26).
Corollary 2 (The meta distribution at the SOC point). As ǫ → 0, the value of the meta
distribution at the SOC point can be simply expressed as
η(θ, ǫ) ∼ e−(1−δ) .

(38)

Proof: From Cor. 1, as ǫ → 0, the SOC can be expressed as
S(θ, ǫ) ∼ λopt popt η(θ, ǫ),

(39)

where λopt = λ0 (given by (37)) and popt = 1 correspond to the SOC point as ǫ → 0. 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 → 0, the mean success
probability at the SOC point can be expressed as
 ǫ δ
ps,opt ∼ 1 −
Γ(1 + δ).
δ

(40)

Proof: Substituting λ = λ0 (given by (37)) and p = 1 in (4) and using e−x ∼ 1 − x as

x → 0 yield the desired expression.
We now provide few remarks pertaining to the high-reliability regime.
Remarks:
•

Letting Cδ =


1 δ e−(1−δ)
,
δ
Γ(1−δ)

the density of transmitters λ∗ , S(θ, ǫ) that maximizes the

density of active links that achieve a reliability at least 1 − ǫ behaves as
 ǫ δ
∗
λ π ∼ Cδ
, ǫ → 0.
θ

(41)

22

The coefficient Cδ depends only on δ. In the practically relevant regime 1/2 ≤ δ < 1, i.e.,

2 < α ≤ 4, Cδ ≈ 1 − δ. In (41), the left side λ∗ π is the mean number of reliable receivers in
a disk of unit radius in the network. Equation (41) reveals an interesting trade-off between

the spectral efficiency (captured by θ) and the reliability (captured by ǫ), where only their
ratio matters. For example, at low rates, ln(1 + θ) ∼ θ; thus, a 10× higher reliability can
be achieved by lowering the rate by a factor of 10.
•

The (potential) transmitter density λopt that achieves the SOC is
 ǫ δ
1
, ǫ → 0.
(42)
λopt ∼
δ
θ πδ Γ(1 − δ)
Here λopt π is the mean number of (potential) transmitters in a disk of unit radius in the
network that achieves the SOC.

•

From (38), it is apparent that, at the SOC point, the fraction of links that satisfy the outage
constraint depends only on the path loss exponent α, as δ , 2/α.

•

The mean success probability ps,opt at the SOC point (given by (40)) allows us to relate the
SOC and the TC. Substituting q ∗ = 1 − ps,opt in [13, (4.29)], we can express the TC as
 ǫ δ
1
, ǫ → 0,
c(θ, ǫ) ∼
δ
θ πδ Γ(1 − δ)
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 ps (θ) = ps,opt instead of
ps (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(ǫδ ), while the original TC defined in [6] scales
linearly in ǫ.

•

For α = 4, the expression of SOC in (35) simplifies to
 ǫ 1/2
S(θ, ǫ) ∼ 0.154
, ǫ → 0,
θ
and the meta distribution gives η ≈ 0.6. In other words, approximately 60% of active links
satisfy the outage constraint if α = 4. Also, for α = 4, the mean success probability at the
√
SOC point is simply given by ps,opt ∼ 1 − 1.2533 ǫ as ǫ → 0. Fig. 8 plots λǫ versus λ and
p for ǫ = 0.007 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. random
variables (which are constant over time).

23

0.05

SOC point
X: 1
Y: 0.0701
Z: 0.04068

0.04
0.03
0.02
0.01
0
1
0.5
0

0.8

0.6

0.4

0.2

0

Fig. 8. Three-dimensional plot of λǫ for ǫ = 0.007, θ = −10 dB, and α = 4. Observe that p = 1 achieves the SOC. The mean
success probability ps (θ) at the SOC point is 0.8964.

A. Network Model
Let Ri denote the random link distance between a transmitter i and its associated receiver in a
√
Poisson bipolar network. We assume that Ri is Rayleigh distributed with mean 1/(2 µ) as it is
the distribution of the nearest-neighbor distance in a PPP of density µ [19].3 This scenario can
be interpreted as the one where an active transmitter tries to communicate to its nearest receiver
in a network where the potential transmitters form a PPP Φt of density λ and the receivers form
an another PPP (independent of Φt ) 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
(Ro , 0), where Ro is the Rayleigh distributed link distance. Under the expectation over the point
process, this link is the typical link.
B. Exact Formulation of the SOC
Lemma 4 (bth moment of the link success probability). For Rayleigh distributed link distances
√
with mean 1/(2 µ), the bth moment of Ps (θ) is
Mb (θ) =
3

µ
.
µ + λθδ Γ(1 + δ)Γ(1 − δ)Db (p, δ)

(43)

Generalizations to other link distance distributions are beyond the scope of this paper. This is because some new techniques

may need to be developed, and it is unclear what other distribution to assume.

24

Proof: See Appendix B.
For b = 1, using D1 (p, δ) = p, we get the expression of the mean success probability ps (θ).
Moreover, for b ∈ N, (43) represents the joint success probability of b transmissions with random
link distance, as obtained in [15, (23)].
As in the deterministic link distance case, using the Gil-Pelaez theorem, we can calculate the
density of reliable links from (9), and the SOC is obtained by taking the supremum of λǫ 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 > 0, the SOC for Rayleigh distributed link
distances is upper bounded as
S(θ, ǫ) ≤





1
1
,
θ δ Γ(1−δ)Γ(1+δ) b(1−ǫ)b
Γ(b)
1
,
θ δ Γ(1−δ) Γ(b+δ)(1−ǫ)b

0 < b ≤ 1,
b > 1.

(44)

Proof: Again using Markov’s inequality, η(θ, ǫ) can be upper bounded as
Mb (θ)
, b > 0,
η(θ, ǫ) ≤
(1 − ǫ)b
1
. Hence for any b > 0, we have
where Mb (θ) = 1+λθδ Γ(1+δ)Γ(1−δ)D
b (p,δ)
S ≤ Su ,
where
Su =

1
λp
.
sup
b
δ
(1 − ǫ) λ,p 1 + λθ Γ(1 + δ)Γ(1 − δ)Db (p, δ)
{z
}
|

(45)

Aλ,p

Aλ,p is maximized at λ = ∞, and it follows that
1
1
Su = δ
sup
,
θ Γ(1 + δ)Γ(1 − δ)b(1 − ǫ)b p 2 F1 (1 − b, 1 − δ; 2; p)
where we have used Db (p, δ) = pb 2 F1 (1 − b, 1 − δ; 2; p) as in (8).

(46)

Notice that the optimization problem in (46) is similar to that in (17). Thus, following the
steps 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 ∈ [0.6321, 1), we can obtain the exact closed-form expression of the tightest Markov
bound as
Sut =

−e ln(1 − ǫ)
.
− δ)Γ(1 + δ)

θδ Γ(1

(47)

25

102

10

1

100

10

-1

0

0.2

0.4

0.6

0.8

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 ∈ [0.6321, 1) and (48)
when ǫ ∈ (0, 0.6321). Observe that the analytical approximation of the SOC upper bound provides a tight upper bound for the
complete range of reliability threshold 1 − ǫ. The curve corresponding to the tightest Markov upper bound obtained analytically
deviates from that obtained numerically at 1 − ǫ = 0.3679 due to the approximation as in (23). The classical Markov bounds
are plotted using (44) for b = 1, b = 2, and b = 4. θ = −10 dB and α = 4.

For ǫ ∈ (0, 0.6321), we can obtain the exact tightest Markov bound numerically. Alternatively,
using the approximation in (23), we get a closed-form expression of the approximate tightest
Markov bound as
Sut

≈



− ln(1 − ǫ)
θ

δ

eδ
.
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 = 0.6321 due to the approximation in (23).
As ǫ becomes smaller, i.e., 1 − ǫ becomes closer to 1, the approximation (48) becomes better.
For ǫ < 0.2, the gap between the approximation of the upper bound and the beta approximation
is less than 0.15 dB.
Theorem 6 (Lower bound on the SOC). The SOC is lower bounded as
√
(1 − 1 − ǫ)2
.
S(θ, ǫ) > δ
ǫθ Γ(1 + δ)Γ(1 − δ)
Proof: The proof follows the proof of Thm. 2 with M1 (θ) =

1
.
1+λpθ δ Γ(1+δ)Γ(1−δ)

(49)

26

Fig. 9 shows lower bounds on the SOC. Similar to the deterministic link distance case, the
Markov lower bounds for b ∈ R+ \ {1} are analytically intractable.
D. High-reliability Regime
Theorem 7 (SOC in the high-reliability regime). For Rayleigh distributed link distances,
 ǫ δ
1
, ǫ → 0,
S(θ, ǫ) ∼
θ Γ(1 + δ)Γ(1 − δ)
and the SOC is achieved at p = 1.

Proof: As in the proof of Thm. 4, let Y = − ln(Ps (θ)) with its Laplace transform as
LY (s) = Ms (θ). Asymptotically,
(a)

LY (s) ∼
where (a) follows from using Db (p, δ) ∼

A
,
sδ

pδ bδ
Γ(1+δ)

|s| → ∞,

(50)

as |b| → ∞ in (43) and thus A = 1/(λθδ pδ Γ(1−

δ)). We claim that the expression in (50) is equivalent to
FY (ǫ) ∼

Aǫδ
,
Γ(1 + δ)

ǫ → 0.

(51)

The proof that (50) and (51) are equivalent is given in Appendix C.
As ǫ → 0, since
FY (ǫ) = P(− ln(Ps (θ)) < ǫ)
∼ P(Ps (θ) > 1 − ǫ)
= η(θ, ǫ),
the density of reliable links in the high-reliability regime can be expressed as
λǫ ∼

ǫδ p1−δ
,
θδ Γ(1 + δ)Γ(1 − δ)

ǫ → 0.

(52)

Here, λǫ is independent of the density λ of (potential) transmitters. As a result, the SOC is
S(θ, ǫ) ∼

ǫδ
sup p1−δ ,
θδ Γ(1 + δ)Γ(1 − δ) p

Setting p = 1 achieves the SOC, which is given as
 ǫ δ
1
S(θ, ǫ) ∼
,
θ Γ(1 + δ)Γ(1 − δ)

ǫ → 0.

ǫ → 0.

27

Similar to the deterministic link distance case, only the ratio of the spectral efficiency and the
reliability matters. As we observe from (52), λǫ does not depend on λ. This is due to the fact
that, in the high-reliability regime, the increase in λ decreases linearly the fraction of reliable
links η(θ, ǫ). For example, a 2× increase in λ decreases η(θ, ǫ) by a factor of 2. Also, the SOC is
a function of just two parameters, the reliability-to-target-SIR ratio and a constant that depends
only 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), which
is the maximum density of concurrently active links that meet a certain constraint on the
success probability. Hence the SOC provides a mathematical foundation for questions of network
densification 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 is
applicable to a wide range of wireless networks.
For Poisson bipolar networks with ALOHA and Rayleigh fading, we provide an exact analytical expression and a simple approximation for the density of reliable links λǫ . The SOC can
be easily calculated numerically as the supremum of λǫ obtained by optimizing over the density
of (potential) transmitters λ and the transmit probability p.
In the high-reliability regime where the target outage probability ǫ of a link goes to 0, we
give a closed-form expression of the SOC which reveals
•

the trade-off between the spectral efficiency and the reliability where only their ratio matters
while calculating the SOC.

•

insights on the scaling behavior of the SOC where, for both deterministic and Rayleigh
distributed link distance cases, we show that the SOC scales in ǫ as Θ(ǫδ ).

Interestingly, p = 1 achieves the SOC in the high-reliability regime. This means that with
ALOHA, all transmitters should be active in order to maximize the number of reliable transmissions in a unit area that succeed with a probability close to one. Hence, in the high-reliability
regime, backing off is not SOC-achieving. This happens because the reduction in the density of
active 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 that
the density of reliable links does not depend on λ as the increase in λ is exactly offset by the

28

fraction of reliable links. To be precise, a t-fold increase in λ decreases the density of reliable
links by a factor of t.
As a future work, it is important to generalize the results obtained for Rayleigh fading to
other fading distributions. However, since the current bounds on the SOC and the high-reliability
regime results exploit a structure induced by Rayleigh fading assumption, one might need to
develop new techniques depending on the fading distribution considered.
A PPENDIX A
P ROOF

L EMMA 3

OF

From (7), we have
Db (p, δ) =


∞  
X
δ−1
b
k

k=1
∞ 
X

=p

k=1

|

By Taylor’s theorem,
Ak (p) =

pk

k−1


δ − 1 k−1
b
.
p
k k−1
{z
}

(53)

Ak (p)

∞
(j)
X
A (1)
k

j!

j=0

(p − 1)j ,

(54)

(j)

where Ak (1) is the jth derivative of Ak (p) at p = 1. Let (k)j , k(k − 1)(k − 2) · · · (k − j + 1)
(j)

denote the falling factorial. Then Ak (1) can be written as

∞  
X
δ−1
b
(j)
(k − 1)j
Ak (1) =
k−1
k
k=1
=
(a)

.

where (a) follows from

Γ(b+δ−j)
Γ(b−j)

Γ(b + δ − j)
(δ − 1)j
Γ(b − j)Γ(1 + δ)
bδ (δ − 1)j
,
Γ(1 + δ)

(55)

. bδ as b → ∞. From (54) and (55),
∞

Ak (p) .

X (δ − 1)j
bδ
(p − 1)j .
Γ(1 + δ) j=0
j!
|
{z
}
pδ−1

From (53) and (56), we get the desired result.

(56)

29

A PPENDIX B
P ROOF

OF

L EMMA 4

Let us denote the random link distance of the typical link by R. Then the probability density
function of R is fR (a) = 2πµa exp(−πµa2 ). Let kzk denote the distance between a receiver
and a potential interferer z ∈ Φt . Given Φt , the conditional link success probability Ps (θ) is

 −α
hR
> θ | Φt = E (1(h > θRα I) | Φt ) ,
Ps (θ) = P
I
where
I=

X

z∈Φt \{zo }

hz kzk−α 1(z ∈ Φt ),

where zo ∈ Φt denotes the desired transmitter. Conditioning on R and then averaging over fading
and ALOHA results in
Ps (θ) | R =
Let f (r) =

Y

z∈Φt \{zo }




1+θ

p


R
kzk



 α + 1 − p .

b
+ 1 − p . Then the bth moment of Ps (θ) is




Y
R 
Mb (θ) = E 
f
kzk
z∈Φt \{zo }


  
Z ∞ 
R
(a)
= ER exp −2πλ
t 1−f
dt
t
0


Z∞
Z∞ 


a
2
(b)
dt e−µπa da
= 2πµ a exp −2πλ t 1 − f
t
0
0


 
Z∞
Z∞ 
1
2
(c)
= 2πµ a exp −2πλa2 y 1 − f
dy  e−µπa da
y

p
1+θr α

0

0

µ
µ + 2λ 0 y (1 − f (1/y)) dy
µ
(d)
=
,

b !
Z∞
pθr α
1− 1−
µ + 2λ
r −3 dr
1 + θr α
{z
}
|0
=

R∞

(57)

Fb

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)

30

follows from the substitution y = 1/r and plugging f (r) back. With 1 as the upper limit of the
integral and µ = λ, (57) reduces to the expression of the bth moment of the success probability
in a Poisson cellular network as in [2, (27)].
With r α = x, the integral in (57) can be expressed as

b !
Z∞
pθx
1
1− 1−
x−δ−1 dx
Fb =
α
1 + θx
0

(e)

=

∞ 
X
k=1

(f)

=

∞

k Z
xk−δ−1
(pθ)
b
(−1)k+1
dx
α
k
(1 + θx)k
0

δ

θ
πδ
Db (p, δ),
2 sin(πδ)

where (e) follows from the binomial expansion of 1 −
follows from

and (−1)k+1

Z∞

xk−δ−1

δ−k



(58)

pθx b
1+θx

and Fubini’s theorem, and (f)

π
Γ(k − δ)
sin(πδ) Γ(k)Γ(1 − δ)



dx = θ
(1 + θx)k
0


k−δ−1
δ−1
=
. Finally, substituting (58) in (57) and using
k−1
k−1

πδ
sin(πδ)

≡ Γ(1 + δ)Γ(1 −

δ), (43) follows.

A PPENDIX C
P ROOF

THAT

(50) AND (51) ARE

EQUIVALENT

The proof uses the Weierstrass approximation theorem that any continuous function f :
[t1 , t2 ] → R can be approximated by a sequence of polynomials from above and below.
In our case, t1 = 0 and t2 = 1. Thus, for any given t > 0, if f (y) is a continuous real-valued
function on [0, 1], for n ≥ 1, there exists a sequence of polynomials Pn (y) and Qn (y) such that
Pn (y) ≤ f (y) ≤ Qn (y) ∀y ∈ [0, 1],
Z 1
(Qn (y) − f (y))dy ≤ t,

(59)
(60)

0

and

Z

1
0

(f (y) − Pn (y))dy ≤ t.

(61)

Even if f (y) has a discontinuity of the first kind, we can still construct polynomials Pn (y) and
Qn (y) that satisfy (59)-(61).4
4

See [20, Sec. 7.53] for the details of the construction of such polynomials.

31

To prove the desired result, we first show that
Z ∞
Z ∞
A
δ
−sy
−sy
lim s
e f (e )dFY (y) =
y δ−1 f (e−y )e−y dy.
s→∞
Γ(δ)
0
0
Pn
k
Let Qn (y) = k=0 ak y with ak ∈ R for k = 0, 1, . . . , n. We then have
Z ∞
Z ∞
δ
−sy
−sy
δ
lim sup s
e f (e ) dFY (y) ≤ lim s
e−sy Qn (e−sy ) dFY (y)
s→∞

s→∞

0

= lim

s→∞

(62)

0

n
X

ak s

k=0

δ

Z

∞

e−(k+1)sy dFY (y)

0

n
X

ak
(k + 1)δ
k=0
Z ∞
(b) A
=
y δ−1 e−y Qn (e−y ) dy
Γ(δ) 0
Z ∞
(c) A
=
y δ−1 e−y f (e−y ) dy,
Γ(δ) 0

(a)

=A

R∞
where (a) follows from lim sδ 0 e−sy dFY (y) = A, (b) follows from the definition of the
s→∞R
∞
gamma function as Γ(δ) , 0 y δ−1 e−y dy, and (c) follows from the dominated convergence
theorem as n → ∞.

By a similar argument for Pn (y), we have
Z ∞
Z ∞
A
−sy
−sy
δ
y δ−1 e−y f (e−y ) dy,
e f (e ) dFY (y) ≥
lim inf s
s→∞
Γ(δ)
0
0

and (62) follows.
Now let


 1,
y
f (y) =
 0,

1
e

≤y≤1

0 ≤ y < 1e .

(63)

Letting s = 1/ǫ in (62) and using (63), we have
Z ∞
−δ
δ
e−sy f (e−sy ) dFY (y)
lim ǫ FY (ǫ) = lim s
ǫ→0
s→∞
0
Z 1
(d) A
=
y δ−1 dy
Γ(δ) 0
A
.
=
Γ(1 + δ)
where (d) follows from (62) and (63).
ACKNOWLEDGMENT
The authors would like to thank Ketan Rajawat and Amrit Singh Bedi for their insights on
the optimization problems in the paper.

32

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