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A New Method for Computing the Transmission
Capacity of non-Poisson Wireless Networks
Radha Krishna Ganti and Jeffrey G. Andrews
Department of Electrical and Computer Engineering
University of Texas at Austin
Austin, TX 78712-0204, USA
Email: rganti@austin.utexas.edu, jandrews@ece.utexas.edu
Abstract—The relative locations of concurrent transmitting
nodes play an important role in the performance of wireless
networks because it largely determines their mutual interference.
In most prior work the set of interfering transmitters has
been modeled by a homogeneous Poisson distribution, which
assumes independence in the transmitting node positions, and
hence precludes intelligent scheduling protocols. One of the main
difficulties in extending the numerous Poisson results is the
absence of an analytical form for the probability generating
functional and the Palm characterization of the underlying spatial
node distribution. In this paper we take an alternative approach
based on the second-order product density of the node distri-
bution, which is asymptotically tight as the outage probability
tends to zero. Unlike the probability generating functional, the
second order product density can be easily obtained for a wide
range of point processes and hence this approach is useful in
analyzing complex wireless networks and MAC protocols. We
use this approach to then provide accurate approximations of
the transmission capacity of wireless ad hoc networks for three
plausible point processes, corresponding to ALOHA, clustering,
and carrier sensing schedulers. The mathematical framework
introduced can be used to analyze other relevant metrics.
I. INTRODUCTION
Interference is a main limiting factor for the performance
of a wireless ad hoc network. The interference in a network is
primarily dictated by the locations of the concurrent transmit-
ters whose location is often modeled by a point process [1],
[2] on the plane. Tools from stochastic geometry and point
process theory have been used to characterize the performance
of various physical layer technologies in an ad hoc network
[3], [4], [5], [6], [7], [8], [9], where interference is a main
limiting factor. But nearly all stochastic geometry work on
wireless networks focuses on the case where the transmitting
nodes are distributed as a Poisson point process (PPP) because
of its tractability. However the PPP does not describe “good”
MAC protocols which attempt to avoid collisions or otherwise
coordinate transmissions. Although the Poisson model has
been valuable in providing tractability and insight into “worst
case” MAC protocols, tractable analytical approaches that go
beyond the Poisson model are sorely needed.
In this paper we provide a new method for looking at the
performance characterization of more general point processes
that could model more sophisticated scheduling approaches
and hence get closer to optimal throughput (good upper bounds
for which are of course unknown for most nontrivial multi-
node networks). We utilize the transmission capacity (TC)
metric that was introduced in [10] and is equal to the maximum
spatial density of simultaneous transmissions possible for a
given outage constraint. The TC of a wireless network is
known only when the underlying nodes are distributed as a
Poisson point process (PPP) or more recently, as a Poisson
cluster process(PCP) [11]. The main difficulty in characteriz-
ing the TC is the evaluation of the outage probability which in
turn requires the conditional probability generating functional
of the underlying node distribution. But unfortunately, the
conditional probability generating functionals are known only
for a PPP [1], [12], PCP [11] and a few variants of the PPP.
As an alternative approach, we consider the TC when the
outage probability is close to zero. We characterize the TC
in this low outage regime using the second order product
density of the spatial distribution of the transmitters, a quantity
that can be analytically evaluated for a large class of point
processes. As an example we look at the TC under three
scenarios: PPP used in modeling ALOHA and networks with
no coordination, PCP used to model sensor networks where
clustering helps in improving the lifetime of the network,
Matern hard-core process used in modeling a CSMA type of
network where there is a strict minimum distance between
neighboring transmissions. Although we emphasize on the TC,
the techniques provided in this paper are general and can be
used to analyze other metrics when the density of concurrent
transmitters is small [13].
II. SYSTEM MODEL
We assume that the nodes are distributed as a stationary
point process [1] Φof density λon the plane. Each node has
its corresponding receiver at a distance din a random direction,
and for a node xits receiver is denoted by r(x). The path-loss
model is denoted by ℓ(x) : R2→[0,∞]and is assumed to be a
non-increasing function of kxkwhich satisfies R∞
δℓ(r)rdr <
∞,for any δ > 0. The small scale fading (power) is denoted
by hxy and is assumed to be i.i.d exponential with unit mean
between any pair of nodes. A node x∈Φcan communicate
with its receiver yif the received signal to interference ratio
(SIR)
SIR(x, y) = hxyℓ(x−y)
I(y, Φ\ {x})≥θ, (1)
where I(y)is the interference at y∈R2and is equal to
I(y, Φ\ {x}) = X
z∈Φ\{x}
hzyℓ(z−y)(2)
Since the process is stationary, the success probability is same
for all transmitters and hence we condition on a node being at
the origin and analyze its success probability. The probability
of success is equal to
Ps=P!o(SIR(o, r(o)) ≥θ),(3)
where P!odenotes the reduced Palm probability [1], [12] of
Φand odenotes the origin (0,0). Transmission capacity is
defined in [4], [10] and given by
Tc(ǫ) = (1 −ǫ) sup λ, (4)
with the constraint
Ps>1−ǫ.
In a strict mathematical sense, the success probability is not a
function of the density of transmitters, i.e., for the same density
of transmitters, the success probability may take multiple
values. For example, consider a cluster point process [1] with
density λ=λp¯c, where λpis the average number of clusters
per unit area, and ¯cis the average number of points in each
cluster. In this case, λp= 1,¯c= 3 will lead to a different Ps
than λp= 3 and ¯c= 1.
III. ASYMPTOTIC TRANSMISSION CAPACITY
As is evident from the definition of TC, we must evalu-
ate the success probability when the transmitting nodes are
spatially distributed as Φ. Since evaluating the exact outage
probability is not possible for many plausible spatial distribu-
tions of nodes, the goal of the paper is to develop new bounds
that are asymptotically tight as λapproaches zero. We begin
by a few definitions. Let f(x)be an integrable function on
the plane. Then
E!o"X
x∈Φ
f(x)#=λ−1ZR2
ρ(2)(x)f(x)dx,
and E!ohPx∈ΦPx6=y
y∈Φf(x)f(y)iis equal to
λ−1ZR2ZR2
ρ(3)(x, y)f(x)f(y)dxdy,
where ρ(2)(x)is the second-order product density [1] of the
point process Φand ρ(3)(x, y)is the third-order product den-
sity. Let G[f(x)] denote the conditional probability generating
functional, i.e.,
G[f(x)] = E!o"Y
x∈Φ
f(x)#.
Lemma 1. The probability of success is bounded by
1−µ≤Ps≤min{1−µ+κ
2,G[exp(−∆(x))]},(5)
where
µ=λ−1ZR2
ρ(2)(x)∆(x)dx,
κ=λ−1ZR2
ρ(3)(x, y)∆(x)∆(y)dxdy,
and
∆(x) = (1 + θ−1ℓ(d)ℓ(x−r(o))−1)−1.
Proof: The success probability is equal to
Ps=P!ohor(o)≥θℓ(d)−1I(r(o))
(a)
=E!oexp(−θℓ(d)−1I(r(o))),
where (a)follows since hor(o)is an exponential random
variable. Taking the expectation with respect to the fading
random variables in the interference we obtain
Ps=E!o"Y
x∈Φ
1−∆(x)#.
Using the inequality 1−Pai≤Q1−ai≤1−Pai+
Pi<j aiaj, and the definition of the n-th order product density
we obtain the lower bound 1−µand the upper bound 1−
µ+κ/2. The other upper bound can be obtained using the
inequality 1−∆(x)≤exp(−∆(x)) and the definition of the
probability generating functional.
The following lemma provides conditions by which the
upper bound can be simplified.
Lemma 2. For any stationary process Φ, if
λ−1RR2ρ(2)(x)∆2(x)dx
κ>2,
then G(exp(−∆(x)))
1−µ+κ/2>1.
Proof: We have,
G(exp(−∆(x)))
=E!o"Y
x∈Φ
1−(1 −exp(−∆(x)))#
(a)
>1−E!oX
x∈Φ
1−exp(−∆(x))
= 1 −λ−1ZR2
ρ(2)(x)(1 −exp(−∆(x)))dx,
where (a)follows from Q1−ai≥1−Pai. Using the
expansion of exp(−x), we have G(exp(−∆(x))) is greater
than
1−µ+
∞
X
m=2
(−1)mλ−1
m!ZR2
ρ(2)(x)∆m(x)dx.
So it is sufficient to prove that the summation is greater than
κ. By the inequality
(x−1) + exp(−x)≥x2
4, x ∈[0,1],
it is adequate to prove
(4λ)−1ZR2
ρ(2)(z)∆2(z)dz≥κ/2.
which follows from the assumption.
It is not necessary that Psapproaches one as the density
approaches zero. For example consider a clustered network
with the average number of clusters per unit area is 1/n and
the average number of nodes per cluster be unity. In this case
even though the intensity of the process approaches zero, the
success probability never approaches one. Since 1−µis a
lower bound on the success probability, it is necessary for
µ→0for Ps→1. It should be observed that µmay take
multiple values for a given density λand hence for each λ
we obtain a set of values of µ.Henceforth in this paper, we
consider only point processes for which
lim
λ→0inf{µ;density =λ} → 0.
The Psfor point processes which do not satisfy the above
condition is always less than one. From now on, by λ→0
we also fix a sequence of parameters of the process, so that
µ→0for this sequence.
Define Tcu(ǫ)to be equal to sup(1 −ǫ)λsubject to the
constraint min{1−µ+κ/2,G[exp(−∆(x))]} ≥ 1−ǫand
Tcl(ǫ)to be equal to sup(1 −ǫ)λsubject to the constraint
µ≤ǫ. We then have
Tcl(ǫ)≤Tc(ǫ)≤Tcu(ǫ).(6)
We now show that Tcl(ǫ)is asymptotically equal to Tcu(ǫ)for
small ǫunder very mild conditions. We first prove that for any
positive density of transmitters λ > 0, the success probability
is strictly less than one.
Lemma 3. The maximum density λso that
1− G(exp(−∆(x))) < ǫ
tends to zero as ǫ→0.
Proof: From the definition of the reduced probability
generating functional,
G(exp(−∆(x)) = E!oexp(−X
x∈Φ
∆(x)) (a)
≥exp(−µ)
where (a)follows from Jensens inequality. So it is sufficient
to prove that sup λwith the constraint exp(−µ)>1−ǫtends
to zero as ǫ→0. But since µis the average of Px∈Φ∆(x)
with respect to the Palm distribution and since ∆(x)>0, a
necessary condition for µ→0is for the density λto tend to
zero.
Theorem 1. If
A.1
lim
λ→0
λ−1RR2ρ(2)(x)∆2(x)dx
κ>2,
A.2 µ= Θ(λγ)for some γ≥1,
A.3 k=o(µ),
as λ→0, the transmission capacity is
Tc(ǫ) = Tcl(ǫ) + o(Tcl(ǫ)), ǫ →0.
Proof: From (6), it suffices to prove Tcu(ǫ) = Tcl(ǫ) +
o(Tcl(ǫ)). The upper bound Tcu(ǫ)is equal to the supremum
value of λso that
min{1−µ+κ/2,G(exp(−∆(x)))}>1−ǫ
For the above condition to be satisfied as ǫgets smaller, it
follows from Lemma 3 that λ→0. Since λshould be small,
it follows from Lemma 2 that
min{1−µ+κ/2,G(exp(−∆(x)))}= 1 −µ+κ/2.
λ1λ2
C1
C2
µ−κ
µ
ǫ
ν
A B C
D
E
Fig. 1. Proof for Theorem 1. Observe that the triangle ABE is congruent
to triangle ACD.
So the upper bound translates to finding the maximum λ
such that µ−κ/2< ǫ. Also by our assumption A.2, µis
locally convex in the neighborhood of λ= 0. See Figure 1
where the upper and the lower bound are illustrated. From the
figure, it suffices to prove limλ→0(λ2−ν)/(λ1+ν) = 0. Also
assumption A.3, implies limλ→0C1/(C1+C2) = 0. Hence
by the congruency of the triangles ABE and ACD it follows
that λ2/λ1tends to zero, hence proving the theorem.
Assumption A.2 is a reasonable assumption. Indeed it is
proved in [13] that
lim
λ→0
µ
λ1−δ=∞,
for any δ > 0under mild assumptions and hence γ≥1in
assumption A.2 is always valid.
IV. EXAMPLES
In this section, we will analyze the asymptotic transmission
capacity using Theorem 1 and compare it with the actual TC.
We consider three different spatial distribution of transmitters:
Poisson point process (PPP), Poisson cluster process (PCP),
Modified Matern hard-core process. These three process ex-
hibit different kind of regularity in terms of the node place-
ment: PPP corresponds to an entirely random arrangement of
nodes, PCP exhibis clustering while the Modified Matern hard-
core process has a minimum distance between the nodes.
A. Poisson point process (PPP)
The success probability of a PPP [1] is equal to
Ps= exp −λZR2
∆(x)dx.
Hence it is easy to observe that
Tc(ǫ) = (1 −ǫ) ln((1 −ǫ)−1)
RR2∆(x)dx.
When ǫis small
Tc(ǫ) = ǫ
RR2∆(x)dx+o(ǫ).(7)
For a PPP ρ(2)(z) = λ2and so
µ=λZR2
∆(x)dx.
Since α > 2,R∆(x)dx < ∞and hence µ= Θ(λ),i.e.,
ν= 1, thus verifying assumption A.2. For a PPP, ρ(3)(x, y) =
λ3and it is easy to verify that κ/µ →0as λ→0and
λ−1κ−1Rρ(2)(x)∆2(x)dx→ ∞, thus verifying assumptions
A.1 and A.3. So by Theorem 1, the asymptotic TC is equal
to sup λsuch that µ < ǫ. In this case it is easy to verify that,
Tcl(ǫ) = ǫ
RR2∆(x)dx,
which agrees with the actual transmission capacity of the PPP
for small ǫin (7).
B. Poisson cluster process (PCP)
A PCP constitutes a Poisson parent process Φpof density
λpand daughter point processes of density ¯c, resulting in
a stationary point process of density ¯cλp. A point of the
daughter point process of a parent point at x∈Φpis
spatially distributed with density f(y−x), y ∈R2. The success
probability in a stationary PCP is provided in [11], and is equal
to
Ps= exp −λpZR2h1−exp(−cβ(y))idy
ZR2
exp(−cβ(y))f(y)dy, (8)
where β(y) = RR2f(y−x)∆(x)dx. It was also proved in [11]
that the transmission capacity of a PCP is equal to
ln((1 −ǫ)−1)
RR2∆(x)dx=ǫ
RR2∆(x)dx+o(ǫ),
when
ǫ < 1−exp −ZR2
f(y)β(y)
sup β(y)dy.
For a PCP, the second order product density is equal to
ρ(2)(z) = λ21 + (f∗f)(z)
λp.
and hence
µ=λZR2
∆(z)dz+ ¯cZR2
(f∗f)(z)∆(z)dz.
Observe that µmay take multiple values for the same λby
choosing different λpand ¯c. We can observe that µ= Θ(λ),
when ¯cis chosen to be small and λpchosen do that ¯cλp=λ.
The expression for ρ(3)(x, y)is very complicated and we will
verify A.3 by simulation. From Figure 2 we observe that κ=
o(µ). It is also easy to verify that
lim
λ→0λ−1µ−1ZR2
ρ(2)(x)∆2(x)dx < ∞,
and hence combined with the fact that κ=o(µ)assumption
A.1 is satisfied. Hence by Theorem 1, the asymptotic TC is
equal to sup λwhen µ < ǫ. Since (f∗f)(z)>0and ∆(z)>
0, it is easy to observe that the supremum value is equal to
Tcl(ǫ) = ǫ
RR2∆(x)dx,
and is obtained by decreasing ¯cto zero and increasing λp.
Even in this case the asymptotic TC matches with the actual
transmission capacity.
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
λ
κ/µ
l(x)=|x|−4, d=1,θ=1
Matern (CSMA)
Poisson cluster process
Fig. 2. κ/µ as a function of the density λfor PCP and Matern hard-
core process. We observe that κ/µ tends to zero as λ→0, thus verifying
assumption A.3.
C. CSMA modelling with Matern Hard core process
We now provide an analysis of the transmission capacity in
a CSMA wireless network. Although the spatial distribution of
the transmitters that concurrently transmit in a CSMA network
is difficult to be determine, the transmitting set can be closely
approximated by a modified Matern hard-core processes [2].
In this subsection we use the model proposed by Baccelli.
et.al. in [2] to model the CSMA process and derive the TC
for a low outage probability.
CSMA Model: We start with a Poisson point process Ψof
unit density. To each node x∈Ψ, we associate a mark
mx, a uniform random variable in [0,1]. The contention
neighborhood of a node xis the set of nodes which result
in an interference power of at least Pat x,i.e.,
¯
N(x) = {y∈Ψ : hyxℓ(y−x)>P}
A node x∈Ψbelongs to the final CSMA transmitting set if
mx< my,∀y∈ N(x).
The average number of nodes in the contention neighborhood
of x∈Ψ, does not depend on the location xby the stationarity
of Ψand is equal to [2]
N=E[¯
N(x)] = 2πZ∞
0
e−Pℓ(r)−1rdr.
The density of the CSMA Matern process Φis equal to
λ=1−exp(−N)
N.
Using the results in [14], [2], the second-order product density
can be shown to be equal to
ρ(2)(r) = 2
b(r)− N λ−1−e−b(r)
b(r)1−e−Pℓ(r)−1,
where
b(r) = 2N−Z∞
0Z2π
0
e−P“ℓ(t)−1+ℓ(√t2+r2−2rt cos(Θ))−1”tdΘdt.
For clarity of exposition, we concentrate on ℓ(x) = kxk−α,
although similar results hold for the non-singular model too.
In the singular case, N=P−2/αC(α), where C(α) =
2πΓ(2/α)α−1.
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0
5
10
15
20
25
30
35
40
λ
Quantity in assumption A.1
l(x)=|x|−4,θ=1,d=1
Fig. 3. The quantity in assumption A.1 versus the density λfor ℓ(x) =
kxk−4,d= 1 and θ= 1. We observe that assumption A.1 holds true in this
case.
0.015 0.02 0.025 0.03 0.035 0.04
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Density λ
1−Ps
l(x)=|x|−4,θ=1, d=1
Simulation
Theory: µ
Theory Asymptote: 18.75λ2
Fig. 4. Outage probability versus density λfor ℓ(x) = kxk−4,d= 1
and θ= 1. We observe that µis a good approximation of 1−Psfor small
lambda. We also observe that 1−Ps= Θ(λ2).
Lemma 4. For small density λ, in the modified Matern hard-
core process
µ∼λα/2γZ∞
0
1−exp(−rα)
C(α)(2C(α)−ξ(r))r−α+1dr, (9)
where γ= 4πC(α)α/2+1 θdα, and
ξ(r) = Z∞
0Z2π
0
e−(tα+(t2+r2−2rt cos(Θ))α/2)tdΘdt
Proof: The result can be obtained by using the sub-
stitution r→P−1/αrin the integral for µand observing
that for small P,λ∼P2/αC(α)−1, and b(rPs−1/α ) =
P−2/α (2C(α)−ξ(r)), and ∆(xP−1/α) = Pθℓ(d)−1kxk−α.
Also observe that the integral in (9) is finite because of the
presence of the term 1−exp(−rα).
Hence µ= Θ(λα/2)and condition A.2 is valid with ν=
α/2>1. As in the previous case we also show the validity of
A.1 and A.3 by simulation. See Figure 2 and Figure 3. From
Figure 4, we observe that µclosely approximates the simulated
1−Psfor small lambda. Hence the asymptotic TC would
also match closely with the actual TC. Since µ= Θ(λα/2)it
follows that Tcl(ǫ) = Θ(ǫ2/α). Hence for α= 4, the TC for
the CSMA is Θ(√ǫ)which is much greater than that of the
the PPP with ALOHA which is equal to Θ(ǫ). In Figure 4, the
asymptote of µfrom Lemma 4, which in the case of α= 2
corresponds to 18.752λ2is also plotted. Using the asymptote
we can also infer that
lim
ǫ→0Tc(ǫ)ǫ−2/α =γZ∞
0
1−exp(−rα)
C(α)(2C(α)−ξ(r))r−α+1dr−2/α
.
V. CONCLUSIONS
In this paper we provide a characterization of the trans-
mission capacity that is order optimal in terms of the outage
constraint. This characterization depends only on the second-
order product density of the spatial distribution of the trans-
mitters, a quantity that can be evaluated analytically for most
point processes. We also showed that the TC evaluated using
the second order product density matches perfectly with the
actual TC when the nodes are distributed as a Poisson point
process and a Poisson cluster process. Using this framework,
we obtained the TC of a CSMA process and showed that it
behaves like Θ(ǫ2/α)where ǫis the outage constraint, unlike
the case of ALOHA where it is Θ(ǫ).
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