Generalized SIR Analysis for Stochastic Heterogeneous Wireless Networks: Theory and Applications
Abstract
This paper provides an analytically tractable framework of investigating the statistical properties of the signaltointerference power ratio (SIR) with a general distribution in a heterogeneous wireless ad hoc network in which there are different types of transmitters (TXs) communicating with their unique intended receiver (RX). The TXs of each type form an independent homogeneous Poisson point process. In the first part of this paper, we introduce a novel approach to deriving the Laplace transform of the reciprocal of the SIR and use it to characterize the distribution of the SIR. Our main findings show that the closedform expression of the distribution of the SIR can be obtained whenever the receive signal power has an Erlang distribution, and an almost closedform expression can be found if the powerlaw pathloss model has a pathloss exponent of four. In the second part of this paper, we aim to apply the derived distribution of the SIR in finding the two important performance metrics: the success probability and ergodic link capacity. For each type of the RXs, the success probability with (without) interference cancellation and that with (without) the proposed stochastic power control are found in a compact form. With the aid of the derived Shannon transform identity, the ergodic link capacities of type RXs are derived with low complexity, and they can be applied to many transmitting scenarios, such as multiantenna communication and stochastic power control. Finally, we analyze the spatial throughput capacity of the heterogeneous network defined based on the derived success probabilities and ergodic link capacities and show the existence of its maximum.
I Introduction
Consider a largescale heterogeneous wireless ad hoc network in which there are different types of transmitters (TXs) and the TXs of each specific type form an independent Poisson point process (PPP) with a certain intensity (density). Each TX has a unique intended receiver (RX) away from it by some (random) distance. Namely, such a heterogeneous wireless network can be viewed as consisting of type TXRX pairs independently scattering on an infinitely large plane . Usually, interference in such a wireless network significantly dominates the transmission performance that is effectively evaluated by the metric of the signaltointerference power ratio (SIR) at RXs. By assuming all TXs in the network transmit narrow band signals and share the same spectrum bandwidth, the SIR of a typical type RX located at the origin, called type SIR, can be written as
(1) 
where denotes the interference power of the typical type RX from all interferers in the network, is the received (desired) signal power of the typical type RX, is the transmit power, is the random channel (power) gain, is the (random) distance between the typical RX and its associated TX, is the pathloss exponent, and is the minimum required SIR for successfully decoding. Note that and are both random variables whose distributions depend upon (random) transmit power, random channel gain as well as pathloss models between TXRX pairs. The SIR pertaining to several important transmitting performance metrics, such as success/outage probability, ergodic link capacity, network capacity, etc. Understanding the statistical properties of the SIR not only helps us realize how the received random signal powers affect the distribution of the SIR, but also provides us a crucial clue indicating the interplay of the transmitting policies and behaviors among many different TXs.
Ia Prior Work and Motivation
Traditionally, the statistical properties of the SIR in a Poissondistributed wireless network are only analytically accessible in very few special cases. Some prior works have already made a good progress on the analysis of the distribution of the SIR by presuming a specific channel gain model (typically see [1, 2, 3, 4, 5]). In reference [1], for example, the closedform success probability, which is essentially the contemporary cumulative density function (CCDF) of the SIR in a singletype Poisson ad hoc network, was firstly found by assuming independent Rayleigh fading channels since the Rayleigh fading channel model gives rise to the solvable Laplace transform of the interference by means of the probability generating function (PGF) of a homogeneous PPP [6, 7, 3]. The outage probability, which is essentially the CDF of the SIR, is studied in [2] without considering random channel gain models and only its bounds are obtained. Although the closedform Laplace transform of the interference with a general random channel gain model is found in [3], it can only be applied to find the CDF/CCDF of the SIR with an exponentialdistributed received signal power. In [5], the bounds on the temporally averaged outage probability are studied specifically for Rayleigh fading channels due to the tractability in mathematical analysis. These prior works aim to study how channel gain randomness affects the success/outage probability so that they simply use constant transmit power and distance while doing analysis.
Since the SIR significantly depends upon the randomness of the received signal and interference powers, some previous works focus on exploiting the SIR randomness by distributed channelaware scheduling and power control in order to improve the success probability is [8, 9, 10, 11, 12, 13]. However, the success/outage probability in these works is only characterized by some lower and upper bounds since it lacks of a tractable Laplace transform of the interference. These bounds may not be always tight in different ranges of the TX intensity even though they are claimed to be asymptotically tight. In addition to the success/outage probability, another important performance metric regarding to the SIR is the ergodic link capacity (rate) of a TX and its analytical results are barely completely and deeply investigated. In the literature, the ergodic link capacity is either characterized by its bounds or obtained by integrating the CCDF of the link capacity that can be written in terms of the success probability[14, 15]. In other words, its accurate and simple expression is never discovered so that many prior works on network capacity (throughput) just simply use a minimum required constant link rate to define their capacity/throughput metrics [1, 2, 3, 4, 5, 8, 16, 9, 10, 11, 12, 13]. Accordingly, the network capacity evaluated in the prior works may be far away from the real fundamental limit of the network capacity.
In the literature, the distribution of the SIR is tractably derived only in the context where the received signal power contains an exponential random variable that creates a natural condition of applying the PGF of homogeneous PPPs to resolve the Laplace transform of the interference. In other words, the distribution of the SIR is not significantly tractable if the receive signal power no longer possesses an exponential random variable. As a result, any transmitting policies adopted by TXs that make the received signal power loose/change its original exponential randomness cannot result in a tractable analysis in the distribution of the SIR. A straightforward example is to let TXs control their transmit power to compensate or cancel the Rayleigh fading gain in their channel and the success probability under this power control cannot be found in closedform despite the fact that the original success probability without power control has a closedform [11]. To tractably study the statistical properties of the SIR with a general distribution, we need to find another way to deal with the interference generated in a Poisson field without utilizing the exponential randomness of the received signal power. Also, the heterogeneity of TXs is hardly modeled in prior works on Poisson ad hoc networks. Such a heterogeneity could exist in the future network of machinetomachine (M2M) communication and internet of things (IoTs) and how it impacts the randomness of the SIR is still fairly unclear [17, 18]. These aforementioned issues foster our motive to develop a generalized framework of analyzing the distribution of the SIR in a heterogeneous wireless network.
IB Main Results and Contributions
In this work, our first main contribution is to derive the integral identity of the Shannon transform as well as devise the novel theoretical framework of tractably analyzing the CDF of the type SIR defined in (1). The main idea behind this framework is to first find the explicitly result of the Laplace transform of the reciprocal of with a general distribution since we can tractably deal with it by the PGL of homogeneous PPPs. Then substituting it into the exploited fundamental identity between the CDF of and the Laplace transform of the reciprocal of . For the analytical framework regarding to the statistical properties of with a general distribution, the following summarizes our main findings.

The general expression of the CDF of without and with interference cancellation is characterized
^{1} , which can be practically evaluated by the numerical inverse Laplace transform. Its nearly closedform result for pathloss exponent is found, whereas its lowcomplexity and tight bounds for an arbitrary are obtained as well. 
We show that the closedform CDF of the exists if and only if received signal power has an Erlang distribution. Namely, any randomness in (from the random transmit power, channel gain and distance) that lets have an Erlang distribution can make the CDF of have a closedform result.

The fractional moment of the without interference cancellation is derived in closedform and that with interference cancellation can be obtained in a neat integral expression.
Due to the generality of the CDF and fractional moment of , they can be used to find the explicitly results of some important performance metrics in many transmitting and receiving scenarios. In this work, our second main contribution is to apply our developed analytical framework to tractably study the success probability and the ergodic link capacity that are the two paramount metrics of evaluating transmission performance. For the success probability, the following are our main findings:

The type success probability without and with interference cancellation can be acquired directly from the CCDF of . Thus, its nearly closedform expression also exists for and its tight bounds for any are also found.

Since the success probability is found with a generaldistributed SIR, it can be used to explicitly evaluate the success probabilities with specified random models involved in the SIR. This fact helps us theoretically show that random channel gains do not necessarily jeopardize/benefit the success probability (relative to constant channel gains) as the TX intensities change and it very likely improves the success probability in a dense network.

Due to the generality of the derived success probability, the success probability with the proposed stochastic power control is tractably characterized by its bounds or found in a nearly closedform expression depending on if is equal to 4 or not. It also reveals how to design the power control scheme to improve the success probability by exploiting the randomness of the received signal power.
The explicit expression of the ergodic link capacity in a Poisson network is hardly found in the literature when the SIR has a general distribution. In this work, we derive a lowcomplexity and general expression of the ergodic link capacity without and with interference cancellation by jointly using the derived integral identity of the Shannon transform and a novel integrating technique. According to the derived general results of the ergodic link capacity and the success probability, we define the spatial throughput capacity of the heterogeneous network that characterizes the area spectrum efficiency in an ergodic sense. We summarize some key observations of the ergodic link capacity and spatial throughput capacity as follows.

Due to the generality of the derived ergodic link capacity, we can easily find the ergodic link capacity without and with channel fading so that we are able to conclude that channel fading does not always reduce or increase the ergodic link capacity (relative to no channel fading) as TX intensities change.

The ergodic link capacity with the proposed stochastic power control is found and its fundamental upper and lower bounds are also characterized. These analytical results help us understand how to design the stochastic power control so that it can benefit the ergodic link capacity.

The spatial throughput capacity proposed in this work can characterize the network capacity with TX heterogeneity, and it is closer to the fundamental limit of the network capacity than other network metrics with a constant link rate in prior works.
The salient trait of the derived CDF of and its corresponding performance metrics, compared with related prior works, is to indicate how they are impacted by the interferences of other types. This provides a very useful clue in optimally deploying the network with some performance constraints. In addition, our main analytical results and findings are correctly validated by numerical simulations so that they can offer a quick and correct approach to evaluating the network performance with a new protocol and/or deployment design.
Ii System Model and Preliminaries
Iia Network Modeling and Performance Metrics
Consider a largescale and interferencelimited heterogeneous wireless ad hoc network on the plane in which there are different types of TXs and the TXs of each type form an independent homogeneous Poisson point process (PPP). Specifically, assume each TX has a unique intended RX and the set consisting of the type TXs with intensity is expressed as
(2) 
where , denotes the th nearest TX of type to the origin and its location, represents the random channel (power) gain from to the typical RX located at the origin induced by fading and/or shadowing effects, is the (random) transmit power of , is the (random) distance between and its receiver. Throughout this paper, all random variables (RVs) with subscript “” are independent for all and and they are i.i.d. for the same . In addition, all channel gains have unit mean for all and . The main variables and symbols used throughout this paper are listed in Table I.
Symbol  Meaning 

()  Homogeneous PPP of the type TXs () 
Intensity of the type TXs  
Pathloss exponent  
Channel gain of TX with unit mean  
Transmit power of TX  
Transmit distance of TX  
Type received signal power (with power control)  
Interference at a type RX (with power control)  
Type signaltointerference power ratio  
Laplace (Inverse Laplace) transform operator  
, Momentgenerating function of RV for  
is the tight lower (upper) bound on  
Type success probability (with power control)  
SIR decoding threshold  
Type ergodic link capacity (with power control)  
Throughput capacity with threshold  
pdf of random variable (RV)  
CDF (CCDF) of RV  
(RV normalized by its mean)  
()  
Power control exponent  
Number of the canceled interferers  
Erlang RV with parameters and  
Variable with interference cancellation  
, Gamma function  
, Upper incomplete Gamma function  
,  
, error function  
, complementary error function 
Assume all TXs adopt the slotted Aloha protocol to access the channel shared in the network so that the type typical RX receives the interference given by
(3) 
where , denotes the Euclidean distance between TXs and , and is the pathloss exponent. Accordingly, the type SIR, as already defined in (1), can be explicitly rewritten as follows
(4) 
Assuming the minimum SIR threshold for successful decoding the received signals at any RXs is , the (transmitting) success probability of a type TX is defined as
(5) 
whereas is called the outage probability of a type TX. Using the definitions of SIR and success probability, we define the type ergodic link capacity as follows.
Definition 1 (Ergodic Link Capacity).
If the capacityapproaching code is used by all TXs, the ergodic link capacity (per unit bandwidth) of a type TXRX pair, called type ergodic link capacity in the heterogeneous wireless ad hoc network, is defined as
(6) 
In prior works, the explicit expression of the ergodic link capacity in Poisson wireless networks was not well studied and derived in a simple and general form. As we will show later, the type ergodic link capacity can be derived in a very neat form by our new proposed mathematical derivation approach. Most importantly, our derived ergodic link capacity is able to explicitly indicate how it is affected by the random channel gain, transmit power and distance models, which provides very useful insight into devising the power control schemes in order to benefit the transmission performance by combating and/or exploiting the randomness in the SIR.
IiB Preliminaries
In this subsection, some preliminary results regarding the multiple independent homogeneous PPPs as well as the integral identity of the Shannon transform are introduced and discussed. These results are the underlying basis of paving a tractable way to analyze the success probability and ergodic link capacity in a very general manner. We first introduce the following theorem.
Theorem 1.
Let be a Borelmeasurable nonincreasing function and it is positively scalable, i.e., for any we have where denotes the inverse function of . Suppose all ’s are independent nonnegative RVs for all and and they are i.i.d. for the same subscript . If for all , the following identity
(7) 
holds for the whole transmitter set where and .
Proof:
First we know the following identity
where following from the assumption that is positively scalable, is obtained by using the probability generation functional (PGF) of independent homogeneous PPPs [6][21], (c) is due to the fact that is nonincreasing and invertible. Thus, carrying out the integral inside in yields the result in (7). ∎
Theorem 1 implicitly reveals an important fact that the biased and transformed supreme distance between the origin and the TXs in has the same distribution as the nearest distance between the origin and a single homogeneous PPP of intensity . To elaborate on this point, letting and gives and we have
which indeed shows that is the biased shortest distance from the origin to and it has the same distribution as the nearest distance from the origin to . As we will see later, Theorem 1 facilitates the derivation processes in the analysis of interference cancellation.
Another important result that needs to be introduced here is the identity of the Shannon transform. The Shannon transform of a nonnegative RV for a nonnegative is defined as [22]
(8) 
which has an integral identity as shown in the following theorem.
Theorem 2 (The integral identity of the Shannon transform).
Consider a nonnegative RV and the Laplace transform of its reciprocal always exists, i.e., . If defined in (8) exists for any , the following identity
(9) 
always holds. Furthermore, we can have
(10) 
if is a nonnegative RV independent of and its Laplace transform exists.
Proof:
The Shannon transformation of nonnegative RV defined in (8) can be rewritten as
If always exists, for any we have
and then substituting this result into yields
which is exactly the result in (9). The result in (10) readily follows from (9) and the definition of the Laplace transform of a nonnegative RV. ∎
The ergodic link capacity in (6) can be expressed in terms of the integral identity of the Shannon transform as
(11) 
since and are independent. This demonstrates that the integral identity of the Shannon transform is very useful in deriving the explicit expression of the ergodic link capacity if the Laplace transforms of the received signal power and interference are analytically tractable. More details about how to use the integral identity of the Shannon transform to find the ergodic link capacity of each type will be demonstrated in the following section. Next, we will first study the general distribution of the type SIR that is regarding the statistical properties under general random channel gain, transmit power and distance models. The fundamental theory pertaining to the general distribution of the type SIR will be established without specifying these random models involved in the SIR so that it is of the modelindependent nature and valid for the distribution of the type SIR with any specific random signal models. Most importantly, the theory not only straightforwardly indicates how different random models involved in the SIR influence the performance metrics regarding the SIR, but also gives us insight into exploiting the model randomness in order to enhance the SIR.
Iii The Statistics of the Type SIR with a General Distribution
Prior works on the distribution of the SIR in Poisson wireless networks are channelmodeldependent and the majority of the prior works reached the closedform distribution of the SIR with assuming communication channel gains are exponentially distributed (i.e., Rayleigh fading), whereas the distribution of the SIR for the channel gains without an exponential distribution is generally intractable. As a result, the prior results cannot thoroughly reveal how the distribution of the SIR is impacted as the random models involved in the SIR are changed. In this section, our goal is to generally characterize the distribution expressions of the type SIR with a general (unknown) distribution. The fundamental approach to fulling our goal is to first study the Laplace transform of the interference of the type RXs since it plays a pivotal role in deriving the general distribution of the type SIR. Surprisingly, we will see that some closedform (or near closedform) expressions of the distribution of the type SIR indeed exist and they intuitively show how the statistical properties of the SIR are influenced by the randomness existing the SIR.
Iiia The Laplace Transforms of Interferences and
Let denote the set of the first strongest interferers in set for the type typical RX. Hence, the type interference can be rewritten as
(12) 
where denotes the residual type interference by removing the first strongest interferers in . For arbitrary random powerlaw channel and transmit power models, the Laplace transforms of and are shown in the following theorem.
Theorem 3.
According to the type interference given in (12), its Laplace transform can be shown in closedform as
(13) 
where , and for is the gamma function. The Laplace transform of the residual interference defined in (12) can be found as
(14) 
where is an Erlang RV with parameters and , , and with and is defined as
(15) 
in which is the upper incomplete gamma function. Also, there exists an for each sample of such that is upperbounded as
(16) 
where .
Proof:
See Appendix A. ∎
There are a couple of crucial implications about in (14) that can be explained in more detail as follows. First, characterizes the statistical property of the interference which is partially cancelable at the RX side, and in for the case of reduces to 0 so that is exactly equal to in (13). In other words, the second term of the right hand side in (14) compensates the Laplace transform of the canceled interference for . Second, although can be theoretically written as a neat result as shown in (14), is actually somewhat complicate in practical applications and calculations. As a result, a lowcomplexity upper bound on is shown in (16), and it is very tight if is small with high probability and it becomes tighter as gets larger since removing more interferers makes increase so that the upper bound gets closer to the Laplace transform of . In the following subsection, we will show how to apply Theorem 3 to characterize the distribution of the type SIR without and with interference cancellation, which is the foundation of developing the generalized analytical approach to the success probability and ergodic link capacity.
IiiB Analysis of the Distributions of and
The definition of the SIR of the type typical RX with interference is already given in (4). Similarly, if the type typical RX is able to cancel the aggregated interference contributed by the first strongest interferers, its SIR in this case is defined as
(17) 
where interference is already defined in (12). In this subsection, our first focus is on the Laplace transforms of the reciprocals of and that play a pivotal role in deriving some important performance metrics of Poisson wireless networks, such as success probability, ergodic link capacity, etc. The explicit distribution expressions regarding to and are summarized in the following theorem.
Theorem 4.
Let and denote the probability density function (pdf) and the cumulative density function (CDF) of RV , respectively. The Laplace transform of the reciprocal of the type SIR defined in (4) can be explicitly expressed as
(18) 
where is given in (13) and is called the type received signal power with unit mean. The CDF of can be shown as
(19) 
where . If each type RX can cancel its first strongest interferers, the Laplace transform of the reciprocal of in (17) can be expressed as
(20) 
Also, the CDF of can be shown as
(21) 
where .
Proof:
See Appendix B. ∎
Theorem 4 demonstrates the general expressions of the Laplace transforms of and as well as the CDFs of and without assuming any specific random channel gain, transmit power and distance models. Although in general the expressions in Theorem 4 cannot be completely found in closedform, they can be calculated by using the numerical inverse Laplace transform. Nonetheless, as shown in the following corollary we still can characterize the lowcomplexity bounds on and and the near closedform of and for without specifying the distribution of .
Corollary 1.
For a general , the CDF of in (19) can be bounded as shown in the following:
(22) 
In particular, if , then can be simply found as
(23) 
in which is the error function and , and thus we have the following closedform bounds on in (23)
(24) 
where . Whereas there exists a lower bound on the CDF of given by
(25) 
Proof:
The CDF of in (19) can be rewritten as
(26) 
Using the inequality for , the upper bound on the result in (26) is
and
where the second inequality holds due to the convexity of for and the final equality follows from solving the inverse Laplace transform for . Therefore, the upper and lower bounds in (22) and (24) are acquired. For , the inverse Laplace transform in (26) can be found in closedform so that we have
(27) 
where the upper bound is obtained by applying Jensen’s inequality to that is concave for . For the CDF of , it can be written as
and its lower bound in (25) is readily obtained by replacing in (16) with . ∎
We can elaborate on a couple of implications of Corollary 1 as follows.

When (e.g., the mean of the interferencetosignal power ratio is fairly small), is accurately approximated by the inverse Laplace transform of the Taylor’s expansion of the term in (26) as
(28) where . Namely, we have for a given as approaches zero
^{4} . In other words, in (28) is very accurate in this case and the bounds in (22) are very tight since they coverage to each other eventually. 
For , the neat expression of and its closedform bounds exist, and they can precisely reveal how much the disparate distributions of affect . Third, in general, the lower bound in (25) is very tight if with high probability so that using for and the Laplace transform table in [23] makes tightly lowerbounded for as
(29) where () denotes that is the tight lower (upper) bound on . This tight lower bound implies as well as the effect of interference cancellation is offset by strong received signal power . Interference cancellation benefits more the RXs with a weaker received signal power.
For the case of received signal power having an Erlang distribution, the closedform results of and in Theorem 4 indeed exist, as shown in following corollary.
Corollary 2.
If the type received signal power with unit mean is an Erlang RV (i.e., where ), then we have
(30) 
and