Generalized SIR Analysis for Stochastic Heterogeneous Wireless Networks: Theory and Applications

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 signal-to-interference 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 closed-form expression of the distribution of the SIR can be obtained whenever the receive signal power has an Erlang distribution, and an almost closed-form expression can be found if the power-law 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 multi-antenna 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.

1Introduction

Consider a large-scale 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 TX-RX 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 signal-to-interference 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

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 TX-RX 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.

1.1Prior Work and Motivation

Traditionally, the statistical properties of the SIR in a Poisson-distributed 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]). In reference [1], for example, the closed-form success probability, which is essentially the contemporary cumulative density function (CCDF) of the SIR in a single-type 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]. 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 closed-form 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 exponential-distributed 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 channel-aware scheduling and power control in order to improve the success probability is [8]. 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]. 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]. 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 closed-form despite the fact that the original success probability without power control has a closed-form [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 machine-to-machine (M2M) communication and internet of things (IoTs) and how it impacts the randomness of the SIR is still fairly unclear [17]. These aforementioned issues foster our motive to develop a generalized framework of analyzing the distribution of the SIR in a heterogeneous wireless network.

1.2Main 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 . 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 characterized1, which can be practically evaluated by the numerical inverse Laplace transform. Its nearly closed-form result for pathloss exponent is found, whereas its low-complexity and tight bounds for an arbitrary are obtained as well.

  • We show that the closed-form 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 closed-form result.

  • The fractional moment of the without interference cancellation is derived in closed-form 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 closed-form expression also exists for and its tight bounds for any are also found.

  • Since the success probability is found with a general-distributed 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 closed-form 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 low-complexity 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.

2System Model and Preliminaries

2.1Network Modeling and Performance Metrics

Consider a large-scale and interference-limited 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

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 1.

Table 1: Notation of Main Variables, Symbols and Functions
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- signal-to-interference power ratio
Laplace (Inverse Laplace) transform operator
, Moment-generating 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 by2

where , denotes the Euclidean distance between TXs and , and is the pathloss exponent. Accordingly, the type- SIR, as already defined in , can be explicitly rewritten as follows

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

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.

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.

2.2Preliminaries

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 ? 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 ? 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]

which has an integral identity as shown in the following theorem.

The ergodic link capacity in can be expressed in terms of the integral identity of the Shannon transform as

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 model-independent 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.

3The Statistics of the Type- SIR with a General Distribution

Prior works on the distribution of the SIR in Poisson wireless networks are channel-model-dependent and the majority of the prior works reached the closed-form 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 closed-form (or near closed-form) 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.

3.1The 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

where denotes the residual type- interference by removing the first strongest interferers in . For arbitrary random power-law channel and transmit power models, the Laplace transforms of and are shown in the following theorem.

There are a couple of crucial implications about in 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 . In other words, the second term of the right hand side in compensates the Laplace transform of the canceled interference for . Second, although can be theoretically written as a neat result as shown in , is actually somewhat complicate in practical applications and calculations. As a result, a low-complexity upper bound on is shown in , 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 ? 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.

3.2Analysis of the Distributions of and

The definition of the SIR of the type- typical RX with interference is already given in . 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

where interference is already defined in . 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 ? 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 ? cannot be completely found in closed-form, they can be calculated by using the numerical inverse Laplace transform. Nonetheless, as shown in the following corollary we still can characterize the low-complexity bounds on and and the near closed-form of and for without specifying the distribution of .

We can elaborate on a couple of implications of Corollary ? as follows.

  • When (e.g., the mean of the interference-to-signal power ratio is fairly small), is accurately approximated by the inverse Laplace transform of the Taylor’s expansion of the term in as

    where . Namely, we have for a given as approaches zero3. In other words, in is very accurate in this case and the bounds in are very tight since they coverage to each other eventually.

  • For , the neat expression of and its closed-form bounds exist, and they can precisely reveal how much the disparate distributions of affect . Third, in general, the lower bound in is very tight if with high probability so that using for and the Laplace transform table in [23] makes tightly lower-bounded for as

    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 closed-form results of and in Theorem ? indeed exist, as shown in following corollary.

For any particular value of , the explicit closed-form expressions of and can be easily found by carrying out the th-order derivatives in and , respectively. For instance, in the special case of , i.e., is an exponential RV with unit mean and variance4, in and in respectively reduce to

and

where denotes the CCDF of . Note that in obviously shows that has a Weibull distribution with parameters and , which was shown in [3] for the network with a single PPP. Accordingly, we can say that has a “modified” Weibull distribution with parameters and since can be expressed in terms of .

Another case that and in Theorem ? can be found in a simpler form is when the received signal power does not possess any randomness, as shown in the following corollary.

Although the inverse Laplace transforms in and in general still cannot be explicitly calculated, they can be evaluated by the numerical inverse Laplace transform for any particular value of . For , the closed-form approximation of also can be inferred from as

Furthermore, for the special case of , has a closed-form expression directly obtained from as

where and is a constant, and the closed-form tight lower bound on for also can be found as

which is directly inferred from . The result in is fascinating (even though it is only valid for ) since it shows that the outage probability with constant transmit power, distance and no channel fading has a closed-form result and it is only characterized by bounds in [2].

Although the CDFs of and in Theorem ? are characterized for an arbitrary distribution of received signal power , directly using them to find the moments of the SIRs cannot acquire a neat or tractable expression. In the following theorem, we show that the fractional moment of the and can be explicitly found in a closed/neat form for a general distribution of .

The results in Theorem ? can be applied to estimate the success probability and the ergodic link capacity in some special contexts, as we will expound these applications in the following sections. Moreover, they also indicate that the fractional moment of the type- SIR increases as long as decreases by maintaining some randomness in the SIR. To clarify this point easily, for example, assuming transmit power is a constant and transmit distance is unity, the term in reduces to which is less than or equal to one based on Jensen’s inequality. This manifests that channel randomness benefits the fractional moment of the SIR. Exploiting the signal randomness in the SIR can increase its fractional moment.

4Application of the SIR Statistics(I): Success Probability

The CDF of the SIR found in the previous section has a paramount application in evaluating the success probability defined in . For the successful decoding threshold , the success probability of a type- RX (called type- success probability) without or with interference cancellation can be simply written as

Now we first specify how to apply the CDF of the SIR to find the success probabilities in some practical application contexts of the random received signal power models. Afterwards, some numerical results are provided to validate our analytical findings.

4.1Success Probability with General Random Models of the Received Signal Power

In general, the success probability cannot be derived in an explicit closed form based on if does not have an Erlang distribution. Nonetheless, the bounds on and can be characterized as shown in the following corollary.

In addition, using the error function’s Maclaurin series can be further written as

where the lower bound in is obtained by applying Jensen’s inequality to the erfc function with a positive argument that is convex. Although the result in is derived by considering the special case of , it is still very important since it is applicable to any random channel gain, transmit power and distance models and able to directly provide some insight into how the randomness of the received signal power affects the success probability. Note that in reduces to if is constant.

In order to have a more tractable result of in practically applicable contexts with a general pathloss exponent, we consider normalized received signal power as a Gamma random variable with mean 1 and variance , i.e., , for . Such a received signal power model is somewhat general because it characterizes the different randomness levels5 of . According to the results in Corollary ? and , without interference cancellation based on for a positive integer can be readily obtained by

whose closed-form expression can be explicitly calculated once is designated. For the special case of and Rayleigh fading interference channels, in reduces to

which reduces to the seminal result firstly shown in [1] for , constant transmit power and distance. The success probability in reveals a very important fact that the closed-form success probability exists as long as the received signal power has an Erlang distribution. This overthrows the traditional impression that the success probability only has a closed-form result for constant transmit power, distance and Rayleigh fading channels.

When the type- RXs can cancel their first strongest interferers, the type- success probability in , i.e., , is readily obtained by in for a general distribution of and has an upper bound as shown in the following corollary.

Furthermore, if with , then can be readily found by as

and then its low-complexity upper bound can be found as

by using Corollary ? and the closed-form upper bound can be found once the value of is designated. For example, if (i.e., ) for all , the results in and reduce to a simple closed form given by

Note that is somewhat complicate in practical computation even though it is the exact expression of without specifying the distribution of , whereas the tight upper bound in presents a low-complex formula of evaluating how different received signal random models impact the success probability with interference cancellation.

4.2Does the Randomness of the Received Signal Power Jeopardize the Success Probability?

The randomness of the received signal power could be induced by random channel gain, transmit distance and power. When it can definitely benefit or jeopardize the SIR has not yet been analytically shown and explained. In the following theorem, we show that the randomness of the received signal power does not necessarily jeopardize the success probability and it even could benefit the success probability if certain condition is satisfied.

Theorem ? also indicates that the randomness of could benefit if . To make this point more understandable, consider a simple example of the success probability with pathloss exponent , constant transmission distance , constant transmit power and all channels with Rayleigh fading, i.e., . In this example, the success probability with Rayleigh fading is inferred from and the success probability without fading is based on so that Rayleigh fading benefits the success probability as , i.e., we have and thus . Another crucial finding shown in Theorem ? is that we are able to improve the success probability by changing the setups of the transmit powers and transmitter intensities based on the channel fading status in the network. That is, if channels do not suffer fading, the setups of the transmit powers and transmitter intensities that make empty help improve the success probability for the given threshold . On the contrary, we should change the setups of the transmit powers and TX intensities that make nonempty to increase the type- success probability if channels suffer (severe) fading.

4.3Success Probability with Stochastic Power Control

In this subsection, we would like to investigate how to improve the success probability by designing distributed stochastic power control schemes that change the distribution of the received signal power. The centralized power control schemes are difficult to be implemented in this heterogeneous ad hoc network since all TXs only know its own local information and cannot optimize their transmit powers jointly in order to maximize their success probabilities. According to the explicit results of the success probability in Section 4.1, the key to maximizing the success probability of the type- TXs is how to minimize the term by optimally devising distributed power control schemes. Since each TX only has its local information available, we specifically propose the stochastic power control scheme for a type- TX as follows

where is the mean of transmit power , is the power control exponent needed to be designed. When there is no power control (i.e., constant transmit power is used), (i.e., ). This power control scheme is motivated by the fractional power control in [11] and the fact that the randomness of the received signal power could improve the success probability, as already pointed out in Section 4.2. We can change to adjust the randomness of to improve the success probability in different network contexts. Therefore, the fundamental problem needed to be firstly studied is how the stochastic power control in changes/benefits the type- success probability. The success probability with stochastic power control was essentially intractable in prior works, whereas it becomes much more tractable if using the success probability results found in Section 4.2. The following theorem presents the type- success probability with the proposed stochastic power control, denoted by .

According to Theorem ?, the stochastic power control scheme with nonzero can reduce the interference since . This also implies that the “randomness” of transmit power always results in less interference no matter if the power depends on the channel gain and/or pathloss. Nonetheless, this does not mean the stochastic power control always benefits the success probability since it may not enhance the received signal power without using a proper value of . To make stochastic power control benefit the type- success probability, this condition must hold, which poses the constraint on the values of that are able to improve the type- success probability. Unfortunately, the explicitly constraints on ’s for all are only tractably to be found for some special cases.

To understand more about the context in which the stochastic power control in definitely benefits the success probability, consider the special case of and the success probability in . To make the stochastic power control benefit in this case, the following inequalities must hold:

By assuming , constant and for all , we have , , ,, , and thus reduces to

The stochastic power control with any that satisfies this inequality outperforms no power control in terms of the success probability. This demonstrates that we need to appropriately use different based on different values of so that stochastic power control can always outperform no power control, which will be validated by the numerical results in Section 4.4.

4.4Numerical Results

Table 2: Network Parameters for Simulation
Parameter TX Type Type 1 Type 2 Type 3
Transmit Power (W) 1 0.5 0.05
Intensity (TXs/m)
Pathloss Exponent
Transmit Distance (m)
Channel Gain
SIR Threshold
Power control exponent

In this subsection, a few numerical results are provided to validate the success probabilities derived in the previous subsections. We consider the heterogeneous wireless ad hoc network consisting of three types of TXs and the simulation parameters for this heterogeneous network are listed in Table 2. First, the simulated and theoretical results of the success probabilities without and with interference cancellation are shown in Figure 1 and is used to calculate the upper bound on the success probability with interference cancellation shown in the figure. As can be seen in Figure 1, the simulation results of the success probabilities without interference cancellation perfectly coincide with their corresponding theoretical results given in . With properly chosen , the upper bounds are also very much close to their corresponding simulated success probabilities so that the tightness of is validated. Thus, our analytical results in Section 4.1 regrading the success probabilities without and with interference cancellation are correct and accurate. Since all transmission distances are the same and the TXs with a higher type number have a much smaller transmit power, the mean of is always smaller than that of so that canceling interference should help to increase much more than , as indicated in . We can easily see this phenomenon by comparing the three subfigures in Figure 1.

Figure 1: Simulation results of the success probabilities without and with interference cancellation: (a) p_1(\theta) and p_{1,L}(\theta) for L= 1, 2, 3. (b) p_2(\theta) and p_{2,L}(\theta) for L=1, 2, 3 (c) p_3(\theta) and p_{3,L}(\theta) for L=1, 2, 3. The upper bound on each type of the success probability is calculated by using .
Figure 1: Simulation results of the success probabilities without and with interference cancellation: (a) and for . (b) and for (c) and for . The upper bound on each type of the success probability is calculated by using .

In Section 4.2, we have pointed out that the randomness of the received signal power does not necessarily jeopardize the success probability. This has been demonstrated in Figure 2 for the success probabilities for channels with or without Rayleigh fading and we certainly observe that Rayleigh fading does not always weaken the success probability under different TX intensities. In a dense network, usually channel randomness helps to improve the success probability since it weakens the interference channels much more than the communication channel. Also, we can exactly find the intensity region in which Rayleigh fading benefits the success probability. For example, in the simulation setting here we have , for Rayleigh fading and for no fading. According to the discussion in the paragraph right after Theorem ?, Rayleigh fading increases when , i.e., , which is accurately illustrated by the curves of in Figure 2.

Figure 2: Simulation results of the success probabilities with and without Rayleigh fading.
Figure 2: Simulation results of the success probabilities with and without Rayleigh fading.
Figure 3: Simulation results of the success probability with 1\times 4 SIMO Rayleigh fading channels. The theoretical results of p_1, p_2 and p_3 are based on the result in  for M_r=4.
Figure 3: Simulation results of the success probability with SIMO Rayleigh fading channels. The theoretical results of , and are based on the result in for .
Figure 4: Simulation results of the success probabilities with and without the stochastic power control: (a) p^{pc}_k(\theta) with \gamma=-0.5 and p_k(\theta)=p^{pc}_k(\theta) with \gamma=0, (b) p^{pc}_k(\theta) with \gamma=-0.5 and p_k(\theta)=p^{pc}_k(\theta) with \gamma=0. The theoretical results of p^{pc}_k are found by using .
Figure 4: Simulation results of the success probabilities with and without the stochastic power control: (a) with and with , (b) with and with . The theoretical results of are found by using .

The success probability found in Section 4.1 is also applicable to the scenario of a TX or RX with multiple antennas. To demonstrate this point, consider the single-input-multiple-out (SIMO) case in which a TX has a single antenna and its intended RX has receive antennas. According to , the success probability for this SIMO channel with independent Rayleigh fading can be written as