In an interference limited environment, the received signal quality at a receiver is typically measured by means of achieved SIR, which is the ratio of the power of the wanted signal to the total residue power of the unwanted signals. let and denote the transmit and received power respectively. Let denote the path gain and is the link gain between the interfering transmitter and the receiver . For the purpose of clarity, unless otherwise stated, a single subscript , or specifies the node, and a double subscript such as specifies the link between node and node . A node is any entity, mobile station (MS) or base station (BS), that is, capable of communicating. For a single interfering user depicted in Figure 1:

Assuming fixed and constant transmit powers, , (1) simplifies to:

where , are the path losses between transmitter and receiver and and respectively.

Like the gain parameter , the loss parameter incorporates effects such as propagation loss, shadowing and multipath fading.

The generalized path loss model for the cross-layer environment is given by

where is an environment specific constant, the constant corresponding to the desired link while corresponds to the interference link. The distance is a constant and is a random variable, is the path loss exponent, is the random component due to shadowing, and is a random variable modeling the channel envelop.

The commonly used path loss equation [2] only accounts for the large-scale path loss with regular cell deployment scenarios, which is incomplete for studying self-organizing networks. The new path loss model proposed in this paper takes into consideration the interaction of the large-scale path loss as well as the small-scale fading. This model is particularly important in studying the performance of self-organizing self-configuring networks.

For the interference scenario described in the system model, the path loss for the desired path and the path loss between the interfering transmitter and the receiver (interfering link) are

where is the path loss model for the desired link and is the path loss model for the interfering link. models the distance between the interference causing transmitter, , and the victim receiver . and are the path loss exponents, and are Gaussian distributed random variables modeling the shadow fading with each zero mean and variances and respectively, and and are the channel envelope modeling the channel fading. For the purpose of clarity, the time and frequency dependencies are not shown. The channel envelope is assumed to follow the Nakagami- distribution. Nakagami distribution is a general statistical model which encompasses Rayleigh distribution as a special case, when the fading parameter , and also approximates the Rician distribution very well. In addition, Nakagami-*m* distribution will also provide the flexibility of choosing different distributions for the desired link and interfering link, such as the Rayleigh for the channel envelope of the desired link, and Rician for the interfering link, or vice versa.

Using (3) and (5), the SIR can be given as

From (7), the SIR has six random variable components, , , , , and . In order to analytically derive the pdf of the SIR, the pdf of the individual components and also their ratios and products need to be determined first.

The following two formulas provide the basic framework for the analysis and will be used throughout the derivation. Given two independent random variables and the pdf of their product where is

Given two independent random variables and the pdf of their ratio where is

### 3.1. pdf of the Ratio of the Propagation Loss

It is assumed that the distance between the interfering transmitter and the receiver, , is uniformly distributed up to a maximum distance of , and that the distance between an interfering transmitter and intended receiver, , is uniformly distributed up to a maximum distance of . Therefore and are both functions of random variables, and their pdfs can be derived using the following random variable transformation [8]:

where and are random variables with pdfs and respectivly, and where is a function of , and are the first derivatives of and respectively.

The mathematical representation of the pdfs of and are

Let and denote the pdfs of and . Then employing the transformation (10), and are derived as

Using (9), the pdf of the ratio of the propagation loss, , is found to be

where , and .

The next step to derive the pdf of the SIR is to find the pdf of the ratio of the lognormal shadowing.

### 3.2. pdf of the Ratio of the Lognormal Shadowing

Given a normally distributed random variable with mean and variance , and a real constant , the product is known to follow a normal distribution with mean and a variance and has a log normal distribution. Since is normally distributed with mean and variance , is a lognormal distributed random variable with mean and variance expressed in terms of the normally distributed , while the mean and variance of are and respectively,

Since the ratio of two independent lognormal random variables is itself a lognormal distributed random variable. Therefore the pdf of is

where

The last components remaining from (7) are the random variables modeling the channel envelop and their ratios.

### 3.3. pdf of the Ratio of the Channel Envelope

In order to accommodate different channel fading distributions, Nakagami- distribution was used to model the channel envelope. Nakagami- distribution is the most general of all distribution known until now [9].

The Nakagami- pdf is given by

where represents the fading figure, is the average received power and is the gamma function given as

Using (8) and (9) the pdf of the ratio of Nakagamai channel evelopes, is

where

for the ratio of the Nakagami-distributed channel is the same as the ratio of two independent Rayleigh distributed envelopes.

The final step in the derivation of the pdf of the SIR is deriving the product of the above obtained pdfs.

### 3.4. pdf of the SIR

As shown in (7) the pdf of the SIR is the product of the three individual random variables, , and . Using the equations presented so far, the final pdf of the SIR is presented in (22):

where , , and denotes :

The final equation does not have a closed form solution but it is possible to solve the integration using numerical methods.