content_image/310623.jpg content_image/310624.jpg content_image/310625.jpg content_image/310626.jpg content_image/310627.jpg
˜ r$_{sel}$ is the distance of the allocated user in the interfering cell l from the BS of interest, r$_{sel}$ is the distance of the allocated user from its serving BS, i.e., (BS l ), θ ∈ [0 , 2 π ] and D = 2 R since we consider universal frequency reuse with one tier of interfering cells. The approach can be extended to any number of tiers in a straightforward manner. In order to determine the PMF of ˜ r$_{sel}$ where ˜ r$_{sel}$ ∈ [ D − R, D + R ] , first of all we compute ˜ r$_{i,k}$ for given θ$_{i}$ and r$_{k}$ using (65) as follows: 

<img src='content_image/310623.jpg'>

where, ˜ r$_{i,k}$ denotes the interfering distance from a specified polar coor dinate ( r$_{k}$, θ$_{i}$ ) in the interfering cell to the BS of interest located at a distance D (see Fig. 1). In addition, it is worth to mention that ˜ r$_{i,k}$ are the points at which P (˜ r$_{sel}$ = ˜ r$_{i,k}$ ) can be defined using (63) as follows: 

<img src='content_image/310624.jpg'>

The two dimensional data set of ˜ r$_{sel}$ , at which P (˜ r$_{sel}$ = ˜ r$_{i,k}$ ) is defined, can then be grouped into M segments of any arbitrary width Δ . This can be done by dividing the distance between D − R and D + R into M equal segments of width Δ and mapping ˜ r$_{i,k}$ accordingly. Clearly, by adding all the probabilities for which ˜ r$_{sel}$ lies in the m th segment we get the probability of ˜ r$_{sel}$ = ˜ r$_{m}$ : 

<img src='content_image/310625.jpg'>

where ˜ r$_{m}$ denotes any discrete value that the RV ˜ r$_{sel}$ can take. Recall X = ¯ K ˜ r − $^{β}$χ , therefore the PDF of X conditioned on ˜ r$_{sel}$ can be determined by RV transformation as follows: 

<img src='content_image/310626.jpg'>

Averaging over the PMF of ˜ r$_{sel}$ , the distribution of the ICI, f$_{X}$ ( x ) , from any cell l can be given as: 

<img src='content_image/310627.jpg'>

It is important to emphasize that the derivation of the distribution of ICI is based on the scheduling decisions of interfering cells at a given time slot. Therefore, the parameter ˜ r$_{m}$ of the ICI distribution varies from one time slot to the other for the location based round robin and greedy round robin schemes. 

