content_image/306289.jpg content_image/306290.jpg content_image/306292.jpg content_image/306294.jpg content_image/306296.jpg
To consider path loss, we now perform a transformation of RVs using (1), γ$_{k}$ = ¯ Kr − β k ζ$_{k}$ , where, γ$_{k}$ is the selected user SNR in each ring k . The CDF and PDF of γ$_{k}$ can then be written as follows: 

<img src='content_image/306289.jpg'>

<img src='content_image/306290.jpg'>

Step 2 ( Selecting the user with maximum SNR among K rings ): In this step, we compute the probability of selecting the k th ring among all other rings. It is important to note that this is equivalent to selecting the ring k which possesses the user with the highest SNR among all rings. Conditioning on γ$_{k}$ , the PDF of r$_{sel}$ can be written explicitly as follows: 

<img src='content_image/306292.jpg'>

By averaging over the distribution of γ$_{k}$ , the final expression for the PMF of r$_{sel}$ is 

<img src='content_image/306294.jpg'>

Using (7), (10) can be written for i.i.d. case as follows: 

<img src='content_image/306296.jpg'>

where r$_{sel}$ ∈ [0 , R ] . The results in (11) are generalized for any shadowing and fading statistics. Even though (11) is not a closed form expression, the integration can be solved accurately using standard mathematical software packages such as MAPLE and MATHEMATICA . 

## B. Proportional Fair Scheduling Scheme 

The proportional fair scheduling scheme allocates the subcarrier to the user with the largest normalized SNR ( γ/ ¯ γ ) [12], where γ and ¯ γ denote the instantaneous SNR and the short term average SNR of a given user, respectively. In other words, the selection criterion is based on selecting a user who has maximum instantaneous SNR relative to its own average SNR. The distribution of r$_{sel}$ can be derived as follows: Step 1 ( Selecting the user with maximum normalized SNR in ring k ): In this step, the performance of proportional fair scheduling scheme is independent of the path loss factor if users are moving relatively 

