7 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: F$_{γ}$$_{k}$ ( γ$_{k}$ ) = u$_{k}$ ∏ i =1 F$_{ζ}$$_{i}$ ( ¯ K − $^{1}$γ$_{k}$ r β $_{k}$) i . i . d = ( F$_{ζ}$ ( γ$_{k}$r β k ¯ K − $^{1}$) ) u$_{k}$ (7) f$_{γ}$$_{k}$ ( γ$_{k}$ ) = 1 r − β k u$_{k}$ ∑ j =1 f$_{ζ}$$_{j}$ ( γ$_{k}$r β k ¯ K − $^{1}$) u$_{k}$ ∏ i =1 ,i = j F$_{ζ}$$_{i}$ ( γ$_{k}$r β k ¯ K − $^{1}$) i . i . d = u$_{k}$ r − β k f$_{ζ}$ ( γ$_{k}$r β k ¯ K − $^{1}$) ( F$_{ζ}$ ( γ$_{k}$r β k ¯ K − $^{1}$) ) u$_{k}$ − 1 (8) Step 2 ( Selecting the user with maximum SNR among K rings ): In this step, we compute the probability of selecting the k th ring among all other rings. It is important to note that this is equivalent to selecting the ring k which possesses the user with the highest SNR among all rings. Conditioning on γ$_{k}$ , the PDF of r$_{sel}$ can be written explicitly as follows: P ( r$_{sel}$ = r$_{k}$ | γ$_{k}$ ) = K ∏ i =1 ,i = k p ( γ$_{i}$ ≤ γ$_{k}$ ) = K ∏ i =1 ,i = k F$_{γ}$$_{i}$ ( γ$_{k}$ ) (9) By averaging over the distribution of γ$_{k}$ , the final expression for the PMF of r$_{sel}$ is P ( r$_{sel}$ = r$_{k}$ ) = ∫ ∞ 0 ( K ∏ i =1 ,i = k F$_{γ}$$_{i}$ ( γ$_{k}$ ) ) f$_{γ}$$_{k}$ ( γ$_{k}$ ) dγ$_{k}$ (10) Using (7), (10) can be written for i.i.d. case as follows: P ( r$_{sel}$ = r$_{k}$ ) = ∫ ∞ 0 K ∏ i =1 ,i = k ( F$_{ζ}$ ( γ$_{k}$r β i ) ) u$_{i}$ u$_{k}$f$_{ζ}$ ( γ$_{k}$r β $_{k}$) r − β k ( F$_{ζ}$ ( γ$_{k}$r β $_{k}$) ) u$_{k}$ − 1 (11) 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