16 VI. PART I: EVALUATION OF IMPORTANT NETWORK PERFORMANCE METRICS In this section, we demonstrate the significance of the deriv ed MGF expressions of the cumulative ICI in quantifying important network performance metrics such as the outage probability P$_{out}$ , ergodic capacity C and average fairness F among users numerically. Evaluation of Outage Probability: The outage probability is typically defined as the probabili ty of the instantaneous interference-to-signal-ratio to exceed a certain threshold. In order to evaluate P$_{out}$ numerically, we use the MGF based technique introduced in [17] for interference limited systems. Firstly, we define a new RV, Z = q ∑ L l =1 X$_{l}$ − X$_{0}$ = qY − X$_{0}$ , where q is the outage threshold and X$_{0}$ is the corresponding signal power of the scheduled user in the central cell. An outage event occurs when p ( Z ≥ 0) , i.e., when the interference exceeds the corresponding signal power. This decision problem is solved in [17] by combining the characteristic function of Z and residue theorem. The characteristic function of Z is defined as φ$_{Z}$ ( jω ) = E [ e $^{jZω}$] . Considering interference Y and signal power X$_{0}$ to be independent, the expression for φ$_{Z}$ ( jω ) can be given as, φ$_{Z}$ ( jω ) = φ$_{Y}$ ( jqω ) φ$_{X}$$_{0}$ ( − jω ) ; where φ$_{Y}$ ( qjω ) can be given by (34), (37), and (40) for different fading models. In general, the characteristic function of X$_{0}$ can be calculated as follows [18]: φ$_{X}$$_{0}$ ( ω ) = E ( e jωx 0 ) = ∫ ∞ 0 e jωx 0 f$_{X}$$_{0}$ ( x$_{0}$ ) dx$_{0}$ = jω ∫ ∞ 0 e jωx 0 F$_{X}$$_{0}$ ( x$_{0}$ ) dx$_{0}$ (47) where F$_{X}$$_{0}$ ( x$_{0}$ ) = ∏ K i =1 F$_{γ}$$_{k}$ ( x$_{0}$ ) for opportunistic schemes and compact closed form expressions of φ$_{X}$$_{0}$ ( jω ) are available in the literature [19, Eq. 19]. For non-opportunistic scheduling schemes φ$_{X}$$_{0}$ ( ω ) = ∑ r$_{K}$ r$_{k}$ = r 1 φ$_{ζ}$$_{|}$$_{r}$$_{k}$ ( jω ) P ( r$_{sel}$ = r$_{k}$ ) , where φ$_{ζ}$$_{|}$$_{r}$$_{k}$ ( jω ) is the characteristic function of ζ in ring k . The outage probability can then be computed by using the classical lemma introduced in [17] as follows: P$_{out}$ = 1 2 + 1 π ∫ ∞ 0 Im ( φ$_{Z}$ ( ω ) ω ) dω (48) where Im( φ$_{Z}$ ( ω )) denotes the imaginary part of φ$_{Z}$ ( ω ) . Using (48), the outage probability can be evaluated using any standard mathematical software packages such as MATHEMATICA . Evaluation of Ergodic Network Capacity: Another important performance evaluation parameter is the network ergodic capacity C , i.e., C = E [ log$_{2}$ ( 1 + X$_{0}$ ∑ L l =1 X$_{l}$ + σ 2 )] (49)