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Proof From convex optimization theory, the utility function in Equation (6.2) is maximized if dUPF 0 rk - rk dU PF = e 0 k Rk for all r where the expectation is taken with respect to all fading realizations k r across many scheduling slots. Hence, if a scheduler can maximize k k for Rk every fading slot, then dUPF 0 and UPF will also be maximized. Hence, the equivalent utility function for proportional fairness is given by K rk U PF2 = e k=1 Rk (6.3)
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where Rk is the average throughput of user k, which depends on the heterogeneous level among users [1, K]. Because of the causality requirement on obtaining Rk, we shall approximate Rk with the moving-average throughput. Letting t denote the tth scheduling instance, the moving-average throughput at the (t + 1)th scheduling instance, Rk(t + 1), is de ned as
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1 1 Rk (t + 1) = 1 R (t ) + rk k tc tc
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where tc is the time constant of the averaging window. For a single-antenna system with TDMA constraint (selecting one user at rk a time), the PF scheduling involves selection of a user with the largest , Rk which is equivalent to the original form of PF scheduler in Qualcomm s HDR [58]. However, this simple scheduling rule does not generalize to multiantenna systems easily. 6.2.2 De nition of Network Capacity
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The network capacity is in general de ned by an average system utility function as discussed in Section 6.2.1. In particular, we shall restrict the average system utility function, U( R1, . . . , RK), to be of the following form U (R1 , . . . , RK ) = e [G(r1 , . . . , rK )] (6.5)
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where G(r1, . . . , rK) is the instantaneous system utility and rk is the instantaneous data rate1 of the kth user. Furthermore, for meaningful optimization, we constrain the instantaneous utility to satisfy G > 0 " rk 0 rk 6.2.3 De nition of Network Coverage (6.6)
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Assuming a sequence of N realizations of fading blocks, the instantaneous achievable data rate(s) of a scheduled mobile user(s) are random variables (functions of the speci c fading realization) in that fading block. In conventional wireless systems where the scheduling is constrained to select one active user at any scheduling instance (fading block), outage is de ned as the event that the instantaneous data rate of the scheduled user r is below a target data rate R0. Let Noutage(R0) be the number of outage events over N realizations of fading blocks. The outage probability is de ned as the likelihood of the outage event, Pout(R0) = Pr[r R0] = limN Noutage(R0)/N. The traditional concept of network coverage is therefore de ned as the maximum distance d between a mobile and a base station such that the outage probability of the scheduled user (at a target bit rate R0) is below a speci ed target e. That is, coverage is given by the maximum distance, d(e, R0), such that
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1 Instantaneous data rate of a user refers to the data rate of a burst transmission at a particular fading block.
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Pout (R0 ) = Pr[r (d) R0 ] e
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where r(d) denotes the instantaneous data rate of any user at a distance d from the base station. This is a commonly employed de nition in cellular systems [117] as well as wireless systems with the scheduler constrained to select one user at a time [25]. However, when the base station has multiple transmit antennas, there are multiple spatial channels as a result of the additional degrees of freedom. This implies that a general multiple-antenna scheduler may select more than one active transmission at any scheduling slot (fading block). Hence, before we can discuss the design of schedulers optimized for network coverage, we must extend the de nition of coverage to include multiple active user transmissions at a time. De nition 6.1 (Utility-Based Outage) Utility-based outage is de ned as the event that the instantaneous coverage utility Gcov(r1, . . . , rK) is below a target utility value G0. The outage probability is de ned as the probability of the occurrence of such event. De nition 6.2 (Worst-Case Outage) Worst-case outage P*ut(G0, d0) is de ned o as the largest possible utility-based outage probability over all possiblemobile locations such that {max(d1, . . . , dK) d0}: * Pout (G0 , d0 ) =
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