Generalised Multi-Receiver Radio Network: Capacity and Asymptotic Stability of Power Control through Banach´s Fixed-Point Theorem

We introduce a model of a generalised multi-receiver radio network with quality-of-service (QoS) constraints. There are two key functions: (1) Ni is non-decreasing and homogeneous and gives i\´s QoS as function of its carrier-to-interference ratios at each of K receivers, (2) nu_ik is a semi-norm that gives the interference experienced by transmitter i at receiver k as function of the power vector. We utilise "norm" concepts and Banach\´s well-known fixed-point theorem to characterise the conditions under which a QoS vector is feasible, and the corresponding power-adjustment process converges. The critical power levels equal a_i/h_i where a_i is the QoS target, and h_i is the \´average\´ channel gain. hi=Ni(h_il,..., h_i,K) where h_ik is the channel gain from transmitter i to receiver k. If the interference experienced by each transmitter i at each receiver k is less than 1 when each power is set to the critical level (i.e., nu_ik(a_l/h_l, a_2/h_2,..., a_N/h_N)<l), then the QoS targets are feasible. Applications of our general result yields simple feasibility formulae for 3 scenarios from Yates (JSAC, 13(7):1341-1348, 1995): (i) fixed base-station assignment, (ii) macro-diversity and, (iii) multiple- connection reception (terminal must maintain acceptable QoS at several receivers), which includes the minimum power assignment as a special case. The 3 formulae have the simple form: largest weighted sum of N-l QoS targets less than one, where the weights are relative channel gains.