In this paper we propose novel and more realistic analytical models for the determination of the peak demand under four power demand control scenarios. Each scenario considers a finite number of appliances installed in a residential area, with diverse power demands and different arrival rates of power requests. We develop recursive formulas for the efficient calculation of the peak demand under each scenario, which take into account the finite population of the appliances. Moreover, we associate each scenario with a proper real-time pricing process in order to derive the social welfare. The proposed analysis is validated through simulations. Moreover, the performance evaluation of the proposed formulas reveals that the absence of the assumption of finite number of appliances could lead to serious peak-demand over-estimations. (C) 2015 Elsevier Ltd. All rights reserved.