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This paper is concerned with the decentralized power allocation problem in random access systems. We propose a scheme that is especially suitable for systems requiring high throughput but with difficulty in establishing centralized control, such as cognitive radio environments. Specifically, we assume successive interference cancellation (SIC) at the receiver for multi-packet reception (MPR). We consider a decentralized random power transmission strategy where each user selects its transmitted power level randomly according to a power distribution conditioned on its own channel state. Our focus is on the design of this distribution such that the system packet throughput is maximized under rate and power constraints. We start from a two-user system. A main finding of this paper is that the supports of the optimal power distributions are of discrete nature. This finding greatly simplifies the distribution optimization …