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In this paper, we develop and investigate a novel power control algorithm (the fair randomized power control algorithm; FRPC) to enhance the throughput level of the slotted Aloha medium access control mechanism. FRPC is developed based on the analysis of the capture effect for which the transmitter captures the channel only if the observed signal-to-interference and noise ratio (SINR) at the receiver is above a certain threshold. We first analyze the sensitivity of the generic randomized power control algorithm with respect to expected value and standard deviation of the power density function. In particular, based on the one-tailed version of Chebyshev's inequality, we derive a closed-form expression for an upper bound on the probability of capture (and an upper bound on the aggregate throughput) in the network. We then apply the latter result to the process of designing the FRPC algorithm for the slotted Aloha MAC mechanism. Our simulation results indicate that the FRPC algorithm leads to significant throughput gain while providing fairness for different mobile users in the system.