Аңдатпа
The problem of monitoring rare events in distributed systems arises in a wide range of applications, including environmental monitoring, industrial automation, and smart infrastructures. In such systems, devices operate independently and without coordination, making random multiple access the only practical and effective approach for utilizing the shared communication channel. We study a rare-event monitoring system consisting of a set of devices and a base station, where devices report events to the base station via a random multiple access scheme. The model accounts for channel noise, which may lead to message loss even in the absence of transmission collisions. Event occurrences are modeled as a family of independent two-state Markov chains. We analyze the impact of repeated transmissions on system performance, aiming to determine the optimal number of retransmissions under selected efficiency criteria and to characterize its dependence on system parameters. Specifically, we investigate (i) the maximum probability that a message from a fixed device is successfully delivered to the base station, and (ii) the maximum frequency at which the base station successfully receives updates about the entire system. For each criterion, we derive explicit expressions as functions of the number of users, the frequency of new messages, and the channel noise. The optimal number of retransmissions is then obtained by optimizing these expressions. By explicitly incorporating channel noise—a factor often neglected but crucial in practice—we show that repeated transmissions naturally arise as part of the optimal design. Our numerical results further support the theoretical analysis, showing that optimal retransmission strategies can substantially enhance overall system performance.