Speed analysis of ALOHA-based random-access algorithm with various slot duration

For multi-user systems of random multiple access to a common communication channel, built on the basis of the ALOHA algorithm, a method for organizing conflict resolution algorithms is considered. In such systems, the time in the common channel is divided into slots equal to the duration of the message transmission time, and subscribers randomly select a slot for transmission. In some systems, the slots have different durations, which under certain conditions can increase the speed of the algorithm. To determine these conditions, options for the influence of the slot duration on the speed of the algorithm are considered. It is shown that the speed of algorithms built on the basis of the ALOHA algorithm can be increased if the relative duration of the empty slot is different from one. An algorithm is proposed that provides the maximum speed when this condition is met. An optimization problem is formulated and solved for choosing the optimal value of the parameter at which the speed of the proposed algorithm is maximum. Similar results are demonstrated for the case when the relative duration of the empty slot is much greater than one.