On ergodic algorithms in systems of multiple access with partial feedback

We consider a model of a multiple access system with a non-standard partial feedback. Time is slotted. Quantities of messages in different time slots are independent and identically distributed random variables. At the beginning of each time slot each message presented in the system is sent to the channel with a certain probability, depending on available system history. If $i\ge1$ messages are being passed simultaneously, each of them is being passed successfully with probability $q_i$, and with probability $1-q_i$ transmission is distorted, the message remains in the system and tries to be sent later. We consider the case when $q_i> 0$ only if $i \le i_0$ for a given $i_0 \ge1$. By the end of the slot we receive information about the quantity of messages that were transmitted successfully (it is the «feedback») — only this information is available. The transmission algorithm (protocol) is a rule of setting transmission probabilities at different times based on the information, available to each moment. In particular, if $q_1 = 1$ and $q_i = 0$ for all $i>1$ then this feedback is called «success-nonsuccess».
In this paper we study the existence of stable algorithms and the rate of convergence. Algorithms determined in this paper are based on additional randomization idea proposed in [3].