## Abstract

We introduce and analyse a new model of a multiple access transmission system with a non-standard «minimal feedback» information. We assume that time is slotted and that arriving messages form a renewal process. At the beginning of any time slot n, each message present in the system makes a transmission attempt with a (common) probability p_{n} that depends on the system information from the past. Given that B_{n} ≥ 1 messages make the attempt, each of them is successfully transmitted and leaves the system with probability q_{B}_{n}, independently of everything else, and stays in the system otherwise. Here fqig is a sequence of probabilities such that q_{i}_{0} > 0 and q_{i} = 0 for i > i0, for some i_{0} ≥ 1. We assume that, at any time slot n, the only information available from the past is whether i0 messages were successfully transmitted or not. We call this the «minimal feedback» (information). In particular, if i_{0} = 1 and q_{1} = 1, then this is the known «success-nonsuccess» feedback. A transmission algorithm, or protocol, is a rule that determines the probabilities {p_{n}}. We analyse conditions for existence of algorithms that stabilise the dynamics of the system. We also estimate the rates of convergence to stability. The proposed protocols implement the idea of 'triple randomization' that develops the idea of 'double randomization' introduced earlier by Foss, Hajek and Turlikov (2016).

Translated title of the contribution | On stability of multiple access systems with minimal feedback |
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Original language | Russian |

Pages (from-to) | 1805-1821 |

Number of pages | 17 |

Journal | Siberian Electronic Mathematical Reports |

Volume | 16 |

DOIs | |

Publication status | Published - 2 Dec 2019 |

## Keywords

- (in)stability
- Binary feedback
- Foster criterion
- Multiple transmission
- Positive recurrence
- Random multiple access

## ASJC Scopus subject areas

- General Mathematics