Bistable Region of Backoff Algorithms with Contention Window in Slotted ALOHA Systems

Abstract—It has been widely recognized that bistable behavior of random access protocols may suddenly deteriorate the performance of the system due to the unexpected slip into an undesired stable operating point. The catastrophe theory succeeds in presenting the bistable region of random access protocols, where packet retransmissions occur in a memoryless manner. In this correspondence, the stability of slotted ALOHA systems using backoff algorithms with the contention window is analyzed without converting the retransmission with the contention window into the memoryless retransmissions. The analysis is based on the catastrophe theory. We first construct a two-dimensional Markovian model, which is equivalent to the model devised for IEEE 802.11 DCF by Bianchi. Then, the balance function of the system is formulated whose zeros provide equilibrium operating points. Finally, we prove that the system is mono-stable for any set of contention window sizes, if the number of allowable retransmissions is less than or equal to eight. Equivalently, the existence of the bistable region is proved if nine or more retransmissions of backlogged packet are permitted.