Basic theorems on the backoff process in 802.11

Since its introduction, the performance of IEEE 802.11 has attracted a lot of research attention and the center of the attention has been the throughput. For throughput analysis, in the seminal paper by Kumar et al. [8], they axiomized several remarkable observations based on a fixed point equation (FPE). Above all, one of the key findings of [8] is that the full interference model, also called the single-cell model [8], in 802.11 networks leads to the backoff synchrony property which implies the backoff process can be completely separated and analyzed through the FPE technique. To date, however, only the uniqueness of the fixed point has been proven to hold under some mild assumptions [8, Theorem 5.1], and we still need an answer to the following fundamental question: Q1:“Exactly under which conditions the fixed point equation technique is valid?” (to be answered in Theorem 1) An intriguing notion, called short-term fairness, has been introduced in some recent works [2,4]. It can be easily seen that this notion pertains to a purely backoff-related argument also owing to the backoff synchrony property in the full interference model [8]. The two papers [2,4] considered the same situation where only two wireless nodes contend for the medium. The former [2] claimed and conjectured through simulations that the summation of the backoff values generated per a packet, denoted by Ω, is uniformly distributed because the initial backoff is uniformly distributed while the latter [4] conjectured based on their experiments that Ω is exponentially distributed in the sense that its coefficient of variation (CV) is one. This left room for misunderstandings about the backoff distribution: Q2:“One of the two works is incorrect?” (to be answered in Theorem 2) In addition, the two works [2,4] defined P[z|ζ] as the probability that other nodes transmits z packets while a tagged node is transmitting ζ packets and acquired the expression of P[z|ζ] valid only for the two node case. It is natural to ask the following pertinent questions: Q3:“Can we develop an analytical model for short-term fairness?” (to be answered in Theorem 3) Q4:“When does the short-term fairness undergo a dramatic