On a probabilistic backoff distribution for IEEE 802.11 DCF networks

. For the DCF protocol model at the data link layer and in a saturated network, the self-similar structure of the probabilistic distribution of a data transmission time is studied, and both exact and approximate expressions for the distribution are obtained. A comparison with the simulation results is carried out.


Introduction
In most of the papers devoted to the analysis of IEEE 802.11 MAC networks, their performance is investigated depending on the modification of the protocol, its parameters, limitations and additional imposed conditions. Less frequently, the time of message transmission ω (backoff d) and their probabilistic-time characteristics (usually the average value and sometimes the variance) are studied. At the same time, the literature shows that the range of ω values relative to the average, at least in the model [1], is large, so that a complete probabilistic description is required for the correct use of the model. In addition, it is necessary when analyzing protocols for latency-critical applications, for example, IEEE 802.11p, where it is required to know the reliability of transmitting a data frame with a backoff of no more than the specified one [2,3].
In the literature [4][5][6], probabilistic models of link layer protocols are constructed and expressions for the generating function of ω are obtained, but their conversion is possible only numerically. In this regard, an analytical expression for the distribution function of ω is of interest, at least for some special cases, which will allow the researchers -to study in detail the self-similar structure of the probability distribution; -to propose approximations of probability distributions of ω and other probabilistic-time characteristics of the protocol; -to evaluate the possibility of using the proposed ratios based on their comparison with the simulation results.
The present work is devoted to solving these problems in order to provide transmitting fixed-duration packets over the DCF protocol in a saturated network using the model [1].

Probabilistic model of data frame transmission time
The results of probabilistic description of the IEEE 802.11 packet service backoffs using the method of generating functions are obtained in [4][5][6]. In these papers, the generating functions of data frame transmission backoff in a saturated network with a distributed control function and a basic type of access in the absence of noise, interference and hidden stations are found. At the same time, almost standard assumptions are made in such cases about the constancy of the probabilities of collision and transmission that are p and τ, respectively, in a time slot (minimum time interval) of a duration of σ.
The methods of finding p and τ differ significantly in the cited studies, however, the final results almost completely coincide up to the notations when they are reduced to the same conditions.
The paper considers the mechanism of carrier sense multiple access (CSMA), the details of which are described, for example, in [1]. For the model under consideration, the package service time ω is represented as the sum [ , and r  is the number of virtual slots that the node waits before the   is true. As for the expression for the generating function of the backoff, it can be represented as [5] Here s p , c p , e p are the probabilities of successful transmission, the collisions of the thirdparty stations and a free channel, respectively, calculated for the number of nodes n based on the known p and τ, and i W is the duration of the backoff after m i  collisions; m is the maximum number of increases in the duration of the backoff. The relation (2) is obtained for an "infinitely persistent user" who indefinitely repeats attempts trying to transmit a frame.
The probability of a data frame transmission backoff exceeding t is described by an additional distribution function , which is obtained by a numerical inversion of the generating function in [4,5]. Figures 1, 2 show graphs of the additional distribution function and the probability distribution of the frequency of data frame transmission backoffs obtained using the developed simulation computer software for the fixed-length frames and the number of stations in the network, varying from 10 to 100. Analytical studies of probabilistic distributions of the data frame backoff, performed both based on a simple probabilistic model [7,8] and using the mechanism of generating functions [4,5], have a number of disadvantages. Analysis of the results of these studies proves that the use of the model [7] does not allow describing the fine structure of probability distributions. Applying the relations (2)-(6) and the numerical implementation of the inverse Z-transformation, taking into account the complexity of the obtained relations and the need for strict accuracy control, is a non-trivial task. In addition, the results obtained are insufficient and inconvenient to use at the same time.
An 802.11 station can only start transmitting at the beginning of each virtual slot. This requirement of the 802.11 protocol is very important, because it leads to the discreteness of the probability distribution of the backoffs. The minimum time unit is the duration σ of an empty virtual slot. Possible backoffs are concentrated in a discrete recurrent set having a selfsimilar structure, a detailed understanding of which is essential for further reasoning.
The following terminology will be used in what follows. The structure and peaks presented in Fig. 2 will be called the structure and peaks of the second order. On a larger timescale, the structure of the first peak presented in Fig. 2 is now reproduced in Fig. 3 (the logarithmic scale along the ordinate axis), based on the simulation results. The corresponding structure and peaks will be called the zero order. The first peak of the second order contains 0 W peaks of zero order, which correspond to the successful transmission of the station from the first attempt, provided that all virtual slots have been empty (there have been no transmissions of third-party stations), i.e. the backoff from the start of the countdown will be of a backoff from this window with a backoff from a collision-free transmission window. Thus, the maximum number of zero-order peaks in the corresponding part of the first-order peak is equal to 1 3 0  W . The following peaks in Fig. 4 have a similar nested self-similar structure.  corresponds to the mixture of random variables [9], i.e. the sum of a random number of the generalized Bernoulli random variables with the generating function (7). The mixture here is a generalized binomial random variable with a random length of a test series, which is approximated by a Gaussian random variable at m r  .
Due to the discreteness of backoffs, for any moment of time N, there is only a finite number of points N t  whose probability is different from zero. The value of the distribution function at the point N can be represented by the expression Each frame service backoff is uniquely determined by -the number of collisions r of the selected station with the probability p; -the number of collisions c of the third-party stations with the probability c p ; -the number of successful transmissions s of the third-party stations with the probability s p ; -the number of empty (transmission-free) slots e with the probability e p . Based on the above, taking into account the distribution of a generalized binomial random variable, the following relation can be written for the backoff distribution function   x P r  is the probability distribution for the number of slots present at r collisions.
The formula (9) can be rewritten in the following, more convenient for practical calculations, form   Figure 6 demonstrates a comparison of additional distribution functions of the frame delivery backoff. An empirical function based on a simulation data sampling is plotted with a solid curve, and the function   N F  1 obtained using (10) is drawn with discrete symbols.
In the calculations for the case when 2  r , the approximation of   x P r  by Gaussian distribution is used. A good coincidence of the results confirms that the application of the assumptions made appears valid.

Conclusion
Thus, in the paper, under the assumptions made, an explicit expression has been obtained for the distribution function of the frame transmission duration at the data link layer of the DCF protocol using the model [1] and the known [4,5] expressions for the generating function. A thin self-similar structure of the distribution function has been studied. Simple analytical relations are obtained for the average value and the variance of the number of virtual slots r  before the next packet transmission attempt. The proposed approximation of the distribution function is proven to be in good agreement with the simulation results. The results obtained make it possible to study the complex structure of the possible transmission backoff magnitudes and offer useful simplified relations and ratios for their probabilistic description.