PROBABILISTIC CHARACTERISTICS OF WIRELESS NETWORKS WITH INFRASTRUCTURE TOPOLOGY

Based on the analysis of the operation of networks IEEE 802.11 DCF, a function is proposed for determining the probability of frame transmission to a central node depending on the number of stations operating in saturation mode. The probabilities of collisions are calculated. Using a polynomial approximations an expression is obtained for the network throughput, which explicitly depends on the number of the simultaneously operating stations.


Introduction.
Wireless networks with infrastructure topology of IEEE 802.11 standard are intensity developed. The data exchange speeds of several gigabits per second, which have already really been reached at the present time, make it possible to use them instead of wired networks, especially on twisted pair. This opens up the possibility of using wireless networks in the shops of industrial enterprises, at production sites. But the high level of electromagnetic interference, which is characteristic for these industrial premises, leads to a distortion of a significant number of frames moving from peripheral stations to the central node. Herewith these stations do not receive confirmations from the central node on sent frames. The collision compensation method CSMA/CA used in this case does not distinguish between the frame damage due to external interference from operating technology equipment and the collision that occurs when frames are mutually damages by two stations operating in the same wireless segment. In both cases the contention windows mechanism is activated, which delays the frames sent by the stations, and thereby significantly reduces the speed of information exchange in a wireless network. Fundamental analysis of IEEE 802.11 network using Distributed Coordination Function (DCF) was carried out in article [1] and later updated in [2]. This analysis is based on the use of bidimentional Markov chain model with ideal channel conditions and saturated load. Each frame needs to wait for a random backoff time before transmitting. The backoff is performed in discrete time units called slots and the stations are synchronized on the slot boundaries. As a result of this analysis, it was obtain that the probability τ that a station transmits in a randomly chosen slot time is where W0=CWmin is minimum contention window, CWmax = 2 m W0.
The probability p that a transmitted frame encounters a collision is defined as the probability that, in a time slot, at least one of the n-1 remaining As can be seen from expressions (1) where n is the number of stations operating in saturation mode, N is the maximum number of stations, W0 is the initial window width, q is the variable parameter.
where E[Fr] is the average frame payload; Ptr is the probability that there is at least one transmission in a considered time interval, i.e. Ptr = 1-(1-τ)n; Ps is the probability that a transmission occurring in the channel is successful, i.e.
η is the number of the empty slots; σ is the slot duration; Ts and Tc are the average times the channel is busy because of successful transmission and collision correspondingly.
Due to DCF δ δ where Tphy is the duration of physical header transmission, R is the information trans-mission rate, SIFS -Short Interframe Space, DIFS -DCF Interframe Space, ACK -Acknowledgment, and δ is the propagation delay.
The calculation results show that expression (6) can be represented as  As follows from the graphs presented in Fig.1 and Fig.2, expressions (4) and (5), as well as coefficients of the approximating polynomials in Tabl.1, the value q = 3 is optimal.
For this value q, the dependence of network throughput S on the number of stations operating in the saturation mode was calculated. The dependency graph is shown in Fig.3. The calculation was carried out using the follow-  The nature of the presented dependence and its main parameters correspond to the throughput validation results obtained using the simulator ns-3 and presented in [7] Conclusions. Based on the analysis of the operation of networks IEEE 802.11 DCF, a function is proposed for determining the probability of frame n S Імовірнісні характеристики роботи бездротових мереж з інфраструктурною топологією Базуючись на аналізі функціонування мереж IEEE 802.11 DCF, запропонована функція для визначення імовірності передачі фрейма центральному вузлу в залежності від кількості станцій, працюючих в режимі насичення. Розраховані імовірності колізій. З використанням поліноміальної апроксимації одержано вираз для смуги пропускання мережі, яка у явному вигляді залежить від кількості одночасно працюючих станцій.