Throughput of IEEE 802.11 DCF in κ - µ Fading Channels

—This letter investigates the throughput performance of the IEEE 802.11 Distributed Coordination Function over κ µ fading environment. The approach considered includes signal capture model with incoherent addition of interfering signals and uniform attenuation for all terminals. The fading model selected allows ﬂexibility and can offer better statistical ﬁtting than traditional ones. Numeric results are presented.

of the unsaturated traffic situation. In [7], the authors propose a perfect power control by considering the signal capture model with coherent and incoherent addition of interfering signals in Nakagami-m, Rice, and Hoyt fading environments. In [8], the analysis of the throughput considers the two-way handshaking, and assumes a more generalized fading model, the η-µ model, which was introduced by Yacoub in [9] and that better represents the channel in a non-line-of-sight (NLOS) scenario. Moreover, the considered signal is composed by clusters of multipath waves in a non-homogeneous environment. Several others works have dealt with the channel throughput in specific protocols (e.g., [10]- [15]). Therefore, in order to expand a scenario not yet explored, this letter investigates the throughput performance of the IEEE 802.11 DCF in a typical small or pico-cell wireless communication scenario, i.e., using the κ-µ fading environment [9]. This is another generic fading model, but able to better represent a line of sight (LOS) channel condition, i.e., a scenario in which there is a direct non-obstructed path between the transmitter and receiver. This conditions are usually found in satellite transmissions [16], infrared communications systems, Li-Fi [17], and also in vehicular communications [18]. The considered approach includes signal capture model with incoherent addition of the interfering signals and uniform attenuation for all terminals.
This letter is organized as follows. Section II offers a brief explanation about the channel model and the MAC sublayer. Section III describes the signal capture effect, the channel throughput and the probabilities involved. Section IV presents channel throughput results. Finally, a conclusion is given in Section V.

A. κ-µ Fading Model
The κ-µ fading model was introduced in [9], and it is a model better suited to represent the small-scale fading variation in the presence of dominant, line-of-sight (LOS), components. The PDF of the squared random variable (RV), i.e., the signal power Ω, is given by in which ω > 0 represents the signal power, Ω = ω/ω is the normalized power,ω = E(Ω) is the average power, I ν (.) is the modified Bessel function of the first kind and order ν [19,Eq. 10.32.4], κ > 0 is the ratio between the powers of the dominant components and the scattered waves, and µ represents the real extension of the number of multipath clusters, and is given by The operators E(.) and V(.) denote expectation and variance, respectively.
As a generic fading model, it can be used to represent more traditional ones, such as Rayleigh, Rice, Nakagami-m, and One-Sided Gaussian models by setting its parameters as follows. For Rayleigh, set µ = 1 and κ → 0; for Rice, set µ = 1; for Nakagami-m, set κ → 0; and for One-Sided Gaussian, set µ = 0.5 and κ → 0.
The PDF and CDF of the ratio of (envelope) RVs κ-µ/κµ can be obtained from [20, Tables 2 and 3]. However, here the interest rests on the squared distribution that represents the signal power. Therefore, the PDF and CDF of the ratio Z = Ω s /Ω n , in which Ω s and Ω n follow (1), are expressed as, respectively, is the Beta function [19, Eq. 5.12.1], 1 F 1 (., ., .) is the Kummer confluent hypergeometric function [19,Eq. 13.2.2], and Q(., .) is the regularized incomplete gamma function [19,Eqs. 8.2.4]. For this letter, the subscripts n and s are used to represent the incoherent addition of the interference signals (the denominator of Z), and the test (desired) signal (the numerator of Z), respectively.

B. Medium Access Control Sublayer
In the IEEE 802.11 protocol, the MAC sublayer (which belongs to the Data Link layer) was designed to support random access to a shared wireless medium. A mechanism based on the CSMA/CA is used with the aim of preventing the occurrence of collisions, i.e., when two or more stations transmit at the same time, commonly causing signal reception failure. When a station is ready to transmit a data frame, it first senses the channel activity. If the channel is sensed idle for at least a period of time defined as Distributed InterFrame Space (DIFS), the transmission happens. If the channel is sensed busy, the transmission is delayed according to the binary exponential backoff rules [5] explained below. A confirmation (ACK) frame is relayed back from the destination to the sender to confirm the correct reception of the data frame. If the ACK frame is not received, the sender assumes that the data frame has been lost and reschedule a retransmission according to the binary exponential backoff rules.
As already mentioned, this letter assumes that DCF mode is used, and therefore no AP is present. Also, two data dialogues are considered: two-way and four-way handshaking [4], which are explained in the following. The simpler twoway handshaking includes only the data frame transmitted by the sender and a positive ACK frame transmitted by the destination. The time interval between these two frames is defined as the Short InterFrame Space (SIFS). If the sender does not receive an ACK within the ACK timeout period, then a retransmission is scheduled in accordance with the binary exponential backoff rules. The four-way handshaking, also known as Request To Send/Clear To Send (RTS/CTS) mechanism, operates similarly to the two-way handshaking. However, before sending a data frame, the transmitting station sends a short RTS control frame. After the receiver station receives the RTS, it waits for a SIFS, and sends a CTS control frame to confirm it is ready to receive the data. Then the data and ACK frames follow as in the two-way handshaking. Note that the RTS and CTS frames are used to inform all stations of an incoming data dialogue and, therefore, all stations are expected not to interfere with the imminent transmission.

C. Binary Exponential Backoff Rules
The binary exponential backoff mechanism [5] is used to spread out in time transmission reattempts, i.e., repeated transmissions due to earlier failures. Whenever a station wants to transmit, it waits for the channel to become idle, transmits its data frame, receives the ACK confirmation frame and moves on. However, if no confirmation is received, a new retransmission is schedule to happen after a randomly chosen backoff time. This transmission procedure can be modeled by a Markov Chain, as depicted in Fig. 1.
Peq Wm A backoff time k (expressed as an integer number of time slots) is uniformly selected from 0 and W i − 1, in which W i = 2 i W 0 is the contention window size for backoff stage i, i ∈ [0, m], with i = 0 for the first retransmission, and W 0 is the initial contention window size.
Before the station can attempt a retransmission, the sending station has to wait the backoff counter to expire. While the channel remains idle, the backoff counter is decremented every slot. However, if the channel is busy, the backoff counter stops decrementing and remains unchanged. When k reaches zero in any state (i, 0) and the medium is idle, the station is allowed to transmit. If successful, then the contention window is set to its initial value W 0 for a new transmission. Also, it is easy to see that the station moves to idle state (I) with probability (1 − q)(1 − P eq ), in which q is the probability that the station buffer has at least one data packet ready to be transmitted, and P eq is the probability of a failed transmission. Otherwise, the backoff stage i is incremented by one and the contention window size doubles until it reaches its maximum size W m = 2 m W 0 . In that case the contention window no longer changes until the data packets is successfully transmitted, or the station reaches its maximum number of retransmissions attempts. In this last case, the transmission is abandoned.

D. Numerical Realization Example
Consider a station that is in idle state (I). Then two packets arrive for transmission, but the channel is busy. The transmission is delayed in accordance to the binary exponential backoff roles. Assume that k = 1 is drawn. The station's Markov chain moves from state (I) to (0, 1). After one timeslot, the state moves (with probability 1) to state (0, 0), in which the station tries to transmit a packet. Assume that the transmission is unsuccessful. Then, a new k is drawn (say k = 3), which moves the state from (0, 0) to (1, 3). Again, the state is moved at each timeslot from (1, 3) to the direction of (1, 0). Then a new retransmission is attempted. Assume that a successful transmission is achieved, and the second packet starts in the Markov chain, i.e., another k is selected (say, k = 2), which moves the state from (1, 0) to (0, 2). The state then moves from (0, 2) in the direction of (0, 0), when the station tries to transmit. Assume that a successful transmission is confirmed, which makes the state to go back to idle (I). This realization example is summarized by (I) → (0, 1) → (0, 0) → (1, 3) → (1, 2) → (1, 1) → (1, 0) → (0, 2) → (0, 1) → (0, 0) → (I).

III. THE CAPTURE EFFECT AND CHANNEL THROUGHPUT
For wireless systems, the radio receiver can capture the desired packet even in the presence of n interfering signals if the power ratio between the desired signal and the joint interference signal at the receiver's antenna is greater than a particular threshold. In this case, the capture threshold can be expressed as Z Ω s Ω n 0, in which Ω s and Ω n represent the random powers of the desired signal and the interference, respectively, also assuming that both are statistically independent. Then, the conditional capture probability for capture threshold z 0 , given the existence of n interfering signals, is given by P capt (z 0 |n) = 1 − F Z (z 0 ), in which F Z (.) is defined in (3).
Let N represent the number of contending stations, and n + 1 the number of interfering data packets. The unconditional capture probability in a generic timeslot can be expressed as in which γ is the probability that a station starts a transmission in a randomly chosen timeslot [7]. The probability that at least one among N existing stations is transmitting in a given timeslot is given as A successful data packet transmission corresponds to one of these events: just one of N station is transmitting, or multiple stations are transmitting but there is a successful signal capture effect. Therefore the probability of a successful data packet transmission is given as [7] A failed transmission due to a collision occurs if at least one other station among the remaining N − 1 is transmitting data packets at the same time, i.e., A failed transmission occurs because of a collision or a channel error. Therefore its probability is given as in which P e the probability of a channel error. As already defined, q corresponds to the probability that the station buffer has at least one data packet ready to be transmitted. Therefore in which λ represents the average data packet generation rate, and γ is the probability that a station starts a transmission in a random time slot, and is given as Finally, E {S ts } is the average data packet length in which σ represents a duration of an empty timeslot, T c , T s and T e correspond to the average time a channel is sensed busy due to collision, the average time of a successful transmission, and the average time of a unsuccessful data transmission, respectively. These time values depend on the handshaking mechanism used. For the two-way handshaking, they can be calculated as in which H, M and PL are the time durations of the PHY header, MAC header and packet payload, respectively, SI FS, DI FS, ACK and ACK timeout are the times specified in the standard [4], and τ is the propagation delay time.
Finally, the normalized throughput S can be represented as the fraction of time the channel is used to successfully transmit user data. The analysis of S results in a non-linear equation system dependent on the probabilities presented above, and given as

IV. RESULTS
Numerical results are presented in the following. In particular, throughput figures for two-way and four-way mechanisms are considered. The network parameters are used according to the IEEE 802.11 protocol: Channel bit rate = 1 Mbps, Packet payload = 1020 bytes, MAC header = 34 bytes, PHY header = 24 bytes, ACK frame = 14 bytes, ACK timeout = 300 µs, Slot time = 20 µs, DIFS = 50 µs, SIFS = 10 µs, τ = 0.2 µs, m = 5 and W min = 8. Numerical results for the throughput are offered in Figs. 2 and 3, with the calculation performed with the help of the mathematical tool Mathematica [21]. Note that without loss of generality, it is assumed that κ = κ s = κ n , µ = µ s = µ n andω 2 s =ω 2 n = 1. Fig. 2 depicts the throughput for two-way access mechanism. It can be observed that the curves are characterized by two well-defined zones, separated by a peak. In the first part, the throughput increases linearly with the increasing packet rate, and in the second part there is a saturation region, where the variation of the data packet rate has little influence over the channel throughput. It can be seen that, in general, a decrease in the channel throughput is observed just after the peak. Varying the number of stations cause the curves to change considerably due to increased traffic. The greater the number of stations N, the steeper the growth of the linear region of the curves, as expected (compare curves 1, 4 and 7 in Fig. 2). The channel error probability P e also influences the channel throughput, as illustrated for the the pairs of curves 2 and 3, and 9 and 10 in Fig. 2. Comparing curves 1 and 2, there is a throughput decrease with the increase of parameters κ and µ. This happens because the larger the values of these parameters, the more deterministic the scenario becomes. On the other hand, if the capture threshold is reduced, a throughput growth is observed. This occurs because reducing the threshold indicates an improvement of the ability of the receiver to detect the desired signal (compare pair of curves 4 and 5, and 7 and 8). As mentioned in the previous section, the κ-µ fading model may represent less generalized distributions. In the figure, curve 11 represents One-Sided Gaussian channel and Nakagami-m channel with m = 0.5, and curve 12 represents a Rice channel.   3 illustrates the throughput results for four-way handshaking access mechanism. In a similar way to the simpler handshaking, the four-way has two well-defines zones (growth and saturation region), but due to the use of control packet (RTS/CTS) the decrease of channel throughput just after the peak is almost non-existent. It can also be observed that: the number of stations change considerably the curves; and the P e is influential only in the saturation region (see curves 2 and 3 in Fig. 3). On the other hand, changing parameters κ, µ and z 0 has a subtle influence over the throughput, as illustrated by the pair of curves 1 and 2, 3 and 4, 5 and 6, and 7 and 8.
It is interesting to note that more severe fading seems to benefit the throughput performance. For instance, comparing results presented here with those appearing in [8] indicate the conclusion above. The reason behind it might be that less interference is produced in such situations.
V. CONCLUSION This letter investigates the throughput of the IEEE 802.11 DCF using two-way and four-way handshaking access mechanisms over κ-µ fading environment with uniform attenuation for all terminals and incoherent addition of interfering signals. It is considered a non-saturated traffic condition, as the one proposed in [7]. Numerical, statistical and analytical methods were employed to obtain the results. The analysis presents the channel throughput for the cited protocol and how it is the influenced by the model's parameters. The use of the κ-µ model in the scenario considered here are unprecedented.