Improving the Throughput in Lightly Disturbed IEEE 802.11 DCF Saturated Network Using the Fragmentation Mechanism Under Basic Access Mode

Abstract

The IEEE 802.11 is the most dominating protocol in the wireless local area networks and in which, the fundamental mechanism to access the channel is called distributed coordination function (DCF). As this mechanism is random, the calculation of the performance metrics for IEEE 802.11 networks is focused on the determination of probabilities such as collision probability and successful transmission probability. Recently, there are performance studies in the literature are based on the mathematical models developed using the Markov chains, which model the DCF mechanism with fragmentation in IEEE 802.11 network under imperfect channel. Yet, these models suffer with high collision probability and low successful transmission probability and as a result, the transmission with fragmentation is not recommended at all in the IEEE 802.11 basic DCF networks when the bit error rate is low $\left( {BER < 5 \times 10^{ - 5} } \right)$. In order to solve this, a new analytical model of the basic DCF mechanism with fragmentation is presented in this paper to accurately evaluate throughput metric in the lightly disturbed IEEE 802.11 network at $BER = 10^{ - 5}$ under saturated traffic. Analytical results show that our model significantly improves the throughput of the IEEE 802.11 DCF network compared with existing model and hence, provides a higher throughput than that is improved with the request to send/clear to send (RTS/CTS) mode, regardless to network size.

1 Introduction

The distributed coordination function (DCF) is a random access scheme based on the carrier sense multiple access with collision avoidance mechanism (CSMA/CA) [1]. It includes two access modes to the channel: the basic mode and the request to send/clear to send (RTS/CTS) mode. It describes two techniques for the packet transmission: transmission without fragmentation and transmission with fragmentation. The effect of the fragmentation on the performance improvement of IEEE 802.11 networks can be explained by the fact that in the case of a corrupted frame (due to a collision or even channel noise), the more the frame is small the more the overhead generated by its failed transmission is small.

The bi-dimensional Markov chain modeling presented in [2] has become a basic method for evaluating the throughput of the IEEE 802.11 DCF network, since many models have extended it to analyze the throughput by considering 802.11 standard details, such as: the retry limit that is introduced in [3, 4], given the packet is abandoned after reaching the retransmission attempts maximum limit, the freezing of backoff counter whose modeling is discussed and improved respectively in [5, 6], given the backoff counter is frozen when the channel is busy and resumed when the channel is detected idle again, the effect of transmission errors that is studied in several research work [7,8,9,10,11] and the impact of ACK timeout and CTS timeout that is first reported in [12, 13].

Recently, various performance studies of the IEEE 802.11 networks and other wireless networks are based on the analytical models ([14, 15]) developed by extending the Markov chain [2] with the packet fragmentation mechanism under saturation or non-saturation traffic conditions. For example, the authors in [16] enhanced and modeled the IEEE 802.11 RTS/CTS scheme in an error-prone channel using the Markov chains and compared its performance with the standard RTS / CTS scheme. The authors in [17] extended the Markov chain models proposed for the IEEE 802.11e-EDCA network under a noisy channel with the packet fragmentation mechanism in order to evaluate the Opportunity Transmission Limit (TXOPLimit) efficiency in improving the saturation throughput. The authors in [18] proposed a Markov chain model including the packet fragmentation to evaluate the overall throughput of the IEEE 802.11e-EDCA network under the impact of Bit Error Rate (BER) and packet length. The authors in [19] presented a simple analytical model and simulation analysis of the algorithm of Binary Exponential Backoff (BEB) with fragmentation to evaluate the impact of fragment size and contention window on the throughput and delay performance of this algorithm over a noisy channel. The authors in [20] modeled using a Markov chain the application of data fragmentation in the IEEE 802.15.4 slotted CSMA/CA protocol to evaluate the normalized throughput under unsaturated traffic.

However, the analytical models ([14, 15]) contain anomalies, resulting in significant degradation of the DCF throughput when applying the fragmentation mechanism in a lightly disturbed IEEE 802.11 network. According to the Markov chain and its interpretation in [14] (respectively, in [15]), the authors distinguished two types of transmission states. The transmission states of type (T, i, 0) (respectively, the transmission states of type (i, 0)) which can encounter a collision or a channel noise and, the transmission states of type (T, i, − 1) (respectively, the transmission state (0, − 1)) which can only encounter a channel noise, because these states represent the transmission of the following fragments of a packet after having reserved the channel. However, the authors in [14] used the transmission probability "$\tau$"calculated only on the states of type (T, i, 0) for expressing the channel occupation probability in the formula (9) and the successful transmission probability in the formula (18). Hence, their channel occupation probability does not represent the channel occupation by the transmission at the states of type (T, i, − 1) and, their successful transmission probability does not express the successful transmission at the states of type (T, i, − 1). Whereas, the authors in [15] expressed their probabilities by the transmission probability "$\tau$" calculated on all the transmission states including the transmission state (0, − 1). However, the transmission state (0, − 1) should not be part of the collision probability formula (8) because it can only encounter channel noise. In addition, it should not be conditioned by the probability that the n − 1 remaining stations defer their transmissions (i.e., $(1 - \tau )^{n - 1}$) in the successful transmission probability formula (13), because the station that reaches to transmit in the state (0, − 1) ensures, through the DCF mechanism, that the channel remains reserved for the transmission of the following fragments of a packet (i.e., in this state, the probability that the remaining stations defer their transmissions is always equal to 1).

In this paper, we propose a new analytical model that improves the analytical model provided in [15], which suffers as explained above with low successful transmission probability and high collision probability when applying the fragmentation mechanism. Our analytic model accurately evaluates the fragmentation effect on the throughput in a lightly disturbed IEEE 802.11 basic DCF saturated network. Our evaluation is made under more realistic assumptions than those of the model in [15], which assumes, on the one hand, that the backoff counter is decremented even when the channel is sensed busy, an assumption that is not consistent with the IEEE 802.11 standard [1], and on the other, that the acknowledgment frames are always successfully received, an assumption that could be applied only in a lightly disturbed IEEE 802.11 network [8] and that is no more applicable when the acknowledgment frames are transmitted at the same bit rate as data frames [9]. The rest of this paper is organized as follows: Sect. 2, presents the CSMA/CA mechanism and the fragmentation. In Sect. 3, we accurately model the basic DCF mechanism with fragmentation using the Markov chains by analyzing the average length of the time slot in order to inspire from the Bianchi’s calculation of the throughput. In Sect. 4, we illustrate the precision and improvement of our model by comparing its analytical results with those obtained from existing models and we conclude the work in Sect. 5.

2 The CSMA/CA Mechanism and the Fragmentation

The distributed mechanism CSMA / CA is performed locally on each station in order to determine the access instants to the channel for transmission. In saturation conditions, each station always has a packet available for transmission after the channel is idle for a definite time duration DIFS (Distributed Inter Frame Space). In this context, to reduce the collision phenomenon, each station must start a backoff procedure and chooses a random number BT "backoff time measured in time slots" in the initial contention window CW_min. The BT is decremented (by 1) whenever the channel is sensed idle for an empty slot time $\sigma$ or is sensed idle for at least a DIFS time duration following a channel occupation. The station whose BT becomes zero is that which immediately sends the packet in the case of basic access mode, while in the case of RTS/CTS access mode, it is the one that sends the RTS frame in order to reserve the channel for the transmission of the packet. The channel will be reserved just after a short inter-frame space (SIFS) time duration, which follows the correct reception of the acknowledgment frame of RTS frame (i.e., the CTS frame). An acknowledgment frame (ACK) is emitted by the receiver to indicate to the sender that no collision or no transmission error is occurred. The ACK frame is emitted after a short inter-frame space (SIFS), which follows the correct reception of the frame that encapsulates the packet. If the ACK frame is not received correctly within an ACK timeout period $ACK\_timeout$, the sender determines that its transmission is failed (due to the collision or channel noise). Then it contends to retransmit the same packet again after a DIFS and new BT (see Fig. 1). Therefore, the $ACK\_timeout$ term should be included in the channel occupation time duration by a failed transmission. The new BT is chosen in the contention window CW, which is doubled after each transmission failure up to the maximum contention window CW_max. Thus, if the transmission fails even after reaching the retransmission attempts maximum limit, the packet is abandoned and the sender starts again the backoff procedure to transmit the next packet.

Fig. 1

Transmission of a packet in basic DCF or in RTS/CTS mode after reserving the channel

We note that, when transmitting packets in basic DCF with fragmentation, the backoff procedures and the carrier sense remain the same ones as in the schema without fragmentation. The difference consists in the IFS (Inter Frame Spacing) used between the transmissions of the fragments of the same packet (see Fig. 2). Only one SIFS is required between the transmissions of the fragments during the transmission of a fragmented packet. Namely, the SIFS time duration is less than the channel idle time duration required to decrement the BTs of other stations. This allows a station, which successfully transmits a fragment of a packet, to keep control of the channel to transmit the next fragment of its packet, and so on until the end of the fragments of the packet. In the basic access mode, each station executes the backoff process in order to reserve the channel for the rest of the fragments (if it exists) by successfully transmitting a fragment of a packet. Thus, if a station fails to reserve the channel up to the retransmission attempts maximum limit then, it abandons the transmission of the fragmented packet and starts the backoff procedure to transmit the next packet. In order to ensure the channel access equity, if a station has failed to receive correctly the acknowledgment (ACK) of a fragment transmitted after having reserved the channel, it retransmits it by performing again the backoff process.

Fig. 2

Event time durations in basic DCF with fragmentation

3 Analytical Modelling

In this section, we model the basic DCF mechanism with fragmentation under disturbed channel and saturated traffic. We exactly model using a bi-dimensional Markov chain the basic DCF service with fragmentation, which is defined from the moment when a packet reaches the head of the transmission queue to start contending for the channel access, until the moment when the packet or fragmented packet either is correctly acknowledged by the receiver or is dropped by the DCF server. Then, we analyze the average time slot length of basic DCF with fragmentation by expressing all the probabilities and time durations of the involved events and then, we are inspired from the Bianchi’s definition [2, 21] to calculate the normalized throughput of disturbed saturated network, which uses the IEEE 802.11 basic DCF with fragmentation.

In our modeling, we assume that: (1) the wireless network is lightly disturbed and fully connected with n contending stations working at saturation condition (i.e. all stations always have a packet available for transmission in single-hop network), (2) the failed transmissions occur due to a collision or channel noise, (3) the average dimension of the generated packet at each station is $MAC\_payload$ (i.e. the average length of MAC payload). For simplicity, we assume that the variance of the packet length is null and the packets are divided into fragments of the same size.

Knowing the fragment dimension L and the packet average dimension $MAC\_payload$, it is possible to obtain the probability $\delta$ of reaching the end of the transmission of the packet.

$$\delta = {\raise0.7ex\hbox{$L$} \!\mathord{\left/ {\vphantom {L {MAC\_payload}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${MAC\_payload}$}}$$

(1)

Therefore, this means that the packet is divided into ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \delta }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\delta $}}$ fragments.

There are four probabilities that are important in our modeling. The collision probability p, which is well identified in all the performance studies of IEEE 802.11 networks, the occupation probability of the channel $p_{b}$ that is introduced by [5], the DATA/ACK exchange error probability $p_{e}$, which is influenced by the channel noise and, the DATA/ACK exchange failure probability $p_{f}$, which is defined by [22] as the combination of the collision probability p and the error probability $p_{e}$.

3.1 Our Markov Chain

Let i (0 ≤ i ≤ m) be the backoff stage, which is incremented by 1 after each transmission failure at the state level (i, 0) as long as it is smaller than the maximum backoff stage m. $W_{i}$ is the size of the contention window at stage i. The expression of $W_{i}$ according to the size of the minimum backoff window $W_{0}$ is given by

$$W_{i} = 2^{i} W_{0} \quad 0 \le i \le m$$

(2)

Let $\left\{ {s\left( t \right),b\left( t \right)} \right\}$ be the bi-dimensional stochastic process, where $s\left( t \right)$ and $b\left( t \right)$ represent the backoff stage i and the backoff time counter respectively for a given station at time t and $b\left( t \right)$ is uniformly chosen in the contention window of its ith backoff stage (0, 1,…,$W_{i} - 1$). Thus, the state of each station is described by (i, j), where i represents the backoff stage, and j represents the backoff time counter and takes values in (0,1,…,$W_{i} - 1$). Except, the state (− 1, − 1) which describes the transmission of the station as soon as it succeeds in reserving the channel and consequently, this transmission can only encounter a channel noise.

Let Fig. 3 be a discrete-time Markov chain, which model the DCF mechanism of the IEEE 802.11 with fragmentation in basic access mode. This chain represents the bi-dimensional stochastic process $\left\{ {s\left( t \right),b\left( t \right)} \right\}$ and the state (− 1, − 1) under the assumption that the probabilities $p_{b}$,$p_{f}$ and $p_{e}$ are independent of the backoff process [2, 5, 8, 23].

Fig. 3

Markov chain for IEEE 802.11 basic DCF with fragmentation

In order to ensure the channel access equity between the stations, each station immediately resumes the execution of the backoff process at the first stage (i.e. the stage 0) on one hand, for the transmission of its next packet whether it is after the successful transmission of its packet (respectively, its fragmented packet) or after the destruction of its packet (respectively, its fragmented packet); and on the other, for the retransmission of a fragment of its packet that is not successfully transmitted after having reserved the channel.

In the scheme without fragmentation of the basic DCF mechanism (i.e. the Markov chain of Fig. 3 with $\delta = 1$), each station starts to transmit its packet if and only if it is in the state (0, 0) (i.e. when its counter chosen at the stage 0, becomes null). If a station fails to receive correctly the ACK of its transmitted packet, it attempts to retransmit it when its counter chosen at its next stage becomes null. And so on until its packet is successfully transmitted or is dropped following its transmission failure at the last stage (i.e. the stage m). In its schema with fragmentation, each station starts to transmit its fragmented packet each time if it is in one of the states of type (i, 0) (i.e. when its counter chosen at the stage i, becomes null). If a station fails to transmit a fragment of its packet up to the last stage (i.e. the stage m), its fragmented packet is dropped. As soon as a station successfully transmits a fragment of its packet in a state of type (i, 0), this station reserves the channel to transmit the next fragment of its packet in the state (− 1, − 1). Then if the fragment is successfully transmitted, the station continues to transmit the next fragment of its packet in the state (− 1, − 1) and so on. If it's not the case, i.e., the fragment/ACK exchange has encountered a channel noise at the state level (− 1, − 1), the station attempts to retransmit this fragment after executing the backoff process again.

Solving the system of balance equations obtained by applying the balance principle between flow leaving and flow entering for each state of the Markov chain of Fig. 3, we obtain the following stationary state probabilities:

$$\begin{aligned} \pi_{i,0} & = p_{f}^{i} \pi_{0,0} \quad 0 \le i \le m, \\ \pi_{i,j} & = \frac{1}{{1 - p_{b} }}\frac{{W_{i} - j}}{{W_{i} }}p_{f}^{i} \pi_{0,0} \quad i \in (0,m),\quad j \in (1,W_{i} - 1)\;{\text{and}} \\ \pi_{ - 1, - 1} & = \frac{{(1 - \delta )(1 - p_{f}^{m + 1} )}}{{\delta (1 - p_{e} ) + p_{e} }}\pi_{0,0} \\ \end{aligned}$$

We apply the condition that the sum of all state probabilities is equal to 1:

$$\pi_{ - 1, - 1} + \sum\limits_{i = 0}^{m} {\sum\limits_{j = 0}^{{W_{i} - 1}} {\pi_{i,j} = 1} } ,$$

from where

$$\pi_{0,0} \left[ {\sum\limits_{i = 0}^{m} {p_{f}^{i} } + \sum\limits_{i = 0}^{m} {\sum\limits_{j = 1}^{{W_{i} - 1}} {\left( {\frac{1}{{1 - p_{b} }}\frac{{W_{i} - j}}{{W_{i} }}p_{f}^{i} } \right)} } + \frac{{(1 - \delta )(1 - p_{f}^{m + 1} )}}{{\delta (1 - p_{e} ) + p_{e} }}} \right] = 1$$

We thus obtain $\pi_{0,0}$ as follows:

$$\pi_{0,0} = \frac{1}{{\sum\limits_{i = 0}^{m} {p_{f}^{i} } + \sum\limits_{i = 0}^{m} {\sum\limits_{j = 1}^{{W_{i} - 1}} {\left( {\frac{1}{{1 - p_{b} }}\frac{{W_{i} - j}}{{W_{i} }}p_{f}^{i} } \right)} } + \frac{{(1 - \delta )(1 - p_{f}^{m + 1} )}}{{\delta (1 - p_{e} ) + p_{e} }}}}$$

Since there are two types of transmissions, which are the transmission at one of the states (i, 0) that can encounter a collision or channel noise and, the transmission at the state (− 1, − 1) that can only encounter a channel noise, we distinguish the two following transmission probabilities $\tau$ and $\tau_{1}$.

We thus express the probability «$\tau$» that a station transmits a packet (respectively, a fragmented packet) in a given time slot, as the sum of all the state probabilities $\pi_{i,0}$.

$$\tau = \sum\limits_{i = 0}^{m} {\pi_{i,0} } = \frac{{1 - p_{f}^{m + 1} }}{{1 - p_{f} }}\pi_{0,0}$$

(3)

In the case of fragmentation, the station that reserved the channel continues to transmit its fragmented packet at the state (− 1, − 1). We thus express the probability «$\tau_{1}$» that a station continues to transmit its fragmented packet in a given time slot, as follows:

$$\tau_{1} = \pi_{ - 1, - 1}$$

(4)

The probability « $p$» that a packet or fragment of a packet encounters a collision thus, is expressed as the probability that at least one of the $n - 1$ remaining stations transmits in a given time slot with the probability «$\tau$». From where:

$$p = 1 - (1 - \tau )^{n - 1}$$

(5)

We express the probability « $p_{b}$ » that a station in the backoff stage detects the channel busy in a given time slot, if at least one of the $n - 1$ remaining stations transmits with the probability «$\tau$» or if one of the $n - 1$ remaining stations transmits with the probability « $\tau_{1}$».

$$p_{b} = 1 - (1 - \tau )^{n - 1} + \, (n - 1) \, r_{1}$$

(6)

In an error-prone environment, the probability « $p_{e}$» that an DATA/ACK exchange encounters a channel noise depends on the bit error rate (BER) and the length of frames involved in this exchange. From where:

$$p_{e} = 1 - \left( {1 - BER} \right)^{{{\text{PHY \_Header }} + {\text{ MAC\_Header }} + \, \delta \cdot {\text{ MAC\_}}payload + ACK}}$$

(7)

We express the failure probability « $p_{f}$» that an DATA/ACK exchange encounters a collision (with a probability $p$) or a channel noise (with a probability $p_{e}$), as follows:

$$p_{f} = p + p_{e}$$

(8)

We deduce $\tau$,$\tau_{1}$,$p$,$p_{b}$ and $p_{f}$ by solving the following nonlinear equations system:

$$\left\{ {\begin{array}{*{20}l} {\tau = \frac{{1 - p_{f}^{m + 1} }}{{1 - p_{f}^{{}} }}\pi_{0,0} } \hfill \\ {r_{1} = \frac{{(1 - \delta )(1 - p_{f}^{m + 1} )}}{{\delta (1 - p_{e} ) + p_{e} }}\pi_{0,0} } \hfill \\ {p = 1 - \left( {1 - \tau } \right)^{n - 1} } \hfill \\ {p_{b} = 1 - (1 - \tau )^{n - 1} + \, (n - 1) \, r_{1} } \hfill \\ {p_{f} = p + p_{e} } \hfill \\ \end{array} } \right.$$

(9)

3.2 Analysis of the Average Length of a Time Slot

At any moment, the channel is in one of the two events: $E_{idle}$ = {the channel is idle} and $E_{busy}$ = {the channel is busy by the transmission of a packet (or a fragmented packet)}.

When transmitting in basic mode without fragmentation, the channel is in one of the two possible events: $E_{suc}$ = {the channel is busy by the successful transmission of a packet} and $E_{failed}$ = {the channel is busy by the failed transmission of a packet}, such as events $E_{idle}$, $E_{suc}$ and $E_{failed}$ are mutually exclusive.

When transmitting in basic mode with fragmentation, we can distinguish the following two possible events: $E_{{{suc\_fragmented\_packet} }} =${the channel is busy by the successful transmission of a fragmented packet} and $E_{failed\_fragment} =${the channel is busy by the failed transmission of an fragment/ACK exchange}, such as events $E_{idle}$, $E_{{{suc\_fragmented\_packet} }}$ and $E_{failed\_fragment}$ are mutually exclusive.

We express the probability « $p_{tr}$» that the channel is busy in a given time slot, if at least one station transmits with the probability «$\tau$» or if one station transmits with the probability « $\tau_{1}$».

$$p_{tr} = 1 - (1 - \tau )^{n} + \, n \, r_{1}$$

(10)

From where, these events have the following probabilities:

$$\Pr \left\{ {{\text{the}}\;{\text{channel}}\;{\text{is}}\;{\text{idle}}} \right\} = P_{idle} = 1 - p_{tr}$$

(11)

$$\begin{aligned} & {\rm Pr} \left\{ {E_{suc\_fragmented\_packet} } \right\} = P_{busy\_suc\_fragmented\_packet} \\ & \quad = {\rm Pr} \left\{ \begin{gathered} {\text{with}}\;{\text{the}}\;{\text{probability}}\;{\text{"r"}}\;{\text{exactly}}\;{\text{one}}\;{\text{station}}\;{\text{successfully}}\;{\text{transmits}}\;{\text{allowing}}\;{\text{the}}\;{\text{reservation}} \hfill \\ {\text{of}}\;{\text{the}}\;{\text{channel}}\;{\text{or,}}\;{\text{with}}\;{\text{the}}\;{\text{probability}}\;{\text{"r1"}}\;{\text{this}}\;{\text{station}}\;{\text{that}}\;{\text{reserved}}\;{\text{the}}\;{\text{channel}}\;{\text{continues}} \hfill \\ {\text{to}}\;{\text{successfully}}\;{\text{transmit}}{.} \hfill \\ \end{gathered} \right\} \\ & \quad = n\tau (1 - \tau )^{n - 1} \left( {1 - p_{e} } \right) + n\tau_{1} \left( {1 - p_{e} } \right) \\ \end{aligned}$$

(12)

$$\Pr \left\{ {E_{failed\_fragment} } \right\} = P_{busy\_failed\_fragment} = 1 - \left( {P_{idle} + P_{busy\_suc\_fragmented\_packet} } \right)$$

(13)

$$\begin{aligned} & \Pr \left\{ {E_{suc} } \right\} = P_{busy\_suc} = P_{busy\_suc\_fragmented\_packet} { (}such\;that\;\delta = 1) \\ & \quad = \Pr \left\{ {{\text{with}}\;{\text{the}}\;{\text{probability}}\;{\text{"r"}}\;{\text{exactly}}\;{\text{one}}\;{\text{station}}\;{\text{successfully}}\;{\text{transmits}}{.}} \right\} \\ & \quad = n\tau (1 - \tau )^{n - 1} \left( {1 - p_{e} } \right) \\ \end{aligned}$$

(14)

$$\Pr \left\{ {E_{failed} } \right\} = P_{busy\_failed} = P_{busy\_failed\_fragment} { (}such\;that\;\delta = 1) = 1 - \left( {P_{idle} + P_{busy\_suc} } \right)$$

(15)

By building on the parameters $T_{suc}^{{}}$ and $T_{failed}^{{}}$ calculated in the literature [24, 25] that respectively represent the time duration of the event $E_{suc}$ and $E_{failed}$, we can determine $T_{suc\_fragmented\_packet}$ and $T_{failed\_fragment}$ which respectively represent the time duration of the event $E_{{{suc\_fragmented\_packet} }}$ and $E_{failed\_fragment}$.

$$T_{suc}^{{}} = T_{H} + \frac{{{\text{MAC\_}}payload}}{R} + \partial + SIFS + T_{ACK} + \partial + DIFS$$

(16)

$$T_{failed}^{{}} = T_{H} + \frac{{{\text{MAC\_}}payload}}{R} + \partial + {\text{ACK\_timeout}} + DIFS$$

(17)

$$T_{suc\_fragmented\_packet} = \frac{1}{\delta } \cdot \left( {T_{H} + \delta \cdot \frac{MAC\_payload}{R} + \partial + SIFS + T_{ACK} + \partial + SIFS} \right) - SIFS + DIFS$$

(18)

$$T_{failed\_fragment} = T_{H} + \delta \cdot \frac{{{\text{MAC\_}}payload}}{R} + \partial + {\text{ACK\_timeout}} + DIFS$$

(19)

The symbols SIFS, DIFS and $T_{ACK}$ respectively represent the time duration of a SIFS, a DIFS and an acknowledgment including the physical header (i.e. $T_{ACK} = \frac{{{\text{ACK}}}}{R}$). $R$ is the channel bit rate, $\partial$ is the propagation delay and $MAC\_payload$ is the average length of MAC payload. $T_{H}$ is the time duration of a physical header and a MAC header (i.e. $T_{H} = \frac{{{\text{ PHY\_Header}} + {\text{MAC\_Header}}}}{R}$). Similar to what has been defined in [10, 22], the ${\text{ACK\_timeout}}$ value is the sum of one SIFS and the time duration of an ACK frame.

The average length of a time slot $T_{avg}^{{}}$ is calculated summing the products of each possible event probability by its time duration. Thus, the average length of a time slot $T_{avg}^{{}}$ is expressed as:

$$\left\{ {\begin{array}{*{20}l} {P_{idle} \sigma + P_{busy\_suc} \, T_{suc} + P_{busy\_failed} \, T_{failed} } \hfill & {if\;\delta = 1\;(i.e.\;without\;fragmentation)} \hfill \\ {P_{idle} \sigma + P_{busy\_suc\_fragmented\_packet} \, T_{suc\_fragmented\_packet} + P_{busy\_failed\_fragment} T_{failed\_fragment} } \hfill & {else} \hfill \\ \end{array} } \right.$$

(20)

The symbol $\sigma$ represents an empty slot time.

$Normalized\_throughput_{{{{{\text{basic}}}} }}$ is the normalized throughput that is calculated as the fraction of average time slot length the channel is used to successfully transmit the MAC payload bits. From where:

$$Normalized\_throughput_{{{\text{basic}}}} = \frac{{P_{busy\_suc\_fragmented\_packet} \, \cdot \frac{{{\text{MAC\_}}payload}}{R}}}{{T_{avg} }}$$

(21)

By putting the value of $\delta$ to 1 in the formula (21), we can deduce $Normalized\_throughput_{{{\text{basic}}}}$ of the transmission without fragmentation obtained in the literature.

4 Numerical Experimentation and Results

As it is mentioned in Sect. 3.1, the equations of our nonlinear system will be solved together using numerical methods. The numerical solution is obtained by using the function Find () in Mathcad software [26]. The obtaining of the unique values for $\tau$,$\tau_{1}$, $p$, $p_{b}$ and $p_{f}$ allows determining the normalized throughput expressed in Sect. 3.2.

The system parameters values used for the results follow the values employed according to IEEE 802.11b standard and are listed in Table 1. As in [15, 22, 23], the both maximum retry limit and minimum retry limit are the same and here, they are put at 5.

Table 1 System parameters

4.1 Our normalized Throughput Results and Comparison

Figure 4 represents the normalized throughput of the IEEE 802.11 saturated network according to the number of stations, which transmit in basic DCF without or with fragmentation or in DCF RTS/CTS mode, when the wireless channel is lightly disturbed at $BER = 10^{ - 5}$ or perfect (i.e.,$BER = 0$). The results concerning the transmission with fragmentation are obtained by assuming that each packet will be divided into two fragments (i.e., $\delta = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}$). In Fig. 4, we illustrate the precision and improvement of our model by comparing its analytical results with those obtained from Yang-Xiao’s model [23] and the model in [15]. In addition, we compare improvement in basic DCF throughput with each of the fragmentation and RTS/CTS mechanisms. We have selected Yang-Xiao’s model, as it is the reference analytical model that analyzes the basic DCF under more realistic assumptions than those in the Bianchi’s model. We have used Hadzi-Velkov’s model of the DCF in RTS/CTS mode [7] as a point of comparison to the models of basic DCF with fragmentation, since it is a classical model and the first model that analytically analyzes the impact of an error-prone channel over the throughput in an IEEE 802.11 saturated network.

Fig. 4

Normalized throughput versus the number of stations

As shown in Fig. 4a, b, the throughput of the basic DCF without or with fragmentation decreases as the number of contending stations increases, while the throughput of the DCF in RTS/CTS mode is almost maintained. We explain this by the fact that the increased number of stations, which participate in the access to the channel, causes the increase of the overhead required to reserve the channel for the transmission in basic mode without or with fragmentation, while the overhead is almost maintained for the transmission in RTS/CTS mode due to the smaller length of the RTS frames, which are used to contend the channel. Under the perfect channel case, Fig. 4a shows that when transmitting in basic DCF without fragmentation, the analytical results from our model, as opposed to those obtained from the model in [15], are close to those obtained from the Yang-Xiao’s model. This means that our analytical model is more accurate than the model in [15] because our model takes into account, as the Yang-Xiao’s model, the probability that the backoff counter is stopped when the channel is sensed busy. As also shown in Fig. 4a, the basic DCF throughput improves with the two mechanisms, the fragmentation or the RTS / CTS. Whereas, the RTS/CTS mechanism is the best solution to improve the throughput of IEEE 802.11 networks under the perfect channel as expected, because it is the mechanism which can confront the collision phenomenon at the lowest cost. Under the lightly disturbed channel at $BER = 10^{ - 5}$, Fig. 4b shows that when transmitting in basic DCF without fragmentation, the analytical model in [15] overestimate the basic DCF throughput because it does not consider the bit error effect on acknowledgment frames, and uses a smaller $T_{failed\_fragment}$ that is denoted by $T^{m}$ in [15]. However, their analytical model underestimates the basic DCF throughput when applying the fragmentation mechanism and consequently, does not recommend the transmission with fragmentation in the lightly disturbed IEEE 802.11 basic DCF network, because their analytical model suffers with high collision probability and low successful transmission probability (see Figs. 5 and 6). Whereas, our accurate analytical model show that the transmission with fragmentation in basic DCF provides a higher throughput value than the transmission in RTS/CTS mode, regardless to network size. This is due to the effect that the fragmentation mechanism has on reducing the error probability occurring during the packet transmission and therefore, increasing the successful transmission probability.

Fig. 5

Collision probability versus the number of stations

Fig. 6

Successful transmission probability versus the number of stations

Figure 7 shows that the fragmentation mechanism significantly improves the basic DCF throughput using our model compared with the analytical model in [15], regardless of the fragment size and number of stations in the lightly disturbed network at $BER = 10^{ - 5}$. Where, throughput improves by about 7.5% and 8.1% when the packet is fragmented into two and three, respectively.

Fig. 7

Normalized throughput versus the number of stations

The interesting observation from Fig. 7 is that the optimal fragment size, which offers the maximum throughput in the lightly disturbed network at $BER = 10^{ - 5}$, is 750 bytes. This is the size that represents the compromise between the benefit of reducing the error probability and the additional time spent on transmission of the fragmented packet (i.e., the time spent on DIFS, SIFS and transmission of PHY and MAC headers).

5 Conclusion

In this paper, we have used the Markov chains to model the basic DCF mechanism with fragmentation in a lightly disturbed IEEE 802.11 network under saturated traffic. Then, we accurately have distinguished all the probabilities and time durations of the possible events involved in determining the average time slot length. Finally, we were inspired from the Bianchi’s calculation to evaluate the throughput of IEEE 802.11 DCF network, which uses the fragmentation in basic access mode at $BER = 10^{ - 5}$. We have found that the throughput obtained using our model is significantly improved compared with the existing model, regardless to network and fragment size by decreasing the collision probability and increasing the successful transmission probability. Our accurate analytical model has determined the optimal fragment size and shown that the transmission with fragmentation in basic DCF mode, which is not recommended in existing studies at $BER = 10^{ - 5}$, improves the throughput more than the transmission in RTS/CTS mode, regardless to IEEE 802.11 network size. As future work, it is interesting to also evaluate the end-to-end delay and to extend this evaluation under unsaturated traffic conditions. Exactly, our analytical model can be reused to improve existing studies in the literature based on the mathematical models developed using the Markov chains, which model the DCF mechanism with fragmentation in IEEE 802.11 network under imperfect channel.