We formulate the analytical evaluation of proposed COSB mechanism with saturation throughput and average delay, on the assumption of ideal channel conditions, i.e. no hidden terminal and capture effects. In the analysis, we assume the fixed number of WEs, each of which is always willing to transmit the data frame, i.e., the network is assumed as a saturated traffic environment. Initially, we study the behavior of a tagged WE with a discrete-time Markov chain model (DTMC) [

8,

9], and we obtain the stationary transmission probability

$\gamma $ for the tagged WE. Since the proposed COSB does not reset the backoff stage to its initial value (that is to zero) after successful transmission, the transmission attempt for every new data frame remains recursive within the backoff stage state dimension. To accurately analyze the performance of COSB, we formulate a recursive discrete-time Markov chain model (R-DTMC). Later, by knowing the exact events that can occur on the communication channel within a randomly selected slot-time, we formulate the normalized throughput and average delay of the proposed COSB mechanism.

#### 4.1. Recursive Discrete-Time Markove Chain (R-DTMC) Model

Consider there are $n$ number of WEs competing for the channel in a WLAN. In the saturated condition, each WE has immediately a data frame available for transmission after each successful transmission. Thus, due to the consecutive data frame transmission, each data frame needs to wait for a random backoff time before transmitting.

Let $b$ be the backoff stage counter for a tagged WE and $m$ be the maximum number of backoff stages $b$ can experience for a data frame, that is $b\in \left(0,m\right)$, such that ${W}_{b}={2}^{b}\times {W}_{min}\times {\omega}^{{p}_{obs}}$ for b^{th} backoff stage and ${W}_{max}={2}^{m}\times {W}_{min}\times {\omega}^{{p}_{obs}}$ for the ${m}^{th}$ backoff stage contention window, where ${W}_{b}$ is the contention window size at b^{th} backoff stage and ${p}_{obs}$ is the observed channel collision probability. Let us adopt the notation ${W}_{b+1}={2}^{b+1}\times {W}_{min}\times {\omega}^{{p}_{obs}}$, for the adaptively scaled-up contention window for $b+1$ backoff stage, when transmission is failed at the ${b}^{\mathrm{th}}$ backoff stage. Similarly, let ${W}_{b-1}={2}^{b-1}\times {W}_{min}\times {\omega}^{{p}_{obs}}$ be the adaptively scaled-down contention window for the $b-1$ backoff stage, when successfully transmitted at the ${b}^{\mathrm{th}}$ backoff stage.

Assume

$\mathrm{\Omega}\left(t\right)$ is the function for the stochastic process representing the backoff counter

$u$ for a tagged WE, where

$u\in \left(0,{W}_{cur}-1\right)$. Since time is discretized as an integer time scale,

$t$ and

$t+1$ correspond to the beginning of two consecutive transmission time slots, and the backoff time counter of each WE decreases at the beginning of each slot time.

Figure 3 illustrates that the backoff time decreases when the communication channel is sensed as idle (

$\sigma $), and it stops when the channel is sensed as busy, which may be due to a successful or unsuccessful transmission of any other WE. Therefore, the time interval between two consecutive slot time beginnings may be much longer and different from the idle slot time size, i.e.,

$\sigma $. Let

$\pi \left(t\right)$ be the stochastic process representing the backoff stages

$\left(0,\text{}1,\text{}2,\text{}\dots ,\text{}m\right)$ of the tagged WE at time

$t.$ The key articulation in our R-DTMC model is that, at each data frame transmission attempt regardless of the number of retransmission attempts, each data frame collides with a practically observed and independent collision probability

${p}_{obs}.$ With these assumptions, COSB can be modeled as the two dimensional process

$\left\{\pi \left(t\right),\text{}\mathrm{\Omega}\left(t\right)\right\}$ with the R-DTMC as depicted in

Figure 5. In this R-DTMC, the transition probabilities are described as follows.

The tagged WE remains at the first backoff stage after a successful transmission on the first backoff stage with the probability,

The backoff counter decreases when the channel is sensed as idle with the probability,

The tagged WE scales-up the current contention window and moves to the next stage

$b$ if a data frame transmission failed on backoff stage

$b-1$ with the probability,

The tagged WE scales-down the current contention window and decreases its backoff stage for the next transmission attempt to

$b-1$ after a successful transmission on backoff stage

$b$ with the probability,

The tagged WE remains at the

${m}^{th}$ backoff stage after an unsuccessful transmission with the probability,

In particular, to the above transition probabilities, as considered in Equation (6), when a data frame transmission is collided at backoff stage

$b-1$, the backoff stage increases to

$b$, and the new backoff value is uniformly chosen from the adaptively scaled-up contention window

${W}_{b}$. On the other hand, Equation (7) describes how when a data frame transmission is successful at backoff stage

$b$, the backoff stage decreases to

$b-1$, and the new backoff value is uniformly chosen from the adaptively scaled-down contention window

${W}_{b-1}$. In case the backoff stage reaches the value

$m$ (that is the maximum backoff value), it is not increased in the subsequent data frame transmission attempt. Let us assume that

${d}_{b,\text{}u}=\underset{t\to \infty}{\mathrm{lim}}P\left\{\pi \left(t\right)=b,\text{}\mathrm{\Omega}\left(t\right)=u\right\},b\in \left(0,m\right),\text{}u\in \left(0,{W}_{b}-1\right)$ be the stationary distribution of the R-DTMC. From

Figure 5, each state transition probability can be written as,

If β =

$\frac{{p}_{obs}}{1-{p}_{obs}}$, the above equation can be written as,

${d}_{1,0}=\mathsf{\beta}\times {d}_{0,0}$. Similarly,

${d}_{b,0}=\mathsf{\beta}\times {d}_{b-1,0}$ where

${d}_{b-1,0}=\mathsf{\beta}\times {d}_{b-2,0}$ till

${d}_{1,0}=\mathsf{\beta}\times {d}_{0,0}$. Therefore, we can write:

Now for the backoff stage

$m$, the

${d}_{m,0}$ can be written as,

Owing to the Markov process-based chain regularities, for each

$u\in \left(1,{W}_{b}-1\right)$, the stationary distribution for {

$\pi \left(t\right),\text{}\mathrm{\Omega}\left(t\right)$} can be written as,

The recursive characteristic of state transition probabilities can be combined as,

From Equations (9)–(11) and (13), Equation (12) can be re-written as,

From Equations (9)–(11) and (14), all the values

${d}_{b,u}$ are expressed as a function of

${d}_{0,0}$ and channel observation-based practical conditional collision probability

${p}_{obs}$.

${d}_{0,0}$ is finally determined by normalizing the R-DTMC states as follows,

From

${W}^{*}={W}_{min}\times {\omega}^{{p}_{obs}}$ and few mathematical steps, the above normalization relation can be written as,

Finally, we get

${d}_{0,0}$ as follows,

Since, a transmission occurs only when the backoff counter of the WE reaches zero regardless of the backoff stage, transmission probability

$\gamma $ can be expressed as follows,

Furthermore, after performing a few mathematical steps to Equation (18) using the value of

${d}_{0,0}$ from Equation (17) we get,

However, in general

$\gamma $ depends on the practical collision probability

${p}_{obs}$, which is always unknown until the channel is observed for the busy slots. A transmitted data frame encounters the collision if at least one of the

$n-1$ remaining WEs transmit. Since each of the transmissions in the system sees this collision in the same state, a steady state can easily be yielded as [

9],

These two ($\gamma $ and ${p}_{obs}$) are monotonic non-linear systems which can be numerically solved for each other.

#### 4.2. Normalized Throughput

Let

θ be the normalized throughput of the network and be defined as the fraction of the communication channel used for successful transmission of the data payload. To compute

θ, let

${\gamma}_{tr}$ be the probability that there is at least one transmission in the considered slot time. Since there are

$n$ number of WEs in the system contending for the medium and each transmits with probability

$\gamma $, the transmission probability

${\gamma}_{tr}$ can be defined as,

If the probability

${\gamma}_{s}$ that a transmission is successful is given by the probability that only one WE transmits in the considered slot time,

${\gamma}_{s}$ can be obtained as,

Thus,

θ can be expressed as the ratio,

Assume

$E\left[P\right]$ is the average data frame payload size (assuming that all the data frames have the same fixed size), then the slot time for transmitting average payload data successfully can be obtained as

${\gamma}_{tr}{\gamma}_{s}E\left[P\right]$, since

${\gamma}_{tr}{\gamma}_{s}$ is the probability for the successful transmission of a data frame in a given slot time. The average length of a given slot time is the sum of three cases; no transmission in a slot time that is

$\left(1-{\gamma}_{tr}\right)\sigma $, a successfully transmitted data frame that is

${\gamma}_{tr}{\gamma}_{s}$, and a collision that is

${\gamma}_{tr}\left(1-{\gamma}_{s}\right)$. Finally, the relation (23) can be written as follows:

where

${T}_{s}$ and

${T}_{c}$ are the average time the communication channel has been busy due to successful transmission and collision, respectively. For analytical evaluation, the values of

$E\left[P\right]$,

${T}_{s},\text{}{T}_{c}$, and idle slot time

$\sigma $ must be expressed with the same time unit. Let

${P}_{hdr}=PH{Y}_{hdr}+MA{C}_{hdr}$ be the time to transmit a data frame header, and

$\delta $ be the channel propagation delay. If ACK is the time to receive an acknowledgement,

${T}_{s}$ and

${T}_{c}$ can be obtained as,

The corresponding values for

${T}_{s}$ and

${T}_{c}$ depend upon the 802.11 standard. The PHY and MAC layer parameters to compute

${T}_{s}$ and

${T}_{c}$ are shown in

Table 1.