# Wireless Networks under a Backoff Attack: A Game Theoretical Perspective
## 3 Network Throughput under Backoff Modification  
### Theoretical Network Throughput  
The 802.11 standard does not provide a way to estimate the network throughput that is achieved. The best-known model to estimate the throughput in a network is [PERSON]'s model [22], which provides expressions both for BA and RTS/CTS mechanisms. The main advantage of this model is that it provides analytical expressions to determine the network throughput. It assumes saturation of the network, that is, that each station always has a packet to transmit. This assumption could be relaxed using more complex models (as [23]). The CSMA/CA mechanism described in Section 2 assumes that all stations will respect the backoff procedure. However, the stations can modify their backoff in such a way that they can obtain a higher throughput, at expense of other stations [4,15]. In order to analyze these effects, we will use [PERSON]'s model [22] to estimate the total network throughput. The results will be used in the posterior sections to study how to enforce network throughput fairness. This model relies on the computation of the following system for each of the \(i\) stations of the network:  
\[\left\{ \begin{array}{l}\tau_{i}=\frac{2}{1+W_{i}+p_{i}W_{i}\sum_{j=0}^{n_{i} -1}(2p_{i})^{j}}\\ p_{i}=1-\prod_{j\
eq i}(1-\tau_{j})\end{array} \right. \tag{1}\]  
where \(p_{i}\) is the collision probability for station \(i\) (the probability that station \(i\) observes a collision while transmitting a packet, which [PERSON]'s model assumes to be constant) and \(\tau_{i}\) is the probability that station \(i\) transmits a packet. The system (1) assumes a binary exponential backoff, where the contention window \(CW\) lies in the interval \([W,CW_{max}]\), where \(m\) is the maximum backoff stage, defined as \(CW_{max}=2^{m}W\) where \(W\) is the minimum size of the contention window. Let us assume that we have a network with \(n\) stations, split into two different classes. There are \(n_{1}\) stations characterized by using a binary exponential backoff as described by IEEE 802.11 standard, and thus, following (1). Also, there are \(n_{2}=n-n_{1}\) stations using a uniformly distributed backoff in the range \([0,W_{2}-1]\), whose expression [22] is:  
\[\tau_{i}=\frac{2}{1+W_{i}} \tag{2}\]The probabilities \(\tau_{i}\) and \(p_{i}\) are the same for all the members of each class. Hence, (1) becomes:  
\[\left\{ \begin{array}{l}\tau_{1}=\frac{2}{1+W_{1}+p_{1}W_{1}\sum_{j=0}^{n_{1}-1}( 2p_{1})^{j}}\\ \tau_{2}=\frac{2}{1+W_{2}}\\ p_{1}=1-(1-\tau_{1})^{n_{1}-1}(1-\tau_{2})^{n_{2}}\\ p_{2}=1-(1-\tau_{1})^{n_{1}}(1-\tau_{2})^{n_{2}-1}\end{array} \right. \tag{3}\]  
where the subscript \(i\) denotes the class of a station. Now, we will obtain the total throughput of the network [22; 23]. The probability that there is at least one station transmitting is denoted as \(P_{tr}\):  
\[P_{tr}=1-\prod_{i=1}^{n}(1-\tau_{i})=1-(1-\tau_{1})^{n_{1}}(1-\tau_{2})^{n_{2}} \tag{4}\]  
and hence, \(1-P_{tr}\) is the probability that no station is transmitting. The probability that there is exactly one station of class \(i\) transmitting, \(P_{s,i}\), is:  
\[\left\{ \begin{array}{l}P_{s,1}=\tau_{1}(1-\tau_{1})^{n_{1}-1}(1-\tau_{2})^{n _{2}}\\ P_{s,2}=\tau_{2}(1-\tau_{1})^{n_{1}}(1-\tau_{2})^{n_{2}-1}\end{array} \right. \tag{5}\]  
and the probability that there are two or more stations transmitting simultaneously (i.e., the collision probability), denoted by \(P_{c}\), is obtained as the total probability minus the probabilities of having exactly none or one station transmitting:  
\[P_{c}=1-\sum_{i}P_{s}-(1-P_{tr})=P_{tr}-n_{1}P_{s,1}-n_{2}P_{s,2} \tag{6}\]  
Now, we obtain the expected duration of a slot time, \(T_{slot}\). We define \(T_{s}\) as the time to count down a backoff unit (i.e., the time that lies between two consecutive calls to the CS method when the channel was sensed idle), \(T_{t}\) as the time duration of a successful transmission and \(T_{c}\) as the time duration of a collision.