## 5 Service Delay Statistical Model  
In this section, the total service delay in 802.11 networks is analyzed to obtain its PGF. We define service delay as the period from the time when the data frame arrives at the HOL of the sending node to the time when it is successfully received by the receiving node. During this period, a node may detect or go through different channel activities, such as whether the node succeeded or failed in sending the data frames and the success or failure of the other \(n-1\) nodes to send data frames and channel leisure [13]. Let \(T_{s}\), \(T_{c_{j}}\), \(T^{\prime}_{s_{j}}\), \(T^{\prime}_{c_{j}}\), and \(T_{idle}\) denote the time when node B successfully transmits a data frame, the node B transmission collision time, the time when the other \(n-1\) nodes successfully transmit the data frames, the other \(n-1\) nodes transmission collision times, and the channel's idle duration, respectively. Then, the service delay, \(T_{w}\), can be expressed as:  
\[T_{w}=T_{s}+\sum\
olimits_{j=1}^{n_{c}}T_{c_{j}}+\sum\
olimits_{j=1}^{n_{c}^{ \prime}}T^{\prime}_{c_{j}}+\sum\
olimits_{j=1}^{n_{s}^{\prime}}T^{\prime}_{s_{ j}}+T_{idle} \tag{18}\]  
where \(n_{c}\), \(n^{\prime}_{c}\), and \(n^{\prime}_{s}\) denote the number of node B retransmissions and the number of collided transmissions and successful transmissions for the other \(n-1\) nodes during \(\overline{B\O}\), respectively. Regarding them, the following conclusions hold:  
**Conclusion 1:**_The number of a node retransmissions, \(n_{c}\), obeys a truncated geometric distribution. Thus,_  
\[p(n_{c}=i)=\frac{{p_{s}(1-{p_{s}})}^{i}}{{1-{(1-{p_{s}})}^{m+1}}}i=0,\ \ \ldots,\ m. \tag{19}\]  
**Conclusion 2:**_If the B node has successfully completed its \((r-1)\)-th transmission and started the backoff procedure, the \(n\) nodes are trying to access the channel at this time. The \(\overline{B\O}\) events and \(k-1\) successful transmissions occur for the remaining \(n-1\) nodes, attempting to access the channel during the node B backoff period. Let the random variable \(v\) denote the number of times that the remaining \(n-1\) nodes attempt to send a data frame before node B successfully transmits its \(r(=k)\)-th data frame. Then, \(v\) obeys the conditional probability distribution:_  
\[p(v|k)=C_{v-1}^{k-1}p_{s}^{\prime k-1}\big{(}1-p_{s}^{\prime}\big{)}^{v-k} \times p_{s}. \tag{20}\]  
_where \(p_{s}^{\prime}=\frac{{(n-1)\times\tau\times(1-\tau)}^{n-2}}{{1-{(1-\tau)}^{n -1}}}\)._  
**Conclusion 3:**_Define \(n^{\prime}_{c}=v-k\), i.e., the number of collisions that occur when the remaining \(n-1\) attempt to send the data frames before the node makes the \(k\)-th successful transmission. Then, the conditional probability distribution of \(n^{\prime}_{c}\) is:_  
\[p(n^{\prime}_{c}|k)=C_{n^{\prime}_{c}+k-1}^{k-1}p_{s}^{\prime k-1}\big{(}1-p _{s}^{\prime}\big{)}^{n^{\prime}_{c}}\times p_{s}. \tag{21}\]  
The following analyzes \(n^{\prime}_{s}\). We find that there must be only busy or idle events in the time period between two consecutive transmissions of a node, and the two events occur alternately in the 802.11 DCF protocol. The length of time of each idle or busy event that occurs depends on the size of the transmitted data frame and the opportunity of the node to transmit the data frame, etc., while the average time for idle events to occur is:  
\[L_{idle}=\frac{1}{p_{b}}-1=\frac{1-p_{b}}{p_{b}}. \tag{22}\]  
Due to channel idle events occurring independently, it obeys the geometric distribution of parameter \(p_{b}\). Let \(N\) represent the number of idle events (counted in a slot time), then:  
\[N=\frac{\overline{B\O}}{L_{idle}}=\frac{\overline{B\O}}{\frac{1}{p_{b}}-1}= \frac{\overline{B\O}\cdot p_{b}}{1-p_{b}} \tag{23}\]Furthermore, since the busy or idle state of the channel in the DCF protocol takes place alternately successively over time, it is impossible for the idle or busy to occur twice in succession.