# Optimized Architecture for Efficient OFDMA Network Design
## 3 Analytical Model for OFDMA Networks
### Scheduling Policies
#### 3.3.1 Conventional Scheduling Policies  
A conventional scheduling policy aims to distribute all resources among the users. As long as there is at least one active mobile (in the transfer state) that is not in failure, all the slots in the current frame will be allocated to that user, ensuring that no resources remain unused. In this study, three conventional scheduling policies corresponding to three specific scheduling regimes are considered. The expression for the departure rate \(\mu(n)\) for the three scheduling policies is given as follows:  
\[\mu n=\frac{\overline{m}(n)N_{S}}{\overline{X}_{ON}T_{F}} \tag{2}\]  
Figure 1: The Multidimensional Continuous-Time Markov Chain. where \(\overline{m}(n)\) is the average number of bits transmitted per slot when there are n active transfers during a TDD frame. The value of \(\overline{m}(n)\) depends on \(K\), the number of MCS (Modulation and Coding Schemes), and \(p_{k}\), the probability vector for each MCS, where \(0\leq k\leq K_{0}\). The calculation of \(\overline{m}(n)\) also depends on \(n\), as the average number of bits per slot must account for all possible combinations of the n mobile users across the \(K+1\) possible coding schemes (including the failure state). Furthermore, \(\overline{m}(n)\) is influenced by the scheduling policy, as the number of slots allocated to each mobile depends on the coding scheme they are using. The expression for \(\overline{m}(n)\), the average number of bits transmitted per slot for n active users, is given by the following equation [13]:  
\[\overline{m}(n) =\sum\
olimits_{(n_{0},\ldots,n_{K})}^{(n_{\ell},\ldots,n)}=(0, \ldots,0)\ \frac{n!}{n_{0}!n_{1}!n_{k}!}\overline{m}(n_{0}|,|\ldots|,|n_{k})\Big{(}\prod \
olimits_{k=0}^{K}p_{k}^{n_{k}}\Big{)} \tag{3}\] \[\qquad\qquad n_{0}+\ldots+n_{K}=n\] \[\qquad\qquad n_{0}\
eq n\]  
\[\overline{m}(n_{0},\ldots,n_{k}) =\sum\
olimits_{k=1}^{K}m_{k}n_{k}x_{k}(n_{0},\ldots,n_{K}) \tag{4}\]  
where \(x_{k}(n_{0}\),...,\(n_{K})\) represents the proportion of resources allocated to a mobile station using MCS\({}_{k}\), given the distribution of the n active users among the \(K+1\) coding schemes (including the failure state), denoted by (\(n_{0}\),...,\(n_{K}\)). 1. Resource fairness: For resource fairness, the scheduling algorithm equally distributes the available slots N\({}_{\text{S}}\) among active users who are not in the failure state during a TDD frame. Thus, for a given distribution (\(n_{0}\),...,\(n_{K}\)) of the active users (\(n=\sum\
olimits_{k=1}^{K}n_{k}\), each of the \(n-n_{0}\) users who are not in the failure state receives an equal share of the total resources. The proportion of resources \(x_{k}(n_{0}\),...,\(n_{K})\) allocated to a user using MCS\({}_{\text{k}}\) is given by \[x_{k}(n_{0},\ldots,n_{K})= \begin{cases}\frac{1}{n-n_{0}}\text{if }k\
eq 0\text{ and }n\
eq n_{0}\\ \qquad\qquad\qquad\qquad\qquad\ 0\text{ else}\end{cases} \] (5) By substituting these proportions into the generic expression (3), the value of \(\overline{m}(n)\), when there are n active users, becomes \[\overline{m}(n) =\sum\
olimits_{(n_{0},\ldots,n)}^{(n_{\ell},\ldots,n_{K})}=(0, \ldots,0)\ \frac{n!}{n-n_{0}}\sum\
olimits_{k=1}^{K}m_{k}n_{k}\Bigg{(}\prod \
olimits_{k=0}^{K}\frac{p_{k}^{n_{k}}}{n_{k}!}\Bigg{)}\] (6) \[\qquad\qquad\qquad n_{0}+\ldots+n_{K}=n\] \[\qquad\qquad\qquad n_{0}\
eq n\]
2. Throughput Fairness: The goal of the throughput fairness policy is to ensure that all active users who are not in failure achieve the same instantaneous throughput. If an active mobile station using MCS\({}_{\text{k}}\) is allocated a proportion of the resources \(x_{k}(n_{0}\),...,\(n_{K})\), its resulting instantaneous throughput will be proportional to \(m_{k}x_{k}(n_{0}\),...,\(n_{K})\).