# An Analytical Framework in OFDM Wireless Networks Servicing Random or Quasi-Random Traffic
## 4 The Quasi-Random P-S Multirate Loss Model with Restricted Accessibility
### Performance Measures Calculation  
Having obtained the unnormalized values of \(q_{\rm fin}(j_{1},j_{2})\), we can calculate the TC probabilities of service-class (\(k\),\(l\)) calls, \(B_{\rm TC}(k,l)\), via:  
\[B_{\rm TC}(k,l)=\sum\limits_{\{(j_{1}+b_{l}>M)\}\cup\{(j_{2}+p_{k}b_{l}>P)\}}G ^{-1}q_{\rm fin}(j_{1},j_{2})pb_{k,l}(j_{1}), \tag{19}\]  
and the CC probabilities of service-class (\(k\),\(l\)) calls via (19) but for a cell with \(N_{kl}-1\) sources.  
Furthermore, we can determine the average number of service-class (\(k\),\(l\)) in-service calls, \(E_{\rm fin}(k,l)\), via:  
\[E_{\rm fin}(k,l)=\sum\limits_{j_{1}=1}^{M}\sum\limits_{j_{2}=1}^{P}G_{\rm fin }^{-1}y_{kl,\rm fin}(j_{1},j_{2})q_{\rm fin}(j_{1},j_{2}), \tag{20}\]  
where \(G_{\rm fin}=\sum\limits_{j_{1}=0}^{M}\sum\limits_{j_{2}=0}^{P}q_{\rm fin}(j_{ 1},j_{2})\) and \(y_{kl,\rm fin}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\) is the mean number of service-class (\(k\),\(l\)) calls in state \((j_{1}-b_{l},j_{2}-p_{k}b_{l})\) calculated via:  
\[y_{kl,\rm fin}(j_{1},j_{2})=\frac{(N_{kl}-n_{kl}+1)\alpha_{kl,\rm dirle}\big{(} 1-pb_{k,l}(j_{1}-b_{l})\big{)}q_{\rm fin}(j_{1}-b_{l},j_{2}-p_{k}b_{l})}{q_{ \rm fin}(j_{1},j_{2})}. \tag{21}\]  
In addition, we can determine the entire system BP based on the TC probabilities of all service-classes, \({\rm BP}_{\rm TC}\), the \({\rm SU}_{\rm fin}\), and the \({\rm PU}_{\rm fin}\), via:  
\[{\rm BP}_{\rm TC}=\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}B_{\rm TC}(k,l)N_ {kl}v_{kl}/\Lambda_{\rm fin},\quad\Lambda_{\rm fin}=\sum\limits_{k=1}^{K}\sum \limits_{l=1}^{L}N_{kl}v_{kl}, \tag{22}\]  
\[{\rm SU}_{\rm fin}=\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}E_{\rm fin}(k,l) b_{l}/M, \tag{23}\]  
\[{\rm PU}_{\rm fin}=\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}p_{k}E_{\rm fin}( k,l)b_{l}/P. \tag{24}\]  
In order to determine the values of \(q_{\rm fin}(j_{1},j_{2})\) according to (18), the unknown values of \(n_{kl}\) are required. These values can be obtained via a stochastic system, with the same parameters and the same states as already described for the proof of (18). However, the state space determination of this system becomes complex due to the large number of service-classes. To this end, we propose an algorithm which is simpler and easy to implement:  
(a) Determine the values of \(q(j_{1},j_{2})\) according to (12) (i.e., via the P-S/res model).  
(b) Determine the values of \(y_{kl}(j_{1},j_{2})\) via the formula:  
\[y_{kl}(j_{1},j_{2})=\frac{\alpha_{kl}\big{(}1-pb_{k,l}(j_{1}-b_{l})\big{)}q(j _{1}-b_{l},j_{2}-p_{k}b_{l})}{q(j_{1},j_{2})}. \tag{25}\]  
(c) Modify (18) to the following formula, where \(y_{kl}(j_{1},j_{2})\) is given by (25):  
\[q_{\rm fin}(j_{1},j_{2})=\left\{\begin{array}{ll}1,&{\rm for}j_{1}=j_{2}=0 \\ \frac{1}{h}\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}(N_{kl}-y_{kl}(j_{1}-b_{ l},j_{2}-p_{k}b_{l}))\alpha_{kl,\rm dirle}b_{l}\big{(}1-pb_{k,l}(j_{1}-b_{l}) \big{)}q_{\rm fin}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\\ {\rm for}j_{1}=1,\ldots,M{\rm and}j_{2}=1,\ldots,P\end{array}\right.. \tag{26}\](d) Determine \(E_{\rm fin}(k,l)\) via (20), where the values of \(y_{kl,{\rm fin}}(j_{1},j_{2})\) are given by:  
\[y_{kl,{\rm fin}}(j_{1},j_{2})=\frac{(N_{kl}-y_{kl}(j_{1}-b_{l},j_{2}-p_{k}b_{l}) )\alpha_{kl,{\rm idle}}\big{(}1-pb_{k,l}(j_{1}-b_{l})\big{)}q_{\rm fin}(j_{1}-b_ {l},j_{2}-p_{k}b_{l})}{q_{\rm fin}(j_{1},j_{2})}. \tag{27}\]  
(e) Determine: (1) the TC probabilities of service-class (\(k\),\(l\)) calls, \(B_{\rm TC}(k,l)\), via (19), and (2) the \({\rm BP}_{\rm TC}\), the \({\rm SU}_{\rm fin}\), and the \({\rm PU}_{\rm fin}\), via (22)-(24), respectively.