# 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  
### The Analytical Model  
Consider again the model of [23] that provides service to \(K\)L service-classes. In the proposed qr-P-S/res model, new calls of service-class (\(k,l\)) are generated from a finite number of sources \(N_{kl}\). The mean arrival rate of idle service-class (\(k,l\)) sources is given by \(\lambda_{kl,\text{idle}}=(N_{kl}-n_{kl})v_{kl}\), where \(n_{kl}\) is the number of in-service calls of service-class (\(k,l\)) and \(v_{kl}\) is the per idle source arrival rate. Based on the above, the per idle source offered traffic-load of service-class (\(k,l\)) is determined by \(\alpha_{kl,\text{idle}}=v_{kl}/\mu\) (in erlang). If \(N_{kl}\rightarrow\infty\) for all service-classes and the total offered traffic-load is constant, then we have the P-S/res model (since the arrival process becomes Poisson). Upon its arrival, a service-class (\(k,l\)) call (\(k=1,\,\ldots,\,K\) and \(l=1,\,\ldots,\,L\)) requires \(b_{l}\) subcarriers. Let the occupied subcarriers and power in the cell be \(j_{1}\) and \(j_{2}\), respectively, when the new call arrives. Then, the admission mechanism for the new call is as follows: (a) if \((M-j_{1}\geq b_{l})\cap(j_{2}+p_{k}b_{l}\leq P)\) then the service-class (\(k,l\)) call is accepted in the cell with probability \(1-pb_{k,l}(j_{1})\). In that case, the service-time is generally distributed with mean \(\mu^{-1}\), (b) if \((M-j_{1}<b_{l})\cup(j_{2}+p_{k}b_{l}>P)\) then the call is blocked due to subcarriers' unavailability. Let \(q_{\text{fin}}(\overrightarrow{j})=q_{\text{fin}}(j_{1},j_{2})\) be the occupancy distribution in the proposed qr-P-S/res model. To prove a formula for the unnormalized values of \(q_{\text{fin}}(j_{1},j_{2})\)'s, we assume that local balance exists between the states \((j_{1}-b_{l},j_{2}-p_{k}b_{l})\) and \((j_{1},j_{2})\). The form of local balance equation is as follows:  
\[\big{(}N_{kl}-y_{kl,\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\big{)}\alpha_{kl,\text{idle}}\big{(}1-pb_{k,l}(j_{1}-b_{l})\big{)}q_{\text{fin}}(j_{1}-b_{l}, j_{2}-p_{k}b_{l})=y_{kl,\text{fin}}(j_{1},j_{2})q_{\text{fin}}(j_{1},j_{2}), \tag{16}\]  
where \(y_{kl,\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\) and \(y_{kl,\text{fin}}(j_{1},j_{2})\) refer to the mean number of service-class (\(k,l\)) calls in states \((j_{1}-b_{l},j_{2}-p_{k}b_{l})\) and \((j_{1},j_{2})\), respectively. Multiplying both sides of (16) by \(b_{l}\) and summing over \(k\) and \(l\) we obtain:  
\[\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}\big{(}N_{kl}-y_{kl,\text{fin}}(j_{ 1}-b_{l},j_{2}-p_{k}b_{l})\big{)}\alpha_{kl,\text{idle}}b\big{(}1-pb_{k,l}(j_{ 1}-b_{l})\big{)}q_{\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})=j_{1}q_{\text{fin} }(j_{1},j_{2}). \tag{17}\]  
The value of \(y_{kl,\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\) in (17) is unknown. To determine it, the following lemma is necessary [59]: Two stochastic systems will be equivalent and result in the same congestion probabilities, if they have (a) the same traffic parameters \((K,L,N_{kl},\alpha_{kl,\text{idle}})\), where \(k=1,\,\ldots,\,K,l=1,\,\ldots,\,L\) and (b) are the same states. Therefore, the purpose is to find a stochastic system, whereby the values of \(y_{kl,\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})\) can be determined. The subcarriers' requirements of calls of all service-classes and the values of \(M\) and \(P\) in the new system are chosen so that both conditions (a) and (b) are valid and the occupancy \((j_{1},j_{2})\) of each state \(\overrightarrow{j}\) is unique. In that case, state \((\overrightarrow{j})=(j_{1},j_{2})\) is reached only via state \((j_{1}-b_{l},j_{2}-p_{k}b_{l})\). Thus, \(y_{kl,\text{fin}}(j_{1}-b_{l},j_{2}-p_{k}b_{l})=n_{kl}-1\). Based on the above, (17) can be written as:  
\[q_{\text{fin}}(j_{1},j_{2})=\left\{