# An Analytical Framework in OFDM Wireless Networks Servicing Random or Quasi-Random Traffic
## 2 The P-S Multirate Loss Model--A Review  
To describe the P-S multirate loss model, consider the downlink of an OFDM-based cell that has \(M\) subcarriers and let \(B\), \(P\), and \(R\) be the system's bandwidth, the cell's available power and the average data rate per subcarrier, respectively. We consider that the range of channel gains (i.e., signal to noise ratios per unit power) is partitioned into \(K\) intervals that are consecutive but do not overlap and denote the average channel gain of each interval \(k\) as \(\gamma_{k}\), \(k=1\),..., \(K\). In addition, we assume \(L\) subcarrier requirements which result in \(KL\) service-classes. Upon its arrival, a service-class (\(k\),\(l\)) call (\(k=1\),..., \(K\) and \(l=1\),..., \(L\)) requires a total of \(b_{l}\) subcarriers. This means that the requirement of each new call in terms of data rate is \(b_{l}R\). In addition, the new call has an (average) channel gain \(\gamma_{k}\). If the \(b_{l}\) subcarriers are not available, then call blocking occurs and the call is lost without affecting the system (i.e., a blocked call cannot retry to be accepted in the cell with the same or less subcarrier requirements). Otherwise, the call is accepted in the cell and remains for a service time which is generally distributed with mean \(\mu^{-1}\). To achieve the data rate \(R\) of a subcarrier assigned to a call (with channel gain \(\gamma_{k}\)), we determine the power \(p_{k}\) via the Shannon theorem: \(R=(B/M)\log_{2}(1+\gamma_{k}p_{k})\). Assuming that service-class (\(k\),\(l\)) calls arrive in the cell according to a Poisson process with arrival rate \(\lambda_{kl}\) and that \(n_{kl}\) is the number of in-service calls of service-class (\(k\),\(l\)), then the steady-state probabilities \(\pi(\mathbf{n})\) are given via a PFS [23]:  
\[\pi(\mathbf{n})=G^{-1}\!\left(\prod\limits_{k=1}^{K}\prod\limits_{l=1}^{L}\alpha_ {kl}^{n_{kl}}\Big{/}n_{kl}!\right), \tag{1}\]  
where \(\mathbf{n}=(n_{11},\ldots,n_{k1},\ldots,n_{k1},\ldots,n_{1L},\ldots,n_{kL},\ldots, n_{kL})\)\(G=\sum\limits_{n\in\mathbf{\Omega}}\left(\prod\limits_{k=1}^{K}\prod\limits_{l=1}^{L} \alpha_{kl}^{n_{kl}}/n_{kl}!\right)\mathbf{\Omega}\) is the system's state space with \(\mathbf{\Omega}=\left\{\mathbf{n}:0\leq\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}n_{ kl}b_{l}\leq M,0\leq\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L}p_{k}n_{kl}b_{l} \leq P\right\}\) and \(\alpha_{kl}=\lambda_{kl}/\mu\) refers to the offered traffic-load (in erlang) of service-class (\(k\),\(l\)) calls. To derive (1), both \(P\) and \(p_{k}\) should be integers. This is achieved by multiplying them by a constant so as to obtain an equivalent representation of \(\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{P}p_{k}^{\prime}n_{kl}b_{l}\leq P^{\prime}\), where \(P^{\prime}\) and \(p_{k}^{\prime}\) are integers. Thus, it can be assumed that \(P\) and \(p_{k}\) are integers, without loss of generality. According to [23], all performance metrics are based on (1). As a specific example, consider the CBP \(B(k\),\(l)\) of service-class (\(k\),\(l\)) calls which is determined via  
\[B(k,l)=1-G(P-p_{k}b_{l},M-b_{l})/G(\mathbf{\Omega}). \tag{2}\]  
However, since \(\mathbf{\Omega}\) grows as \((M\!P)^{KL}\), the applicability of (1) (and consequently of (2)) can only be limited to moderate size systems and therefore is not adequate for procedures related to network planning. In [23], the determination of \(G(P-p_{k}b_{l},M-b_{l})\) (and consequently the determination of CBP via (2)) is based on the algorithms of [39; 40]. These algorithms were initially proposed for the CBP calculation in circuit-switched networks [41; 42]. The algorithms of [39] are based on mean-value analysis and z-transforms. The algorithm of [40] is based on the quite complex approach of numerical inversion of generating functions [43]. Both algorithms are applied to loss models that have a PFSand are impractical compared to the formula of Kaufman-Roberts (K-R) [44; 45].