# Probabilistic Retry and Threshold Multirate Loss Models for Impatient Calls
## 2 The Probabilistic Retry Loss Model  
In the probabilistic retry loss model, a link of capacity \(C\) b.u. services calls from \(K\) service-classes. Service-class \(k\) (\(k=1,\,\ldots,\,K\)) calls arrive in the link according to a Poisson process with an arrival rate \(\lambda_{k}\) and have an initial (peak) bandwidth requirement of \(b_{k}\) b.u. If these \(b_{k}\) b.u. are available, then a new service-class \(k\) call remains in the link for a service-time, which is exponentially distributed with mean \(\mu_{k}^{-1}\). If the available link b.u. are less than \(b_{k}\), then the new call is blocked and may: (1) Abandon the link with probability \(1-p_{kr}\) or (2) immediately retry to be connected in the system with probability \(p_{kr}\), reduced bandwidth requirement \(b_{kr}<b_{k}\) and increased service time \(\mu_{kr}^{-1}>\mu_{k}^{-1}\) given by \(\mu_{kr}^{-1}=b_{k}\mu_{k}^{-1}/b_{kr}\) (see Figure 1, where the call admission mechanism of [39] does not include the grey parts). The steady state probabilities in the probabilistic retry loss model cannot be determined via a PFS. However, it is proved in [45] that the computation of the link occupancy distribution, \(q(j)\), with \(j=0,\,1,\ldots,\,C\), is based on the following recursive formula:  
\[q(j)=\left\{ \begin{array}{l}1,\,\,\text{if}\,\,j=0\\ \frac{1}{i}\left(\sum\limits_{k=1}^{K}\alpha_{k}b_{k}q(j-b_{k})+\sum\limits_{ k=1}^{K}\alpha_{kr}p_{kr}p_{kr}\gamma_{k}(j)q(j-b_{kr})\right)\\ j=1,\ldots,C\\ 0,\,\,\text{otherwise}\end{array} \right. \tag{1}\]  
where \(\alpha_{k}=\lambda_{k}\mu_{k}^{-1}\) is the offered traffic-load of service-class \(k\) calls, \(\alpha_{kr}=\lambda_{k}\mu_{kr}^{-1}\) is the retry offered traffic-load of service-class \(k\) calls accepted in the link with their reduced bandwidthrequirement (\(b_{kr}\) b.u. ), \(p_{kr}\) is the retry probability of blocked calls of service-class \(k\) while \(\gamma_{k}(j)=1\) when \(j>C-(b_{k}-b_{kr})\). In order to derive (1), we assume that: (1) Local balance exists between adjacent states, which is correct in models whose steady state probabilities have a PFS and (2) retry calls do not exist in the link (i.e., the occupied link bandwidth from such calls is considered to be negligible) when the link occupancy \(j\) does not exceed the retry boundary, i.e., when \(j\leq C-(b_{k}-b_{kr})\). The second assumption is expressed in (1) via \(\gamma_{k}(j)\). Having computed, via (1), the unnormalized values of \(q(j)\)'s, we can now calculate the CBP of service-class \(k\) calls with their reduced bandwidth requirement of \(b_{kr}\) b.u., \(B_{kr}\), as follows:  
\[B_{kr}=(1-p_{kr})\sum_{j=C-b_{k}+1}^{C-b_{kr}}G^{-1}q(j)+\sum_{j=C-b_{kr}+1}^{ C}G^{-1}q(j) \tag{2}\]  
where \(G=\sum\limits_{j=0}^{C}q(j)\) is the normalization constant. Similarly, we may also obtain the CBP of service-class \(k\) calls with their initial (peak) bandwidth requirement of \(b_{k}\) b.u., \(B_{k}\), and the conditional CBP denoted as \(B_{kr}^{*}=\text{Prob}\{B_{kr}|j>C-b_{k}\}\) via (3) and (4), respectively:  
\[B_{k}=\sum_{j=C-b_{k}+1}^{C}G^{-1}q(j) \tag{3}\]  
\[B_{kr}^{*}=\text{Prob}\{B_{kr}|j>C-b_{k}\}=\frac{B_{kr}}{B_{k}} \tag{4}\]  
Additionally, we can determine the mean number of in-service calls of service-class \(k\), \(\overline{n}_{k}\), accepted in the link with \(b_{k}\) b.u. via the formula:  
\[\overline{n}_{k}=\sum_{j=1}^{C}y_{k}(j)\frac{q(j)}{G} \tag{5}\]  
where \(y_{k}(j)\) refers to the mean number of service-class \(k\) calls with \(b_{k}\) b.u. in state \(j\). The latter can be determined via the formula:  
\[y_{k}(j)=\frac{\alpha_{k}q(j-b_{k})}{q(j)} \tag{6}\]  
Figure 1: The call admission mechanism for a new call in the probabilistic retry loss model. Similarly, we can compute the mean number of in-service calls of service-class \(k\), \(\overline{n}_{kr}\), accepted in the link with \(b_{kr}\) b.u. via the formula:  
\[\overline{n}_{kr}=\sum_{j=1}^{C}y_{kr}(j)\frac{q(j)}{G} \tag{7}\]  
where \(y_{kr}(j)\) refers to the mean number of service-class \(k\) calls with \(b_{kr}\) b.u.