# Probabilistic Retry and Threshold Multirate Loss Models for Impatient Calls
## 4 Performance Evaluation  
In this section, we consider an application example of a single link and present both analytical and simulation CBP results of the proposed probabilistic threshold loss model as well as the corresponding analytical and simulation results of the probabilistic retry loss model. For reference we also present the corresponding analytical CBP results of the models of [39, 40] together with those obtained via the EMLM. The simulation results presented in this section are mean values of seven runs and are based on Simscript III [46]. In each run, 1 million calls are generated while the initial 2% of these generated calls is not considered in the blocking probability results, so as to account for a warm-up period. Regarding reliability ranges, they are less than two orders of magnitude and therefore are not presented in the following figures. In our application example, two service-classes are considered whose calls are accommodated in a link of 32 b.u. Calls of the first service-class have an initial bandwidth requirement of 6 b.u. and they do not have the option to retry with a reduced bandwidth requirement. On the other hand, calls of the second service-class have an initial (peak) bandwidth requirement of 24 b.u. In the case of the probabilistic retry loss model, blocked calls of this service-class retry with probability \(p_{2r}\) and a reduced bandwidth requirement of 8 b.u. In the case of the probabilistic threshold model, calls of this service-class try to be connected in the link with probability \(p_{2c}=p_{2r}\) and a reduced bandwidth requirement of 8 b.u., when \(j>J_{0}\). The value of the threshold \(J_{0}\) is set to 4 b.u. Regarding the values of \(p\) we consider the cases of: (a) \(p_{2r}=p_{2c}=0.2\) and (b) \(p_{2r}=p_{2c}=0.6\). Calls of both service-classes arrive in the link according to a Poisson process with arrival rates \(\lambda_{1}=0.8\) and \(\lambda_{2}=0.1\) while their service time is exponentially distributed with \(\mu_{1}^{-1}=\mu_{2}^{-1}=1.0\), and \(\mu_{2r}^{-1}=\mu_{2c}^{-1}=3.0\). Based on these values, the initial offered traffic-load values (in erl) are the following: \(\alpha_{1}=0.8\), \(\alpha_{2}=0.1\) and \(\alpha_{2r}=\alpha_{2c}=0.3\). In the \(x\)-axis of Figures 3-6, we consider an increase in the value of \(\alpha_{1}\) in steps of 0.2 erl. Thus, point 1 (in the \(x\)-axis) is: \(\alpha_{1},\alpha_{2},\alpha_{2r}=(0.8,0.1,0.3)\) and point 7 is: \(\alpha_{1},\alpha_{2},\alpha_{2r}=(2.0,0.1,0.3)\). Figure 3 presents the analytical together with the simulation CBP of the first service-class calls assuming the probabilistic retry loss model (p-RM in Figure 3) for \(p_{2r}=0.2\) and \(p_{2r}=0.6\) and the corresponding analytical CBP for the retry loss model of [39] (RM in Figure 3) and the EMLM. On the same hand, in Figure 4, we present the corresponding CBP results of \(B_{2r}\). Both figures show that the analytical results obtained via the probabilistic retry loss model are quite close to the corresponding CBP simulation results. Regarding Figure 3, we observe that decreasing \(p_{2r}\) results in a CBP decrease for the first service-class calls compared to the values of [39] or even the values of the EMLM (where retries are not permitted). As far as Figure 4 is concerned, we observe that a decrease of \(p_{2r}\), significantly increases the CBP of calls of the second service-class in the case of the probabilistic retry loss model, compared to the case of the model of [39]. Regarding the EMLM, it is obvious that it cannot capture the behavior of the proposed model or the model of [39]. Figure 4: CBP—Second service-class (probabilistic retry loss model). Figure 3: CBP—First service-class (probabilistic retry loss model). Figure 5: CBP–First service-class (probabilistic threshold loss model). Figure 6: CBP–Second service-class (probabilistic threshold loss model).