# Probabilistic Retry and Threshold Multirate Loss Models for Impatient Calls
## 3 The Proposed Probabilistic Threshold Loss Model  
In the proposed probabilistic threshold loss model, a link of capacity \(C\) b.u. services calls from \(K\) service-classes. Service-class \(k\) (\(k\) = 1,..., \(K\)) 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. and a reduced bandwidth requirement of \(b_{kc}\) b.u. Contrary to the probabilistic retry loss model, in this model, it is the value of \(j\) (the occupied bandwidth of the link) together with the value of the threshold \(J_{0}\leq C-b_{k}\) that are closely related to the bandwidth requirement (initial or reduced) a new call will use in order to enter the system. More specifically, the call admission mechanism applied in the case of a new service-class \(k\) call is the following: (a) Assuming that \(j\leq J_{0}\) then the call will be accepted in the link with \(b_{k}\) b.u. and will be serviced for an exponentially distributed service time with mean \(\mu_{k}^{-1}\), and (b) assuming that \(j>J_{0}\) then the call will: (b1) abandon the link with probability \(1\)-\(p_{kc}\) or (b2) try to be connected in the system with probability \(p_{kc}\), reduced bandwidth requirement \(b_{kc}\)\(<\)\(b_{k}\) and increased service time \(\mu_{kc}^{-1}>\mu_{k}^{-1}\) computed as \(\mu_{kc}^{-1}=b_{k}\mu_{k}^{-1}/b_{kc}\). Needless to say, if the \(b_{kc}\) b.u. are not available then the call will be blocked (see Figure 2, where the call admission mechanism of [38] does not include the grey parts). To determine the unnormalized values of \(q(j)\)'s, we initially consider the case of three service-classes whose calls are serviced by a link of capacity \(C\) and have the following call arrival parameters \((\lambda_{1},\lambda_{2},\lambda_{3})\) and service parameters \((b_{1},b_{2},b_{3})\), \((\mu_{1},\mu_{2},\mu_{3})\). Assuming that \(j>J_{0}\), then calls of the third service-class have the choice to enter the system with probability \(p_{3c}\), reduced bandwidth requirement \(b_{3c}<b_{3}\), and increased service time \(\mu_{3c}^{-1}>\mu_{3}^{-1}\) computed as \(\mu_{3c}^{-1}=b_{3}\mu_{3}^{-1}/b_{3c}\). Contrary to the calls of the third service-class, calls of the first and second service-classes do not have the option to enter the system with a reduced bandwidth requirement. The derivation of a recursive formula for the computation of \(q(j)\)'s, requires the assumption (approximation) that local balance exists between the adjacent states: (a) \(j-b_{k}\) and \(j\) for \(k=1\), \(2\), \(3\) and (\(b\)) \(j-b_{kc}\) and \(j\) for \(k=3\). More precisely, we consider the following local balance equations for calls of the first two service-classes and \(j=1\),..., \(C\):  
\[\lambda_{1}b_{1}q(j-b_{1}) =y_{1}(j)b_{1}\mu_{1}q(j) \tag{9}\] \[\lambda_{2}b_{2}q(j-b_{2}) =y_{2}(j)b_{2}\mu_{2}q(j) \tag{10}\]  
where \(y_{1}(j)\) and \(y_{2}(j)\) refer to the average number of first and second service-class calls, respectively, that exist in the system in state \(j\). Regarding the local balance equations for calls of the third service-class, we have the following two equations for calls accepted with their initial bandwidth requirement and calls accepted with their reduced bandwidth requirement:  
\[\lambda_{3}b_{3}q(j-b_{3})=y_{3}(j)b_{3}\mu_{3}q(j),\text{ for }j-b_{3}\leq J_{0} \tag{11}\]  
\[\lambda_{3}p_{3c}b_{3c}q(j-b_{3c})=y_{3c}(j)b_{3c}\mu_{3}q(j),\text{ for }j-b_{3c}>J_{0}, \tag{12}\]  
where \(y_{3}(j)\) refers to the average number of the third service-class calls with \(b_{3}\) b.u. that exist in state \(j\) while \(y_{3c}(j)\) refers to the average number of the third service-class calls with \(b_{3c}\) b.u. that exist in state \(j\).