# A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources
## 3 Non-Availability Probability of Strictly Defined Resources  
The non-availability probability \(H(i,x_{d})\) of \(x_{d}\) strictly determined resources determines the probability of an event in which each resource from the \(x_{d}\) selected resources does not have enough \(t_{i}\) free AUs to set up a connection of class \(i\). This probability can be determined with respect to the occupancy of AUs, i.e., directly at the occupancy level of AUs. Thus, we have:  
\[H(i,x_{d})=\sum_{n=(f-t_{i}+1)x_{d}}^{V}H(i,x_{d}|n)[P(n)]_{V}, \tag{6}\]  
where \(H(i,x_{d}|n)\) is the conditional non-availability probability of a selected number of \(x_{d}\) resources for a call of class \(i\), determined under the assumption that the total number of busy AUs in the LAG is equal to \(n\). The lower limit of the sum in Equation (6) determines the minimum number of AUs that can induce non-availability for a selected number of \(x_{d}\) resources.  
The conditional distribution \(H(i,x_{d}|n)\) can be determined on the basis of the following reasoning. The selected \(x_{d}\) resources are unavailable to a call of class \(i\) if \(n_{d}\) AUs (\(n_{d}\leq n\)) are arranged in these \(x_{d}\) resources in such a way that none of them has \(t_{i}\) free AUs. On the basis of Equation (5), the number of such arrangements is \(F(n_{d},x_{d},f,f-t_{i}+1)\). The remaining \(n-n_{d}\) busy AUs can be arranged (accommodated) in any way within the remaining \(k-x_{d}\) resources. The number of such arrangements is \(F(n-n_{d},k-x_{d},f,0)\). Since the total number of arrangements of \(n\) AUs in \(k\) resources is \(F(n,k,f,0)\), the probability \(H(i,x_{d}|n)\) can be determined in the following way:  
\[H(i,x_{d}|n)=\sum_{n_{d}=(f-t_{i}+1)x_{d}}^{\min(n,f_{x_{d}})}\frac{F(n_{d},x_ {d},f,f-t_{i}+1)F(n-n_{d},k-x_{d},f,0)}{F(n,k,f,0)}. \tag{7}\]  
Equation (6) derived in the present article for multi-service systems is the counterpart of the Palm-Jacobaeus loss formula (Equation (1)) derived for single-service systems [24]. The original Palm-Jacobaeus formula is a substantial extension of the Erlang model for the full-availability group (Erlang B-formula [49]): The Erlang B-formula makes it possible to determine the blocking probability in a system with a capacity of \(k\) resources, whereas the Palm-Jacobaeus loss formula [24] allows us to determine the probability \(H(x_{d})\) of occupancy (non-availability) of \(x_{d}\) strictly determined resources (from a set containing \(k\) resources). In a similar way, Equation (6), developed in the present article, is a substantial extension of the model proposed in [48]. Equations (2) and (3) allow us to determine the occupancy distribution and the blocking probability in a multi-service system with a capacity of \(V=kf\) AUs. The model proposed in the article makes it possible to determine the probability of an event in which the selected \(x_{d}\) (from among all the \(k\) resources) resources of a system can serve a call of a given class. Equation (6) proposed in the article is then in the same relation to the LAG model as the Palm-Jacobaeus formula is to the basic Erlang model and constitutes a substantial extension of the practical capabilities of the model in engineering applications.