# A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources  
[PERSON]  
1 Faculty of Computing and Telecommunications, Poznan University of Technology, ul. Polanka 3, 60-965 Poznan, Poland; [EMAIL_ADDRESS] (A.K. ); [EMAIL_ADDRESS] (M.S. )1 Faculty of Computing and Telecommunications, Poznan University of Technology, ul. Polanka 3, 60-965 Poznan, Poland; [EMAIL_ADDRESS] (A.K. ); maciej.stasiak@put.poznan. Regardless of the objective and technique used to optimize resource management, however, the area covered by a specific group of cells can be treated--from the point of view of traffic engineering--as a multi-service system in which multi-service requests are handled using \"separated\" resources (cells), creating network resources [10]. An analysis of the state of research on the analytical modeling of cellular systems indicates that multi-service (multi-rate) models are most often used to determine the traffic characteristics of such systems at the call level [14; 15; 16; 17; 18; 19; 20]. In most of the work on modeling cellular systems at the call level, only single cells [16; 17; 18; 21] were taken into account. For the purpose of analyzing a group of cells (resources) as a single system, it is necessary to use more complex multi-service models, including the so-called limited-availability group model [22] and the non-full-availability group model [10]. In these studies [10; 22], groups of cells with a connection handover mechanism for load balancing were analyzed. The developed methods take into account only information about the amount of occupied resources in a group of cells (resources), without the possibility of determining the occupancy states of specific (indicated) cells (resources) in the group. This assumption did not directly affect the determination of the probability of blocking in a group of cells (resources); however, it did not allow for taking into account separate call admission restrictions in individual cells. In many new tasks posed to, among others, SON mechanisms and network slicing in 5G systems, the ability to determine the occupancy (non-availability) of precisely indicated (and not any) cells can be very important. The development of a general and effective analytical method for the determination of the non-availability of strictly determined/defined resources in a certain set of them for defined classes of traffic streams will make it possible for the method to be further parameterized, e.g., within the context of the aforementioned traffic distribution in a group of cells, the distribution of resources for network slicing, or the assessment of the number of necessary links in trunks (e.g., EtherChannels). In the case of single-service systems, the Palm-Jacobaeus formula provided us with the ability to determine the probability of blocking the indicated resources within their group. A method for the determination of the blocking probability \(H(x_{d})\) of \(x_{d}\) strictly determined resources (from a set containing \(k\) resources) is proposed in [23], while in [24] this method is presented in the form of the following equation, known as the Palm-Jacobaeus loss formula:  
\[H(x_{d})=\frac{E_{k}(A)}{E_{k-x_{d}}(A)}, \tag{1}\]  
where \(E_{k}(A)\), the so-called Erlang B-formula, is the blocking probability in a single-service group composed of \(k\) resources to which traffic with an intensity of \(A\) Erlangs is offered. The parameter \(x_{d}\) defines the number of strictly determined resources to be investigated. In traffic engineering, Equation (1) has been used, directly or in the form of the so-called modified Palm-Jacobaeus loss formula [25; 26], to model complex non-full-availability systems, single-service resource groups [25; 27], and single-service multi-stage switching networks, e.g., [28; 29; 30; 31; 32]. A counterpart (corresponding equivalent) of the group of single-service resources is the group of separated multi-service resources--the limited availability group (LAG).