# A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources

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## Abstract

**:**

## 1. Introduction

## 2. Model of the Group with Limited Availability

## 3. Non-Availability Probability of Strictly Defined Resources

## 4. Results and Discussion

#### 4.1. General Assumptions

#### 4.2. Simulator

- the number k of resources,
- the capacity f of a single resource,
- the number M of traffic stream classes,
- the number ${t}_{i}$ of demanded allocation units for a stream of class i ($1\le i\le M$),
- the intensity ${\mu}_{i}$ of a service stream of class i,
- the proportions of offered traffic ${A}_{1}{t}_{1}:{A}_{2}{t}_{2}:\dots :{A}_{M}{t}_{M}=x:y:\dots :z$, i.e., the values of the parameters $x,y,\dots ,z$,
- the average value a of traffic offered to a single allocation unit in the system, where:$$a=\frac{{\sum}_{i=1}^{M}{A}_{i}{t}_{i}}{kf}.$$

#### 4.3. Accuracy of the Model

- System 1: $k=3$, $f=20$ AUs, $V=60$ AUs, $M=3$, ${t}_{1}=1$ AU, ${t}_{2}=2$ AUs, ${t}_{3}=3$ AUs, ${A}_{1}{t}_{1}:{A}_{2}:{t}_{2}:{A}_{3}{t}_{3}=1:1:1$, ${\forall}_{1\le i\le M}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}=1$;
- System 2: $k=5$, $f=20$ AUs, $V=100$ AUs, $M=3$, ${t}_{1}=1$ AU, ${t}_{2}=2$ AUs, ${t}_{3}=5$ AUs, ${A}_{1}{t}_{1}:{A}_{2}:{t}_{2}:{A}_{3}{t}_{3}=1:1:1$, ${\forall}_{1\le i\le M}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}=1$;
- System 3: $k=5$, $f=50$ AUs, $V=250$ AUs, $M=3$, ${t}_{1}=1$ AU, ${t}_{2}=3$ AUs, ${t}_{3}=7$ AUs, ${A}_{1}{t}_{1}:{A}_{2}:{t}_{2}:{A}_{3}{t}_{3}=1:1:1$, ${\forall}_{1\le i\le M}\phantom{\rule{0.166667em}{0ex}}{\mu}_{i}=1$.

- The analysis of the considered system is carried out from the microstate level (multi-dimensional Markov process) to the macrostate level (one-dimensional Markov process)—a detailed analysis of this problem is presented in [55].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Probability of the non-availability of x strictly determined resources for class 1 calls (${t}_{1}=1$) in System 1, where x is the set of strictly determined resources in the limited-availability group.

**Figure 2.**Probability of the non-availability of x strictly determined resources for class 2 calls (${t}_{2}=2$) in System 1, where x is the set of strictly determined resources in the limited-availability group.

**Figure 3.**Probability of the non-availability of x strictly determined resources for class 3 calls (${t}_{3}=3$) in System 1, where x is the set of strictly determined resources in the limited-availability group.

**Figure 4.**Probability of the non-availability of x strictly determined resources for class 1 calls (${t}_{1}=1$) in System 2, where x is the set of strictly determined resources in the limited-availability group.

**Figure 5.**Probability of the non-availability of x strictly determined resources for class 2 calls (${t}_{2}=2$) in System 2, where x is the set of strictly determined resources in the limited-availability group.

**Figure 6.**Probability of the non-availability of x strictly determined resources for class 3 calls (${t}_{3}=5$) in System 2, where x is the set of strictly determined resources in the limited-availability group.

**Figure 7.**Probability of the non-availability of x strictly determined resources for class 1 calls (${t}_{1}=1$) in System 3, where x is the set of strictly determined resources in the limited-availability group.

**Figure 8.**Probability of the non-availability of x strictly determined resources for class 2 calls (${t}_{2}=3$) in System 3, where x is the set of strictly determined resources in the limited-availability group.

**Figure 9.**Probability of the non-availability of x strictly determined resources for class 3 calls (${t}_{3}=7$) in System 3, where x is the set of strictly determined resources in the limited-availability group.

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**MDPI and ACS Style**

Głąbowski, M.; Kaliszan, A.; Stasiak, M.
A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources. *Appl. Sci.* **2020**, *10*, 4019.
https://doi.org/10.3390/app10114019

**AMA Style**

Głąbowski M, Kaliszan A, Stasiak M.
A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources. *Applied Sciences*. 2020; 10(11):4019.
https://doi.org/10.3390/app10114019

**Chicago/Turabian Style**

Głąbowski, Mariusz, Adam Kaliszan, and Maciej Stasiak.
2020. "A Palm-Jacobaeus Loss Formula for Multi-Service Systems with Separated Resources" *Applied Sciences* 10, no. 11: 4019.
https://doi.org/10.3390/app10114019