# Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches

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## Abstract

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## 1. Introduction and an Overview of Existing Results

## 2. Diffusion Single Station Models

#### 2.1. First-In-First-Out G/G/1/N Station

#### 2.2. Preemptive-Resume G/G/1/N/PRIOR Station

- $K=0$: we consider the highest priority class $k=0$ alone and use the single class model presented in the previous section. The customers of lower classes are transparent for $k=0$ class; therefore, the solution is correct. ${f}^{\left(0\right)}(n,t;{x}_{0})={p}^{\left(0\right)}(n,t)={v}^{\left(0\right)}\left(t\right)$.
- $K=1$: we consider two classes, $k=0,1$, determine ${\alpha}^{\left(1\right)}$, ${\beta}^{\left(1\right)}$ following (14), solve the diffusion equation with these parameters to obtain ${f}^{\left(1\right)}(x,t;{x}_{0})$, which approximates the distribution ${p}^{1}(n,t)$ of the joint number of customers of classes $k=0$ and $k=1$; we then compute ${v}^{\left(1\right)}(n,t)$.
- $K=2$: we consider the system with three classes, $k=0,1,2$ to determine the parameters ${\alpha}^{\left(2\right)}$, ${\beta}^{\left(2\right)}$ following (14), solve the diffusion equation to obtain ${f}^{2}(x,t;{x}_{0})$ and ${p}^{2}(n,t)$, then, using ${p}^{\left(1\right)}(n,t)$ of the previous step, compute ${v}^{\left(2\right)}(n,t)$, etc., until $K=L$.

## 3. Validation of the Priority Model

#### 3.1. Two Priorities, Low Input Intensities

#### 3.2. Two Priorities, Medium Input Intensities

#### 3.3. Two Priorities, High Input Intensities

#### 3.4. Three Priorities; Mean Service Time Depends on Priority

#### 3.5. Two Priorities; General Interarrival and Service Time Distributions

## 4. Network of Priority and Non-Priority Queues

#### 4.1. The Output Stream at the FIFO Station

#### 4.2. The Output Stream at the Priority Station

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- The next customer in the class k is in the system (this occurs with probability $\frac{{\varrho}^{\left(k\right)}}{1-{R}^{(k-1)}}$) and will leave it after its completion time;
- –
- There are no customers of this class in the system, and we shall wait for the time described by ${F}_{A}^{\left(k\right)}\left(x\right)$ until it appears and enters the server;
- –
- No customer of class k is present in the system, and a customer of higher class comes before him, so the busy period ${\gamma}^{(k-1)}$ must first be terminated.

## 5. The SDN Switch

- Proposing the diffusion model of a multiclass G/G/1/N/Priority station, i.e., a station with general interarrival and service time distributions, limited buffer, and with preemptive-resume priority queues. Each class of customers has its specified priority level and its own parameters of the interarrival and service time distribution. Within one priority class, the scheduling is based on the FIFO algorithm. The model covers transient and steady-state analysis.
- Validation of this model by comparison with discrete-event simulation for various loads and interarrival and service time distributions, and discussion of errors;
- The model of an open network with any topology integrationg priority and FIFO service stations;
- A model of SDN switch exchanging packets with undetermined routing with SDN controller and validation of this model.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Section 3.1, low load: total mean queue length as a function of time for both classes (P0 + P1) taken together.

**Figure 2.**Section 3.1, low load: mean queue lengths as a function of time for priority (P0) and non-priority (P1) classes, and diffusion and simulation results.

**Figure 3.**Section 3.1, low load: probabilities of empty queues for priority (P0) and non-priority (P1) classes.

**Figure 4.**Section 3.1, low load: probabilities of saturated queues as a function of time for priority (P0) and non-priority (P1) classes.

**Figure 5.**Section 3.2, medium load: mean queue lengths of priority (P0) and non-priority (P1) classes.

**Figure 6.**Section 3.2, medium load: total mean queue length for both classes together.

**Figure 7.**Section 3.2, medium load: probabilities of empty queue ${p}_{0}\left(t\right)$ for priority (P0) and non-priority (P1) classes.

**Figure 8.**Section 3.2, medium load: probabilities of saturated queues ${p}_{N}\left(t\right)$ for priority (P0) and non-priority (P1) classes.

**Figure 9.**Section 3.3, high load: mean queue lengths of priority (P0) and non-priority (P1) classes.

**Figure 10.**Section 3.3, high load: total mean queue length of both priority classes.

**Figure 11.**Section 3.3, high load: probabilities of empty queues for priority (P0) and non-priority (P1) classes.

**Figure 12.**Section 3.3, high load: probabilities of saturated queues for priority (P0) and non-priority (P1) classes.

**Figure 13.**Section 3.4, three priorities: mean queue lengths as a function of time for priority (P0), medium-priority (P1), and low-priority classes.

**Figure 14.**Section 3.4, three priorities: total mean queue length as a function of time of three priority classes.

**Figure 15.**Section 3.4, three priorities; probabilities of empty queues as a function of time for three priority classes (P0), (P1), (P2).

**Figure 16.**Section 3.4, three priorities: probabilities of saturated queues as a function of time; in case of P(0) only diffusion results are available, the simulations were too short to give such small values.

**Figure 17.**Section 3.5: mean queue lengths of priority (P0) and non-priority (P1) classes for very high values of ${C}_{A}^{2}$, ${C}_{B}^{2}$.

**Figure 18.**Section 3.5: total mean queue length of both priority classes for very high values of ${C}_{A}^{2}$, ${C}_{B}^{2}$.

**Figure 19.**Section 3.5: probabilities of empty queues for priority (P0) and non-priority (P1) classes for very high values of ${C}_{A}^{2}$, ${C}_{B}^{2}$.

**Figure 20.**Section 3.5: probabilities of saturated queues for priority (P0) and non-priority (P1) classes for very high values of ${C}_{A}^{2}$, ${C}_{B}^{2}$.

**Figure 22.**The input flow $\lambda $ (packets per second) to the SDN switch, considered interval of 1 s.

**Figure 23.**Mean queue length at the switch for priority (P0) and non—priority (P1) packets as a function of time, $\u03f5=0.2$, diffusion, and simulation results.

**Figure 24.**Mean queue length at the switch for priority (P0) and non—priority (P1) packets as a function of time, $\u03f5=0.5$, diffusion and simulation results.

**Figure 25.**Mean response time at the switch as a function of time, $\u03f5=0.2$, diffusion, and simulation results.

**Figure 26.**Mean response time at the switch as a function of time, $\u03f5=0.5$, diffusion, and simulation results.

**Figure 27.**Mean delay introduced by the communication with the controller as a function of time, $\u03f5=0.2$, diffusion, and simulation results.

**Figure 28.**Mean delay introduced by the communication with the controller as a function of time, $\u03f5=0.5$, diffusion, and simulation results.

**Figure 29.**Loss probability due to the queue saturation as a function of time, $\u03f5=0.2$, diffusion, and simulation results.

**Figure 30.**Loss probability due to the queue saturation as a function of time, $\u03f5=0.5$, diffusion, and simulation results.

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**MDPI and ACS Style**

Nycz, T.; Czachórski, T.; Nycz, M.
Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches. *Sensors* **2021**, *21*, 5042.
https://doi.org/10.3390/s21155042

**AMA Style**

Nycz T, Czachórski T, Nycz M.
Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches. *Sensors*. 2021; 21(15):5042.
https://doi.org/10.3390/s21155042

**Chicago/Turabian Style**

Nycz, Tomasz, Tadeusz Czachórski, and Monika Nycz.
2021. "Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches" *Sensors* 21, no. 15: 5042.
https://doi.org/10.3390/s21155042