# Fluid-Flow Approximation in the Analysis of Vast Energy-Aware Networks

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- (i)
- we develop a queuing model of an extensive computer network. The model captures the dynamics of flows changing due to the nature of transmitted traffic and the network’s control algorithms. We consider the TCP Reno algorithm cooperating with the RED algorithm on IP level, but they may be easily replaced by other algorithms. The model is based on the well-known fluid-flow approximation. Its original part is the implementation of the whole structure inside the database system. This is the way to overcome the limitations of storing and using large amounts of data resulting from the size of the model and the analysis of transients. This unconventional solution allows us to model previously inaccessible topologies related to the Internet.
- (ii)
- we apply this model in the quantitative analysis of the impact of an energy-saving algorithm used in routers on the network’s performance. We discover that it reduces network congestion and save energy but significantly lowers network throughput. The studied network has a realistic hierarchical, nonhomogenous structure with routers and links of various throughputs and buffer sizes and is a copy of an existing part of the Internet; it was taken from a map developed in the Opte project in [29].

## 2. Related Works

## 3. Fluid-Flow Approximation

#### 3.1. Basic Model

- -
- ${W}_{i}\left(t\right)$ is the size of congestion window for connection i, i.e., the number of packets (or more precisely blocks) the sender is allowed to dispatch without waiting for acknowledgment of reception from the receiver;
- -
- ${R}_{i}\left({q}_{i}\left(t\right)\right)$ is the round trip time for a flow i, i.e, the mean time after which such acknowledgment is received.

- -
- ${q}_{i}$ is the set of mean queues in connection i,
- -
- ${V}_{i}$ is the set of nodes belonging to the connection i,
- -
- ${T}_{j}$ is the propagation time between two nodes j, $j+1$,
- -
- ${L}_{i}$ is the set of links between the nodes belonging to connection i.

#### 3.2. Fluid-Flow Approximation of an Energy-Efficient Node

- When the actual queue length q decreases and reaches the lower threshold of ${T}_{A}$, the router service is switched off. This results in the growth of the router queue.
- When the actual queue q increases and reaches the upper threshold of ${T}_{B}>{T}_{A}$, the router service is switched on. It is the moment when we assume that the router has a long enough queue to start transmitting.
- In other cases, the state of the router does not change.

## 4. Implementation and Numerical Results

Algorithm 1: Fluid-flow approximation written as a database procedure |

Algorithm 2: Energy-saving modification of fluid-flow approximation |

- 0% (red)—the router is always transmitting,
- 50% (yellow)—the router is transmitting for at least 50% of the time,
- 100% (green)—the router is never transmitting,
- between the above values, there are intermediate colors.

- (a)
- $C{R}_{1i}\u2a7e\mathrm{10,000}$ circa 0.0048% of nodes,
- (b)
- $1000<C{R}_{1i}\u2a7d\mathrm{10,000}$ circa 0.054% of nodes,
- (c)
- $100<C{R}_{1i}\u2a7d1000$ circa 0.38% of nodes,
- (d)
- $10<C{R}_{1i}\u2a7d100$ circa 4.9% of nodes,
- (e)
- $1<C{R}_{1i}\u2a7d10$ circa 93.33% of nodes,
- (f)
- $C{R}_{1i}\u2a7d1$ circa 1.33% of nodes.

- 0.558—violet,
- 14,110.012—white,
- 28,219.465—green,
- between the above values, there are intermediate colors.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Comparison of the idle times in the longest flow, basic and energy-aware models. The horizontal axis displays the unique numbers of network nodes that form the path of the flow.

**Figure 7.**The number of routers having current queue lengths greater than 20% and 25% of their buffer sizes in basic and energy-saving models.

**Figure 8.**The number of routers having the current queue length ⩾50% and 75% of the buffer for basic and energy-saving models. The curves for energy-saving mode are close to zero and practically invisible on the linear scale.

**Figure 9.**Average occupancy of router queues (percentage of buffer volume) as a function of time for basic and energy-saving models.

**Figure 10.**Average throughput per link in the network as a function of time for basic and energy-saving models.

Element | Parameter | Meaning |
---|---|---|

Node | N | Total number of nodes |

${q}_{j}$ | Current queue size at j-th node | |

${C}_{j}$ | Output flow of j-th node | |

${p}_{ij}$ | Loss probability at j-th node for i-th flow | |

${x}_{j}$ | Weighted average queue length at j-th node | |

${t}_{mi{n}_{j}}$ | Lower threshold for ${x}_{j}$ queue | |

${t}_{ma{x}_{j}}$ | Upper threshold for ${x}_{j}$ queue | |

${p}_{ma{x}_{j}}$ | Maximum loss probability at upper threshold for ${x}_{j}$ queue | |

${w}_{j}$ | Weight for ${x}_{j}$ queue | |

${T}_{A}$ | Lower threshold that switch off router’s service | |

${T}_{B}$ | Upper threshold that switch on router’s service | |

Flow | K | Total number of flows |

${K}_{j}$ | Set of flows crossing node j | |

${l}_{j}$ | Throughput of input stream of j-th node | |

${W}_{i}$ | Congestion window size of i-th flow | |

${R}_{i}$ | Round trip time of i-th flow | |

${q}_{i}$ | Set of current queues at i-th connection | |

${V}_{i}$ | Set of nodes for i-th connection | |

${T}_{i}$ | Propagation time between i-th and $(i+1)$-th nodes | |

${L}_{i}$ | Set of links for i-th connection | |

${N}_{i}$ | Number of identical flows at j-th node | |

Time | t | t-th time step |

$\Delta t$ | Time length of a time step | |

$\tau $ | Time when the loss occurred | |

Mode | $N-mode$ | Normal mode |

$ES-mode$ | Energy-saving mode |

**Table 2.**Comparison of the number of rows, memory, and disk size for the basic (B) and energy-saving (E) models.

Table | No of Entries | Memory Size (KB) | Disk Size (KB) |
---|---|---|---|

FLOWS_B | 50,000 | 4201 | 4460 |

FLOWS_E | 50,000 | 4217 | 4468 |

FLOWS_HISTORY_B | 50,050,000 | 3,939,782 | 3,863,984 |

FLOWS_HISTORY_E | 50,050,000 | 3,997,634 | 3,832,560 |

ROUTERS_B | 134,023 | 4706 | 4800 |

ROUTERS_E | 134,023 | 4806 | 4944 |

ROUTERS_HISTORY_B | 134,157,023 | 3,698,226 | 3,567,340 |

ROUTERS_HISTORY_E | 134,157,023 | 3,967,444 | 3,788,272 |

LOSSES_B | 44,559,219 | 1,905,595 | 1900 128 |

LOSSES_E | 45,721,414 | 2,020,755 | 1,936,288 |

**Table 3.**Comparison of the number of nodes in an idle state (non-transmit) between the basic (B) and energy-saving (E) models.

State | Idle Time (%) | Nodes in N-Mode (%) | Nodes in ES-Mode (%) |
---|---|---|---|

Highly active | 0–25 | 3.52 | 0.0024 |

Active | 25–50 | 94.17 | 0.84 |

Mostly idle | 50–75 | 2.31 | 99.15 |

Idle | 75–100 | 0 | 0.006 |

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Nycz, M.; Nycz, T.; Czachórski, T.
Fluid-Flow Approximation in the Analysis of Vast Energy-Aware Networks. *Mathematics* **2021**, *9*, 3279.
https://doi.org/10.3390/math9243279

**AMA Style**

Nycz M, Nycz T, Czachórski T.
Fluid-Flow Approximation in the Analysis of Vast Energy-Aware Networks. *Mathematics*. 2021; 9(24):3279.
https://doi.org/10.3390/math9243279

**Chicago/Turabian Style**

Nycz, Monika, Tomasz Nycz, and Tadeusz Czachórski.
2021. "Fluid-Flow Approximation in the Analysis of Vast Energy-Aware Networks" *Mathematics* 9, no. 24: 3279.
https://doi.org/10.3390/math9243279