3.1 Network Interruption Fault Identification based on Small Time Scale Networks

Small time scale network interruption fault identification refers to the fault diagnosis of network systems on a small time scale. By analyzing network system signals, possible faults in the network system are identified. Usually, the characteristics are analyzed mainly from the aspects of small time scale characteristics of signals, characteristics of network interruption signals, network system faults, communication distance and signal transmission. The small time scale characteristics of signals refer to the fact that network signals often have very small time resolution on small time scales, which makes fault diagnosis more difficult. Therefore, the small time scale characteristics of the signal need to be considered for better fault diagnosis. Network interruption signals often have some characteristics, such as sudden rapid changes, continuous slow changes. These features can better achieve fault diagnosis. Network system failure refers to the possibility that a network system may be affected by some failures on a small time scale, such as equipment aging, system updates, human errors, and so on. Therefore, the impact of these faults on the network system should be considered for better fault diagnosis. On a small time scale, communication distance and signal transmission may be subject to some interference, such as noise ^[20,21,22]. Therefore, the impact of these interferences and factors on communication distance and signal transmission should be considered. Based on bottleneck links, small-scale network fault identification has been studied. Firstly, the case where there is only one bottleneck link in a connected backhaul is considered. The transmission rate of the output link in the transmission direction of a connection is $c_{s} $. The effective bandwidth allocated by a bottleneck link for this connection is $c_{r} $. If $c_{s} > c_{r} $, then the bandwidth allocated to all other links on the return path of this connection is greater than the maximum sum of $c_{s} $ and $c_{r} $. The round-trip link model structure of a single bottleneck link is shown in Fig. 1.

In practical applications, there is more than one bottleneck link on the round-trip path of the connection. The maximum transmission rate of a connection on the output link $j_{0} $ is $c_{0} $. The effective bandwidth allocated by bottleneck link $j_{n} $ for this connection is $c_{n} $. The effective bandwidth allocated by bottleneck links $j_{n} $ and $j_{n-1} $ for this connection is $c_{n-1} > c_{n} $. Any bottleneck free link $j$ between bottleneck links $j_{n-1} $ and must meet the effective bandwidth $c_{j} \ge c_{n-1} $. There are $n$ bottleneck links on one link. The effective bandwidth of this link for this connection is $c_{0} >c_{1}> c_{n} $. From 3.1, it can be seen that the rate of confirming the packet returning to the sender is $c_{n} $. The maximum transmission rate of the sender is the minimum value between $c_{0} $ and $(1+a)c_{n} $. $a$ is a progressive increment. Further assuming $c_{0} > (1+a)c_{n} $, the bottleneck links $j_{i} $ and $j_{i+1} $ on the connected round-trip path satisfy $c_{i} > (1+a)_{cn} > c_{i+1} $. So the number of bottleneck links on the round-trip path of the connection is reduced to $r$, that is, link $j_{i+1} $, $j_{i+2} $, ..., $j_{n} $. Therefore, the round-trip link model structure of the connection is shown in Fig. 2.

Fig. 1. Round-trip link model structure diagram for a single bottleneck link.

Fig. 2. Architecture diagram of a multi-bottleneck link.

From the above analysis, it can be seen that when a small-scale network is connected, it can utilize several bottleneck links to achieve communication round-trip. When a long buffer appears on a bottleneck link, the validity value of the buffer is greater than the interrupt failure window. At this point, the average value of the maximum packet in the buffer is the standby state. This buffer is a buffer for multiple bottleneck links. The average value of the maximum group is represented by Formula (1).

In Formula (1), $A$ represents the average rate of achievement. $w_{i} $ represents the initial value in the connection state. $n$ represents the number of links. $i=1$, $2$, ..., $m-1$.

According to the example of the $i$-th communication return, two links each transmit $w_{i} $ back-to-back data packets, both of which are transmitted to the bottleneck link. The data packets belonging to the transmitting party are transmitted to link 1. Packets belonging to the bottleneck are transmitted to link 2. At this point, the data packet is transmitted to the waiting connection state. The value in buffered state can be calculated using Formula (2).

In Formula (2), $a$ represents the packet level increment of congestion windows. $c_{n} $ represents effective broadband. On this basis, the average value of the maximum number of groups can be calculated using Formula (3).

In Formula (3), $w_{ i-1} $ represents the value at the moment before the connection state. To ensure that the allocation of links meets the requirements, the width of the connection is allocated. The standard value of the distribution is calculated is shown in Formula (4).

In Formula (4), $c$ represents the bandwidth on the path. Based on Formula (4) analysis, if $n=1$ is present on the communication round-trip path, the minimized bandwidth link must exist on the round-trip path and be effective. If $n=2$, then $c$ represents the effective bandwidth on that path. At this point, the bottleneck bandwidth is shown in Formula (5).

In Formula (5), when $0<c_{2} <c{}_{\max } $, it indicates that there must be a bottleneck link. The possibility of becoming a bottleneck link is shown in (6).

In Formula (6), $k$ represents the width rate.

3.2 Construction of a Mathematical Model for Fault Tolerant Identification of Network Interruptions

A reconfigurable service carrying network (RSCN) link fault-tolerant mathematical model has been constructed for identified link failures, which can achieve small-scale network interruption fault repair. Normal communication is ensured in the event of failure. The physical network description of the reconfigurable service bearer network model is completed by a $G^{u} =(N^{u} ,E^{u} ,C^{u} )$ undirected graph. $C^{u} $ represents the carrying capacity, provided by the network. $N^{u} $ represents a physical node. $E^{u} $ represents the set of links. The construction standard for the RSCN model is $G^{r} =(N^{r} ,E^{r} ,C^{r} )$. $N^{r} $ represents a virtual node and belongs to $N^{u} $. $E^{r} $ represents a virtual link and belongs to $E^{u} $. $R^{r} $ represents the demand for bearing capacity. The construction of the RSCN model can be regarded as a mapping process, represented by $G^{s} $. This process is between subsets $G^{r} $ and $G^{u} $. On the basis of meeting the constraint conditions in $G^{r} $, it can be represented by Formula (7).

In Formula (7), $N^{s} \subset N^{u} $, $E^{s} \subset E^{u} $. The mathematical model flow commonly used for network fault identification is shown in Fig. 3.

Fig. 3. Mathematical model process commonly used for network fault identification.

To describe the importance of resources in the network, the resource urgency measure $e$ is introduced into the model. Represented by nodes, many links in a network pass through them. Once $e$ malfunctions, it will cause widespread paralysis of the entire network, greatly affecting its connectivity. Therefore, the influence of node $e$ also needs to be considered. $n_{i} $ represents a node. The degree in $G^{u} $ is represented by $d_{i} $. $NI$ represents the degree of influence vector, belonging to $n_{i} $ and its adjacent nodes. $CF$ represents a resource node that belongs to a physical network. It can be used to describe the degree of segmentation, which is caused by the failure of the remaining network in $n_{i} $. The adjacency matrix is represented by $A(G^{u} )$. $G^{u} $ is a subset. After normalizing the node connectivity, it is the degree of node connectivity. The processing formula is shown in (8).

In Formula (8), $n_{i} $ represents the connectivity of the node. The strength of link connectivity can be expressed by the degree of adjacent point connectivity, as shown in Formula (9).

In Formula (9), $d_{i} $ represents the specific degree value in the connection. $d_{j} $ represents the connection value in connectivity. When the network is interrupted, the RSCN affected can be described by saturation, as shown in Formula (10).

In Formula (10), $x$ represents the resources in the node and link. $k$ represents the number of nodes. $M_{x} $ represents the assigned resource value. The degree of resource urgency is shown in Formula (11).

In Formula (11), $\alpha $ represents the regulatory factor 1. $\beta $ represents regulatory factor 2. $\alpha +\beta =1$. $SF(x)$ represents the degree of impact caused by a malfunction. The RSCN link fault-tolerant mathematical model has been constructed. While measuring the construction cost, ensure that the number of RSCNs affected by resource failures is minimized by reducing the occupancy rate of resources with higher urgency. After the requirement decomposition of the model is completed, it is transformed into the sequential solution of each requirement. In $s,t$, the determination of the connection path in $G^{u} $ represents the construction of the model, represented by $V_{s,t} $. The values that need to be met are shown in Formula (12).

In Formula (12), $b(c)$ represents the bandwidth value in the connecting link. $d_{e} $ represents the measurement value in connectivity. To verify that the constructed attribute model can perform network fault tolerance identification, the constructed mathematical model is compared with random network models and scale-free network models. The results are shown in Fig. 4.

As shown in Fig. 4, the fault tolerance recognition matching rate of the random network model is 93.25%. The fault-tolerant recognition matching rate of the mathematical model is 96.72%. The fault tolerance recognition matching rate of the scale-free network model is 89.97%. The fault-tolerant recognition matching rate of the mathematical model is significantly higher than the other two models, indicating that the constructed mathematical model can be used for experiments. The link recovery mechanism is used to map and migrate terminal failed links. Based on the goal of minimizing adjustment costs, fault-tolerant analysis of interrupt failures is implemented. The mathematical model construction process for fault tolerance identification of network interruption faults is shown in Fig. 5.

Fig. 4. Matching of the three models.

Fig. 5. Mathematical construction process for fault-tolerant identification of network outages based on small time scales.

4.1 Performance Analysis of Mathematical Models based on Fault Tolerance Identification

The fault-tolerant performance of the mathematical model in the case of interrupt faults with different window sizes was tested. The accuracy, loss rate accuracy, and absolute error of the four methods are used as evaluation indicators. Fig. 6 shows the accuracy and loss rate results of these four methods.

Fig. 6. Comparison of the accuracy and loss rate results of the four methods.

Fig. 7. Accuracy and mean absolute error results of the four methods.

As shown in Fig. 6, as the number of iterations increases, the accuracy of all four methods shows an upward trend. The loss rates of all four methods show a decreasing trend. The final accuracy of the mathematical model proposed in the study is 90.9%. The final accuracy values of DNN, RNN, and CNN are 87.35%, 83.71%, and 82.46%, respectively. The loss rate of the mathematical model proposed in the study is 2.43%. The loss rates of DNN, RNN, and CNN are 2.47%, 2.49%, and 2.53%, respectively. This indicates that judging from the two dimensions of accuracy and loss rate, the mathematical model proposed in the study performs better than the comparison method. Fig. 7 shows the accuracy and Mean absolute error results of the four methods.

Fig. 7(a) shows the accuracy comparison results of the four methods. As shown in Fig. 7(a), with the increase of iteration, the accuracy of all four methods shows an upward trend. The final accuracy of the mathematical model proposed in the study is 91.53%. The final accuracy of DNN, RNN, and CNN were 86.26%, 81.96%, and 78.81%, respectively. Fig. 7(b) shows the comparison results of Mean absolute error of the four methods. It can be seen from Fig. 7(b) that with the increase of iteration, the Mean absolute error of the four methods shows a downward trend. The Mean absolute error of the proposed mathematical model is 0.0004. The Mean absolute error values of DNN, RNN and CNN are 0.008, 0.0061 and 0.0069 respectively. This shows that the performance of the mathematical model proposed in the study is also better than the comparison method from the perspective of accuracy and Mean absolute error. The running time of the four methods is compared. In order to prevent the occurrence of accidental errors, the running time of the four methods is Threefold repetition repeated experiments. The running time results of the four methods are shown in Table 1.

It can be seen from Table 1 that in the threefold repetition experiments, the proposed mathematical model has the lowest time consumption. And the average time consumption of the method in the three repeated experiments is 232 seconds. The average time consumption for DNN, RNN, and CNN is 280 s, 271 s, and 291 s. This indicates that the running time of the mathematical model is less than the other three methods. Therefore, from the perspective of running time, the proposed mathematical model performs better than the comparison method. Based on the above, the comparison results from multiple dimensions indicate that the mathematical model has high stability and feasibility.

4.2 Application Analysis of Mathematical Models based on Fault Tolerance Identification

In order to test the application performance and effectiveness of the research and construction model, a scalable noxim simulator was used to simulate small-scale networks to complete simulation testing. During testing, random network mode and hotspot mode are used as comparative methods to determine the application effectiveness of the mathematical model.

From Fig. 8(a), it can be seen that in random mode, as the interrupt fault window increases, there is no increase in network band and number. The network status is in a small band state. In hotspot mode, the situation that occurs when the interrupt fault window increases is roughly the same as in random mode, with small fluctuations. But they all meet the design requirements. This indicates that the constructed mathematical model has good fault-tolerant performance in both types of networks. From Fig. 8(b), it can be seen that in the random model, the success rate of the mathematical model shows a decreasing trend. The overall success rate is 89.63% and will eventually stabilize.

In the hotspot model, the success rate of the mathematical model decreases first and then slightly increases as the proportion increases. The successful operation rate when it finally stabilizes is 81.95%. This indicates that the constructed attribute model can meet the design requirements. In order to further verify the fault-tolerant effect of the mathematical model, interrupt faults in small-scale time networks are simulated in random mode and hotspot mode. The results are shown in Fig. 9.

Fig. 8. Fault tolerance and long-term operational success test results.

Fig. 9. Node connectivity test

Fig. 10. Average link utilization and average packet energy consumption tests

As shown in Fig. 9, in the event of a node failure, it is mapped to achieve link migration. It can also achieve effective connection and information transmission between neighboring and failed nodes. The mapping results of at least two adjacent nodes near each failed node are ensured. The higher the utilization rate of physical links, the higher the effectiveness value of network resource utilization, indicating better fault tolerance performance of the system. This indicates that there is a correlation between the utilization of physical links and fault tolerant identification. In order to further verify the fault-tolerant ability of the mathematical model, the link utilization rate and average energy consumption of data packets are used to determine the effectiveness of fault-tolerant recognition. The results are shown in Fig. 10.

As shown in Fig. 10(a), as the ratio continues to increase, the utilization rate of the links in both modes fluctuates. The average utilization rate in random mode is 87.53%. The average utilization rate in hotspot mode is 88.21%. The difference in fault tolerance utilization between the two is very small, and both are above 85%. This indicates that the mathematical model has good fault-tolerant recognition rate in both modes. As shown in Fig. 10(b), after applying the mathematical model constructed in the research, the number of interrupts increases in random mode. The average energy consumption of data packets is 0.67. When the mathematical model is not used, the average energy consumption of the data packet is 0.71. After using the mathematical model, the average energy consumption of data packets in hotspot mode is 0.49.

The average energy consumption of the data packet before using the mathematical model is 0.77. This indicates that the average energy consumption of the data packet significantly decreases after using the mathematical model. Using mathematical models under the same conditions can reduce energy consumption and complete data transmission. To verify the recognition and classification ability of mathematical models for different network interruption faults, two modes were applied to dataset testing. A total of 6 types of network interruption data were used in the dataset, as shown in Fig. 11. The patterns of different shapes and colors in the figure represent different types of network interruption faults. The same type of network interruption fault is the same. Fig. 11(a) shows the classification effect of random patterns. Fig. 11(b) shows the classification effect of hotspot patterns.

Fig. 11. Comparison of network outage fault classification in two modes.

The classification of network interruption fault data can be used to determine the fault tolerance ability of mathematical models. As shown in the figure, both modes can recognize and classify six types of network interruption data. However, there are differences in the classification and location labeling of the same network interruption fault type between the two. This indicates that there are differences in the judgment of fault-tolerant nodes between the two modes in the constructed mathematical models. There are still differences in the number of scatter points used to identify and classify faults, but the difference is not significant. This indicates that both models can effectively classify interrupt networks in the constructed mathematical models, and the results are very good. The reliability of the constructed mathematical model has been verified.

Fig. 12. Comparison of fault tolerance identification results for small and medium time scale networks in transmission control protocol.

In order to further validate the performance of the research constructed model, the study applies it to the Internet router buffer design platform for performance verification. One of the goals of the Internet router switching design is to ensure that the link utilization can reach 100%, however, the emergence of small time scale networks can greatly affect the link utilization, in order to validate the performance of the model in the small time scale network fault tolerance identification, the study is conducted in the Internet router buffer design platform for fault tolerance fault identification. The comparison results of small time scale network fault tolerance identification for transmission control protocols are shown in Fig. 12.

As analyzed in Fig. 12, in the Internet router buffer 500 M throughput buffer platform, the throughput designed for 500 M can only provide about 253 M network bandwidth before the fault-tolerant fault identification process. After the introduction of fault-tolerant fault identification model, the overall bandwidth are significantly improved, and can provide about 436 M network bandwidth. Although there is still a gap with the real value, the bandwidth is significantly improved compared to the bandwidth when there is a fault. This indicates that the mathematical model constructed in the study for fault tolerance identification of network outages at small time scales can significantly improve the bandwidth flux in the fault tolerance of the Internet router buffer design platform, and provide practical results for the enhancement of network traffic.