Mathematical prediction model construction of network packet loss rate and nonlinear mapping user experience under the Internet of Things

3.1 Establish network data transmission module

First, the network data transmission module is established, and the network PLR prediction process is developed based on IoTs. Second, the prediction framework of PLR is designed to obtain more accurate prior information; finally, the mathematical model of network PLR prediction is constructed to further improve the prediction accuracy. The network data transmission module is an important part of the prediction of the network PLR and adopts a task-driven mode to process parameters [11]. The realization mechanism of the data transmission module is divided into four steps, which are loading socket library, server initialization, Listen Thread Func monitoring thread, and Test Task thread. First, use the Init InstanceO function in the App to enable the loaded socket library to receive network data, add the WSA Startup function, create a socket, and initialize it before initialization. Second, use CServer D1g to increase the msocket variable into SOCKET, privatize the access authority of the socket library, increase the return value of the InitSocket function, and initialize the network server. When the socket generated by the msocket function is the same as the socket library, it means that the server is initialized successfully. Third, use the listen function to monitor the client, that is, the listening thread Listen Thread Func, and change the parameters through the listening thread to achieve the goal of monitoring multiple clients. In addition, the accept function in the listening thread can generate a new socket; at this time, the accept function is in the successful execution state, and the m_ISOK function becomes the TRUE function to determine the connection state between each client and the server. Fourth, each client sends a test task to the Test Task thread after a successful connection with the server, and its structure is RECVPARAM, which can pass parameters and sockets to transmission control protocol (TCP) to facilitate the use of related functions [12]. Since the network PLR is associated with the threshold of the data transmission module, after the threshold of the data packet is input, the test task is sent, and it can be judged whether the relevant parameters are input successfully. After success, start the communication connection between the server and the client. Through the aforementioned mechanism, data transmission is completed.

3.2 Develop a network PLR prediction process based on IoTs

IoTs is used to train and estimate network data in the network packet loss estimation phase. The process of calculating the amount of network packet loss is as follows: first, create a training-level estimate, obtain the network control data by calculating the background of IoTs, and pass the network connection in a slide step. Second, the prior information about the transmission probability of each link is used to obtain the predicted initial information of the network. In addition, the authors detect the package through sendback-to-back in IoT, the transmission probability of each network path is transmitted to the next sliding link, and the corresponding link transmission probability is found. The prior information obtained from it can be used as the input premise of the prediction step to achieve the goal of prediction. Third, in order to find the loss situation at a specific time point, resampling according to the obtained particle set, directly predict the target state, find the link transmission probability in the network sliding stage, and then obtain the state of the network transfer packet. Fourth, according to the probability value of each link in the sliding phase and the transmission probability of each link at the previous moment, predict the network PLR.

In the prediction stage of the network PLR, IoTs is used to train and predict the network data. The procedure for predicting the network PLR is as follows. First, the prediction training phase is created. Network observation data is obtained by statistics of the background traffic of IoTs, and the transmission probability of network links is obtained by sliding steps. Second, the initial particle set of network prediction is obtained by using the prior information of transmission probability of each link. In addition, this study transmits the transmission probability of each network path to the next sliding link through the sendback-to-back detection packet in IoTs to find the corresponding link transmission probability. The prior information obtained from it can be used as the input premise of the prediction step, thus achieving the goal of prediction. Third, in order to find the loss situation at a specific time point, resampling is conducted according to the obtained particle set to directly predict the target state, find the link transmission probability of the network sliding stage, and then obtain the state of the network transfer packet. Fourth, according to the probability value of each link in the sliding stage and the transmission probability of each link at the previous moment, the network PLR is predicted.

Using two test cases, verify that the data transfer module can be used when encountering obstacles – no normal transmission. Firstly, confirm that the server and client connections are normal based on the startup status of the transmission server and client. Second, the preconditions are the same as the expected results: enter the destination client address, click the start button, and wait for the result. The specific processes are as follows:

1. Data collection: collect information related to network data packets generated by IoT devices or sensors. This information may include packet transmission timestamp, packet size, transmission path, and network topology.

2. Feature extraction: extract features related to the network PLR from the collected data. For example, metrics such as packet transmission delay, loss count, and link load can be calculated as features.

3. Data preprocessing: preprocess the extracted features to eliminate noise and outlier, making the data more suitable for model training. This can include data cleansing, feature normalization, missing value processing, and other steps.

4. Data partitioning: divide the dataset into training and testing sets. The training set is used for model learning and parameter estimation, while the testing set is used to evaluate the performance of the model.

5. Model selection: select a suitable prediction model to model the network PLR. Machine learning algorithms (such as regression, decision tree, and Random Forest) or statistical models (such as time-series analysis and Bayesian network) can be considered.

6. Model training: use the training set to train the selected model. Depending on the selected model, it may be necessary to adjust the hyperparameters of the model and use appropriate optimization algorithms for model parameter estimation.

7. Model evaluation: use a test set to evaluate the performance of the model. Various indicators can be used, such as root-mean-square deviation, mean absolute error, and coefficient of determination (R²), to measure the fit and accuracy of the model.

8. Model optimization: based on the evaluation results, optimize and improve the model. This may include adjusting the model structure, improving feature extraction, and adding more training data.

9. Prediction: use the trained model to predict the future network PLR. Predictions can be updated based on real-time collected data, and network configuration and resource allocation can be adjusted in a timely manner.

10. Monitoring and adjustment: continuously monitor the prediction performance of the network PLR, and adjust and optimize the model and system as required to improve the prediction accuracy and network performance.

3.3 Design the frame of PLR prediction

In order to solve the problem of packet loss in the network, a prediction method is developed and a method for calculating the link PLR is proposed. When modeling, consider network conditions: some predictive models consider changes in network conditions during the modeling process. They may use variables such as latency and bandwidth as features and incorporate them into the model for modeling. Such a model can improve prediction performance by establishing more accurate correlations with changes in network conditions. Dynamic update models: some models can dynamically update predictions based on the real-time observed network conditions. For example, if latency or bandwidth changes, the model can adapt to new network conditions by retraining or adjusting parameters. This method can improve the adaptability and accuracy of the model. Figure 1 shows the link PLR predictions based on this model.

Figure 1

Frame of PLR prediction.

As can be seen from Figure 1, the prediction framework is mainly composed of two parts: the first part is the training part, which uses the network measurement method to obtain prior information, and the second part is mainly for prediction, without sending data packets; the data packets can be sent to the destination node through the background traffic, and the network link PLR can be predicted by the acquired prior information [13]. If the network is unstable, it will lead to errors in the prediction results; therefore, the authors adopt the active transmission method to obtain the detection data packets of the prior information in advance. This process does not need to send detection packets, and the observation data can be obtained by relying on the background flow. Therefore, the logic tree topology T composed of nodes and links represented by V and E is first considered in the prediction, and T = ( V , E ) is obtained at this time, and the path from the source node S to the destination node D i can be represented by P ( S , D i ) . Since the sum of the link PLRs is 1, the link PLR can be obtained only by preliminary prediction of the link transmission probability. From M target nodes and N links, M equations can be obtained from source and target nodes containing N variables. The calculation model is as follows:

where P i is the end-to-end path prediction result between the source node S and the destination node D i , K is the total number of network links, and α n is the predicted probability of link n. If M < N , the network packet loss prediction equation has no unique solution.

3.4 Establish the mapping between PLR and user experience quality QoE

The mathematical prediction model for network PLR may have some limitations and assumptions in its application. The following are some common limitations and assumptions: steady-state assumption – many mathematical prediction models assume that the network PLR is predicted under steady-state conditions. This means that the model assumes that the characteristics of the network (such as latency, bandwidth, and traffic) remain unchanged during the prediction period. However, in actual networks, these features may change, which may affect the accuracy of the model. Assume a specific distribution: some prediction models may assume that the arrival time and service time of data packets follow a specific probability distribution, such as Poisson distribution or exponential distribution. However, the distribution of actual network data may be more complex and may not fully comply with these assumptions. This may have a certain impact on the accuracy of the model. Based on the study of the impact of packet loss on the quality of user experience QoE, the mapping of PLR to video quality QoE is established. As we all know, the evaluation methods of QoE mainly include subjective methods and objective methods, which are very difficult to operate. Therefore, the objective QoE evaluation method will map the measured PLR to the user experience quality QoE, which becomes a new idea.

According to the scatter diagram of the relationship between the PLR and the peak signal-to-noise ratio (PSNR) value, it is found that there is a cubic relationship between the PLR and the PSNR value after matching the linear, square, mixed, logarithmic, power, cubic, and other models. Form: y = ax + bx2 + cx3 + d. Here, the regression equation is established: y = ax3 + bx2 + cx + d. From this expression, we can see that this is a univariate nonlinear equation, and the coefficient of the equation can be obtained using the univariate nonlinear fitting method. It can also be seen as the fitting of univariate third-order polynomial.

3.5 Build a mathematical model for network PLR prediction

The bit-level errors generated in the simulation can not only effectively improve the resilience of network packets, but also improve the prediction reliability. However, the interval of bit-level errors is long, which requires long-term observation, and the amount of computation increases accordingly. In order to further reduce the amount of calculation, the mathematical symbol of the packet loss state is defined, if the network packet is lost during transmission, its mathematical symbol is 0, and if the receiving end loses the packet, the mathematical symbol is 1. In order to reduce the computational cost, a hidden Markov model (HMM) is used to analyze the packet loss state of digital signals and allocate the estimated packet loss state, and the calculation model is as follows:

where a 1 k represents the network packet loss prediction state, d 1 + d 2 + + d N represents the HMM length corresponding to the error-free state, and d k represents the HMM length corresponding to the burst state k [14]. Substitute the packet loss state distribution model in (1) to obtain the network PLR prediction model as shown in Figure 2.

Figure 2

Prediction model of network PLR.

3.6 PLR and retransmission rate

TCP provides a reliable transport protocol, and one way to use it is to accept data received from the other end. But both the data set and the confirmation message will disappear. TCP solves this problem by setting the time to send gas. If no notification is received when the timer expires, the data will be sent again. When multiple packets of the same window are lost, TCP transmission performance may be greatly affected. TCP uses a cumulative acknowledgment mechanism, forcing the sender to wait a round-trip time to find the lost packets. Although retransmitting unnecessary packets that may have been received can reduce overall throughput, most TCP congestion control algorithms, including the improved congestion control mechanism based on traditional TCP-Reno used by Windows and the default BIC and improved CUBIC used by Linux, do not use the selective acknowledgement (SACK) mechanism, that is, the data receiver can notify the sender of all successfully sent data packets, and the sender only needs to send the real lost data. Therefore, the phenomenon of retransmission still exists to a large extent. This is also the basis for most of the research on the PLR and the author’s use of retransmission to determine packet loss. This is because in the TCP transmission process, the transmission of payment information can be bidirectional or unidirectional. Therefore, a situation arises where the client only responds to the information sent by the server [15]. At this point, the number of contiguous packets sent by the client is the same, but the response number is different. Therefore, it cannot be judged that the message is retransmitted only based on the sequence number matching. Therefore, the retransmission message is defined as follows:

Definition 1: in a unidirectional TCP flow, a message with completely consistent message information is called a retransmission message.

Definition 2: the retransmission rate is the ratio of the number of retransmitted packets to all packets in a unidirectional TCP flow.

The specific formula is as follows (3):

where N r represents the number of retransmitted packets, and N s represents the total number of packets.

In the TCP transmission process, when the receiver receives an out-of-sequence segment, it will immediately generate an acknowledge character (ACK) (repeated ACK) to let the sender know that it has received an out-of-sequence segment and tell the sender the serial number it wants to receive. Therefore, the sender does not know whether a duplicate ACK is caused by a missing segment or just a segment reordering, which may only result in 1–2 duplicate ACKs. The repeated ACKs at this time are not due to packet loss. If the sender receives three or more ACKs, it determines that the packet has been lost and resends the lost data. This process is a TCP fast retransmission mechanism. At this point, the data packet returned by the sender is subject to packet loss, so the ACK packet returned by the receiver cannot be considered as packet loss. Also, if the P packet is dropped before the indicator, then the P packet and the subsequent return packet will appear on the indicator only once, and the drop of the P packet will be a packet return; if the packet P is lost after the indicator, the return packet P and the return packet can be found on the indicator; and if the packet is returned, the lost packet P can be found at this time. So there are the following points.

Definition 3: at the receiving end, after sorting a unidirectional TCP stream by sequence number, if the non-pure ACK packet P 2 is retransmitted after the packet P 1 , and the P 1 is not a retransmitted packet, the P 1 or P 2 is a packet loss packet.

According to this definition, if a retransmitted packet occurs after a non-retransmitted packet is observed, and the retransmitted packet is not a pure ACK packet, it is impossible to determine which packet is the lost packet, but it can be determined that the retransmitted packet is not a pure ACK packet – packet loss.

Packet loss will cause retransmission; therefore, it is a commonly used research method to estimate the PLR by counting retransmitted packets. Many previous studies were based on PLR; strictly speaking, it is actually based on the retransmission rate. Cllsen evaluates the PLR by comparing the number of retransmitted packets and the total number of packets; it also shows that for the TCP algorithm that does not use the SACK mechanism, this method, which is evaluated in terms of retransmission rate, may overestimate the PLR [16]. Under normal circumstances, when a packet loss occurs, the TCP sender will retransmit all the sent packets after the packet loss. Obviously, the number of retransmitted packets will be larger than the number of lost packets. Therefore, the retransmission rate should be related to the basic PLR, but there are different measures. Therefore, the authors start from the comparison between the retransmission rate and the PLR and, specifically, studies the relationship between the retransmission rate and the PLR.

Network data transmission module is an important part of network PLR prediction. Parameters are processed in task-driven mode using data transmission module implementation machine. The system is divided into four steps, namely, loading socket library, server initialization, ListenThreadFunc that listens to the wiretap, and TestTask wiretap. Receive Leigh Load socket library with InitInstanceO function in App Network data, add WSAStartup function, create socket before initialization Word, and initialize it. Second, use CServerD1g to increase msocket change Volume into SOCKET, privatize socket library access, and increase return Value InitSocket to initialize the network server. msocket generated by the function is the same as the socket library, representing server initialization Success. Thirdly, use the listening thread “ListenThreadFunc” to listen to clients, and achieve the goal of listening to multiple clients by changing the listening thread parameters. In addition, listen to the accept function in the thread, which can generate a new. The accept function is in the successful execution state, m_ISOK function. The change to TRUE function determines the connection status of each client to the server. Fourth, after each client successfully connects to the server, it sends the test task to the TestTask thread. Its RECVPARAM structure passes parameters and sockets to TCP, and sends test tasks after inputting packet thresholds. You can determine whether the related parameters are entered successfully. After the connection is successful, the connection between the server and the client is enabled – letter connection. Through the aforementioned mechanism, data transmission is completed.

3.7 Other related lateral degrees

3.7.1 First packet loss measurement

If the result of transition from a congestion-free network state to a congested state is to be evaluated, the object of interest is the loss of the first packet. It has been proven that the network is congested after the first packet is lost, and theoretically, more packet losses occur. Therefore, comparing the first PLR with the total PLR, it is possible to study the effectiveness of packet loss regulation, that is, theoretically, the total PLR will be greater than the initial PLR. Therefore, the authors will use this measurement, the original packet loss measurement, and compare it to the overall packet loss [17]. The specific definitions are as follows:

Definition 4: for a TCP flow with packet loss, the first packet loss is called the first packet loss.

Definition 5: the total number of packets sent before the first packet is lost is the number of packets sent before the first packet is lost.

According to Definition 5, if a TCP flow has packet loss, the packet forwarded before the first packet loss is the packet forwarded before the first packet loss; if no packet loss occurs in a TCP flow, the forwarded packets before the first packet loss are all the packets of this flow.

Definition 6: The ratio of the total number of the first packet loss to the total number of forwarded packets before the first packet loss is called the first PLR.

The specific formula is as follows:

(4) p fl = p firstloss = N f N fs ,

where N f is the total number of packets lost for the first time, and N fs is the total number of packets before the first packet loss, including the sum of the total number of packets in the flow without packet loss and the number of packets before the packet loss in the flow with packet loss.

For example, the following is two complete end-to-end TCP transmission processes with packet loss, as shown in Figure 3. Among them, sequence 1 is the TCP flow without packet loss, and the number of packets is n ₁; sequence 2 is the TCP flow with packet loss, and the number of packets is n ₂. Assuming that unordered factors are not taken into account, the sequence of TCP packets observed at the receiving end (i.e. sorted by time) can be distinguished by the following method:

Figure 3

Sequence of messages sorted by time.

In sequence 2 of Figure 3, underlined messages indicate retransmission messages. The packet sequence obtained by sorting it according to the sequence number is shown in Figure 4.

Figure 4

Message sequence sorted by message sequence number.

In sequence 2 of Figure 4, by definition 3, since t 4 is retransmitted and t 3 is not retransmitted, if t 4 is not a pure ACK packet, t 3 is a packet loss packet, which is represented by a*. If the observed objects are generally these two TCP streams, then according to definitions 2–3, the retransmission rate is 2 n 1 + n 2 ; according to definition 6, the first PLR parameter N f is 1, and the parameter N fs is all the packets marked with O, namely, n 1 + 3 , so the first PLR is 1 n 1 + n 3 .

4.1 Test process

Using two test cases, verify that the data transfer module can transfer normally when it encounters obstacles. First, the precondition is to perform a piling test on the client, the data transmission server and the client are both enabled, and the link between the server and the client is normal. Second, the prerequisites are the same as expected: increase the amount of data to 5,000, enter the address of the target user, click the start button, and wait for the result [20]. The authors modeled the slow data transmission in the complex network environment in the platform as shown in Figure 5.

Figure 5

Data transmission delay in complex network environment.

Under normal circumstances, the data transmission delay can ensure the real-time performance of network data packet transmission within 5 s. As shown in Figure 5, the delays of network nodes are all within 5 s, so the real-time nature of data transmission can be guaranteed.

4.2 Test results

To analyze the network packet loss problem of the model developed by the authors, the data loss rate (PLR) is calculated and the model is as follows:

where PLR represents the packet loss rate of data packets, the number of data packets lost in the estimated path, and the total data text packet in a predictable manner. The test results are shown in Table 1.

Based on the data in Table 1, we can observe that the predicted PLR of the mathematical model designed in this study is consistent with the actual PLR when the total number of packets is the same as the number of lost packets. This consistent result indicates that the mathematical model designed in this study has precise characteristics in predicting network PLR. This consistent result means that our mathematical model can accurately capture the factors that cause network packet loss and effectively predict packet loss events. The consistency between the predicted results of the model and the actual observed values proves the effectiveness and accuracy of the model.

When the total data packets and the number of lost data packets are the same, the PLR predicted by the mathematical model is consistent with the actual PLR. The mathematical model designed in this study has the characteristic of accurate prediction effect.

The mathematical prediction model for network PLR may perform better in specific scenarios or network configurations, while it may not reach its optimal state in other situations. The following are some examples of specific scenarios and network configurations that may cause the model to fail to reach the optimal state: complex network topology. If the network has a complex topology, such as large-scale distributed systems or multi-level networks, traditional mathematical prediction models may not accurately capture the changes and propagation of network PLR. This is because these models are usually based on simplified assumptions and topological structures, while ignoring the complexity of network topology. In a highly dynamic network environment, traditional mathematical prediction models may not be able to adapt to changes in a timely manner, such as significant fluctuations in link quality and sudden increases in traffic. This is because these models are typically based on steady-state assumptions and historical data for prediction, and in highly dynamic environments, models may need to be updated and adjusted more frequently. When facing these specific scenarios or network configurations, targeted adjustments or selection of more suitable prediction models may be necessary. This may include adopting more complex models, considering more network features, adding more training data, or using models specifically tailored to specific scenarios. Taking into account network conditions and model characteristics, selecting a suitable prediction model can improve the accuracy and reliability of predictions.