Method for maximum-plus stochastic network calculus analysis of delay performance of cyber-physical systems

Abstract

The application discloses a method for analyzing delay performance of a cyber-physical system by maximum plus stochastic network calculus, and relates to the technical field of cyber-physical systems. The method comprises the following steps: dividing the cyber-physical system into a plurality of network nodes; performing mathematical modeling on each network node; defining an arrival curve and a service curve respectively; performing equivalent deformation on an inequality in the definition of the service curve to obtain an inequality in the same form as the arrival curve; deducing the arrival curve and a corresponding boundary function thereof and the service curve and a corresponding boundary function thereof; introducing two lemmas and combining the obtained parameters to calculate a final inequality; and analyzing end-to-end delay of the cyber-physical system through the inequality. The application models a wireless channel by using a new model, simplifies the calculation of parameters, performs equivalent deformation on the inequality of the service curve to facilitate the calculation of the service curve without changing the original meaning, and verifies the effectiveness of the result through experiments and analyzes the influence of different parameters on system delay.

Description

Technical Field

[0001] This invention belongs to the field of cyber-physical systems technology, and particularly relates to a method for analyzing the latency performance of cyber-physical systems using maximum plus random network calculus. Background Technology

[0002] Cyber-physical systems are complex systems that integrate multiple technologies such as sensing, computing, communication, and control, with sensing technology being the most fundamental technology. For example... Figure 1 As shown, it is a cyber-physical system with multiple distributed sensors and capable of data fusion through edge computing. That is, the sensors distributed in different areas send data to the edge server. The edge server performs data fusion processing on the received information to reduce data redundancy, and then transmits the processed data to the remote control terminal. The remote control terminal processes the information, converts it into execution commands, and then sends them back to the sensors, thus forming a control loop.

[0003] To ensure the stability of cyber-physical systems, studying network latency is essential. Currently, network calculus serves as a valuable tool for investigating network latency boundaries. Network calculus defines arrival and service curves to describe the basic characteristics of the input flow and the server's processing capacity, respectively, and then uses mathematical tools to derive the latency boundaries. Network calculus is divided into deterministic and stochastic network calculus. Because the capacity of wireless channels varies randomly over time, stochastic network calculus is more suitable for handling wireless channels than deterministic network calculus. The mathematical tools used in network calculus are also divided into minimum-plus and maximum-plus methods. Minimum-plus network calculus has been applied to the analysis of various network performances and has a relatively complete theoretical framework. However, it is difficult to model networks with packet retransmission and that provide services to users probabilistically. Furthermore, serial systems with edge computing capabilities can lead to non-conservation of traffic, which contradicts the essence of minimum-plus network calculus.

[0004] Currently, most analyses of end-to-end latency are based on minimum plus random network calculus. The problem with minimum plus random network calculus is that it is not applicable to systems where flow is not conserved. However, cyber-physical systems do have the problem of flow non-conservation, so minimum plus random network calculus is not applicable to cyber-physical systems. Secondly, minimum plus network calculus is difficult to simulate for network nodes with retransmission mechanisms. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a method for analyzing the latency performance of cyber-physical systems using maximum addition random network calculus. By introducing the concept of maximum addition, this invention uses maximum addition random network calculus to study and analyze the end-to-end latency performance of cyber-physical systems with edge computing capabilities and multiple distributed sensors.

[0006] The objective of this invention can be achieved through the following technical solutions:

[0007] This application provides a method for analyzing the latency performance of a cyber-physical system using maximum random network calculus, including the following steps:

[0008] S1 divides the cyber-physical system into several network nodes;

[0009] S2, Perform mathematical modeling on each network node to obtain corresponding parameters;

[0010] The network nodes include general wireless channels and parallel wireless channels; the general wireless channel is modeled using the pp model to obtain parameter σ; the parallel wireless channel is modeled using a multi-queue forked connection model to obtain parameter ρ.

[0011] S3 defines the arrival curve and the service curve respectively;

[0012] S4, perform equivalent transformations on the inequalities in the service curve definition to obtain a transformation inequality of the same form as the arrival curve;

[0013] S5, Derive the arrival curve and its corresponding boundary function for the arrival process, and the service curve and its corresponding boundary function for the service process;

[0014] S6. Introduce two lemmas and calculate using the parameters to obtain the final inequality;

[0015] S7, Analyze the final inequality to obtain the end-to-end latency of the cyber-physical system;

[0016] The final inequality is used to reflect the relationship between the delay magnitude and the violation probability.

[0017] Preferably, the cyber-physical system includes: a plurality of distributed sensors, edge computing servers, remote control servers, and actuators.

[0018] Preferably, the end-to-end latency includes: a parallel wireless channel portion S1 from the sensor to the edge computing server, a processing portion S2 of the edge computing server, a wireless channel portion S3 from the edge computing server to the remote control server, a processing portion S4 of the remote control server, and a wireless channel portion S5 from the remote control server to the actuator.

[0019] Preferably, the processing part S2 of the edge computing server includes data fusion and feature extraction.

[0020] Preferably, data from several of the sensors are collected and transmitted simultaneously, and all data packets transmitted by each of the sensors at the same time are set as a whole.

[0021] Preferably, the arrival curve for each node is obtained using the following formula:

[0022]

[0023] Where r A To reach the slope of the curve, r A <ρ A (-θ A );

[0024] Where a(n) represents the arrival time of the data packet n≥1, and its coefficient is λ. A The Poisson process.

[0025] Preferably, the service curve for each node is obtained using the following formula: Where r is the slope of the service curve, and r > ρ(θ).

[0026] Preferably, the service model for the parallel channel is obtained, and its formula is:

[0027] β f (n)=r f n

[0028] Where, r f >ρ f (θ f ).

[0029] Preferably, the lemma includes Lemma 1 and Lemma 2; Lemma 1 is a delay bound for multi-node cascades; Lemma 2 is: for any positive number a... k b k , k = 1,...,K and for any x ≥ 0, we have: in,

[0030] Preferably, the inequality is obtained by formulating:

[0031]

[0032] Wherein, θ1, θ2 and θ3 are the parameter values ​​of the network nodes S1, S3 and S5, respectively.

[0033] The beneficial effects of this invention are as follows:

[0034] (1) By introducing maximum plus random network calculus, the impact of non-conservation of traffic is avoided, which facilitates the modeling of the retransmission mechanism.

[0035] (2) The PP model is used to model the wireless channel, which realizes a reasonable and uncomplicated model of the wireless channel and facilitates the acquisition of parameters.

[0036] (3) The parallel wireless channel is modeled using a multi-queue bifurcation connection model, which simplifies the complex multi-channel transmission problem into a parallel server structure. Furthermore, this simplification allows the parameter acquisition to reuse the general wireless channel method.

[0037] (4) Redefine the service curve by making equivalent changes to the inequalities in the definition of the service curve, so as to simplify the determination of the service curve and the corresponding boundary function without changing the original meaning. Attached Figure Description

[0038] To better understand and implement this application, the technical solution is described in detail below with reference to the accompanying drawings.

[0039] Figure 1 This is a schematic diagram of the structure of a traditional cyber-physical system;

[0040] Figure 2 A flowchart illustrating the steps of a method for analyzing the latency performance of a cyber-physical system using maximum random network calculus, as provided in this application embodiment;

[0041] Figure 3 A schematic diagram illustrating the end-to-end time delay from sensor to actuator provided for embodiments of this application;

[0042] Figure 4 A schematic diagram illustrating the relationship between the violation probability and the delay limit for different numbers of parallel channels k in embodiments of this application;

[0043] Figure 5 A schematic diagram illustrating the relationship between the probability of violating the end-to-end delay limit and the delay limit, which is the probability of successfully transmitting data packets p-value provided in this application embodiment;

[0044] Figure 6 This is a schematic diagram illustrating the relationship between the parameter θ value and the delay limit provided in the embodiments of this application. Detailed Implementation

[0045] To further illustrate the technical means and effects adopted by the present invention to achieve its intended purpose, exemplary embodiments will be described in detail below, examples of which are illustrated in the accompanying drawings. In the following description relating to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of methods and systems consistent with some aspects of this application as detailed in the appended claims.

[0046] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. The singular forms “a,” “the,” and “the” used in this application and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to any and all possible combinations comprising one or more of the associated listed items.

[0047] The following detailed description of the specific implementation methods, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided in detail.

[0048] Example 1

[0049] Please see Figure 2 This invention provides a method for analyzing the latency performance of a cyber-physical system using maximum random network calculus, comprising the following steps:

[0050] S1 divides the cyber-physical system into several network nodes;

[0051] S2, Perform mathematical modeling on each network node to obtain corresponding parameters;

[0052] The network nodes include general wireless channels and parallel wireless channels; the general wireless channel is modeled using the pp model to obtain parameter σ; the parallel wireless channel is modeled using a multi-queue forked connection model to obtain parameter ρ.

[0053] S3 defines the arrival curve and the service curve respectively;

[0054] S4, perform equivalent transformations on the inequalities in the service curve definition to obtain a transformation inequality of the same form as the arrival curve;

[0055] S5, Derive the arrival curve and its corresponding boundary function for the arrival process, and the service curve and its corresponding boundary function for the service process;

[0056] S6. Introduce two lemmas and calculate using the parameters to obtain the final inequality;

[0057] S7, Analyze the final inequality to obtain the end-to-end latency of the cyber-physical system;

[0058] The final inequality is used to reflect the relationship between the delay magnitude and the violation probability.

[0059] Furthermore, the cyber-physical system includes: a number of distributed sensors, edge computing servers, remote control servers, and actuators.

[0060] Furthermore, the end-to-end latency includes: a parallel wireless channel portion S1 from the sensor to the edge computing server, a processing portion S2 of the edge computing server, a wireless channel portion S3 from the edge computing server to the remote control server, a processing portion S4 of the remote control server, and a wireless channel portion S5 from the remote control server to the actuator.

[0061] Furthermore, the processing part S2 of the edge computing server includes data fusion and feature extraction.

[0062] Furthermore, data from several of the sensors are collected and transmitted simultaneously, and all data packets transmitted by each sensor at the same time are set as a whole.

[0063] Furthermore, the arrival curve for each node is obtained using the following formula:

[0064]

[0065] Where r A To reach the slope of the curve, r A <ρ A (-θ A );

[0066] Where a(n) represents the arrival time of the data packet n≥1, and its coefficient is λ. A The Poisson process.

[0067] Furthermore, the service curve for each node is obtained using the following formula: Where r is the slope of the service curve, and r > ρ(θ).

[0068] Furthermore, the service model of the parallel channel is derived, and its formula is:

[0069] β f (n)=r f n

[0070] Where, r f >ρ f (θ f ).

[0071] Furthermore, the lemma includes Lemma 1 and Lemma 2; Lemma 1 is the delay bound of a multi-node cascade; Lemma 2 is: for any positive number a... k b k , k = 1,...,K and for any x ≥ 0, we have: in,

[0072] Furthermore, the inequality is obtained, and its formula is:

[0073]

[0074] Wherein, θ1, θ2 and θ3 are the parameter values ​​of the network nodes S1, S3 and S5, respectively.

[0075] Specifically, the entire analysis method consists of three steps. First, mathematical modeling is performed on each network node of the entire network system. This application uses the pp model to model a general wireless channel and a multi-queue bifurcation connection model to model a parallel wireless channel, obtaining the corresponding parameters σ and ρ. Second, the inequalities in the service curve definition are transformed into equivalent forms, resulting in the same form as the arrival curve. The arrival curve and its corresponding first boundary function for the arrival process, as well as the service curve and its corresponding second boundary function for the service process, are derived. Third, using two lemmas and the parameters obtained in the first two steps, the final inequality is calculated. This final inequality reflects the relationship between latency and violation probability, and its application allows for the analysis of the latency performance of the cyber-physical system.

[0076] Please see Figure 3 The end-to-end latency includes a parallel wireless channel section S1 from the sensor to the edge computing server, an edge computing section S2 which includes data fusion and feature extraction, a wireless channel section S3 from the edge computing server to the remote control server, a remote control processing section S4, and a wireless channel section S5 from the remote control to the actuator.

[0077] Since data from multiple sensors are collected and transmitted simultaneously, all data packets transmitted by each sensor at the same moment can be considered as a whole. Therefore, it is unnecessary to discuss the arrival process of each sensor's data; only the overall arrival process needs to be discussed to represent the sensors. The data packets mentioned below refer to the data packets in this overall concept. Assume that the arrival process a(n) represents the arrival time of the n≥1th data packet, which is a coefficient λ. A The Poisson process is such that the sequence of arrival time intervals for each data packet is an independent and identically distributed exponential distribution with parameter λ. A In this embodiment, the arrival and service processes are (σ,ρ) constrained processes. Constraint processes are divided into lower-constraint and upper-constraint types. The definition of a constraint process is given here:

[0078] The lower constraint type is defined as follows:

[0079] Assume the function g(n) is of the lower bound type. If n ≥ m ≥ 1 and θ > 0, the following inequality is satisfied:

[0080] E(e -θ[g(n)-g(m)] )≤e-θ[ρ(θ)(n-m)-σ(θ)] (1)

[0081] The upper constraint type is defined as follows:

[0082] Assume the function g(n) is of upper constraint type, and assume n ≥ m ≥ 1 and θ > 0, satisfying the inequality:

[0083] E(e θ[g(n)-g(m)] )≤e θ[ρ(θ)(n-m+1)+σ(θ)] (2)

[0084] The parameters σ and ρ can be regarded as the intercept and slope of the affine function; θ is a constant that meets specific requirements. The magnitude of this constant directly determines the tightness of the final delay limit. We will use different subscripts to indicate the ownership of θ, but in essence, they are all constants that are set in advance by the analyst.

[0085] In this embodiment of the application, the arrival process is a (σ A (θ A ),ρ A (θ A The lower constraint type of )) is that which satisfies Suppose that {a(n+1)-a(n), n=1,2,3,...} are independent and identically distributed time intervals, which satisfy... Where n≥1, the time interval is λ. A The characteristics of the exponential distribution can be summarized as follows:

[0086]

[0087] σ A (-θ A )=0 (3)

[0088] Where, θ A >0.

[0089] Because the arrival process a(n) is (σ A (θ A ),ρ A (θ A If the lower constraint type is given, then its corresponding random arrival curve α(n) and boundary function f(x) can be obtained by the following formula:

[0090]

[0091] Where r A To reach the slope of the curve, r must be satisfied. A <ρ A (-θ A ).

[0092] The above describes the process of obtaining the arrival curve. Next, we will discuss the service curves of each node.

[0093] First, all nodes in the system, including parallel wireless channels, general wireless channels, edge computing, and remote control service processes, are of upper constraint type (σ(θ), ρ(θ)), that is, for n≥m≥1 and θ>0, the following inequality is satisfied:

[0094]

[0095] Where s(n,m) represents the service time experienced from the m-th data packet to the n-th data packet.

[0096] If the arrival process s(m,n) is of lower constraint type (σ(θ),ρ(θ)), then its corresponding random arrival curve β(n) and boundary function g(x) are, where

[0097]

[0098] Where r is the slope of the service curve, which must satisfy r > ρ(θ).

[0099] This application uses the following subscripts to distinguish the parameters of each service process: subscript w represents a general wireless channel service process; subscript f represents a parallel wireless channel service process; and subscripts b and c represent edge computing service processes and remote control service processes.

[0100] This application first describes the service model of a general wireless channel, and some conclusions from this section will be used in the subsequent parallel channel service model. The wireless channel is modeled using the PP (Probabilistic Priority) model, which stipulates that only one data packet can be transmitted within a time slot τ, and only at the beginning of the time slot. Assuming the probability of successfully transmitting a data packet within a time slot is p, and the probability of failure is q = 1-p, it is clear that the time length for transmitting a packet follows a geometric distribution, so its expected value is...

[0101] Let the wireless channel service process be (σ w (θ w ),ρ w (θ w The upper constraint type is )). Therefore, for n≥m≥1 and θ Sw >0 holds the following inequalities:

[0102]

[0103] Similarly, consider {s w The groups (v,v), v=1,2,3,...} are independent and identically distributed, and all follow a geometric distribution. Therefore, we have Therefore, substituting into the formula, we can deduce:

[0104]

[0105] σ w (θ w )=0 (8)

[0106] in,

[0107] Combining inequality (8) and inequality (6), we get:

[0108] β w (n)=r w n

[0109]

[0110] Where, r w >ρ w (θ w ).

[0111] Next, the service model for parallel channels is explained, using a multi-queue forked connection system to solve the parallel channel problem. Assuming multiple sensors simultaneously collect and transmit data, all data packets at the same moment can be considered as a whole. The process of them arriving at the edge computing end through different wireless channels can be seen as a forking process; the process of waiting for all data packets to arrive at the edge computing end before further data processing is seen as a connecting process. Treating the wireless channel as a parallel server, the entire parallel channel module can be treated as a multi-queue forked connection system.

[0112] Q i (n) represents the service duration of the i-th data packet in the n-th total data packet. Assuming all parallel channels are independent and identically distributed, any channel can be treated as a separate wireless channel, and any wireless channel is (σ... Q (θ f ),ρ Q (θ f The upper constraint type of )) where σ Q (θ f ) and ρ Q (θ f The parameters are obtained according to equation (8). The formula for the parallel channel parameters is derived from the single-channel parameters, namely:

[0113]

[0114] ρ f (θ f )=ρ Q (θ f (10)

[0115] Where k represents the number of parallel operations. Substituting σ... Q (θ f ) and ρ Q (θ f From this, we can obtain:

[0116]

[0117]

[0118] in,

[0119] Combining inequality (11) and inequality (6), we get:

[0120] β f (n)=r f n

[0121]

[0122] Where, r f >ρ f (θ f ).

[0123] The edge computing module and the remote control module have the same model, so they can be described together. Since processing time depends only on the data packet size, this application assumes the data packet size is constant. Therefore, the data processing time for edge computing and remote control is also constant, denoted as T. b and T c That is, its delay D b (n)=T b D c (n)=T c .

[0124] Before conducting the delay analysis, this application introduces two lemmas.

[0125] Lemma 1 is a delay bound for a multi-node cascaded system. Assume the input stream passes through an N-node cascaded system, and the random service curve β of the input stream for each independent node... k ∈G(k=1,2,...,N), and the boundary function g k ∈G(k||1,2,...,N), and furthermore, the random arrival curve α∈G of the input stream, and the boundary function f∈G, let D(n) be the delay of the nth > 0th data packet. For have:

[0126]

[0127] in,

[0128] Lemma 2 states: For any positive number a k b k , k = 1,...,K and for any x ≥ 0, we have:

[0129]

[0130] in,

[0131] Based on the two lemmas above, the following final inequality can be given:

[0132] For the end-to-end delay D(n) of a cyber-physical system with multiple distributed sensors and edge computing, it has:

[0133]

[0134] Where θ1, θ2 and θ3 represent the parameter values ​​of network nodes S1, S3 and S5 respectively, and p and q are similar.

[0135] The present invention will now evaluate the latency performance of a cyber-physical system with multi-sensor and edge computing based on the analysis results obtained above, and compare these results with simulation results using MATLAB software. Considering the system topology and... Figure 1 Without loss of generality, we assume that all wireless channels, including the parallel portion, have the same probability of successfully transmitting a data packet, and that the time slot τ for each transmission is also the same, where τ = 0.002s, and the edge computation delay T is... b =0.005s, remote control delay duration T c =0.007s. For the simulation set, the present invention processes the data packet input parameters as described above in the system 5000 times and records the delay information.

[0136] exist Figure 4 and Figure 5 The diagram shows the relationship between the violation probability and the delay limit for different numbers of parallel channels k and the probability of successful data packet transmission p, and compares these limits with simulation results. It can be seen that the violation probability decreases exponentially with increasing delay. Figure 4 and Figure 5 The comparison shows that both increasing the value of k and decreasing the value of p will increase the delay, but the influence of the value of p is slightly greater than that of the value of k. In other words, decreasing the value of p by a small amount will cause a large delay.

[0137] Figure 6 It displays the relationship between the parameter θ and the delay limit, showing that regardless of whether θ is reduced... i , {i=1,3,5} and θ ABoth of these factors will make the delay bound curve more relaxed, resulting in a more relaxed upper bound for the delay. In other words, to obtain a tighter bound, a larger θ must be chosen. i , {i=1,3,5} and θ A .

[0138] In summary, this invention provides a method for analyzing the latency performance of cyber-physical systems (CPS) with multiple sensors and edge computing based on maximum-plus-random network calculus (MPL). Currently, most end-to-end latency analyses are based on minimum-plus-random network calculus (MPL). However, MPL is unsuitable for flow-nonconservative systems, which are precisely what CPS systems are. Furthermore, MPL is difficult to simulate network nodes with retransmission mechanisms. To address these issues, we introduce the concept of maximum addition and use MPL-based random network calculus to study and analyze the end-to-end latency performance of CPS systems with edge computing capabilities and multiple distributed sensors. Moreover, we employ a novel model for wireless channel modeling, simplifying parameter calculation; we innovatively perform equivalent transformations of the service curve inequalities, facilitating service curve calculation without altering the original meaning; and finally, experiments verify the effectiveness of the results, analyzing the influence of different parameters on system latency.

[0139] In summary, the present invention solves the following technical problems:

[0140] (1) It solves the problems that minimum plus random network calculus is not applicable to systems with non-conservative flow and is difficult to simulate network nodes with retransmission mechanisms. Typically, cyber-physical systems (CPS) have multiple data processing modules, leading to non-conservative flow, where the flow at input nodes is not equal to the flow at output nodes. Most network systems also have retransmission mechanisms. Therefore, using traditional minimum plus random network calculus to analyze the latency of CPS is unsuitable. Maximum plus random network calculus, on the other hand, studies the latency of a single data packet, avoiding the impact of non-conservative flow, and also facilitates the modeling of retransmission mechanisms.

[0141] (2) The modeling problem of general wireless channels and parallel wireless channels based on maximum plus random network calculus has been solved. The most critical and difficult part of network calculus is how to model network nodes. As far as I know, current wireless channel modeling is based on minimum plus random network calculus, which is not applicable to maximum plus random network calculus.

[0142] (3) The problem of difficulty in obtaining the service curve has been solved. The service curve is an important parameter for obtaining the final delay limit. The general way to obtain it is through definition and complex probability analysis. However, there is a simple method for obtaining the arrival curve. We compared the definitions of the two and found that they have many similarities. By making an equivalent transformation of the service curve definition and using the method for obtaining the arrival curve to obtain the service curve, the difficulty of obtaining it is greatly reduced.

[0143] Therefore, the present invention has the following beneficial effects:

[0144] (1) By introducing maximum plus random network calculus, the impact of non-conservation of traffic is avoided, which facilitates the modeling of the retransmission mechanism.

[0145] (2) The PP model is used to model the wireless channel, which realizes a reasonable and uncomplicated model of the wireless channel and facilitates the acquisition of parameters.

[0146] (3) The parallel wireless channel is modeled using a multi-queue bifurcation connection model, which simplifies the complex multi-channel transmission problem into a parallel server structure. Furthermore, this simplification allows the parameter acquisition to reuse the general wireless channel method.

[0147] (4) Redefine the service curve by making equivalent changes to the inequalities in the definition of the service curve, so as to simplify the determination of the service curve and the corresponding boundary function without changing the original meaning.

[0148] In the above embodiments, the descriptions of each embodiment have different focuses. For parts that are not described in detail or recorded in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0149] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for analyzing the latency performance of a cyber-physical system using maximum summed random network calculus, characterized in that: Includes the following steps: S1 divides the cyber-physical system into several network nodes; S2, Perform mathematical modeling on each network node to obtain corresponding parameters; The network nodes include general wireless channels and parallel wireless channels; the general wireless channel is modeled using a pp model to obtain parameters. The parallel wireless channel is modeled using a multi-queue bifurcation connection model to obtain parameters. ; S3 defines the arrival curve and the service curve respectively; S4, perform equivalent transformations on the inequalities in the service curve definition to obtain a transformation inequality of the same form as the arrival curve; S5, Derive the arrival curve and its corresponding boundary function for the arrival process, and the service curve and its corresponding boundary function for the service process; S6. Introduce two lemmas and calculate using the parameters to obtain the final inequality; S7, Analyze the final inequality to obtain the end-to-end latency of the cyber-physical system; The final inequality is used to reflect the relationship between the delay magnitude and the violation probability. The cyber-physical system includes: a number of distributed sensors, edge computing servers, remote control servers, and actuators; Simultaneously collect and transmit data from several of the aforementioned sensors, and set all data packets transmitted by each of the aforementioned sensors at the same time as a whole; The service curve for each node is obtained using the following formula: ,in For the slope of the service curve, ; The service model for the parallel channel is derived from the following formula: ,in, .

2. The method for analyzing the latency performance of a cyber-physical system using maximum added random network calculus according to claim 1, characterized in that: The end-to-end latency includes: the parallel wireless channel portion from the sensor to the edge computing server. The processing part of the edge computing server The wireless channel portion from the edge computing server to the remote control server The processing part of the remote control server and the wireless channel portion from the remote control server to the actuator. .

3. The method for analyzing the latency performance of a cyber-physical system using maximum added random network calculus according to claim 2, characterized in that: The arrival curve for each node is obtained using the following formula: ; in To reach the slope of the curve, ; in, For the arrival process, it represents the first... The arrival time of the data packet, which is a coefficient. The Poisson process.

4. The method for analyzing the latency performance of a cyber-physical system using maximum added random network calculus according to claim 1, characterized in that: The lemmas include Lemma 1 and Lemma 2; Lemma 1 is a delay bound for multi-node cascaded circuits; Lemma 2 is: for any positive number... , , And for any ,have: ,in, .

5. The method for analyzing the latency performance of a cyber-physical system using maximum added random network calculus according to claim 4, characterized in that: The final inequality is obtained, and its formula is: ; in, , and The network nodes are respectively , and The parameter value.

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