CN113961439A - DAG task WCRT calculation method based on SMT method - Google Patents

Abstract

The invention belongs to the field of embedded real-time systems, and particularly relates to a DAG task WCRT calculation method based on an SMT method. The method comprises the steps of firstly generating a DAG task; then establishing a DAG model; establishing an objective function; establishing conditional constraints of the nodes; to ensure that a node must begin execution after its predecessors have completed execution, a constraint node is not preemptible at execution. Establishing load retention constraints; at any moment, the execution condition is met at the node, and no idle core exists; and finally establishing a source point and starting to execute time constraint. By establishing the constraint for realizing the continuous working characteristics, the method ensures that for any node and critical predecessors thereof, if the length of a time interval is greater than 0, the executed load is equal to the computing resources which can be provided in the time interval, and finally realizes the conversion from the response time analysis problem to the optimization problem.

Description

DAG task WCRT calculation method based on SMT method

Technical Field

The invention belongs to the field of embedded real-time systems, and particularly relates to a DAG task WCRT calculation method based on an SMT method.

Background

Multi-core processors are gradually replacing traditional single-core processors and become the hardware basis for most computer systems. With the development of multi-core real-time systems, the demand for computing power is increasing. To fully utilize the computational resources of a multi-core processor and meet the rapidly growing demands for high performance computing and low power consumption, real-time software must support parallelism. The real-time system is widely applied to safety-critical systems such as automobiles and airplanes, and the fields have extremely high requirements on safety, so that the real-time system is very important for modeling and analyzing. The key to ensure the safety of the real-time system is to ensure that tasks in the system strictly meet real-time constraints, and the Worst Case Response Time (WCRT) analysis of the real-time tasks is an important method for ensuring the safety of the real-time system. The model of the real-time task is usually an abstract and simple model, so the analysis of the model is often efficient and low in complexity, but the model cannot be used for describing a complex real-time system, so the more the model of the real-time system can be described, the higher the difficulty of the analysis of the model is.

Directed Acyclic Graph (DAG) is a very expressive model in the field of real-time systems, and is a common model for modeling parallel programs. For the single task model, each node in the graph represents a code segment to be executed sequentially, and the edges between the nodes represent the dependency relationship of the nodes. In the real-time field, the famous scholars Graham proposes a classic computation method of WCRT boundaries for a general DAG model, and the general DAG model means that a graph does not contain a special structure. All studies in this respect are based on the Graham world since then. However, the Graham boundary is an inaccurate upper time boundary, and with the development of information technology, the requirement for the accuracy of the task response time boundary in a real-time system is higher and higher, so that the research based on the Graham boundary has certain limitations. The invention can be used to compute general DAG model-accurate WCRT.

Disclosure of Invention

The invention provides a method for systematically calculating general DAG accurate WCRT by utilizing Satisfiability Model Theory (SMT) in the field of operational research. The method solves the limitation problem that the time upper bound of the Graham bound is inaccurate in the prior art. The invention provides a computing method of a DAG task WCRT based on an SMT method, which comprises the following steps:

the method comprises the following steps: generating a DAG task;

step two: establishing a DAG model;

step three: establishing an objective function;

step four: establishing conditional constraints of the nodes;

step five: establishing load retention constraints;

step six: establishing a source point starting execution time constraint;

step seven: and solving the objective function.

Further, the objective function formula in step three is as follows:

max(f_sk-s_sc) (2)

further, the conditional constraints in step four include: the execution time constraint of the nodes, the priority constraint of the nodes and the non-preemptive constraint of the nodes ensure that a feasible solution exists in the target function;

further, the execution time constraint formula of the node is as follows:

further, the priority constraint formula of the node is as follows:

further, the node non-preemptive constraint formula is as follows:

further, in the load holding constraint in the fifth step, in order to satisfy the execution condition, there is no idle core, which is defined as follows:

for any node u_bAnd critical point thereofPrecursor u_aLogical expression Π in equation (6)₁Representing a time interval f_a,s_b]Is strictly greater than 0.

For any node u_bAnd its critical precursor u_aIf the following condition is satisfied, it is called node u_x∈P_ar(u_a) Is contained in the edge (u)_a,u_b) Relevant time interval [ f_a,s_b]Of (1), i.e. node u_xIs the interval [ f_a,s_b]Is provided with H_abRepresents a time interval [ f_a,s_b]A set of all the included nodes within.

Π₂:s_x＜s_b∧f_x＞f_a (7)

For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) B is a Boolean variable_abxThe definition of (A) is as follows,

further, the arbitrary side (u)_a,u_b) E-containing node set H_abDefinition of H_abTwo types of subsets of (a):

left subset H_L: h satisfying the following conditions_abNode u in_xBelonging to the left subset H_L，

Π₃:s_x＜f_a (12)

Right subset H_R: h satisfying the following conditions_abNode u in_xBelong to the right subset H_R，

Π₄:f_x＞s_b (13)

For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) Define a Boolean variable L_abx(R_abx) To represent u_xWhether or not it is contained in H_L(H_R) In (1), the definition is as follows,

further, the arbitrary side (u)_a,u_b) E and any node u_x∈P_ar(u_b) Establishing constraints for implementing the continuous operating characteristics:

the invention ensures that the node can be executed only after the precursor of the node is executed by establishing the DAG model, the objective function and the conditional constraint of the node, and the constraint node can not be preempted during the execution; establishing load holding constraint, and meeting execution conditions at a node at any time without idle cores; and establishing a constraint for realizing continuous working characteristics, ensuring that for any node and critical predecessor thereof, if the length of a time interval is greater than 0, the executed load is equal to the computing resource which can be provided in the time interval, and finally realizing the conversion from a response time analysis problem to an optimization problem.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a graph comparing experimental data according to the present invention.

Detailed Description

The constants and variables used in the SMT model were defined before implementing the invention:

-m: the number of cores included in the multi-core platform C;

-U: a set of nodes in graph G;

-E: set of edges in graph G;

-W_a: node u_aE, the worst case execution time of U;

-P_re(u_a): node u_aE is the set of all the predecessors of U;

-S_uc(u_a): node u_aBelonging to the set of all successors of U;

-P_ar(u_a): node u_aBelonging to the set of all parallel nodes of U;

-s_a: representing a node u_aE, starting execution time of U;

-f_a: representing a node u_aE, completing the execution time of U;

-e_a: representing a node u_aE is the execution time of U;

-B_abx: boolean type variable, edge (u)_a,u_b) E and node u_x∈P_ar(u_a) Is used to index the variable.

-L_abx: a Boolean type variable represented by [ f [ ]_a,s_b]Whether the start execution time of the inner node is less than f_a。

-R_abx: a Boolean type variable represented by [ f [ ]_a,s_b]Whether the completion execution time of the inner node is greater than s_b。

The invention discloses a real-time DAG task WCRT (worst case response time calculation) method based on SMT (surface mount technology), which aims to obtain the worst case response time of a general DAG task and provides a calculation method based on Satisfiability Model Theory (SMT), wherein the method can calculate the accurate worst case response time of the general DAG task; and expressing the dependency relationship between the nodes in the DAG model by using an inequality or an equality, and realizing the conversion from a response time analysis problem to an optimization problem by restricting the time behavior of the nodes. The method comprises the following specific steps:

the method comprises the following steps: a DAG task is generated. The user can set parameters of the DAG task and generate an information text of the DAG task.

Step two: and establishing a DAG model. In the task model, a real-time task is represented by DAGG ═ (U, E), where U represents a set of nodes of a Directed Acyclic Graph (DAG), and E is a set of edges connecting the nodes. Each node U in U_aRepresenting a continuous piece of executable code, and each node corresponding to an execution time e_aRepresenting the time required to execute the segment of code, each node has an execution time constraint called worst case execution time defined as e_aWill not exceed its worst case execution time W_aI.e. e_a≤W_a. Each side (u) in E_a,u_b) Representing node u_aAnd u_bConstraint relationship between, at the edge (u)_a,u_b) In China, call u_aIs u_bOne precursor of (a), (b), (c), (d)_bIs u_aThe existence of the predecessor successor relation limits the execution sequence of the nodes, namely the node u_bMust be at u_aExecution can only be started after execution is completed, defining P_re(u_a) Represents u_aSet of all predecessors, S_uc(u_a) Represents u_aAll subsequent sets. Furthermore, if u_aIs u_bIs a precursor of or u_bThe precursor of the precursor is called u_aIs u_bAncestor of u_bIs u_aDefine the symbol A_nc(u_a) Represents u_aSet of all ancestors, D_es(u_a) Represents u_aSet of all offspring, P_ar(u_a) Represents a sum of u_aThe set of all nodes executed in parallel, namely:

P_ar(u_a)＝U-A_nc(u_a)-D_ec(u_a)-{u_a} (1)

in addition, a node without an ancestor is defined as a source, and a node without a descendant is defined as an endpoint. To avoid loss of generality, the invention assumes that the DAG task G has only one source point u_scAnd an end point u_skThe method is also suitable for multi-source and multi-destination tasks, where there are multiple sources and destinations in task GIn this case, it is necessary to add one virtual source point and one virtual destination point having an execution time of 0, and establish edges pointing from the virtual source point to all the source points and edges pointing from all the destination points to the virtual destination points.

Step three: and establishing an objective function, namely formula (2), wherein the objective function value is the worst-case response time (WCRT) of the model.

max(f_sk-s_sc) (2)

Step four: establishing conditional constraints of the nodes;

and establishing the execution time constraint of the node. For any node u_aIt executes time e_aAnd cannot be greater than its worst case execution time. The constraint is as in equation (3),

a priority constraint for the node is established. For any edge (u) in E_a,u_b) Constructing the priority relationship between the nodes with the constraint of formula (4) to ensure the node u_bMust be in its predecessor u_aThe execution can only be started after the execution is completed,

and establishing node non-preemptive constraints. For any node U in U_aThe constraint represented by equation (5) is node u_aAre not preemptible at the time of execution,

step five: load retention constraints are established. Load maintenance is an important property of DAG tasks, i.e., at any time, if a node u_bIf the execution condition is satisfied, then there will be no idle core in C. First, some definitions and notations are given.

For any node u_bAnd its critical precursor u_aLogical expression Π in equation (6)₁Representing a time interval f_a,s_b]Is strictly greater than 0.

Wherein, P_re(u_a) Represented node u_aThe set of predecessors of.

For any node u_bAnd its critical precursor u_aIf the following condition is satisfied, it is called node u_x∈P_ar(u_a) Is contained in the edge (u)_a,u_b) Relevant time interval [ f_a,s_b]Of (1), i.e. node u_xIs the interval [ f_a,s_b]Is provided with H_abRepresents a time interval [ f_a,s_b]A set of all the included nodes within.

Π₂:s_x＜s_b∧f_x＞f_a (7)

For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) B is a Boolean variable_abxThe definition of (A) is as follows,

the formulas (9) to (11) ensure the Boolean variable B_abxThe condition of equation (8) is followed.

For any edge (u)_a,u_b) E-containing node set H_abFurther define H_abAre used to select the two types of subsets of,

left subset H_L: h satisfying the following conditions_abNode u in_xBelonging to the left subset H_L，

Π₃:s_x＜f_a (12)

Right subset H_R: h satisfying the following conditions_abNode u in_xBelong to the right subset H_R，

Π₄:f_x＞s_b (13)

For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) Define a Boolean variable L_abx(R_abx) To represent u_xWhether or not it is contained in H_L(H_R) In (1), the definition is as follows,

to implement equations (14) and (15) in an SMT program, the following inequalities are defined. For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b)，

Equation (16) ensures that for interval [ f_a,s_b]Arbitrary contained node u of_x(i.e. B)_abx1) if u_xBelonging to the left subset H_L(Π₃1), then L_abxEqual to 1. Similarly, formula (17) ensures that if u_xBelong to the right subset H_RThen R_abxEqual to 1.

For any node u_bAnd its critical precursor u_aTime interval [ f_a,s_b]Internal execution load W_abThe way of calculating (a) is as follows,

establishing constraints for achieving sustained operating characteristics: for edge (u)_a,u_b) E and any node u_x∈P_ar(u_b)，

Equation (19) ensures that u is arbitrary_bAnd its critical precursor u_aIf the time interval [ f_a,s_b]Is greater than 0 (i.e. pi)₁1) then in the interval f_a,s_b]The load internally performed is equal to the load at f_a,s_b]Within which computing resources may be provided.

Step six: and establishing a source point starting execution time constraint. Source point u_scExecuting on a certain kernel of C, constraints like equation (20),

s_sc＝0 (20)

step seven: and solving the objective function.

Experimental results prove that the SMT method can calculate the accurate worst-case response time of the DAG task.

In the first step, the parameters of the task are specified when the task is generated, and the meaning of the parameters is as follows, wherein n represents the number of nodes contained in the task, m represents the number of kernels, d represents the output degree of the nodes, and e represents the execution time of the nodes. For example, the experimental data obtained in fig. 2(a) is generated according to the parameter settings in which the number of kernels m is fixed to 6, the degree of departure d of the node is fixed to 3, the execution time e of each node is 10, and the variable is the number of nodes n, i.e., the horizontal axis in fig. 2 (a).

For each generated task, the solving time t of the SMT model is calculated, and in addition, the Graham boundary is calculated by using a classical method and is used for a comparison experiment. Defining a boundary difference DIF to compare Graham boundaries R_GAnd the precise time boundary R obtained by the SMT model_E。

The left vertical coordinate of each sub-graph in fig. 2 is the average bound difference of each experiment, and the right vertical coordinate is the average computation time t of the SMT model of each experiment. As can be seen from the figure, the average accuracy of the SMT model is about 10% in the average case, and the solution efficiency is ideal in most cases.

It is to be understood that the present invention has been described with reference to certain embodiments, and that various changes in the features and embodiments, or equivalent substitutions may be made therein by those skilled in the art without departing from the spirit and scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.

Claims (9)

1. A DAG task WCRT calculation method based on an SMT method is characterized by comprising the following specific steps:

the method comprises the following steps: generating a DAG task;

step two: establishing a DAG model;

step three: establishing an objective function;

step four: establishing conditional constraints of the nodes;

step five: establishing load retention constraints;

step six: establishing a source point starting execution time constraint;

step seven: and solving the objective function.

2. An SMT method based WCRT calculation method for DAG tasks according to claim 1, wherein: the objective function formula in the third step is as follows:

max(f_sk-s_sc) (2)

3. an SMT method based WCRT calculation method for DAG tasks according to claim 1, wherein: the conditional constraints in the fourth step include: the execution time constraint of the nodes, the priority constraint of the nodes and the non-preemptive constraint of the nodes ensure that a feasible solution exists in the objective function.

4. An SMT method based WCRT calculation method for DAG tasks according to claim 3, wherein: the execution time constraint formula of the node is as follows:

5. an SMT method based WCRT calculation method for DAG tasks according to claim 3, wherein: the priority constraint formula of the node is as follows:

6. an SMT method based WCRT calculation method for DAG tasks according to claim 3, wherein: the node non-preemptive constraint formula is as follows:

7. an SMT method based WCRT calculation method for DAG tasks according to claim 1, wherein: in the load holding constraint in the step five, in order to satisfy the execution condition, no idle core exists, which is defined as follows:

for any node u_bAnd its critical precursor u_aLogical expression Π in equation (6)₁Representing a time interval f_a,s_b]Whether the length of (a) is strictly greater than 0;

for any node u_bAnd its critical precursor u_aIf the following condition is satisfied, it is called node u_x∈P_ar(u_a) Is contained in the edge (u)_a,u_b) Relevant time interval [ f_a,s_b]Of (1), i.e. node u_xIs the interval [ f_a,s_b]Is provided with H_abRepresents a time interval [ f_a,s_b]A set of all the included nodes within;

Π₂:s_x＜s_b∧f_x＞f_a (7)

for any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) B is a Boolean variable_abxThe definition of (A) is as follows,

8. an SMT method based WCRT calculation method for DAG tasks according to claim 7, wherein: said arbitrary side (u)_a,u_b) E-containing node set H_abDefinition of H_abTwo types of subsets of (a):

left subset H_L: h satisfying the following conditions_abNode u in_xBelonging to the left subset H_L，

Π₃:s_x＜f_a (12)

Right subset H_R: h satisfying the following conditions_abNode u in_xBelong to the right subset H_R，

Π₄:f_x＞s_b (13)

For any edge (u)_a,u_b) E and any node u_x∈P_ar(u_b) Define a Boolean variable L_abx(R_abx) To represent u_xWhether or not it is contained in H_L(H_R) In (1), the definition is as follows,

9. an SMT method based WCRT calculation method for DAG tasks according to claim 8, wherein: said arbitrary side (u)_a,u_b) E and any node u_x∈P_ar(u_b) Establishing constraints for implementing the continuous operating characteristics:

Images