CN114609589A

CN114609589A - Heuristic backtracking-based real-time phased array radar beam resident scheduling method

Info

Publication number: CN114609589A
Application number: CN202210222565.9A
Authority: CN
Inventors: 程婷; 李中柱; 王元卿; 侯子林; 李立夫
Original assignee: University of Electronic Science and Technology of China
Current assignee: University of Electronic Science and Technology of China
Priority date: 2022-03-09
Filing date: 2022-03-09
Publication date: 2022-06-10
Anticipated expiration: 2042-03-09
Also published as: CN114609589B

Abstract

The invention belongs to the field of radar system resource management, and particularly relates to a novel real-time adaptive resident scheduling method for a phased array radar. The invention firstly determines the actual execution task queue, the delay task queue and the deletion task queue in the scheduling interval through the time pointer, thereby ensuring the importance and the urgency criterion in the beam resident scheduling, and then the invention adjusts the actual execution time of the tasks in the actual execution task queue by a heuristic backtracking method, thereby reducing the deviation degree between the actual execution time of each task and the expected execution time thereof, and ensuring the expected execution time criterion in the beam resident scheduling. In addition, the actual execution task queue is partitioned and then backtracked, so that the calculation complexity of the method is greatly reduced, and the real-time performance of the method is ensured.

Description

Heuristic backtracking-based real-time phased array radar beam resident scheduling method

Technical Field

The invention belongs to the field of radar system resource management, and particularly relates to a method for adaptive resident scheduling of a phased array radar.

Background

Since the transmission beam direction of a Phased Array Radar (PAR) can be changed rapidly in a short time, the phased array radar can perform different types of tasks during its operation time to realize a multifunctional characteristic. In order to realize reasonable distribution of limited system resources among multiple tasks, an efficient beam residence scheduling method with real-time performance needs to be designed.

In recent years, a beam-dwell scheduling method, particularly, an adaptive beam-dwell scheduling method that can match a time-variable task request in an online scheduling process, has been widely studied. According to the method for solving the parking scheduling optimization problem, the adaptive beam parking scheduling method can be divided into a heuristic-based beam parking scheduling method and an intelligent optimization-based beam parking scheduling method (see the documents: Zhang, H., Xie, J., Ge J., et al.: A hybrid adaptive genetic algorithm for task scheduling protocol in the phase array, European Journal of Operational Research,2019,272, (3), pp.868-878)

The beam residence scheduling method based on the heuristic rule solves the beam residence scheduling problem by adopting the heuristic rule. The literature (Orman, A.J., Potts, C.N., Shahani, A.K., et al.: Scheduling for a multi-functional phased array radio system', European Journal of operational Research,1996,90, (1), pp.13-25) uses the working mode priority of the resident tasks as the Scheduling criteria and designs the corresponding Scheduling method according to the heuristic rules. The literature (Haritsa, J.R., Livny, M., Carey, M.J.: Earlie de-adaptive scheduling for Real-Time database Systems', Proceedings two-Time Systems Symphos, San Antonio, TX, USA, Decumber 1991, pp.232-242) uses the deadline of the resident task as the scheduling criterion, and uses the First In First Out (FIFO) for tasks with the same deadline. The document (lujian, kawaing, yuwenxian.: real-time task scheduling research of multifunctional phased array radar', academic press, 2006,34, (4), pp.732-736) sets a comprehensive priority based on the priority of a working mode and a task deadline, and obtains a scheduling task sequence by introducing a time pointer in the scheduling analysis process. The literature (Zhang, h., Xie, j., Zong, b., et al.: Dynamic priority scheduling method for the air-default phase array front', ietrad air & Navigation,2017,11, (7), pp.1140-1146) further considers the threat level of the target in the setting of the overall priority. The scheduling methods proposed in the above documents have the characteristics of high time utilization rate and good real-time performance, but none of them considers the expected execution time criterion, which requires that the actual execution time of the resident task should be as close as possible to the expected execution time of the task. (see the literature: great light, fenugreek, lujian, zhonghui.: multifunctional phased array radar adaptive scheduling simulation ', systems simulations proceedings, 2004,16, (9), pp.2026-2029) based on this problem, the literature (Cheng, t., He, z., Tang, t.: Dwell scheduling algorithm for multifunctionalized array radars based on the scheduling gain', Journal systems Engineering and Electronics,2008,19, (03), pp.479-485) proposes a beam Dwell scheduling method based on scheduling interval analysis. The method sequences tasks according to task scheduling gains, and distributes actual execution time to each resident task in sequence according to an expected time criterion. The method can effectively consider the expected execution time criterion, but has the problems of low time utilization rate, poor real-time performance and the like.

In order to solve the problem that a beam residence scheduling method based on a heuristic method cannot simultaneously consider the system time utilization rate and the expected time criterion, a series of beam residence scheduling methods based on an intelligent optimization algorithm are proposed. The method establishes a beam residence scheduling problem as an optimization problem, and introduces a series of optimization algorithms according to the characteristics of the optimization problem to obtain an optimal actual execution task sequence and an optimal actual execution time of an actual execution task. The literature (L, Hao., X, Yang., S, Hu.: 'Task scheduling of improved time shifting base on genetic algorithm for phase array front', 2016IEEE 13th International Conference on Signal Processing (ICSP), Chengdu, China, November 2016, pp.1655-1660) designs a fitness function according to the importance, urgency and expected execution time criteria of a Task, and uses a genetic algorithm to obtain an optimal actual execution Task queue. In the literature (Yang, S., Tian, K., Liu, R.: Task scheduling algorithm Optimization for anti-mistact PhasedAlrraray Radar', IET Radar Sonar & Navigation,2019,13, (11), pp.1883-1889.), a fitness function is designed according to the specific characteristics of the problem of the anti-derivative Radar scheduling, and the problem is solved by a genetic algorithm. The literature (Meng, F., Tian, K., 'phase-Array radio Task Scheduling Method for Hypersonic-peptide Vehicles', IEEE Access,2020,8, pp.221288-221298) adopts a particle swarm-simulated annealing mixing algorithm to solve the beam residence Scheduling problem under the supersonic target tracking Task. In the literature (Shaghaghi, M., Adve, R.S., Zhen, D.: multiple Cognitive radio Task Scheduling Using Monte Carlo Tree Search and Policy Networks', IET radio resource & Navigation,2018,12, (12), pp.1437-1447) a target function of a multifunctional Cognitive Radar Scheduling problem is designed according to the urgency of a Task and an expected time criterion, and an optimal actual execution Task sequence and an optimal actual execution time are obtained by adopting a Monte Carlo Search Tree optimization method. The problem is solved by taking the minimum loss rate of the Task as an objective function and adopting a Reinforcement Learning method in the literature (Xu, L., Zhang, T.: requirement Learning based Dynamic Task Scheduling for multifunctionality Radar Network', IEEE Radar Conference, Florence, Italy,2020, pp.1-5). However, the beam dwell scheduling method based on the optimization algorithm has poor real-time performance, and therefore cannot be used in actual radar beam dwell scheduling.

Based on the problems, the invention provides a heuristic backtracking-based real-time phased array radar beam resident scheduling method. The method firstly adopts a scheduling method in documents (Lujian and bin, Wagneng, Yuwenxian: 'multifunctional phased array radar real-time task scheduling research', electronic newspaper, 2006,34, (4), pp.732-736) to determine the actual execution task, the delay task and the lost task in the current scheduling interval. And then, traversing all feasible execution sequences of the actually executed tasks, and performing backtracking optimization on the actual execution time of the tasks corresponding to each possible execution sequence so as to take the expected execution time criterion into consideration. And finally, calculating the time offset of all feasible task execution sequences, and selecting the scheduling sequence corresponding to the task execution sequence with the minimum time offset as the optimized scheduling sequence. In addition, the method carries out block processing on the scheduling task sequence in the current scheduling interval, thereby reducing the computational complexity of the method. Simulation results show that compared with the existing method, the method can give consideration to the priority, the deadline and the expected execution time criterion, and can realize real-time beam residence scheduling.

Disclosure of Invention

The invention provides a novel beam residence scheduling method based on heuristic backtracking, which is characterized by comprising the following steps:

suppose that the current scheduling interval t₀,t₀+t_SI]With N resident tasks T ═ T₁,T₂,...,T_N]Applying for scheduling, where t₀For the start time of the current scheduling interval, t_SIFor the duration of the scheduling interval, the resident task model is T_i＝{W_i,rt_i,st_i,l_i,dw_iIn which W_iTo work mode priority, rt_iTo expect the execution time, st_iFor the actual execution time,/_iIs a time window, dw_iIs the dwell time. The phased array radar beam residence scheduling method based on heuristic backtracking comprises the following steps:

step 1: initialization time pointer tp ═ t₀Initializing the queue T of the actual execution task in the scheduling interval_exDelaying the task queue T_dlAnd deleting the task queue T_drIs empty.

Step 2: deleting the application scheduling task satisfying the formula (1) in the T and adding the task to the T_drIn (1).

rt_i+l_i＜tp (1)

And step 3: all tasks satisfying the formula (2) are selected from the T, and a schedulable task set T at the current moment is generated_cd＝T_cd,1,T_cd,2,...,T_cd,QWherein Q is T_cdAnd (4) the number of internal tasks.

Calculating the comprehensive priority p corresponding to each task according to the formula (3)_iIn which Np_iIs T_cdThe sequence numbers of the middle tasks in the order from big to small according to the priority of the working mode, Nd_iIs T_cdAnd the sequence numbers of the middle tasks are sorted from small to large according to the task deadline.

And 4, step 4: from T_cdElecting to have the greatest composite priority

Task of (2)

Delete it from T and add it to T_exIn (1).

And 5: update tp ═ tp + dw_i. If tp>t₀+t_SIOr T is null, let T_dlT, and go to step 6; otherwise, go back to step 2.

Step 6: suppose that

The number of tasks in (1) is N₁Will T_exIs divided into

Sub-execution task queue T_l,l＝1,2,...,N₁K, each sub-execution task queue contains at most K tasks, wherein the symbols

Representing a rounding up. First sub-execution task queue T₁＝[T₁,T₂,...,T_K]The second sub-execution task queue T₂＝[T_K+1,T_K+2,...,T_2K]By analogy, the first sub-execution task queue T_l＝[T_(l-1)K+1,T_(l-1)K+2,...,T_lK]. Each sub-execution task queue T_lCorresponding to a sub-scheduling interval [ t ]_a,l,t_b,l]Wherein t is_a,lAnd t_b,lThe calculation of (c) is as follows:

and 7: all possible execution orders of the execution tasks in the sub-execution queue are traversed. By T_lFor example, define T_lThe j execution order in (1) is [ T ]_(l-1)K+1,j,T_(l-1)K+2,j,...,T_(l-1)K+i,j,....,T_lK,j]J ═ 1, 2.. K! . E.g. T_(l-1)K+i,jFor sub-execution queue T_lThe ith task in the jth execution order. Allocating actual execution time for the execution tasks in each execution sequence of each sub-execution task queue according to the following formula:

the actual execution time of the allocation is then determined according to:

if the conditions are met, the execution sequence is a feasible execution sequence, and the actual execution time of the tasks corresponding to the execution sequence is obtained, otherwise, the permutation and combination is an infeasible execution sequence.

And 8: and backtracking the actual execution time of the actual execution tasks in all the sub-execution task queues according to all the feasible execution sequences. In the execution order T₁＝[T₁,T₂,...,T_K]For example, the specific backtracking steps are as follows:

step 8.1: and performing backtracking processing on the tasks in sequence from short to long according to the time window. Suppose that the current task to be backtraced is T_iThe desired execution time is rt_iThe actual execution time is st_i. If st_i-rt_iIf the value is more than 0, the step 8.2 is executed; if st_i-rt_iIf the value is less than 0, the step 8.4 is carried out; if st_i-rt_iIf 0, go to step 8.6.

Step 8.2: by T_iTo start, the first task that has been traced back is found forward. Suppose the task is T_jDefinition of t_end＝st_j+dw_j(ii) a If T_iThe task which is not backtracked in the front is t_end＝t_a,1. If at [ t_end,st_i]In total, P tasks [ T_P,1,T_P,2,...,T_P,P]Optimizing T according to_iActual execution time of (2):

if at [ t_end,st_i]Has no task, optimizes T according to the following formula_iActual execution time of (2):

st_i＝max(t_end,rt_i) (9)

step 8.3: updating [ T ] according to_P,1,T_P,2,...,T_P,P]Actual execution time of (2):

and go to step 8.6.

Step 8.4: by T_iAs a starting point, the first task that has been traced back is found backwards. Assume that the task is T_jDefinition of t_end＝st_j. If T_iThe task which is not backtracked later, t_end＝t_b,1. If at [ st_i,t_end]In total, P tasks [ T_P,1,T_P,2,...,T_P,P]Optimizing T according to_iActual execution time of (2):

if at [ st_i,t_end]Has no task, optimizes T according to the following formula_iActual execution time of (2):

st_i＝min(t_end-dw_i,rt_i) (12)

step 8.5: updating [ T ] according to_P,1,T_P,2,...,T_P,P]Actual execution time of (2):

step 8.6: task recording T_iThe backtracking has been completed. And if all the tasks are backtracked, turning to the step 9, otherwise, turning to the step 8.1.

And step 9: and calculating the time offset degree of each sub-execution task queue under all feasible execution sequences. By T₁The sorting result of the j-th order is taken as an example, the time offset bias_jThe calculation formula is as follows:

wherein b is_i,jThe calculation formula is as follows:

and selecting the task execution time corresponding to the task execution sequence with the lowest time offset degree from the sub-execution queues as the actual execution time of the tasks, and integrating all the sub-execution queues to obtain the actual execution task queue again. And finishing the analysis of the scheduling process.

Principle of the invention

The invention guarantees the importance criterion and the urgency criterion of scheduling when determining the actual execution task, the delay task and the deletion task, and guarantees the expected execution time criterion of scheduling by backtracking the actual execution time of the actual execution task. The principle of which is explained below.

Suppose there are N tasks to apply for scheduling in the scheduling interval, which is denoted as T ═ T₁,T₂,...,T_N]. When scheduling, the importance criterion and the urgency criterion require that tasks with high work mode priority and tasks with early deadlines should be executed as soon as possible, respectively. Equation (3) embodies the two criteria described above. The desired execution time criterion requires that the actual execution time of the actual executed task should be as close as possible to the desired execution time. Equation (14) embodies this criterion. Since the search task does not include the desired execution time, the search task b is represented by equation (15)_iIs 0.

According to the three rules of scheduling, the objective function of the beam residence scheduling problem is established as follows

G_i(st_i,rt_i,l_i,w_i,t₀)＝g₁(w_i)g₂(rt_i,l_i,t₀)g₃(st_i,rt_i,l_i) (16)

Wherein, g₁(w_i)＝w_iDue to g₁(w_i) The priority of the task work mode is increased, so that the importance criterion of the scheduling is embodied;

wherein c is₁Is a normal number due to g₂(rt_i,l_i,t₀) The deadline of the task is increased along with the reduction of the deadline of the task, so the item embodies the urgency criterion of scheduling;

wherein c is₂Is a normal number due to g₃(st_i,rt_i,l_i) The value of (c) decreases as the deviation between the actual execution time and the expected execution time increases, so this term embodies the expected execution time criterion of the schedule. The objective function and the constraint condition in the beam residence scheduling problem are comprehensively considered, and the mathematical model of the beam residence scheduling problem is established as follows：

Wherein N is₂And N₃The number of the delayed tasks and the number of the deleted tasks are respectively, and obviously N is equal to N₁+N₂+N₃. In the optimization model, the last two inequalities reflect the conditions that the delay task and the delete task should satisfy.

The invention adopts a method based on heuristic backtracking to solve the optimization problem, and the specific analysis method comprises the following steps:

in the scheduling process, in order to maximize the time utilization rate and schedule high-priority tasks as much as possible, the invention adopts a beam resident scheduling method based on a time pointer to confirm the actual execution tasks in the scheduling interval, delay the tasks and delete the tasks. The present invention then considers backtracking to optimize the actual execution time of each task to minimize the deviation of the desired execution time from it. According to the invention, all possible task execution sequences are traversed, screened and infeasible task execution sequences are removed to ensure that actual execution time which does not meet the constraint condition does not occur in the subsequent backtracking process. The actual execution time of the task is then traced back. From g₃(st_i,rt_i,l_i) Can be seen in the time offset of | rt_i-st_iIn the case of the same value of l, the time window l_iG for smaller tasks₃(st_i,rt_i,l_i) And the method is smaller, so that the invention backtracks the tasks with shorter time windows. As shown in FIG. 1, assume T_kHas a minimum time window of T_iSecond, T_jIs the largest, the method first applies to T_kBacktracking to make it execute the time st_kAs close as possible to its desired execution time rt_k. Then the method is to T_iBacktracking to make it execute the time st_iAs close as possible to its desired execution time rt_i. At adjustment st_iWhile preventing T_iAnd T_jOverlap in residence time of T_jActual execution time st_jAnd should be adjusted accordingly. When T is_jAdjusted to T_kExecution completion time, T_iAdjusted to T_jWhen the execution is completed, T is the previous backtracking result cannot be changed_iThe backtracking of (2) is completed. When T is_iAnd T_kAfter the backtracking is completed, T_jThe actual execution time of (a) cannot be further adjusted, so the method considers that the backtracking process is completed. In addition, step 6 reduces the computational complexity of the method by blocking the actual execution task queue.

Drawings

FIG. 1 is a schematic diagram of backtracking

FIG. 2 is TDR for four methods;

FIG. 3 is the HVR of four methods;

FIG. 4 is TUR for four methods;

fig. 5 is ATSR for four methods.

FIG. 6 is a run length of four methods

FIG. 7 is a run length of method A and the inventive method

Detailed Description

TABLE 1 Radar Beam dwell task parameter Table

Five tasks of verification, precise tracking, common tracking, horizon search and airspace search are considered in the simulation scene. The simulation time length is 12s, the scheduling interval time length is set to be 50ms, the ratio of the number of precision tracking targets to the number of common tasks is 1:4, and K is set to be 5 in step 6. The radar mission parameters are shown in table 1.

In order to fully evaluate the performance of the present invention, this section adopts Task Drop Rate (TDR), achievement Value rate (HVR), Time Utilization rate (Time availability Ratio), Average Time migration rate (ATSR) and running Time length as performance evaluation indexes. The above criteria are defined as follows:

the task loss rate (TDR) is the ratio of the number of lost tasks to the number of tasks applying for scheduling in the simulation time length:

TDR＝N_drop/N_total (18)

wherein N is_dropIndicating the number of lost tasks, N_totalIndicating the number of tasks to be applied for scheduling;

the achievement value rate (HVR) is the ratio of the sum of the working mode priorities of the actually executed tasks to the sum of the working mode priorities of the application scheduling tasks in the simulation duration:

wherein N is_exeIndicating the number of tasks actually performed. The index is used for reflecting the proportion of the high-priority task which is successfully scheduled;

time Utilization Rate (TUR): the ratio of the sum of the actual execution task residence time to the total simulation time is defined as:

wherein, t_totalIs the total simulation duration.

Average time offset ratio (ATSR): reflecting the average offset rate between the actual execution time and the expected execution time of all the tracking tasks:

wherein N is_traIndicating the number of trace tasks actually performed.

The operation time is as follows: a running time duration within one scheduling interval.

The beam residence scheduling method based on heuristic backtracking for the real-time phased array radar is adopted for beam residence scheduling, and performance comparison is carried out with the method A, the method B and the method C. Wherein, the method A is a phased array radar beam residence scheduling method based on time pointer (see the documents: Lu Jian, Hu Wei, Yu Wen., 'multifunctional phased array radar real-time Task scheduling research', electronic newspaper, 2006,34, (4), pp.732-736), the method B is a phased array radar beam residence scheduling method based on scheduling interval analysis (see the documents: Cheng, T., He, Z., Tang, T., Dwell scheduling algorithm for multi-functional phased array radar beam basis on the scheduling gain ', Journal of Systems Engineering and Electronics,2008,19, (03), pp.479-485), the method C is a phased array radar beam residence scheduling method based on genetic algorithm (see the documents: L, Hao o, X, Yang., S, schema, Hu.:' timing of phased array radar beam residence, inter-Processing, and electronic plan 2016), chengdu, China, November 2016, pp.1655-1660). The simulation platform is MATLAB R2016a, the computer processor is Core i7-10700, and the memory is 16G. Fig. 2 to 6 show the statistics of 100 monte carlo under different indexes.

Fig. 2 is a task loss rate curve. When the target number is increased to 30, the method C starts to lose the task first. This is because method C performs scheduling by using the idea of assigning each task an actual execution time closest to the expected execution time, which inevitably results in an underutilized time gap within the scheduling interval, and therefore the scheduling sequence obtained by method C has the highest task loss rate. Compared with the method C, the method B directly optimizes the actual execution time of each task by adopting a genetic algorithm, has a relatively low task loss rate, and still cannot completely eliminate the time gap in the scheduling interval. The method A introduces the time pointer for scheduling, so that the tasks in the scheduling interval can be closely arranged, and the lowest task loss rate is achieved. The invention only adjusts the actual execution time of the task on the basis of the actual execution task queue obtained by the method A, and does not change the scheduled task. Therefore, the task loss rate obtained by the method is completely consistent with that obtained by the method A.

FIG. 3 is a graph of an implementation of a merit rate that reflects whether tasks with higher mode priority are executed as much as possible. It can be seen that the realized value rate curve and the task loss rate curve obtained by the four methods have opposite trends, that is, when the task loss rate is higher, the realized value rate is lower.

Fig. 4 is a time utilization curve. When the number of the targets is less, the time utilization rate curve linearly increases along with the increase of the number of the targets; when the target number is increased to a certain degree, the task is lost, and the time utilization rate of the system tends to be saturated. The method A and the method provided by the invention have the maximum time utilization rate.

Fig. 5 is a graph of average time shift rate. It can be seen that the time offset rate is higher for method a, and lower for methods B and C. This is because both method B and method C consider the desired time criterion when scheduling, while method a does not consider the desired time criterion when scheduling. Compared with the time migration rate obtained by the method A, the method provided by the invention obtains the feasible actual task execution time by traversing all the execution task sequences, and further backtracks and optimizes the feasible actual execution time, thereby greatly reducing the time migration rate.

FIG. 6 shows the operating time of the four methods. The genetic algorithm needs to repeatedly calculate the fitness functions of a large number of individuals, so the method C has the largest calculation amount and the longest running time; method B has a run time superior to method C but inferior to the present invention dispense and method a. It can be seen that the operation time of both method B and method C exceeds one scheduling interval (50ms), so neither method has real-time performance.

FIG. 7 shows the run length of method A and the inventive method. It can be seen that method a has the lowest run length. Because the method adds the backtracking step on the basis of the method A, the method has longer running time compared with the method A. However, the running time of the method of the invention is far shorter than the scheduling interval duration (50ms), so that the method and the method A of the invention have real-time performance.

In summary, compared with the scheduling method based on the time pointer, the method provided by the invention can give consideration to the time offset rate of the task under the condition of realizing the completely consistent task loss rate and the realization value rate; compared with a scheduling method based on scheduling interval analysis and a genetic algorithm, the method has lower task loss rate and higher realization value rate, and ensures the real-time performance of scheduling analysis. The invention relates to a real-time beam residence scheduling algorithm which gives consideration to the priority, the deadline and the expected execution time criterion.

Claims

1. A novel beam residence scheduling method based on heuristic backtracking is characterized in that:

suppose that at the current scheduling interval t₀,t₀+t_SI]With N resident tasks T ═ T₁,T₂,...,T_N]Applying for scheduling, where t₀For the start time of the current scheduling interval, t_SIFor the duration of the scheduling interval, the resident task model is T_i＝{W_i,rt_i,st_i,l_i,dw_iIn which W_iTo work mode priority, rt_iTo expect the execution time, st_iFor the actual execution time,/_iIs a time window, dw_iIs the dwell time. The phased array radar beam residence scheduling method based on heuristic backtracking comprises the following steps:

Step 2: deleting the satisfied application scheduling task in the T and adding the satisfied application scheduling task to the T_drIn (1).

rt_i+l_i＜tp (1)

And step 3: all tasks meeting the formula are selected from the T, and a schedulable task set T at the current moment is generated_cd＝T_cd,1,T_cd,2,...,T_cd,QWherein Q is T_cdThe number of internal tasks.

Calculating the comprehensive priority p corresponding to each task according to the formula_iWhereinNp_iIs T_cdThe sequence numbers of the middle tasks in the order from big to small according to the priority of the working mode, Nd_iIs T_cdAnd the sequence numbers of the middle tasks are sorted from small to large according to the task deadline.

And 4, step 4: from T_cdElecting to have the greatest composite priority

Task T of_i ^*It is deleted from T and added to T_exIn (1).

Step 6: suppose that

The number of tasks in (1) is N₁Will T_exIs divided into

Representing a rounding up. First sub-execution task queue T₁＝[T₁,T₂,...,T_K]The second sub-execution task queue T₂＝[T_K+1,T_K+2,...,T_2K]By analogy, the first sub-execution task queue T_l＝[T_(l-1)K+1,T_(l-1)K+2,...,T_lK]. Each sub-execution task queue T_lCorresponding to one sub-scheduling interval

Wherein t is_a,lAnd t_b,lThe calculation of (c) is as follows:

the actual execution time of the allocation is then determined according to:

And 8: and backtracking the actual execution time of the actual execution tasks in all the sub-execution task queues according to all the feasible execution sequences. In the execution order T₁＝[T₁,T₂,...,T_K]For example, asThe specific backtracking steps are as follows:

step 8.1: and performing backtracking processing on the tasks in sequence from short to long according to the time window. Suppose that the current task to be backtracked is T_iThe desired execution time is rt_iThe actual execution time is st_i. If st_i-rt_iIf the value is more than 0, the step 8.2 is executed; if st_i-rt_iIf the value is less than 0, the step 8.4 is carried out; if st_i-rt_iIf 0, go to step 8.6.

Step 8.2: by T_iAs a starting point, the first task that has been traced back is found forward. Assume that the task is T_jDefinition of t_end＝st_j+dw_j(ii) a If T_iThe task which is not backtracked in the front is t_end＝t_a,1. If at [ t_end,st_i]In total, P tasks [ T_P,1,T_P,2,...,T_P,P]Optimizing T according to_iActual execution time of (2):

st_i＝max(t_end,rt_i) (9)

and go to step 8.6.

Step 8.4: by T_iAs a starting point, the first task that has been traced back is found backwards. Assume that the task is T_jDefinition of t_end＝st_j. If T_iThe task which is not backtracked later, then t_end＝t_b,1. If at [ st_i,t_end]In total, P tasks [ T_P,1,T_P,2,...,T_P,P]Optimizing T according to_iActual execution time of (2):

if in [ st_i,t_end]Has no task, optimizes T according to the following formula_iActual execution time of (2):

st_i＝min(t_end-dw_i,rt_i) (12)

step 8.6: memory task T_iThe backtracking has been completed. If all tasks have completed backtracking, go to step 9, otherwise go to step 8.1.

wherein b is_i,jThe calculation formula is as follows:

and selecting the task execution time corresponding to the task execution sequence with the lowest time offset from the sub execution queues as the actual execution time of the tasks, and integrating all the sub execution queues to obtain the actual execution task queue again. And finishing the analysis of the scheduling process.