Adaptive Task Scheduling Algorithm for Multifunction Integrated System with Joint Radar–Communications Waveform

Abstract

Joint radar–communications (JRC) waveform could enable simultaneously radar sensing and communication functions in a multifunction integrated system (MFIS). Because of the special JRC waveform characteristics of MFIS, the traditional task scheduling algorithm for multifunction phased array radar cannot be applied to the MFIS to release its full potential. Therefore, a novel adaptive task scheduling algorithm for JRC waveform enabled MFIS is proposed in the paper. Firstly, the modified JRC waveform scheduling criteria for MFIS are remodeled after establishing the multi-task models. Secondly, the task scheduling optimization problem, which can reflect the scheduling capability of the JRC waveform, is reformulated, and the proposed novel adaptive task scheduling algorithm based on the JRC waveform effectively schedules different kinds of tasks. Finally, the simulation results show that the proposed algorithm not only has a high JRC waveform scheduling capability but also provides better performance compared with the traditional ones.

1. Introduction

With the development of electronic technology, it is difficult for a single electronic hardware to account for the realistic demand in today’s complex multi-task scenario, which promotes the development of electronic systems, such as detection, communication and interference, to the multifunctional integration system (MFIS). MFIS can bring many advantages, such as system volume decrease, power consumption reduction, electromagnetic interference suppression, and the improvement of spectrum resource congestion, which have been widely studied [1,2,3,4,5,6,7]. As two widely available functions in MFIS, radar sensing and wireless communication have many similarities in the system structure and working mechanism, which makes the integration of them an inevitable development direction, and the research and application of joint radar–communications (JRC) waveform is very important. The versatility of MFIS makes it a multi-task system. How to allocate system resources reasonably and manage it adaptively in the complex multi-task scenario is of practical and important significance for improving system efficiency and exerting its full work potential.

MFIS serves multiple functions simultaneously, such as confirmation, tracking, search, interference and communication, so it requires the effective allocation of limited time resources. The study of how multiple tasks are sorted chronologically and executed leads to the task scheduling problem, which is usually complicated and non-polynomial time (NP) hard [8]. Therefore, it is necessary to explore an effective task scheduling algorithm. In recent years, research on task scheduling has been the focus of concentrated research [9]. Generally, the common task scheduling algorithms of radar systems mainly include the fixed template algorithm, multi-template algorithm, partial template algorithm and adaptive scheduling algorithm [10]. The adaptive scheduling algorithm is the most flexible and widely adopted under multitasking scenarios. Most of its scheduling strategies are studied based on priority, deadline, threat level, time window, pulse interleaving, dwell time and scheduling interval [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The authors in [11] proposed the task mode priority-based scheduling algorithm. Each type of task was preset with a mode priority, and the task with the highest mode priority was preferentially scheduled. The authors in [12] proposed the earliest deadline first (EDF) algorithm, where the task with the earliest deadline had the highest priority and was preferentially scheduled in the scheduling process, which was one of the typical dynamic priority algorithms. These two scheduling algorithms both use a single parameter to determine the task priority, which is insufficient in practice because the task priority is not uniquely determined by the task working mode. The deadline is also a very important factor affecting the task priority, which reflects the urgency of the task. Therefore, the authors in [13,14,15,16] proposed the highest mode priority and EDF (HPEDF) algorithm, which mapped the task mode priority and deadline to the same layer to construct the task synthetic priority, and then sorted the task requests according to the synthetic priority, and scheduled the tasks through the time pointer. Compared with the traditional algorithms, it is very effective. Further, to make full use of the prior information of the target, the authors in [17,18,19] determined the synthetic priority of the task based on the target threat degree and deadline, and the authors in [20] proposed a task scheduling algorithm based on value optimization under a synthetic priority for anti-missile phased array radar. The concept of time window was proposed in [21]. It represents the effective range of task execution time that can move around the request time, which makes it possible to schedule more tasks. The authors in [22] used a time window-based backtracking adaptive algorithm to solve the conflict tasks at the same time, where the actual execution time of conflicting tasks on the timeline could be adjusted reasonably within their allowed time window. Authors in [21,22,23] indicated that the task scheduling algorithms based on the time window could effectively improve the successful scheduling ratio and time utilization ratio. The pulse interleaving technique is used to transmit or receive other tasks during the waiting period of a task. It is introduced to enhance the time utilization ratio, and much research has been contributed [24,25,26,27,28,29]. In [30,31], the task dwell time was not a fixed value but was modeled as a fuzzy set to allow for increased radar schedule flexibility. The authors in [32] took the coherent processing interval (CPI) as the pace and dynamically changed the scheduling interval to propose an adaptive scheduling algorithm based on CPI and the impact of tasks under multiple target conditions.

The above research on task scheduling are mostly aimed at the traditional multifunctional phased array radar system. For the MFIS, the authors in [33] proposed a multiple task parallel EDF algorithm to achieve the adaptive scheduling of the system according to the conventional phased array radar scheduling strategy. The authors in [34] proposed a task-scheduling scheme based on the greedy algorithm, and the authors in [35] established a resource scheduling mathematical model that meets the urgency principles of task execution, and proposed an integrated system resource scheduling method based on an improved genetic-particle swarm optimization algorithm. The existing MFIS task scheduling does not consider the application of JRC waveforms as far as the authors’ knowledge. In MFIS, JRC waveform has both radar detection and communication functions [4,5], i.e., transmitting one waveform can perform two tasks at the same time, which means that the successful execution of two tasks can be achieved by consuming the dwell time of a single task on the timeline. Therefore, the application of JRC waveforms may result in better task scheduling performance for MFIS and realize the full potential of MFIS. However, the application of JRC waveform also brings new problems and challenges for the MFIS task scheduling, such as when radar and communication tasks can be executed based on JRC waveform, i.e., JRC waveform scheduling criteria, and MFIS task scheduling algorithm based on JRC waveform, which motivates our research on MFIS task scheduling in the paper.

In this paper, the application of JRC waveform to MFIS task scheduling is studied. Firstly, the JRC waveform scheduling criteria are proposed based on the task model, which is a unique analysis of MFIS task scheduling different from the traditional phased array radar task scheduling. Considering that a radar task and a communication task can be executed based on JRC waveform only at the same beam direction, it is necessary to introduce beam direction into the task model. For the radar task and the communication task requesting the same beam direction, whether the two tasks can be executed based on JRC waveform is judged according to the time resources requested by the two tasks, and the JRC waveform scheduling criteria are given. Secondly, the MFIS task scheduling optimization model is established according to JRC waveform scheduling criteria, scheduling principles and time resource constraints, wherein the successful scheduling ratio of communication tasks is introduced, which can reflect the scheduling capability of the task scheduling algorithm for JRC waveforms. Finally, an adaptive scheduling algorithm based on JRC waveform is proposed. The performance of the algorithm, including the successful scheduling ratio, high value ratio and time utilization ratio are analyzed by simulation experiments.

The rest of this paper is organized as follows. The task model, time resource constraint, JRC waveform scheduling criteria and optimization problem model are given in Section 2. In Section 3, an efficient adaptive task scheduling algorithm based on JRC waveform is presented. The performance of the proposed algorithm is analyzed in Section 4. Finally, Section 5 summarizes the conclusions.

Notation: ⋂ denotes the intersection of two sets. ∅ and $N^{+}$ denote the empty set and the positive integer set, respectively.

2. Problem Formulation

The typical and generic MFIS block diagram is shown in Figure 1. MFIS can not only realize multiple kinds of individual and dedicated functions such as radar, jamming and communication but also realize the integration of two functions at the same time, such as the integration of radar and communications. The system realizes these functions by actively controlling the beam direction, dwell time, waveform generation and so on. All kinds of tasks and their functions are uniformly scheduled by the task scheduler in the system. In the waveform library of MFIS, a variety of waveforms for radar, interference, communication and JRC can be selected. According to the scheduling analysis of the task scheduler, if two tasks can be executed based on JRC waveform, the system will generate a JRC waveform to realize the two functions simultaneously; if not, a dedicated waveform will be used for execution.

In recent years, the research on JRC has attracted extensive attention and considerable research has been devoted to designing JRC waveforms that can be applied to both radar sensing and communication transmission, especially the linear frequency modulated (LFM) pulse in radar waveform and orthogonal frequency-division multiplexing (OFDM) signal in communication waveforms [4,5,36,37,38,39,40,41,42,43,44,45]. The LFM pulse used to design the JRC waveform usually faces the disadvantage of a low communication rate. As one of the best candidates for JRC waveforms, OFDM has the advantages of flexible subcarrier modulation, easy implementation with fast Fourier transform (FFT), high spectral efficiency, and the availability of processing gain at the receiver [4], which can provide high radar and communication performance. It is worth mentioning that the application of JRC waveforms may result in the partial loss of radar and communication performance [40,41], but this effect can be effectively reduced by excellent JRC waveform design technology [38,39,40,41,42,43]. Therefore, the scheduling algorithm in this paper does not consider the task performance loss caused by JRC waveforms.

The overall MFIS task scheduling structure is shown in Figure 2. When all the task requests arrive, the scheduling algorithm will determine whether the tasks can be executed according to resource constraints. The executable tasks will be added to the task execution queue, the tasks whose deadlines have expired will be abandoned, and the tasks that need to be executed later will be added to the delayed queue and re-enter the request task pool for resource allocation. In the following, we will first establish the task model and give the time resource constraints, then propose the JRC waveform scheduling criteria, and finally give the task scheduling optimization problem studied in this paper.

2.1. Task Model

The typical task model of MFIS includes priority, request time, dwell time, execution time, time window, deadline, waveform and beam direction of every dwell request. Thus, the $k\mathrm{th}$ task model can be described as

$$T_k=\{P_k,t_{\mathrm{a}k},t_{\mathrm{dw}k},t_{\mathrm{e}k},w_k,t_{\mathrm{d}k},s_k,P_{\mathrm{os}k}\},$$

(1)

the task parameters are listed in

Table 1. Task parameters.

$T_k$	The $k\mathrm{th}$ task
$P_k$	Mode priority of the $k\mathrm{th}$ task
$t_{\mathrm{a}k}$	Request time of the $k\mathrm{th}$ task
$t_{\mathrm{dw}k}$	Dwell time of the $k\mathrm{th}$ task
$t_{\mathrm{e}k}$	Execution time of the $k\mathrm{th}$ task
$w_k$	Time window of the $k\mathrm{th}$ task
$t_{\mathrm{d}k}$	Deadline of the $k\mathrm{th}$ task
$s_k$	Waveform used for the $k\mathrm{th}$ task execution
$P_{\mathrm{os}k}$	Beam direction of the $k\mathrm{th}$ task

. The deadline of the $k\mathrm{th}$ task satisfies

$$t_{\mathrm{d}k}=t_{\mathrm{a}k}+w_k.$$

(2)

Each task should be executed before its deadline; otherwise, the task dwell request will be considered to have failed.

2.2. Time Resource Constraint

The task scheduling of MFIS is based on the scheduling interval (SI). The system processes the echo signal in the previous SI and determines the task execution sequence in the next SI. According to (1), tasks to be executed need to consider time resources, such as task request time, dwell time, time window and deadline. In an SI, all the tasks to be executed must be scheduled within the duration of the SI. If a task is not successfully executed in the SI, it will be delayed to the next SI or deleted. The time constraints to be satisfied are

$$\max \{t_{\mathrm{start}},t_{\mathrm{a}k}-w_k\}\le t_{\mathrm{e}k}\le \min \{t_{\mathrm{d}k},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}k}\},$$

(3)

where $k=1,\cdots ,N_1$, $N_1$ is the number of tasks to be executed, $t_{\mathrm{start}}$ is the start time of the SI, and $t_{\mathrm{SI}}$ is the duration of SI.

The timepiece of the dwell time of successfully executed task $T_k$ is $(t_{\mathrm{e}k},t_{\mathrm{e}k}+t_{\mathrm{dw}k})$. Within $t_{\mathrm{SI}}$, the occupied timepieces cannot be preempted by other tasks; otherwise, the preempted tasks will fail to execute. Therefore, the occupied time pieces should satisfy

$$\bigcap _{k=i_1}^{i_M}(t_{\mathrm{e}k},t_{\mathrm{e}k}+t_{\mathrm{dw}k})=\varnothing ,i_m\in [1,N_1]\cap N^{+},$$

(4)

where $m=1,2,\cdots ,M$, $M\le N_1$ is the number of occupied timepieces in an SI. At the same time, the total length of timepieces occupied by these $N_1$ tasks must not be longer than $t_{\mathrm{SI}}$, i.e., it must satisfy

$$\sum _{k=i_1}^{i_M}{t}_{\mathrm{dw}k}\le t_{\mathrm{SI}}.$$

(5)

The above time constraints (3)–(5) are for all tasks to be executed in an SI. In the following, in addition to satisfying these constraints, the specific conditions under which the radar and communication tasks can be executed simultaneously based on the JRC waveforms will be analyzed.

2.3. JRC Waveform Scheduling Criteria

In MFIS, JRC waveforms have both radar and communication capabilities. The fact that a single waveform can be used to execute radar task and communication task at the same time is a unique characteristic of MFIS and is related to the task scheduling performance. For example, there are three tasks to be executed shown in Figure 3, where the horizontal is the timeline, and the vertical is the task mode priority. The traditional multi-function phased array radar schedules tasks according to scheduling mode 1, i.e., the time resources occupied by one task cannot be preempted by other tasks during execution, otherwise, the task scheduling fails. In MFIS, scheduling mode 1 will cause a great waste of time resources because it does not play the advantage of radar communication integration. If both the radar task and communication task can be executed based on JRC waveforms, the two tasks can be combined into a single task, which not only saves the time resources occupied by one task but also increases the number of tasks successfully scheduled as shown in scheduling mode 2.

Hence, when a radar task and a communication task can be executed simultaneously based on JRC waveform is a critical issue that needs to be paid attention to. To this end, the conditions that radar and communication tasks can be executed simultaneously based on JRC waveforms, i.e., JRC waveform scheduling criteria will be proposed in this paper.

During the MFIS task scheduling, beam directions are also considered in addition to time resources for tasks to be executed since tasks may be highly directional, such as tracking tasks and communication tasks. Therefore, the task time requirement and beam direction request are the aspects that need to be considered in this paper to determine whether radar task and communication task can be executed simultaneously based on the JRC waveform.

In view of the unique advantages of JRC waveform applied to MFIS task scheduling, JRC waveform scheduling criteria are reintroduced as follows:

Basically, assume that task $T_i$ and task $T_j$ are the radar task (communication task) and communication task (radar task) to be executed, respectively, and task $T_i$ assigns system resources first.

(1)

Intersection criterion: the constraint ranges of two task execution times intersect, i.e.,

$$\begin{array}{cc}\hfill & \left[\max \{t_{\mathrm{start}},t_{\mathrm{a}i}-w_i\},\min \{t_{\mathrm{d}i},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}i}\}\right]\bigcap \hfill \\ \hfill & \left[\max \{t_{\mathrm{start}},t_{\mathrm{a}j}-w_j\},\min \{t_{\mathrm{d}j},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}j}\}\right]\ne \varnothing .\hfill \end{array}$$

(6)

An empty intersection means that task $T_j$ starts to execute after the deadline of task $T_i$. Obviously, the two tasks cannot be executed simultaneously based on the JRC waveform.

(2)

Dwell time criterion: the dwell time of task $T_j$ is no longer than that of task $T_i$, i.e.,

$$t_{\mathrm{dw}i}\ge t_{\mathrm{dw}j}.$$

(7)

The task scheduling algorithm in this paper allocates system resources according to the task priority (which reflects the task importance), i.e., the more important task has the priority to allocate time resources. If (7) is not satisfied and task $T_i$ and task $T_j$ can be executed simultaneously based on the JRC waveform, the final task dwell time is equal to the dwell time of task $T_j$. Thus, the available time resources are reduced due to the execution of potentially less important tasks $T_j$, which will affect the execution of subsequent important tasks. Therefore, constraint (7) is very necessary.

(3)

Beam direction criterion: the beam directions requested by the two tasks are consistent, i.e.,

$$P_{\mathrm{os}i}=P_{\mathrm{os}j}.$$

(8)

The radar task and communication task to be executed must satisfy the above three criteria at the same time before they can be executed simultaneously based on the JRC waveform. At this point, these two different kinds of tasks are considered as one task, which is a non-preempted one, or both tasks will fail to execute.

2.4. Optimization Problem Modeling

In the process of MFIS scheduling tasks, the following principles should be followed: (1) importance principle, i.e., prioritizing the most important tasks; (2) urgency principle, i.e., the closer the deadline of the task is to the beginning of the SI, the more urgent the task is. The importance and deadline are the inherent characteristics of a task, which are determined before the scheduling algorithm is carried out. In addition to following the above principles, the successful scheduling ratio of communication tasks, which embodies the interactive characteristics between the scheduling algorithm and the task, needs to be paid attention to during the actual MFIS task scheduling because it can reflect the task scheduling capability of JRC waveform. From the above characteristics, the first two principles are juxtapositional. Following the two principles, task scheduling results can be obtained by an interaction between the scheduling algorithm and the task. Thus, the successful scheduling ratio of communication tasks is closely related to the two principles. Therefore, in order to highlight the two principles and reflect the task scheduling capability of JRC waveform, the objective function is established as

$$h(P,t_{\mathrm{a}},w,t_{\mathrm{start}},N_{succ},N_{totc})=[h_1\left(P\right)+h_2(t_{\mathrm{a}},w,t_{\mathrm{start}})]·h_3(N_{succ},N_{totc}),$$

(9)

where $h_1\left(P\right)$ is the increasing function of task priority P, reflecting the importance of the task. The more important the task, the greater the $h_1\left(P\right)$. $t_{\mathrm{a}}$ and w are the request time and time window of the task, respectively. $h_2(t_{\mathrm{a}},w,t_{\mathrm{start}})$ is the increasing function of the relative distance between the task deadline $(t_{\mathrm{a}}+w)$ and the start time $t_{\mathrm{start}}$ in SI, reflecting the urgency of the task. The more urgent the task, the greater the $h_2(t_{\mathrm{a}},w,t_{\mathrm{start}})$.

$N_{succ}$ and $N_{totc}$ are the number of communication tasks executed successfully and the total number of communication tasks in the SI, respectively. $h_3(N_{succ},N_{totc})$ is the increasing function of successful scheduling ratio of communication tasks $N_{succ}/N_{totc}$, reflecting the task scheduling capability of the JRC waveform. The stronger the scheduling capability, the greater the $h_3(N_{succ},N_{totc})$. It can be seen from (9) that the objective function is the integration of multiple objectives, which guarantees the performance of the scheduling algorithm in many aspects.

Assume that there are N task requests in an SI, and the number of executed tasks, delayed tasks, deleted tasks and communication tasks are $N_1$, $N_2$, $N_3$ and $N_{totc}$, respectively. The time axis in SI is occupied by the M timepieces, and the corresponding tasks are marked as $T_{i_1},\cdots ,T_{i_M}$. Obviously, $N=N_1+N_2+N_3$. $2(N_1-M)$ tasks in $N_1$ tasks are regarded as $N_1-M$ tasks based on the JRC waveform, and in $2(N_1-M)$ tasks, radar tasks (communication tasks) and communication tasks (radar tasks) are respectively recorded as $\{T_{j_1},\cdots ,T_{j_{N_1-M}}\}$ and $\{T_{j_1^{\prime }},\cdots ,T_{j_{N_1-M}^{\prime }}\}$. Then, the MFIS task scheduling optimization problem can be described as

$$\begin{array}{cc}\hfill & \max \sum _{k=1}^{N_1}h(P,t_{\mathrm{a}},w,t_{\mathrm{start}},N_{succ},N_{totc})\hfill \\ \hfill & s.t.\hfill \\ \hfill & \left\{\begin{array}{c}{t}_{\mathrm{d}{k}_1}=t_{\mathrm{a}{k}_1}+w_{k_1},\hfill \\ \max \{t_{\mathrm{start}},t_{\mathrm{a}{k}_1}-w_{k_1}\}\le t_{\mathrm{e}{k}_1}\le \min \{t_{\mathrm{d}{k}_1},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}{k}_1}\},\hfill \\ \left[\max \{t_{\mathrm{start}},t_{\mathrm{a}j}-w_j\},\min \{t_{\mathrm{d}j},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}j}\}\right]\bigcap \hfill \\ \left[\max \{t_{\mathrm{start}},t_{\mathrm{a}{j}^{\prime }}-w_{j^{\prime }}\},\min \{t_{\mathrm{d}{j}^{\prime }},t_{\mathrm{start}}+t_{\mathrm{SI}}-t_{\mathrm{dw}{j}^{\prime }}\}\right]\ne \varnothing ,\hfill \\ t_{\mathrm{dw}j}\ge t_{\mathrm{dw}{j}^{\prime }},\hfill \\ P_{\mathrm{os}j}=P_{\mathrm{os}{j}^{\prime }},\hfill \\ \bigcap _{k_1=i_1}^{i_M}(t_{\mathrm{e}{k}_1},t_{\mathrm{e}{k}_1}+t_{\mathrm{dw}{k}_1})=\varnothing ,\hfill \\ \sum _{k=i_1}^{i_M}{t}_{\mathrm{dw}k}\le t_{\mathrm{SI}},\hfill \\ t_{\mathrm{a}{k}_2}+w_{k_2}\ge t_{\mathrm{end}},\hfill \\ t_{\mathrm{a}{k}_3}+w_{k_3}<t_{\mathrm{end}},\hfill \end{array}\right.\hfill \end{array}$$

(10)

where $k_1=1,\cdots ,N_1$, $j=j_1,\cdots ,j_{N_1-M}$, $j^{\prime }=j_1^{\prime },\cdots ,j_{N_1-M}^{\prime }$, $k_2=1,\cdots ,N_2$, $k_3$ = 1,⋯,$N_3$, and $t_{\mathrm{end}}$ is the end time of SI. The constraints of (10) contain nine terms. The first two items are common to all tasks to be executed. The third item to the fifth item is the scheduling criteria for JRC waveforms. A radar task and a communication task that simultaneously satisfy these three constraints can be considered as a single task to be executed. The timepieces occupied by the dwell time of all execution tasks are independent of each other and their total length is no longer than $t_{\mathrm{SI}}$, as shown in items six and seven. The last two items are for the $N_2$ tasks to be delayed and for the $N_3$ tasks to be deleted, respectively. As can be seen from (10), the MFIS task scheduling problem based on the JRC waveform is a multi-constraint optimization problem, which is NP-hard and requires an effective scheduling algorithm to solve.

3. Adaptive Scheduling Algorithm Based on JRC Waveform

3.1. Task Priority

In the design of task priority, it is necessary to consider not only the task mode priority but also the deadline. Assume that there are K task requests currently. Sorting the K tasks according to their mode priority from high to low and deadline from early to late, we obtain the sequence numbers of the kth task $T_k$ in the two sortings are $N_{{\mathrm{p}}_k}$ and $N_{{\mathrm{d}}_k}$, respectively. Obviously, these two sequence numbers satisfy $N_{{\mathrm{p}}_k},N_{{\mathrm{d}}_k}\in [1,K]\cap N^{+}$. Therefore, the synthetic priority of task $T_k$ is expressed as [14,15,16]

$$y\left(k\right)=\frac{\beta N_{{\mathrm{p}}_k}+(K+2-\beta )N_{{\mathrm{d}}_k}}{K+1},$$

(11)

where $\beta \in [1,Q+1]$ is the weight coefficient, which is used to control the influence of task mode priority and deadline on the synthetic priority.

3.2. Scheduling Algorithm

In order to make full use of time resources, time pointer $t_p$ is introduced to schedule system tasks. Assume that there are N task requests in an SI, in order of synthetic priority from highest to lowest. The duration of SI is $[t_{\mathrm{start}},t_{\mathrm{end}}]$, and $t_{\mathrm{end}}$ satisfies $t_{\mathrm{end}}=t_{\mathrm{start}}+t_{\mathrm{SI}}$.

Suppose the task to be scheduled at $t_p$ is task 0. Firstly, the task type of task 0 is judged, and then the task to be executed at $t_p$ is determined. According to JRC waveform scheduling criteria and task types, task scheduling analysis at $t_p$ can be divided into three cases:

(1)

If task 0 is a radar task or a communications task, according to (6)–(8), there are radar tasks or communication tasks of different types from task 0, and the task with the highest synthetic priority is denoted as task 1. Then task 1 and task 0 can be executed simultaneously at $t_p$ based on the JRC waveform. The two tasks are regarded as a single task, and the task parameters are consistent. Then, update $t_p$ as

$$t_p=t_p+t_{\mathrm{dw}0}.$$

(12)

(2)

If task 0 is a radar task or communication task, and according to (6)–(8) there is no radar task or communication task of a different type from task 0, or task 0 is neither a radar task nor a communication task, only task 0 is to be executed at $t_p$, and $t_p$ is updated as shown in (12). The difference between case 1 and case 2 is that at $t_p$, the former has two tasks to be executed simultaneously, while the latter has only one task to be executed.

(3)

If there is no task 0 to be executed at $t_p$, then slide the time pointer and update $t_p$ as

$$t_p=t_p+\Delta t_p,$$

(13)

where $\Delta t_p$ is the preset smallest sliding step for $t_p$.

Next, the above task scheduling analysis is cycled for the remaining task requests. Note that tasks that might be executed across SI will be delayed to the next SI for scheduling analysis, while tasks whose deadline is less than $t_p$ will be deleted. Therefore, with Figure 4, the adaptive scheduling algorithm based on JRC waveform for the MFIS can be established according to the following steps:

• Step 1: Initialize the parameters in the SI: the number of task requests is N, the time pointer is $t_p$, the end time of the SI is $t_{\mathrm{end}}$, set $i=0$.
• Step 2: Find the Q tasks whose deadlines are earlier than $t_p$ and delete them, and set $i=i+Q$.
• Step 3: Select all the tasks that can be executed earlier than $t_p$. If the task does not exist, update the $t_p$ according to (13) and then turn to Step 2. Otherwise, turn to Step 4.
• Step 4: Calculate the synthetic priorities of all selected tasks according to (11) and sort them from highest to lowest, then the resulting new task queue is $\{T_k,k=1,2,\cdots ,K\}$, where K is the number of selected tasks. Then, choose the task $T_1$ with the highest synthetic priority.
• Step 5: Determine whether $T_1$ needs to schedule analysis in the next SI. If so, send it to the task delay queue; otherwise, perform scheduling analysis on it, as shown in Step 6–8.
• Step 6: Check whether $T_1$ is a radar task or a communication task. If so, turn to Step 7; otherwise, send it to the task execution queue and update $t_p$ according to (12). Then, set $i=i+1$.
• Step 7: Test whether the task following the JRC waveform scheduling criterion with $T_1$ can be selected from the tasks other than $T_1$. If so, turn to Step 8; otherwise, send $T_1$ to the task execution queue and update $t_p$ according to (12). Then, set $i=i+1$.
• Step 8: Select the task $T_k$, which has the highest synthetic priority. Send $T_1$ and $T_k$ to the task execution queue and update $t_p$ according to (12). Then, set $i=i+2$.
• Step 9: When $t_p\ge t_{\mathrm{end}}$ or $i\ge N$, turn to Step 10; otherwise, turn to Step 2.
• Step 10: Check whether the remaining tasks can be delayed to the next SI. If so, send them to the delay queue; otherwise, they are deleted.
• Step 11: The scheduling process in SI ends, and the task execution queue, delay queue and deletion queue are finally obtained.

3.3. Performance Metric Index

In this paper, the successful scheduling ratio of communication tasks, successful scheduling ratio, high value ratio and time utilization ratio are selected as the performance evaluation index.

• The successful scheduling ratio of communication tasks (SSRCT). It is the ratio of the number of successfully scheduled communication tasks to that of all requested communication tasks, i.e.,

  $$\mathrm{SSRCT}=N_{succ}/N_{totc},$$

  (14)

  where $N_{succ}$ is the number of successfully scheduled communication tasks, and $N_{totc}$ is the total number of communication tasks. In the proposed algorithm, it can reflect the task scheduling capability of JRC waveform and the consumption of limited resources. The higher the SSRCT, the more communication tasks are successfully scheduled, where some communication tasks are executed based on the JRC waveform and the same number of radar tasks are also executed at the same time, which greatly reduces the consumption of time resources.
• The successful scheduling ratio (SSR) [14,15,29]. It is the ratio of the number of successfully scheduled tasks to that of all task requests, i.e.,

  $$\mathrm{SSR}=N_{suc}/N_{tot},$$

  (15)

  where $N_{suc}$ is the number of successfully scheduled tasks, and $N_{tot}$ is the total number of tasks. It reflects the urgency principle and the performance of the algorithm for task scheduling. Under the limited resource constraint, the higher the SSR, the better the algorithm performance.
• The high value ratio (HVR) [29]. It is the ratio of the total value of successfully scheduled tasks to that of all task requests, i.e.,

  $$\mathrm{HVR}=\sum _{k=1}^{N_{suc}}{P}_k/\sum _{k=1}^{N_{tot}}{P}_k.$$

  (16)
  It can reflect SSR and how many important tasks are prioritized by the algorithm.
• The time utilization ratio (TUR) [14,15,29]. It is the ratio of the time consumption of successfully scheduled tasks to the total available time resource, i.e.,

  $$\mathrm{TUR}=\sum _{k=1}^{N_{suc}^{\prime }}\frac{t_{\mathrm{dw}k}}{T_{tot}},$$

  (17)

  where $T_{tot}$ is the total time resource, $N_{suc}^{\prime }\le N_{suc}$ is the number of timepieces occupied by successfully scheduled tasks in the entire timeline. It is used to reflect the time resource utilization of task scheduling. When scheduling tasks, the scheduling algorithms should effectively use the limited time resource to schedule as many tasks as possible, while ensuring that the system has enough time to rest to prevent it from overheating and depleting the system performance.

4. Simulation Results and Analysis

4.1. Simulation Parameters

The simulation scenario includes eight kinds of tasks: confirmation, high precision tracking, precision tracking, communication, electronic jamming, normal tracking, tracking loss and search. Both tracking and search tasks are generated periodically at their sampling intervals from the specified time, and the ratio between the number of high precision tracking targets, precision tracking targets and normal tracking targets is 2:3:5. Other task requests, such as confirmation, communication, electronic jamming and tracking loss are randomly generated during the simulation time. Communication tasks are classified according to the data rate as Communication 1 (high data rate) and Communication 2 (low data rate). The ratio of the number of tracking targets to that of communication tasks is 2:1, and the number of Communication 1 is the same as that of Communication 2.

The available waveform types in the system waveform library are radar waveform, communication waveform, interference waveform and JRC waveform. A total of 10 beam directions are requested for the scenario. The beam directions requested by the tracking tasks are determined at the beginning of the simulation, and that requested by the other tasks are random during the simulation. $t_{SI}=50$ ms, $\Delta t_p=1$ ms. The whole simulation duration is 10 s, and the number of targets increases from 10 to 120 in order to denote different time load situations of MFIS. For each additional 10 targets, we perform 100 Monte Carlo simulations and take the average of the experimental results. The specific task parameters in the simulation are listed in

Table 2. Specific task parameters.

Task Type	Task Mode	Dwell Time (ms)	Time Window (ms)	Sample Interval (ms)
Confirmation	6	4	30	-
High precision tracking	5	2	30	150
Precision tracking	4	3	30	250
Communication	3	4	200	-
Electronic jamming	3	10	200	-
Normal tracking	3	4	30	500
Tracking loss	2	4	50	-
Search	1	4	100	10

. In order to analyze the performance of the proposed scheduling algorithm, three conventional task scheduling algorithms, the EDF algorithm [12], HPEDF algorithm [14] and time window-based backtracking adaptive algorithm [22], are used to compare with the proposed algorithm.

4.2. Performance Analysis

A simulation scenario is randomly generated according to the task parameters in Table 2, where the number of targets is 50. Taking high precision tracking, precision tracking, normal tracking, search and communication tasks as examples, the distribution of their request times along the time axis is shown in Figure 5, where the horizontal is the timeline, showing only 0.3 s, the vertical is the task mode priority and a colored rectangle represents the dwell time of a task. The function of the black rectangular box is to circle the communication and radar tasks whose dwell times overlap partially or completely. Along the time axis, there are many radar tasks and communication tasks whose dwell times overlap, which brings great possibilities for the application of JRC waveform in MFIS task scheduling.

Figure 6 shows the task requests and the scheduled tasks by the proposed algorithm from 5.1 s to 5.5 s, where the red rectangle indicates that the task is executed based on the JRC waveform. As can be seen from the scheduled tasks by the proposed algorithm, some radar tasks occupy the same time pieces as some communications tasks, which indicates that the proposed algorithm tries to schedule as many tasks as possible to enhance the task scheduling capabilities of the system.

Figure 7 shows the successful scheduling ratio of the requested communication tasks. As can be seen from the figure, SSRCT gradually decreases with the increase in the number of communication task requests for the EDF algorithm, HPEDF algorithm and the time window-based backtracking adaptive algorithm, while SSRCT is in a fully successful scheduling state until the number of communication task requests reaches 30, and gradually decreases after the number reaches 35 for the proposed algorithm. The reason for the curve decline phenomenon is that the scheduling algorithm is more likely to prioritize the radar tasks with high mode priority with the overload of task requests, while the scheduling ability of the communication tasks with relatively low mode priority is greatly decreased. Especially when the number of communication task requests reaches 35, the tasks successfully scheduled by the traditional algorithms are basically the radar tasks with high mode priority. However, since the proposed algorithm has the opportunity to schedule the same number of communication tasks based on JRC waveforms when scheduling radar tasks, the possibility of communication tasks being scheduled is greatly improved compared with the three traditional algorithms, and the SSRCT can still reach 36% when the number of communication task requests is 60. This shows that the proposed algorithm has strong JRC waveform scheduling capability.

Figure 8 shows the comparison of the successful scheduling ratio of the four algorithms. It can be seen from the figure that the SSRs of the four algorithms decrease as the number of targets increases. Among them, the EDF algorithm, HPEDF algorithm and time window-based backtracking adaptive algorithm drop the task requests when the number of targets is 10, while the proposed algorithm starts to drop the task requests when the number of targets is over 20. Moreover, the SSR of the proposed algorithm is significantly higher than the other three algorithms, and reaches 89% when the target number is 70, while the other three algorithms only reach 64%. This is because some radar tasks and communication tasks can be combined in pairs, and then the paired two tasks are integrated into one based on the JRC waveform in the timeline, which greatly reduces time consumption and greatly increases the number of successfully scheduled tasks.

Figure 9 shows the comparison of the high value ratio of the four algorithms. It can be observed that the three traditional algorithms in MFIS have worse results, while the proposed algorithm provides the best performance with outstanding advantages. Since the proposed algorithm completely starts from the special JRC waveform characteristics of MFIS, the potential of MFIS is deeply explored based on JRC waveforms. In addition, the high value ratio of the proposed algorithm decreases very slowly with the increase of the number of targets, which indicates that all tasks scheduled generally have very high values in the case of task overload, i.e., tasks with high mode priority are always preferentially scheduled in the timeline. This is also reflected in Figure 7.

Figure 10 shows the comparison of the time utilization ratio of the four algorithms. It can be seen that the time utilization ratio of the four algorithms gradually tends to 1 with the increase of the number of targets, i.e., the time resources are gradually saturated. Importantly, it can be observed with Figure 8 that when the number of targets is less than 40, the proposed algorithm always achieves a higher successful scheduling ratio with the least time-resource consumption compared with the traditional algorithms. As a result, the MFIS has more cooling time to ensure its high working performance. As the number of targets continues to increase, the time resources can be further enriched while maintaining a highly successful scheduling ratio by the proposed algorithm because of the unique JRC waveform scheduling capability.

In summary, the proposed algorithm applies the advantages of integration in MFIS, i.e., the MFIS task scheduling algorithm can simultaneously schedule radar tasks and communication tasks based on JRC waveforms under the proposed JRC waveform scheduling criteria, which effectively saves time resources and increases the number of tasks scheduled. Compared with the traditional algorithms, it shows strong JRC waveform scheduling capability and has a more successful scheduling ratio, high value ratio and time utilization ratio.

5. Conclusions

MFIS with JRC waveform is the inevitable trend of future development. The MFIS needs an efficient scheduling algorithm to fully release its multifunctional integration potential. Aiming for the implementation of the JRC waveform, a novel adaptive task scheduling algorithm is proposed in this paper. The application of the JRC waveform enables the system to possibly execute one task simultaneously with another one, which not only obtains the decrease of time resource consumption of a single task but also increases the magnitude of executed tasks within the same time span. The proposed algorithm analyzes the task requests by JRC waveform scheduling criteria and then schedules the tasks based on the time resource constraints. The simulations show that the proposed algorithm has very good scheduling capability of JRC waveform and achieves better performance compared with the EDF algorithm, HPEDF algorithm and time window-based backtracking adaptive algorithm in aspects of the successful scheduling ratio of communication tasks, successful scheduling ratio, high value ratio and time utilization ratio. In the future, testing a robust optimization algorithm to solve the scheduling problem in a real MFIS will be another research hotspot.