## 3 The Scheduling Method Based on Time Shifting Constraints and Pulse Interleaving
### The Description of Proposed Methodology
Find all tasks from \(T\) that satisfy \(t_{e}\)+\(w<t_{p0}\), which can be executed at time \(t_{p0}\), to form a task set \(R=[R_{1},R_{2},..,R_{N}]\), and calculate the priority of tasks in \(R\). The method for calculating priority follows an improved MHPF (Modified Highest Priority First) criterion: Let the work mode priority of each task in set \(R\) be denoted as \(P=[P_{1},P_{2},...,P_{L}]\), the deadlines as \(t_{d}=[t_{d1},t_{d2},...,t_{dL}]\), and the threat degrees as \(Thre=[Thre_{1},\ Thre_{2},...,\ Thre_{L}]\). Task \(i\) is considered to have a higher priority than task \(j\) if \(P_{i}>P_{j}\), or if \(P_{i}=P_{j}\), \(Thre_{i}>Thre_{j}\), or if \(P_{i}=P_{j}\), \(Thre_{i}=Thre_{j}\), \(t_{di}<t_{dij}\). 5. Select all tasks from the set \(R\) that satisfy the shifting constraint given in (18) to form a new set \(Q=[Q_{1},Q_{2},...,Q_{n}]\). If \(Q\) is empty, meaning that none of the tasks in \(Q\) satisfy the shifting constraints of (18), proceed to Step 6; otherwise, proceed to Step 7. 6. Update time pointers \(t_{p0}\), \(t_{p1}\), and total system energy consumption: when \(t_{p0}=t_{p1}\): \[t_{p0}=t_{p0}+l,t_{p1}=t_{p1}+l,\] (21) otherwise: \[t_{p0}=t_{p0}+l,E_{0}=E_{0}e^{-l/\tau},\] (22) where \(l\) represents the minimum step length for the time pointer to slide; proceed to Step 8. 7. Find the task with the highest priority from set \(Q\): 1. If \(flag=0\), it indicates that task \(Q_{k}\) cannot be interleaved with the previous task. In this case, schedule task \(Q_{k}\) at time \(t_{p0}\) and move \(Q_{k}\) into the execution task queue. If task \(Q_{k}\) can be interleaved, then update parameters according to (23); if interleaving is not possible, then update parameters \(t_{p0}\) and tp1 according to (24). Based on the type of task \(Q_{k}\), update \(E_{0}\) and \(flag\), and increment \(i\) by 1. \[t_{p0}=t_{p0}+t_{xb},t_{p1}=t_{p1}+\Delta T_{k},\] (23) \[t_{p0}=t_{p0}+\Delta T_{k},t_{p1}=t_{p0}.\] 2. If \(flag=1\) and task \(Q_{k}\) satisfies the constraints of (3), (4), (7), and (17), then task \(Q_{k}\) is moved into the execution task queue, \(i\) is incremented by 1, and the parameters are updated according to (8) to (15). If task \(Q_{k}\) does not satisfy the constraints, the parameters are updated according to (25) to (28). when \(t_{p0}+l<t_{p1}-t_{r1}\), the following applies: \[t_{p0}=t_{p0}+t_{xb},t_{p1}=t_{p1}+\Delta T_{k},\] (25) \[t_{w1}=t_{w1}-l,E_{0}=E_{0}e^{-l/\tau}.\] (26) In other cases: \[t_{ek}+w_{k}<t_{p},k=1,\ldots,N_{\text{delete}}, \tag{20}\]0 (27) \[t_{ek}+w_{k}<t_{p},k=1,\ldots,N_{\text{delete}}, \tag{20}\]1 (28) Check whether \(t_{p0}\geq t_{end}\) or \(i\geq N\). If either condition is true, the task scheduling for the current interval is complete. Delete the time pointers, and output the execution queue and deletion queue for this scheduling interval. Otherwise, proceed to Step 3. The proposed scheduling algorithm primarily involves task interleaving determination and time-shifting arrangement. In each scheduling window, only the currently ready task set (denoted as \(q\)) is considered. Among them, a subset of tasks (typically tracking tasks,denoted as \(n\)) require interleaving conflict checking. For each such task, the algorithm checks all other overlapping tasks to determine temporal compatibility, resulting in an interleaving complexity of \(\mathcal{O}(n^{2})\) per cycle. Additionally, priority assignment and feasible interval extraction are conducted before interleaving. The priority computation is \(\mathcal{O}(q)\), while the interval check is bounded and does not scale with \(q\) or the total number of tasks, \(N\). As a result, the per-cycle complexity is dominated by the \(\mathcal{O}(n^{2})\) term. Assuming the number of scheduling cycles is \(T\), the total worst-case complexity becomes \(\mathcal{O}(T\cdot n^{2})\). In practice, the number of tracking tasks, \(n\), is typically much smaller than \(N\), and the number of ready tasks, \(q\), is also constrained due to radar resource limits.