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Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain
Journal of Systems Engineering and Electronics Vol. 19, No. 3, 2008, pp.479–485
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain Cheng Ting, He Zishu & Tang Ting School of Electronic Engineering, Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received November 27, 2006)
Abstract: A real-time dwell scheduling model, which takes the time and energy constraints into account is founded from the viewpoint of scheduling gain. Scheduling design is turned into a nonlinear programming procedure. The real-time dwell scheduling algorithm based on the scheduling gain is presented with the help of two heuristic rules. The simulation results demonstrate that compared with the conventional adaptive scheduling method, the algorithm proposed not only increases the scheduling gain and the time utility but also decreases the task drop rate.
Keywords: multifunction phased array radar; dwell scheduling; pulse interleaving; time and energy constraint.
1. Introduction The beam direction of phased array radars can be switched rapidly, which provides the multifunction and self-adaptive characteristics. To make full use of the phased array radars, it is necessary to manage their resources effectively. The radar dwell scheduling design is a crucial problem in resource management. It refers to how to choose an optimal scheduling sequence to balance different kinds of radar events under the time and energy constraints[1]. The template-based[2−4] and adaptive scheduling algorithms[5] are two kinds of typical methods. In the template-based scheduling algorithm, the templates are designed offline. Since the states of the targets and the working condition of the radar are changing dynamically, the template library is usually unable to match the radar task load correctly. Although Ref. [6] introduced the online template-based scheduling algorithm, it is difficult to decide the horizon parameter. Therefore, the adaptive scheduling algorithm is considered to be the most effective scheduling policy. To improve the scheduling performance of the radar systems, the pulse interleaving technique was put forward[7]. The energy constraint must be considered when interleaving pulses. To avoid the design complexity, the pulse interleaving is not used in the
conventional adaptive scheduling algorithm, which takes only the time constraint into account. The model of the real-time scheduling problem is established from the point of scheduling gain in this article. The pulse interleaving technique, and the time and energy constraints are considered during the modeling process. The dwell scheduling algorithm based on the scheduling gain is given based on the founded model. The scheduling interval is discretized to simplify the time and energy constraint analysis, which guarantees the real-time of the algorithm.
2. Model of the dwell scheduling problem Along with the computer technique being embedded into the phased array radars, modern phased array radars can be newly viewed to be composed of two parts: the control computer and the radar front. The communication between them is based on a time interval, which is called the scheduling interval (SI). During an SI, the control computer must offer the scheduling sequence of the tasks scheduled in the next SI and process the data returned from the previous SI. The task model of a dwell can be given as follows T = {w, rt, tx, wa, rx, pt, pr, l}
(1)
where w is the priority level of the task, rt is the de-
480
Cheng Ting, He Zishu & Tang Ting
sired execution time, tx is the transmission duration, wa is the waiting duration, rx is the receiving duration, pt and pr are the transmission and receiving powers, respectively, and l is the time window[8] of the task. The waiting duration of a searching dwell is zero and that of a tracking dwell is calculated according to the estimated range of the target. When the task is scheduled in [rt − l, rt + l], it is said to be scheduled in time. 2.1
Scheduling constraints
Although the pulse interleaving technique can improve the time utility of the radar system, it will prolong the transmission duration and easily drive the system hot enough to be damaged. The energy constraint must be taken into consideration when interleaving pulses. Usually, the consumed energy of the system can be expressed as follows E(t) =
t
P (x)e−
t−x τ
dx
(2)
0
where P (x) is the power function, and τ is the lookback period. Equation (2) demonstrates that the system energy consumption is a weighted sum and its weight function gives more weight to the recent values and less weight to the past values[2] . The sustainable system energy consumption is described with a threshold Eth . The energy constraint is satisfied as long as the following inequation holds E(t) Eth , ∀t
(3)
Besides, the time constraint should also be satisfied. When a dwell task is accomplished successfully and in time, the time constraint holds. A successfully accomplished task indicates that its transmission and receiving durations are not corrupted with other tasks. 2.2
Formulation of the objective
The scheduling gain of an SI is chosen as the objective function of the algorithm, which depicts the utility brought to the user when the tasks are scheduled in the SI according to the obtained scheduling sequence from the algorithm. It is the sum of scheduling gains of the tasks scheduled in the SI. A function with several variables is introduced to depict the gain of a
scheduled task, which can be denoted as follows g(st, rt, l, w, t0 , s) = g1 (w, s)g2 (rt, l, t0 )g3 (st, rt, l) (4) where st is the actual execution time of the task, t0 is the starting time of the current SI, and s is the set of the typical environment parameters. The function is constructed from three aspects. These are the importance of the task, the urgency of the task, and the validity of scheduling it. Firstly, g1 (w, s) depicts the importance of the task. The set of the typical environment parameters for a tracking task is constructed with the state variables of the target, which express its maneuvering level. For a searching task, s describes the region to be searched. The value of this function increases as the priority level of the task and the maneuvering level of the target or the essentiality of the searched region steps up. Secondly, g2 (rt, l, t0 ) depicts the urgency of the task. The last feasible execution time of the task is rt + l; the smaller the distance between t0 and rt + l, the higher is the urgency of the task. Finally, g3 (st, rt, l) depicts the validity of scheduling the task; the nearer the actual execution time to the desired execution time, the higher is the validity. The forms of the above three functions are set according to the actual situation as long as their values can describe the importance, the urgency of the task, and the validity of scheduling it reasonably. The gain of a scheduled task introduced here can not only express the importance of the task but also its urgency, which is different from the utility value proposed by Samuel S Blackman[9]. Furthermore, the values of g1 (w, s) and g2 (rt, l, t0 ) are solely determined by the task’s characteristics and the starting time of the current SI. By combining them, we get g1 (w, s)g2 (rt, l, t0 ). It can be regarded as the synthetic priority of the task in the current SI. 2.3
The model of dwell scheduling problem
Assume N dwell tasks apply for scheduling in [t0 , t0 + L]. The number of scheduled, delayed, and deleted tasks are N1 , N2 , and N3 respectively. Obviously, N1 + N2 + N3 = N . Each of the scheduled tasks is assigned with an actual execution time sti1 , i1 = 1, 2, . . . , N1 , which composes the scheduling sequence
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain of this SI. When a task cannot be scheduled in the interval, we need to check whether its last feasible scheduling instant rt + l is larger than t0 + L. In that case, the task will be delayed. Otherwise, it will be deleted. So far, we can get the model of the dwell scheduling problem as follows max
N1
culated as g1 (w, s)g2 (rt, l, t0 ). It corresponds to rule one. The second block rearranges the feasible execution time of the task according to the distances to the
corresponds to rule two. The judging block guarantees the whole scheduling process to satisfy the time and energy constraints.
t0 + L), i1 = 1, . . . , N1
i1=1
range” block. In the first block, the tasks are arranged according to their synthetic priorities, which are cal-
gins with the element with the minimal distance. It
s.t. max(rti1 − li1 , t0 ) sti1 < min(rti1 + li1 ,
[sti1 , sti1 + txi1 ] ∪ [sti1 + txi1 + wai1 ,
blocks, namely, the “arrange” block and the “rear-
desired execution time, and the searching process be-
g(st, rt, l, w, t0 , s)
i1=1
N1
481
During the realization of the algorithm, the SI is dis(5)
sti1 + txi1 + wai1 + rxi1 ] = φ E(t) Eth , t ∈ [t0 , t0 + L) rti2 + li2 t0 + L, i2 = 1, . . . , N2 rti3 + li3 < t0 + L, i3 = 1, . . . , N3 As mentioned before, the objective of our problem is to maximize the scheduling gain of an SI, which is the sum of the scheduling gains of the scheduled tasks. There are totally five constraints in the model. The first two correspond to in time and successful scheduling, the third one is the energy constraint, and the last two are the conditions for delaying and deleting tasks.
cretized into several time slots. The granularity of the discretization is 1ms, which is too small to lead large ∆g3 (st, rt, l). Once the SI is discretized, we can resort to a recursive method to analyze the changes of the time and energy states when a task is added into the scheduling queue. As seen in Fig. 1, S and E describe the time and energy states of the SI, respectively; their length is L. S is initialized as a zero vector and the elements in E can be initialized as follows E(i) = E0 e−i/τ , i = 1, . . . , L
When a task is newly scheduled, it will induce the changes of the two state vectors, which are denoted as ∆S and ∆E ∆S(i − t0 ) = 1, i ∈ (st, st + tx]∪
3. Dwell scheduling algorithm based on the scheduling gain According to Eq. (5), the design of the scheduling algorithm turns to be a non-linear programming problem. It is difficult to get an optimal solution for it. Here, we resort to some heuristic rules to get a suboptimal solution. From the expression of the objective function, it is found that its value will increase when the importance and urgency levels of the scheduled tasks step up. Furthermore, the distance between the desired execution time and the actual execution time of a scheduled task should be as small as possible. Based on these rules, the dwell scheduling algorithm based on the scheduling gain can be obtained. Figure 1 depicts its flow chart. It contains two crucial
(6)
(st + wa, st + wa + rx] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ pt
∆E(i − t0 ) = 0
(7)
ek/τ , i ∈ (st, st + tx]
k=st+1−i
⎪ ⎪ ⎪ ⎪ ⎩ ∆E(st + tx − t0 )est+tx−i , i ∈ (st + tx, L] Equation (7) demonstrates that the first st − 1 elements of ∆S and ∆E are zeros; the newly scheduled task influences only a partial state of the SI. Therefore, once a new task is scheduled, only the elements of index larger than st − 1 are updated. The updated time and energy states are the initial states to the following task scheduling analysis. In this way, the complexity of constraints analysis is reduced dramatically.
482
Cheng Ting, He Zishu & Tang Ting
Fig. 1
Flow chart of the scheduling algorithm
4. Simulation 4.1
Simulation parameters
Six types of radar tasks are considered in the simulation. The speeds of the targets are randomly distributed in 0−4 Ma while their accelerations are randomly distributed in 0−5 g. Three simulation scenarios are set. These correspond to totally 20, 40, and 80 targets appearing in the surveillance domain, where the proportion among the three kinds of tracking tasks is 1:2:2. These scenarios represent low level, common
level, and high level of the task load, respectively. The whole simulation time is 3 s and the length of an SI is set to be 50 ms, τ = 200 ms, Eth = 250 J. The arrival of the tasks is modeled as a Poisson process where λ = 100. The task parameters refer to Refs. [2−4, 10, 11], and are shown in Table 1. The functions constructing the gain of a scheduled task are set as: g1 (w, s) = w for searching tasks (there is only one region assumed to be searched), g1 (w, s) = w + k · sT for other tasks, rt+l−t0 g2 (rt, l, t0 ) = e−c L , g3 (st, rt, l) = 1 − |st−rt| , l where s = [v, a], k = [0.4, 0.6], and c = 0.2. The
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain Table 1
483
Parameters of the tasks
Task type
Priority level
Dwell time/ms
Transmission power/kW
Period/ms
Time window/ms
Confirmation
6
5
5
-
50
High-precision track
5
3
5
150
15
High priority search
4
5
4
100
50
Precision track
3
4
4
250
20
Normal track
2
5
4
500
20
Low priority search
1
4
3
50
100
whole simulation work is realized on a PC with 2.5G CPU and 256M memory. 4.2
Simulation results
Both the scheduling algorithm proposed here and the conventional adaptive scheduling one[10] are simulated in the three scenarios mentioned above. Their performances are compared, including the scheduling gain, task drop rate, and time utility of the system. The results are averaged over 100 Monte Carlo simulations. The scheduling gain curves of the conventional scheduling algorithm and the proposed one in different scenarios are shown in Figs. 2−4, where the new scheduling algorithm represents the scheduling algorithm based on the scheduling gain. The average gains, task drop rates, and time utilities of the algorithms are compared in Tables 2−4. The average run time (ART) of the algorithms is also listed in the tables. It is seen that the scheduling algorithm based on the scheduling gain not only increases the scheduling gain of the system but also improves the time utility and the task drop rate. Furthermore, to inspect the performance improvement of the scheduling algorithm solely caused by the introduction of the task’s scheduling gain, the drop rate curves of the adaptive scheduling algorithm based on scheduling gain without interleaving and the conventional one are compared; this is shown in Fig. 5. Owing to the introduction of the task’s scheduling gain, the scheduling algorithm takes both the importance and the urgency of the task into account. It therefore decreases the task drop rate effectively. The interleaving technique can further enhance the time utility of the system, and the drop rate is then improved considerably. By comparing the results in the three scenarios, the superiority of the adaptive scheduling algorithm based on
scheduling gain will be more obvious when the task load grows. Finally, owing to the discretization of the SI, the complexity of the scheduling algorithm based on scheduling gain does not raise because of the pulse interleaving technique and the energy constraint analysis, which guarantees the real-time of the algorithm.
Fig. 2
The comparison of the scheduling gain curves in scenario 1
Fig. 3
The comparison of the scheduling gain curves in scenario 2
484
Cheng Ting, He Zishu & Tang Ting Table 4
The performance comparison in scenario 3
Algorithm
Average gain Drop rate Time utility ART/s
Conventional
23.187 7
23.23%
47.53%
0.004 0
New
35.711 9
12.17%
53.67%
0.005 5
5. Conclusions
Fig. 4
The comparison of the scheduling gain curves in scenario 3
The resources of multifunction phased array radar should be managed effectively to fully exert its performance. The design of the radar dwell scheduling algorithm is a key in it. The model of the scheduling problem is founded from the viewpoint of scheduling gain, and with the help of two heuristic rules, the scheduling algorithm based on the scheduling gain is provided in this article. Firstly, the algorithm resorts to the discretization and a recursive method to simplify the time and energy constraints analysis, which reduces the complexity and therefore guarantees the real-time of the algorithm. Secondly, the “arrange” and “rearrange” blocks in the algorithm heighten the scheduling gain effectively. At last, the pulse interleaving technique makes full use of the waiting duration of a dwell. More tasks can be scheduled in a fixed interval, the time utility is increased, and the drop rate is decreased at the same time.
References [1] Baugh R A. Computer control of modern radars.
New
York: RCA Corporation, 1973: 49-51. [2] Shih C S, Gopalakrishnan S, Ganti P, et al. Templatebased real-time dwell scheduling with energy constraint. Fig. 5
The comparison of the drop rate curves
Proceedings of the 9th IEEE Real-Time and Embedded Technology and Applications Symposium. Toronto: IEEE
Table 2
The performance comparison in scenario 1
Algorithm
Computer Society, 2003: 19-27. [3] Shih C S, Gopalakrishnan S, Ganti P, et al. Scheduling
Average gain Drop rate Time utility ART/s
real-time dwells using tasks with synthetic periods. Pro-
Conventional
9.238 4
1.26%
26.29%
0.001 0
ceedings of the 24th IEEE International Real-Time Sys-
New
12.887 9
0.01%
26.51%
0.001 7
tems Symposium. Mexico: IEEE Computer Society, 2003: 210-219.
Table 3
The performance comparison in scenario 2
Algorithm
[4] Shih C S, Ganti P, Gopalakrishnan S, et al. Synthesizing task periods for dwells in multi-function phased ar-
Average gain Drop rate Time utility ART/s
ray radars. Proceedings of the IEEE Radar Conference.
Conventional
15.146 7
5.39%
37.42%
0.001 8
Philadelphia: The IEEE Aerospace and Electronic Systems
New
22.560 5
0.10%
39.25%
0.002 6
Society, 2004: 145-150.
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain [5] CAI Qingyu, Xue Yi, Zhang Boyan. Phased array radar data processing and its simulation techniques.
Beijing:
National Defence Industry Press, 1997: 226-257. [6] Gopalakrishnan S, Caccamo M, Shih C S, et al. Finite-
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[11] Lu Jianbin, Hu Weidong, Yu Wenxian. Adaptive scheduling algorithm for real-time dwells in multifunction phased array radars. Systems Engineering and Electronics, 2005, 27(12): 1981-1987. (in Chinese)
horizon scheduling of radar dwells with online template construction. Proceedings of the 25th IEEE International
Cheng Ting was born in1982. She is a Ph. D. candi-
Real-Time Systems Symposium.
Portugal: IEEE Com-
date in University of Electronic Science and Technology. Her research interests include resource manage-
[7] Farina A, Neri P. Multitarget interleaved tracking for
ment of phased array radar and multi-target tracking. E-mail: [email protected]
puter Society, 2004: 23-33.
phased array radar. IEE Proc. Communication, Radar and Signal Processing. Part F, 1980, 127(4): 312-318. [8] Huizing A G, Bloemen A A F. An efficient scheduling algorithm for a multifunction radar. Proceedings of the IEEE International Symposium on Phased Array Systems and Technology. Boston, 1996: 359-364.
He Zishu was born in 1962. He is a professor in University of Electronic Science and Technology. His research interests include MIMO systems, signal processing of radar, and so on.
[9] Blackman S S. Multiple-target tracking with radar application. Dedham: MA, 1986. [10] Zeng Guang, Hu Weidong, Lu Jianbin, et al. The simulation on adaptive scheduling for multifunction phased array radars. Journal of System Simulation, 2004, 16(9): 20262029.
Tang Ting was born in 1983. She is a master candidate in University of Electronic Science and Technology. Her research interests include the resource management of phased array radar.
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain Cheng Ting, He Zishu & Tang Ting School of Electronic Engineering, Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received November 27, 2006)
Abstract: A real-time dwell scheduling model, which takes the time and energy constraints into account is founded from the viewpoint of scheduling gain. Scheduling design is turned into a nonlinear programming procedure. The real-time dwell scheduling algorithm based on the scheduling gain is presented with the help of two heuristic rules. The simulation results demonstrate that compared with the conventional adaptive scheduling method, the algorithm proposed not only increases the scheduling gain and the time utility but also decreases the task drop rate.
Keywords: multifunction phased array radar; dwell scheduling; pulse interleaving; time and energy constraint.
1. Introduction The beam direction of phased array radars can be switched rapidly, which provides the multifunction and self-adaptive characteristics. To make full use of the phased array radars, it is necessary to manage their resources effectively. The radar dwell scheduling design is a crucial problem in resource management. It refers to how to choose an optimal scheduling sequence to balance different kinds of radar events under the time and energy constraints[1]. The template-based[2−4] and adaptive scheduling algorithms[5] are two kinds of typical methods. In the template-based scheduling algorithm, the templates are designed offline. Since the states of the targets and the working condition of the radar are changing dynamically, the template library is usually unable to match the radar task load correctly. Although Ref. [6] introduced the online template-based scheduling algorithm, it is difficult to decide the horizon parameter. Therefore, the adaptive scheduling algorithm is considered to be the most effective scheduling policy. To improve the scheduling performance of the radar systems, the pulse interleaving technique was put forward[7]. The energy constraint must be considered when interleaving pulses. To avoid the design complexity, the pulse interleaving is not used in the
conventional adaptive scheduling algorithm, which takes only the time constraint into account. The model of the real-time scheduling problem is established from the point of scheduling gain in this article. The pulse interleaving technique, and the time and energy constraints are considered during the modeling process. The dwell scheduling algorithm based on the scheduling gain is given based on the founded model. The scheduling interval is discretized to simplify the time and energy constraint analysis, which guarantees the real-time of the algorithm.
2. Model of the dwell scheduling problem Along with the computer technique being embedded into the phased array radars, modern phased array radars can be newly viewed to be composed of two parts: the control computer and the radar front. The communication between them is based on a time interval, which is called the scheduling interval (SI). During an SI, the control computer must offer the scheduling sequence of the tasks scheduled in the next SI and process the data returned from the previous SI. The task model of a dwell can be given as follows T = {w, rt, tx, wa, rx, pt, pr, l}
(1)
where w is the priority level of the task, rt is the de-
480
Cheng Ting, He Zishu & Tang Ting
sired execution time, tx is the transmission duration, wa is the waiting duration, rx is the receiving duration, pt and pr are the transmission and receiving powers, respectively, and l is the time window[8] of the task. The waiting duration of a searching dwell is zero and that of a tracking dwell is calculated according to the estimated range of the target. When the task is scheduled in [rt − l, rt + l], it is said to be scheduled in time. 2.1
Scheduling constraints
Although the pulse interleaving technique can improve the time utility of the radar system, it will prolong the transmission duration and easily drive the system hot enough to be damaged. The energy constraint must be taken into consideration when interleaving pulses. Usually, the consumed energy of the system can be expressed as follows E(t) =
t
P (x)e−
t−x τ
dx
(2)
0
where P (x) is the power function, and τ is the lookback period. Equation (2) demonstrates that the system energy consumption is a weighted sum and its weight function gives more weight to the recent values and less weight to the past values[2] . The sustainable system energy consumption is described with a threshold Eth . The energy constraint is satisfied as long as the following inequation holds E(t) Eth , ∀t
(3)
Besides, the time constraint should also be satisfied. When a dwell task is accomplished successfully and in time, the time constraint holds. A successfully accomplished task indicates that its transmission and receiving durations are not corrupted with other tasks. 2.2
Formulation of the objective
The scheduling gain of an SI is chosen as the objective function of the algorithm, which depicts the utility brought to the user when the tasks are scheduled in the SI according to the obtained scheduling sequence from the algorithm. It is the sum of scheduling gains of the tasks scheduled in the SI. A function with several variables is introduced to depict the gain of a
scheduled task, which can be denoted as follows g(st, rt, l, w, t0 , s) = g1 (w, s)g2 (rt, l, t0 )g3 (st, rt, l) (4) where st is the actual execution time of the task, t0 is the starting time of the current SI, and s is the set of the typical environment parameters. The function is constructed from three aspects. These are the importance of the task, the urgency of the task, and the validity of scheduling it. Firstly, g1 (w, s) depicts the importance of the task. The set of the typical environment parameters for a tracking task is constructed with the state variables of the target, which express its maneuvering level. For a searching task, s describes the region to be searched. The value of this function increases as the priority level of the task and the maneuvering level of the target or the essentiality of the searched region steps up. Secondly, g2 (rt, l, t0 ) depicts the urgency of the task. The last feasible execution time of the task is rt + l; the smaller the distance between t0 and rt + l, the higher is the urgency of the task. Finally, g3 (st, rt, l) depicts the validity of scheduling the task; the nearer the actual execution time to the desired execution time, the higher is the validity. The forms of the above three functions are set according to the actual situation as long as their values can describe the importance, the urgency of the task, and the validity of scheduling it reasonably. The gain of a scheduled task introduced here can not only express the importance of the task but also its urgency, which is different from the utility value proposed by Samuel S Blackman[9]. Furthermore, the values of g1 (w, s) and g2 (rt, l, t0 ) are solely determined by the task’s characteristics and the starting time of the current SI. By combining them, we get g1 (w, s)g2 (rt, l, t0 ). It can be regarded as the synthetic priority of the task in the current SI. 2.3
The model of dwell scheduling problem
Assume N dwell tasks apply for scheduling in [t0 , t0 + L]. The number of scheduled, delayed, and deleted tasks are N1 , N2 , and N3 respectively. Obviously, N1 + N2 + N3 = N . Each of the scheduled tasks is assigned with an actual execution time sti1 , i1 = 1, 2, . . . , N1 , which composes the scheduling sequence
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain of this SI. When a task cannot be scheduled in the interval, we need to check whether its last feasible scheduling instant rt + l is larger than t0 + L. In that case, the task will be delayed. Otherwise, it will be deleted. So far, we can get the model of the dwell scheduling problem as follows max
N1
culated as g1 (w, s)g2 (rt, l, t0 ). It corresponds to rule one. The second block rearranges the feasible execution time of the task according to the distances to the
corresponds to rule two. The judging block guarantees the whole scheduling process to satisfy the time and energy constraints.
t0 + L), i1 = 1, . . . , N1
i1=1
range” block. In the first block, the tasks are arranged according to their synthetic priorities, which are cal-
gins with the element with the minimal distance. It
s.t. max(rti1 − li1 , t0 ) sti1 < min(rti1 + li1 ,
[sti1 , sti1 + txi1 ] ∪ [sti1 + txi1 + wai1 ,
blocks, namely, the “arrange” block and the “rear-
desired execution time, and the searching process be-
g(st, rt, l, w, t0 , s)
i1=1
N1
481
During the realization of the algorithm, the SI is dis(5)
sti1 + txi1 + wai1 + rxi1 ] = φ E(t) Eth , t ∈ [t0 , t0 + L) rti2 + li2 t0 + L, i2 = 1, . . . , N2 rti3 + li3 < t0 + L, i3 = 1, . . . , N3 As mentioned before, the objective of our problem is to maximize the scheduling gain of an SI, which is the sum of the scheduling gains of the scheduled tasks. There are totally five constraints in the model. The first two correspond to in time and successful scheduling, the third one is the energy constraint, and the last two are the conditions for delaying and deleting tasks.
cretized into several time slots. The granularity of the discretization is 1ms, which is too small to lead large ∆g3 (st, rt, l). Once the SI is discretized, we can resort to a recursive method to analyze the changes of the time and energy states when a task is added into the scheduling queue. As seen in Fig. 1, S and E describe the time and energy states of the SI, respectively; their length is L. S is initialized as a zero vector and the elements in E can be initialized as follows E(i) = E0 e−i/τ , i = 1, . . . , L
When a task is newly scheduled, it will induce the changes of the two state vectors, which are denoted as ∆S and ∆E ∆S(i − t0 ) = 1, i ∈ (st, st + tx]∪
3. Dwell scheduling algorithm based on the scheduling gain According to Eq. (5), the design of the scheduling algorithm turns to be a non-linear programming problem. It is difficult to get an optimal solution for it. Here, we resort to some heuristic rules to get a suboptimal solution. From the expression of the objective function, it is found that its value will increase when the importance and urgency levels of the scheduled tasks step up. Furthermore, the distance between the desired execution time and the actual execution time of a scheduled task should be as small as possible. Based on these rules, the dwell scheduling algorithm based on the scheduling gain can be obtained. Figure 1 depicts its flow chart. It contains two crucial
(6)
(st + wa, st + wa + rx] ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ pt
∆E(i − t0 ) = 0
(7)
ek/τ , i ∈ (st, st + tx]
k=st+1−i
⎪ ⎪ ⎪ ⎪ ⎩ ∆E(st + tx − t0 )est+tx−i , i ∈ (st + tx, L] Equation (7) demonstrates that the first st − 1 elements of ∆S and ∆E are zeros; the newly scheduled task influences only a partial state of the SI. Therefore, once a new task is scheduled, only the elements of index larger than st − 1 are updated. The updated time and energy states are the initial states to the following task scheduling analysis. In this way, the complexity of constraints analysis is reduced dramatically.
482
Cheng Ting, He Zishu & Tang Ting
Fig. 1
Flow chart of the scheduling algorithm
4. Simulation 4.1
Simulation parameters
Six types of radar tasks are considered in the simulation. The speeds of the targets are randomly distributed in 0−4 Ma while their accelerations are randomly distributed in 0−5 g. Three simulation scenarios are set. These correspond to totally 20, 40, and 80 targets appearing in the surveillance domain, where the proportion among the three kinds of tracking tasks is 1:2:2. These scenarios represent low level, common
level, and high level of the task load, respectively. The whole simulation time is 3 s and the length of an SI is set to be 50 ms, τ = 200 ms, Eth = 250 J. The arrival of the tasks is modeled as a Poisson process where λ = 100. The task parameters refer to Refs. [2−4, 10, 11], and are shown in Table 1. The functions constructing the gain of a scheduled task are set as: g1 (w, s) = w for searching tasks (there is only one region assumed to be searched), g1 (w, s) = w + k · sT for other tasks, rt+l−t0 g2 (rt, l, t0 ) = e−c L , g3 (st, rt, l) = 1 − |st−rt| , l where s = [v, a], k = [0.4, 0.6], and c = 0.2. The
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain Table 1
483
Parameters of the tasks
Task type
Priority level
Dwell time/ms
Transmission power/kW
Period/ms
Time window/ms
Confirmation
6
5
5
-
50
High-precision track
5
3
5
150
15
High priority search
4
5
4
100
50
Precision track
3
4
4
250
20
Normal track
2
5
4
500
20
Low priority search
1
4
3
50
100
whole simulation work is realized on a PC with 2.5G CPU and 256M memory. 4.2
Simulation results
Both the scheduling algorithm proposed here and the conventional adaptive scheduling one[10] are simulated in the three scenarios mentioned above. Their performances are compared, including the scheduling gain, task drop rate, and time utility of the system. The results are averaged over 100 Monte Carlo simulations. The scheduling gain curves of the conventional scheduling algorithm and the proposed one in different scenarios are shown in Figs. 2−4, where the new scheduling algorithm represents the scheduling algorithm based on the scheduling gain. The average gains, task drop rates, and time utilities of the algorithms are compared in Tables 2−4. The average run time (ART) of the algorithms is also listed in the tables. It is seen that the scheduling algorithm based on the scheduling gain not only increases the scheduling gain of the system but also improves the time utility and the task drop rate. Furthermore, to inspect the performance improvement of the scheduling algorithm solely caused by the introduction of the task’s scheduling gain, the drop rate curves of the adaptive scheduling algorithm based on scheduling gain without interleaving and the conventional one are compared; this is shown in Fig. 5. Owing to the introduction of the task’s scheduling gain, the scheduling algorithm takes both the importance and the urgency of the task into account. It therefore decreases the task drop rate effectively. The interleaving technique can further enhance the time utility of the system, and the drop rate is then improved considerably. By comparing the results in the three scenarios, the superiority of the adaptive scheduling algorithm based on
scheduling gain will be more obvious when the task load grows. Finally, owing to the discretization of the SI, the complexity of the scheduling algorithm based on scheduling gain does not raise because of the pulse interleaving technique and the energy constraint analysis, which guarantees the real-time of the algorithm.
Fig. 2
The comparison of the scheduling gain curves in scenario 1
Fig. 3
The comparison of the scheduling gain curves in scenario 2
484
Cheng Ting, He Zishu & Tang Ting Table 4
The performance comparison in scenario 3
Algorithm
Average gain Drop rate Time utility ART/s
Conventional
23.187 7
23.23%
47.53%
0.004 0
New
35.711 9
12.17%
53.67%
0.005 5
5. Conclusions
Fig. 4
The comparison of the scheduling gain curves in scenario 3
The resources of multifunction phased array radar should be managed effectively to fully exert its performance. The design of the radar dwell scheduling algorithm is a key in it. The model of the scheduling problem is founded from the viewpoint of scheduling gain, and with the help of two heuristic rules, the scheduling algorithm based on the scheduling gain is provided in this article. Firstly, the algorithm resorts to the discretization and a recursive method to simplify the time and energy constraints analysis, which reduces the complexity and therefore guarantees the real-time of the algorithm. Secondly, the “arrange” and “rearrange” blocks in the algorithm heighten the scheduling gain effectively. At last, the pulse interleaving technique makes full use of the waiting duration of a dwell. More tasks can be scheduled in a fixed interval, the time utility is increased, and the drop rate is decreased at the same time.
References [1] Baugh R A. Computer control of modern radars.
New
York: RCA Corporation, 1973: 49-51. [2] Shih C S, Gopalakrishnan S, Ganti P, et al. Templatebased real-time dwell scheduling with energy constraint. Fig. 5
The comparison of the drop rate curves
Proceedings of the 9th IEEE Real-Time and Embedded Technology and Applications Symposium. Toronto: IEEE
Table 2
The performance comparison in scenario 1
Algorithm
Computer Society, 2003: 19-27. [3] Shih C S, Gopalakrishnan S, Ganti P, et al. Scheduling
Average gain Drop rate Time utility ART/s
real-time dwells using tasks with synthetic periods. Pro-
Conventional
9.238 4
1.26%
26.29%
0.001 0
ceedings of the 24th IEEE International Real-Time Sys-
New
12.887 9
0.01%
26.51%
0.001 7
tems Symposium. Mexico: IEEE Computer Society, 2003: 210-219.
Table 3
The performance comparison in scenario 2
Algorithm
[4] Shih C S, Ganti P, Gopalakrishnan S, et al. Synthesizing task periods for dwells in multi-function phased ar-
Average gain Drop rate Time utility ART/s
ray radars. Proceedings of the IEEE Radar Conference.
Conventional
15.146 7
5.39%
37.42%
0.001 8
Philadelphia: The IEEE Aerospace and Electronic Systems
New
22.560 5
0.10%
39.25%
0.002 6
Society, 2004: 145-150.
Dwell scheduling algorithm for multifunction phased array radars based on the scheduling gain [5] CAI Qingyu, Xue Yi, Zhang Boyan. Phased array radar data processing and its simulation techniques.
Beijing:
National Defence Industry Press, 1997: 226-257. [6] Gopalakrishnan S, Caccamo M, Shih C S, et al. Finite-
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[11] Lu Jianbin, Hu Weidong, Yu Wenxian. Adaptive scheduling algorithm for real-time dwells in multifunction phased array radars. Systems Engineering and Electronics, 2005, 27(12): 1981-1987. (in Chinese)
horizon scheduling of radar dwells with online template construction. Proceedings of the 25th IEEE International
Cheng Ting was born in1982. She is a Ph. D. candi-
Real-Time Systems Symposium.
Portugal: IEEE Com-
date in University of Electronic Science and Technology. Her research interests include resource manage-
[7] Farina A, Neri P. Multitarget interleaved tracking for
ment of phased array radar and multi-target tracking. E-mail: [email protected]
puter Society, 2004: 23-33.
phased array radar. IEE Proc. Communication, Radar and Signal Processing. Part F, 1980, 127(4): 312-318. [8] Huizing A G, Bloemen A A F. An efficient scheduling algorithm for a multifunction radar. Proceedings of the IEEE International Symposium on Phased Array Systems and Technology. Boston, 1996: 359-364.
He Zishu was born in 1962. He is a professor in University of Electronic Science and Technology. His research interests include MIMO systems, signal processing of radar, and so on.
[9] Blackman S S. Multiple-target tracking with radar application. Dedham: MA, 1986. [10] Zeng Guang, Hu Weidong, Lu Jianbin, et al. The simulation on adaptive scheduling for multifunction phased array radars. Journal of System Simulation, 2004, 16(9): 20262029.
Tang Ting was born in 1983. She is a master candidate in University of Electronic Science and Technology. Her research interests include the resource management of phased array radar.