M. R. Garey, D.R. Johnson, B. B. Simons, R. E. Tarjan, "Scheduling UnitTime Tasks with Arbitrary Release Times and Deadlines", SIAM J. Comput., 10 (1981), pp. 256-269.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for instance [7] In the case of distinct release dates, Dessouky, Lageweg, Lenstra and van de Velde show ( 7] that both the problem of minimizing makespan and the problem of minimizing the sum of the completion times (for a xed number m of uniform machines) are polynomial. It is shown in [5,6,9] that the problem of minimizing the number of late jobs (1jr i j P U i ) is polynomial as soon as processing times are equal, while the general problem is NP hard [8] Baptiste ( 2] shows that minimizing the weighted number of late jobs to be scheduled on a single machine can be done in ....

Michael. R. Garey, David. S. Johnson, Barbara B. Simons and R.E. Tarjan, Scheduling unit-time tasks with arbitrary release times and deadlines, SIAM Journal of Computing, 10 (1981) 256-269. 14

....types of tasks, with periodic tasks with complex constraints. We will present a method for complexity reduction and online scheduling to ensure these transformed constraints. 3. 1 Offline complexity reduction Finding optimal solutions to most sets of complex constraints is an NP hard problem [9]. Consequently algorithms will be heuristic and produce suboptimal solutions only. Performing the complexity reduction offline, however, allows for elaborate methods, improvement of results and modifications in the non successfull case. The transformation method should be flexible to include new ....

M. Garey, D. Johnson, B. Simons, and R. Tarjan. Scheduling unit-time tasks with arbitrary release times and dead lines. IEEE Transactions on Software Engineering, 10(2):256-- 269, May 1981.

....[2] and mixed task sets of non preemptive sporadic and periodic tasks, 11] Joint scheduling of periodic and aperiodic tasks with unknown and unlimited arrival times has been addressed in the context of dynamic scheduling for single processors. Schedulability tests of O(N 2 ) were presented in [10] The issue of using resources unused by the periodic tasks on single processors has been studied in the context of earliest deadline first scheduling, 4] 3] gives online acceptance tests of O(N ) Efficient server algorithms for earliest deadline first have been presented in [22] Server ....

....larger than that of the other tasks, the number of intervals we have to go through, i.e. up to the aperiodic s deadline, will be unaffected by the maximum N . It is shown in [9] that this acceptance test has equivalent results but with simpler run time handling as to the ones presented in [10] and [3] which are optimal for single processors. Due to space limitations we do not give proofs here. The interested reader is referred to [9] 4.2 On Line Scheduling On line scheduling is performed locally for each node. We have the set of tasks T i , consisting of static, guaranteed, and ....

M.R. Garey, D.S. Johnson, B.B. Simons, and R.E. Tarjan. Scheduling unit-time tasks with arbitrary release times and dead lines. IEEE Transactions on Software Engineering, 10(2):256--269, May 1981.

....of a task, the spare capacities are updated to reflect the decrease in available resources. This guarantee algorithm is O(N) N being the number of intervals. It is shown in [7] that this acceptance test has equivalent results but with simpler run time handling as to the ones presented in [9] and [6] which are optimal for single processors. On line scheduling On line scheduling is performed locally for each node. If the spare capacities of the current interval sc(I c ) 0, EDF is applied on the set of ready tasks. sc(I c ) 0 indicates that a guaranteed task has to be executed or ....

M. Garey., D. Johnson, B. Simons, and R. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. IEEE Trans. on Soft. Eng., May 1981.

....medium 2D mesh Point to point links in 2 dim. mesh n cube Point to point links in k ary n cube Approximate None No cost assumed All Full cost always assumed Random Full no cost assumed in 50 of the cases Deadline Period Deadline equal to period [41] MinMax Overlapping execution windows [16, 10] Slicing Non overlapping execution windows [13, 27] Cluster Explicit Clustering as provided by TaskSet class Pairwise Pairwise clustering [43] Period Period based clustering [1] Replicate Explicit Replication as provided by TaskSet class Allocate Exhaustive Branch and bound strategy [35, ....

M. R. Garey, D. S. Johnson, B. B. Simons, and R. E. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. SIAM Journal on Computing, 10(2):256--269, May 1981.

....that is, a relatively low amount of interprocessor communication in the system. 5. 4 Deadline assignment To assign individual arrival times and deadlines to the tasks in the task graph, we have used two deadline assignment techniques that guarantee that arrival times and deadlines are consistent [23] with the partial order over T, that is, if i OE j then a i a j and D i D j . The first technique, herein called the MINMAX technique [23, 24] assigns consistent arrival times and deadlines by processing the tasks once in topological order, assigning a j max(fa j g [ fa i c i : i ....

....deadlines to the tasks in the task graph, we have used two deadline assignment techniques that guarantee that arrival times and deadlines are consistent [23] with the partial order over T, that is, if i OE j then a i a j and D i D j . The first technique, herein called the MINMAX technique [23, 24], assigns consistent arrival times and deadlines by processing the tasks once in topological order, assigning a j max(fa j g [ fa i c i : i OE Delta j g) and once in reverse topological order assigning D i min(fD i g [ fD j Gamma c j : i OE Delta j g) This will assign the largest ....

M. R. Garey, D. S. Johnson, B. B. Simons, and R. E. Tarjan, "Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines," SIAM Journal on Computing, vol. 10, no. 2, pp. 256--269, May 1981.

....for those associated with the initial and the final tasks) was shown to be NP hard by Garey and Johnson [9] Carlier [4] proposed an efficient branch and bound algorithm for solving the problem. When all the task processing times are equal, Simons [16] and Garey, Johnson, Simons and Tarjan [10] proposed polynomial algorithms for the optimal solution. In the preemptive case, however, simple polynomial algorithms solve the problem for makespan minimization subject to release and delivery times. Indeed, as observed by Garey et al. 10] the presence of precedence constraints (with zero ....

....Simons [16] and Garey, Johnson, Simons and Tarjan [10] proposed polynomial algorithms for the optimal solution. In the preemptive case, however, simple polynomial algorithms solve the problem for makespan minimization subject to release and delivery times. Indeed, as observed by Garey et al. [10], the presence of precedence constraints (with zero delay) is essentially irrelevant in this case: One can first modify the release and delivery times so that they become consistent with the precedence relations, and then apply the Largest Delivery Time policy, see Horn [13] It is easily seen ....

M. R. Garey, D.R. Johnson, B. B. Simons, R. E. Tarjan, "Scheduling UnitTime Tasks with Arbitrary Release Times and Deadlines", SIAM J. Comput., 10 (1981), pp. 256-269.

....of unit length packets with hard deadlines by a single server in slotted time. We adopt the use of the word packet in this context, as is common in the literature on scheduling in communication networks Ling and Shro 1996, Peha and Tobagi 1990, Pingali and Kurose 1991, Chipalkatti et al. 1989, Garey et al. 1981, whereas the word job, task, or call is used in other literature. By hard deadlines is meant that if a packet is not scheduled before its deadline, it is discarded from the system. By on line is meant that whatever the arrivals up to the current time slot, the future arrivals are ....

Garey, M.R., D. S. Johnson, B. B. Simons, R. E. Tarjan. 1981. Scheduling unit-time task with arbitrary release times and deadlines. SIAM J. Comput. 10 259-269.

.... times and the sum of the tardiness, the corresponding nonpreemptive problem Pmjp i = p; r i jF can be solved in O(n 3m 4 ) Again, a generalization to the preemptive case of these algorithms, or of other algorithms, initially designed for the non preemptive scheduling of equal length jobs [8, 9, 10, 13, 20], seems unlikely. 3. Dominance properties. We now consider the parallel machine scheduling problem Pmjpmtn; p i = m; r i j P w i U i . Since when a job is late it can be arbitrarily late, we look for schedules on which a maximum weighted number of jobs are ontime, i.e. are scheduled between ....

M. R. Garey, D. S. Johnson, B. B. Simons and R. E. Tarjan, Scheduling unit-time tasks with arbitrary release times and deadlines, SIAM Journal of Computing, 10 (1981), pp. 256-269.

....for those associated with the initial and the final tasks) was shown to be NP hard by Garey and Johnson [10] Carlier [4] proposed an efficient branch and bound algorithm for solving the problem. When all the task processing times are equal, Simons [17] and Garey, Johnson, Simons and Tarjan [11] proposed polynomial algorithms for the optimal solution. In the preemptive case, however, simple polynomial algorithms solve the problem for makespan minimization subject to release and delivery times. Indeed, as observed by Garey et al. 11] the presence of precedence constraints (with zero ....

....Simons [17] and Garey, Johnson, Simons and Tarjan [11] proposed polynomial algorithms for the optimal solution. In the preemptive case, however, simple polynomial algorithms solve the problem for makespan minimization subject to release and delivery times. Indeed, as observed by Garey et al. [11], the presence of precedence constraints (with zero delay) is essentially irrelevant in this case: One can first modify the release and delivery times so that they become consistent with the precedence relations, and then apply the Largest DeliveryTime policy, see Horn [14] It is easily seen from ....

M. R. Garey, D.R. Johnson, B. B. Simons, R. E. Tarjan, "Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines", SIAM J. Comput., 10 (1981), pp. 256-269.

....and deadlines. Unfortunately, it seems difficult to apply these techniques to schedule AND OR task systems. Deadline Modification. The technique of deadline modification is used in some optimal algorithms for scheduling AND only task systems to meet deadlines on one or two processors [Garey77] [Garey81]. The one processor algorithm proceeds in two steps. 1) The deadlines are modified repeatedly according to the following rule: if a task T j has a predecessor T i , then d i min(d i , d j p j ) 2) The precedence constraints are discarded and the task system is scheduled according to the ....

....NPcomplete. Later, multiprocessor scheduling is considered and it is shown that for general graphs and for in trees, the problem of minimizing completion time is also NP complete. 3.1.1. Scheduling to Meet Deadlines on a Single Processor There are well known polynomial time algorithms [Garey77] [Garey81] for scheduling tasks with AND only precedence constraints, identical processing times, and arbitrary deadlines on one or two processors. It is natural to ask whether the corresponding AND OR scheduling problems may be solved in polynomial time. Unfortunately, this extended problem is NP 19 ....

[Article contains additional citation context not shown here]

Garey, M. R., D. S. Johnson, B. B. Simons and R. E. Tarjan. Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines. SIAM J. Computing (May 1981) vol. 10, no. 2, pp. 256-269.

.... pipelines to maximize throughput are described in [9, 14] The general problem of scheduling to meet deadlines on identical multiprocessor systems is also NP hard [5, 10] However, polynomial algorithms for optimally scheduling tasks with identical processing times on one or two processors exist [4, 7]. Our algorithms make use of one of them. We review the traditional flow shop model in Section 2. The flow shop with recurrence and the periodic flow shop model are described. Section 3 summarizes our earlier results on optimally scheduling homogeneous tasks with identical processing times on each ....

....completion. This scheduling decision is propagated on to the subsequent processors; whenever T ij completes on P j , T i(j 1) starts on P j 1 . The scheduling decision is more complicated if release times and deadlines are arbitrary rational numbers, that is, not multiples of . Garey et al. [7] introduce the concept of forbidden regions during which tasks are not allowed to start execution. The release times of selected tasks are postponed to insert the necessary idle times to make an EEDF schedule optimal. We call the release times generated from the effective release times by this ....

M. R. Garey, D. S. Johnson, B. Simons, and R. E. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. SIAM J. Comput., 10-2:256--269, 1981.

....are not known to be NP hard. For two problems that are known to be NP hard, we give heuristic algorithms to minimize completion time. The algorithms have small running time and good worst case performance. Our work is related to some previous work on deterministic scheduling to meet deadlines [6] [8] and to minimize completion time [9] 10] 13] 14] We were inspired by an AND OR model that was proposed as a means of modeling distributed systems for real time control [18] Two recent systems incorporated AND OR precedence constraints of some sort in their implementation [16] 19] The ....

....heuristic to minimize completion time on m processors. We then explain why no priority driven heuristic can provide a better worst case performance bound than the one presented here. 3.1. Scheduling to Meet Deadlines on a Single Processor. There are wellknown polynomial time algorithms [6] [8] for scheduling tasks with AND only precedence constraints, identical processing times, and arbitrary deadlines on one or two processors. It is natural to ask whether the corresponding AND OR scheduling problems may be solved in polynomial time. Unfortunately, this extended problem is NP complete, ....

[Article contains additional citation context not shown here]

M. R. Garey, D. S. Johnson, B. B. Simons, and R. E. Tarjan, Scheduling unit-time tasks with arbitrary release times and deadlines, SIAM J. Comput. 10 (1981), pp. 256-269.

.... of scheduling no wait flow shops is reviewed in [15] The general problem of scheduling to meet deadlines on identical multiprocessor systems is also NP hard [11, 25] However, polynomial algorithms for optimally scheduling tasks with identical processing times on one or two processors exist [10, 13]. Our algorithms in Chapter 4 and Chapter 5 make use of one of them. 3.2 Pipeline Scheduling Lawler et al. 24] identify three differences between the pipeline scheduling and the flow shop scheduling problems: 1) As opposed to flow shops, no buffering of subtasks is allowed between stages in ....

....on to the subsequent processors; whenever the subtask T ij completes on P j , T i(j 1) starts on P j 1 . The scheduling decision on the first processor is slightly more complicated if release times and deadlines are arbitrary rational numbers, that is, not multiples of . Garey et al. [13] introduced the concept of forbidden regions during which tasks are not allowed to start execution. Their algorithm postpones the release times of selected tasks. This is done to adequately insert the necessary idle times to make an EEDF schedule optimal. We call the release times generated from ....

[Article contains additional citation context not shown here]

M. R. Garey, D. S. Johnson, B. Simons, and R. E. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. SIAM J. Comput., 10-2:256--269, 1981.

.... jobs with arbitrary release times, that is, 1 j nopmtn; r j ; p j = 1 j L max ; is easy [18] Moreover, if we add precedence constraints and we want to minimize the maximum completion time (makespan) that is, we want to solve 1 j nopmtn; prec; r j ; p j = 1 j C max ; the problem is still easy [11]. The algorithm that solves it makes use of forbidden regions, intervals of time during which no task can start if the schedule is to be feasible. The idea is that because of the nonpreemption, scheduling a task at a certain point in time could force some other late task to miss its deadline. At ....

M.R. Garey, D.S. Johnson, B.B. Simons, and R.E. Tarjan, "Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines," SIAM Journal Comput., 10(2), May 1981.

....but not with shared resources have appeared. Blazewicz [2] shows the optimality of a preemptive earliest deadline first (EDF) scheduler assuming the release times and the deadlines are modified according to the partial order among the tasks. The same technique is used by Garey at al. [6] to optimally schedule unit time tasks. In [4] Chetto et al. show sufficient conditions for the EDF schedulability of a set of tasks, assuming the release times and the deadlines are modified as above. Our main contributions are: an exact characterisation of EDF like schedulers that can be used ....

....that can be used to correctly schedule precedence constrained tasks, and showing how preemptive algorithms, even those that deal with shared resources, can be easily extended to deal with precedence constraints too. We do this by inventing the notion of quasi normality, which is an extension to [6]. Furthermore, while the formal results are general, we also present a straightforward application of these results to the Priority Ceiling Protocol (PCP) and the Stack Resource Policy (SRP) developing schedulability formulas that are valid when the SRP is extended to handle both shared resources ....

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M.R. Garey, D.S. Johnson, B.B. Simons and R.E. Tarjan, "Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines," SIAM Journal Comput., 10(2), May 1981.

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M. R. Garey, D.R. Johnson, B. B. Simons, R. E. Tarjan, "Scheduling UnitTime Tasks with Arbitrary Release Times and Deadlines", SIAM J. Comput., 10 (1981), pp. 256-269.

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M.R. Garey, D.S. Johnson, B.B. Simons, and R.E. Tarjan. Scheduling UnitTime Tasks with Arbitrary Release Times and Dead lines. IEEE Transactions on Software Engineering, 10(2):256--269, May 1981. 90

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M. Garey, D. Johnson, B. Simons, and R. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. SIAM Journal on Computing, 10(2):256-269, 1981.

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M. Garey, D. Johnson, B. Simons, and R. Tarjan. Scheduling unit-time tasks with arbitrary release times and deadlines. SIAM Journal on Computing, 10(2):256--269, 1981.

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M. Garey, et al., " Scheduling Unit-Time Tasks w ith Arbitrary Release Times and Deadlines," SIAM Journal on Computing, Vol. 10, No. 2, May 1981, 256-269.

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Garey, M.R., Johnson, D.S., Simons, B.B., and Tarjan, R.E. (1981). Scheduling Unit-Time Tasks with Arbitrary Release Times and Deadlines, SIAM J.

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