W.A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, (21), 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....solution. Clearly systems in which P n Gamma1 x=0 x:w m cannot be scheduled. If resource sharing is allowed, those in which P n Gamma1 x=0 x:w m can be scheduled by the resource sharing algorithm mentioned above. Baruah, Howell, and Rosier [1] used this fact, the network reduction of Horn [7], and the Ford Fulkerson algorithm [4] to show that there are solutions to the periodic scheduling problem. Thus, the decision problem for such a periodic task system reduces to checking that P n Gamma1 x=0 x:w m. A method similar to that of Baruah, Howell, and Rosier will be used in Section ....

W. A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....O(mn log n) method to compute JPPS used in this paper, please refer to [16] MaxFlow Maximum Flow Intervals String Rep.O(n 4 ) Omega Gamma n log n) The MaxFlow Bound is a graph formulation of the problem of preemptively scheduling n operations on m machines, first proposed by W.A. Horn [14]. The maximum flow through this graph indicates whether or not a feasible preemptive schedule exists with a length less than or equal to a trial number. The implemented method to solve the maximum flow problem in the graph is based on the Layered Network approach [2] This method has a worst case ....

W.A. Horn, Some simple scheduling algorithms, Naval Research Logistics Quarterly 21 (1974) 177-185.

....When a burst of requests arrive at the same time, it is the deadline of the tasks that should be taken into consideration. The scheduler, in this case, schedules on the basis of earliestdeadline first. To implement this scheme, the scheduler must know the deadlines of all the tasks. Horn [12] developed an algorithm to schedule the tasks based upon the earliestdeadline first policy. Tasks with earlier deadlines and earlier ready times were chosen to execute before tasks with later deadlines and ready times.The assumptions made in the earliest deadline first policy are: tasks require ....

....monotonic scheduling. H ST C MI P AT Scheduling on Multiprocessors Horn developed an algorithm for scheduling independent, arbitrary timed, preemptable tasks on a multiprocessor. Little appeared to be known about simple minimization algorithms for multimachines at the time this paper was written [12]. The assumptions made are: all machines are identical, and there is no constraint on finish time. The method which ensures minimum delay is as follows. Assuming that there are m identical machines tasks are ordered according to increasing deadlines. Each of the first m tasks is assigned to ....

Horn, W.A., Some Simple Scheduling Algorithms, Naval Research Logistics Quarterly, 21, 1974

.... can do even better if we know a priori that l = o(n) This result directly follows from the results of [14] and the details appear in [35] We conclude by noting that the bipartite parametric flow problem has many applications including multiprocessor scheduling with release times and deadlines [21, 24], 0 1 integer programming problems [29, 30] maximum subgraph density [21] finding a maximum size set of edge disjoint spanning trees in an undirected graph [28, 29, 30] network vulnerability [9, 19] partitioning a data base between fast and slow memory [11] and the sportswriter s ....

W. Horn, Some simple scheduling algorithms, Naval Research Logistics Quarterly, 21 (1974), pp. 177--185.

....shortest remaining processing time, preempting when jobs of shorter processing time are released. We also generalize EDD to, upon the release of jobs with earlier dues dates than the job currently being processed, preempt the current job and process the job with the earliest due date. Theorem 2. 4 ([Bak74, Hor74]) SRPT is an optimal algorithm for 1jr j ; pmtnj P C j , and EDD is an optimal algorithm for 1jr j ; pmtnjL max Proof: As before, we argue by contradiction, using a similar greedy exchange argument. However, instead of exchanging entire jobs, we exchange pieces of jobs, which is now allowed in ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....objective is to minimize the weighted completion time P j w j C S j ; if all w j are 1=n, the objective becomes the average completion time. For the single machine case, if the release dates are 0 for all jobs, then the weighted completion time problem can be solved optimally in polynomial time [18]. We are interested in a more general setting with release dates and precedence constraints and multiple machines, any of which makes the problem NP hard [21] Thus, we will consider approximation algorithms, or, in an on line setting, competitive ratios. An important motivation for studying ....

W.A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177-- 185, 1974.

....shortest remaining processing time, preempting when jobs of shorter processing time are released. We also generalize EDD: upon the release of jobs with earlier dues dates than the job currently being processed, preempt the current job and process the job with the earliest due date. Theorem 2. 4 ([Bak74, Hor74]) SRPT is an exact algorithm for 1jr j ; pmtn j P C j , and EDD is an exact algorithm for 1jr j ; pmtn jL max Proof: As before, we argue by contradiction, using a similar greedy exchange argument. However, instead of exchanging entire jobs, we exchange pieces of jobs, which is now allowed in ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....is to minimize the weighted completion time P j w j C S j ; if all w j are 1=n, the objective becomes the average completion time. For the single machine case, if the release dates are 0 for all jobs, then the weighted completion time problem can be solved optimally in polynomial time [17]. We are interested in a more general setting with release dates and precedence constraints and multiple machines, any of which makes the problem NP hard [20] Thus, we will consider approximation algorithms, or, in an on line setting, competitive ratios. An important motivation for studying ....

W.A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185 (1974).

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W.A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, (21), 1974.

....if i j. As it is shown in [2] such an order always exists. It can be trivially shown by the exchange argument that for any problem in the class there exists an optimal schedule where i j ) C i C j . Let us consider completion times C j for all j = 1; 2; n as deadlines. It is known [13] that a feasible schedule for the decision problem P jpmtn with deadlines C j exits if and only if i=1 maxf0; p i maxf0; C i C j gg C j m and C j p j : Under the conditions p j = p and i j ) C i C j this predicate is maxf0; p C i C j g mC j jp and C j p: Introducing ....

W. Horn, Some simple scheduling problems, Naval Research Logistics Quarterly 21 (1974) 177-185.

....in the EDD algorithm. The optimal assignment algorithm has a complexity of O(nkmax logk max nlogn) B. Aperiodic EDF Algorithm Earliest Deadline First (EDF) is a dynamic scheduling algorithm that at any instant executes the task with the earliest absolute deadline among all the ready tasks [23]. Preemption is allowed; all tasks consist of a single job, and have di#erent arrival times, computation times and deadlines. The guarantee test is k=1 # k (t) #j = 1, n. The complexity of the algorithm is O(n) per task, since inserting the newly arrived task into an ordered queue (the ....

W. Horn, Some simple scheduling algorithms, Naval Research Logistics Quarterly, 21, 1974, pp. 177-185.

....the SRPT only when the processing of the current job is completed or when a new job is released (at n points during the schedule) 1 1.2 Earliest Due Date (EDD) Rule EDD Rule: Sort the job in non decreasing order of their due dates. Process the jobs according to this order. Theorem 1. 4 ([4]) EDD rule is optimal for 1jr j ; pmtnjLmax . Proof: The proof is similar to the proof of Theorem 1.1. Consider a schedule, S 1 , which is not according to EDD. Assume that on time t, there exist two jobs j; k, such that r j ; r k t, d j d k , and the job k is processed. Consider all the time ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185,1974.

....on a task set is defined to be the sum of the values obtained on eachtaskin the set. 3 Constraints on the Execution Environment Given a set of n tasks, determining whether the mandatory portions of all these tasks can be scheduled to completion on a single processor takes Theta(n log n) time [3]. In a distributed on line environment, however, in which the individual tasks may be generated by different processes executing on physically separated nodes, it is often the case that neither the number of tasks, nor their individual characteristics, are a priori known. If no restrictions are ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....integer valued parameters, there is a valid discrete schedule for T on one processor iff there is a valid continuous schedule for T on one processor. Proof: From Lemma 2.1, wemay assume without loss of generality that T is a complete task system. Using a technique similar to that given by Horn [Hor74] we will construct, from T , a flow network G and an integer K suchthatvalid schedules of T on one processor correspond to maximum flows of K in G. Furthermore, it will be the case that all capacities in G will be integers. Since there are maximal network flow algorithms that never introduce ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....m identical parallel machines. By choosing the time line intervals to be appropriately small, suchaschedule can be constructed with both its minimum and maximum lag arbitrarily close to zero. With the IBC, however, the problem becomes much more challenging. By using a network flow argument, Horn [7]proved that Condition (1) is a necessary and sufficient condition for feasibilityeven in the presence of the IBC. The problem of efficiently constructing such schedules at runtime (Horn s network flow argument translates into an exponential time algorithm) was first posed by Liu in 1969 [10] Liu ....

W. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....from time 0 until time L. We also prove that the lag of each tasks is zero at time L in this sub schedule, so it can be repeated infinitely to acquire a complete schedule. We then show that G has a flow of size mL in which edges may carry a fractional flow. Together with the results of [11], this implies that G also has a flow of size mL in which all edges carry a flow of 0 or 1, thereby proving that instance Phi has a pfair schedule. Before defining G, we first give some preliminary definitions and prove some properties about tasks. The lemma below follows easily from Definition ....

....3.2 and Definition 3.2, this implies that in(f; h4; r; ii) 1. If A 6= 0, then by Definitions 3.1 and 3.2, MU (r) P x2M w x = 1 Gamma FU (r) so in(f; h4; r; ii) 1. Lemma 3.25 implies the existence of a flow of size mL in G. This implies the existence of a unitary flow of size mL through G [11]. Together with Lemma 3.21, this yields the following theorem. Theorem 3.1: A pfair schedule exists for any instance Phi that satisfies Assumptions 1 and 2. 4 The Supertasks Approach In this section, we describe an idea for scheduling mixed task sets by breaking the problem into two easier ....

W. A. Horn. Some Simple Scheduling Algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....on the methods used to generate preemptive real time schedules for multiprocessors and on fault tolerant scheduling techniques for periodic real time tasks. Earliest deadline first (EDF) is well known to be optimal scheduling algorithm for tasks with deadlines on a uniprocessor system. Horn [Hor74] showed that the problem of scheduling tasks with release times and deadlines on identical processors can be reduced to the network flow problem which can be solved in O(n 3 ) time. Gonzalez and Sahni [GS78] have an O(n m log n) algorithm when all release times and deadlines are equal but the ....

....Sahni [GS78] have an O(n m log n) algorithm when all release times and deadlines are equal but the processors have different speeds while Sahni and Cho [SC80] give a O(n log n mn) algorithm for the same problem while also allowing variable deadlines. Martel [Mar82] extends the solution given in [Hor74] to the case where the processors are not identical. It may be noted that the case of fixed release times and variable deadlines is symmetrical to the case of variable release times and a fixed common deadline. Any algorithm for the former can be used to generate schedules for the latter and vice ....

W. A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....in [LW73] where it is shown that for feasible scheduling of a set of periodic tasks, the condition U 1 is both necessary and sufficient. Optimal preemptive schedules minimizing maximum lateness and total delay for the single machine case and for a restricted multimachine case have been shown in [Ho74]. However, optimal preemptive schedules for the general case of n processors is still open (cf. LW73] DM89] A new notion of temporal fairness based on proportionate progress of real time tasks called proportionate fairness or P fairness has been described in [BCPV93] BGP95] The resources ....

W.A. Horn, Some Simple Scheduling Algorithms, Naval Research Logistics Quarterly, Vol. 21, No.1 (1974) 177 - 185.

....of all instances of all the tasks can be guaranteed. If such a guarantee can be given, the task set is accepted, and scheduling decisions are made at run time. Frame based scheduling Frame based scheduling relies on the aperiodic scheduling problem, which can be solved exactly in polynomial time [9]. Each task in the frame is aperiodic. The task set and timeline are transformed into an instance of a maximum network flow problem, the solution to which provides a schedule for the frame if and only if one exists. Because one can specify the earliest and latest times each task can be scheduled ....

....from the set of all schedules for T 0 , if one exists. Our heuristic (labelled H1) is quite simple: select any schedule for T 0 . To find a candidate schedule for S 0 from the modified task set, and to verify that S 1 ; Sn can be derived from it, we use the maximum network flow method [9]. To implement this method of fault tolerant scheduling, one would generate all n 1 schedules off line, and switch between them as needed dynamically. 4.3 Evaluation of the heuristic Our simulator produces random task sets with utilization ranging from 0 to 100 in the fault free case. The ....

W. A. Horn. Some Simple Scheduling Algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

....to time i 1. Constraint 2: No task may be allocated more than one copy of the resource in any single slot. The problem of constructing periodic schedules for such task systems was discussed by Liu in 1969 [5] and is called the (multiple resource) periodic scheduling problem. It has been shown [2, 4] that an instance of the periodic scheduling problem is feasible (i.e. has a periodic schedule) if and only if ( P x2 Gamma x:e=x:p) m. Given a feasible instance of the periodic scheduling problem, a schedule generation algorithm performs a (possibly empty) pre processing phase followed by ....

W. A. Horn. Some simple scheduling algorithms. Naval Research Logistics Quarterly, 21:177--185, 1974.

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W.A. Horn [1974] Some Simple Scheduling Algorithms, Naval Research Logistics Quarterly 21, pp. 177-185.

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