D. Gillies and J. Liu. Scheduling tasks with and/or precedence constraints. In Proc. 2nd IEEE Conf. on Parallel and Distribute Processing, Dec. 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....over shape 1 Benefit over shape 2 (b) FIG. 8. Performance impact of tunability in non malleable model (a) and in malleable model (b) as job arrival interval and laxity are varied. earliest deadline first (EDF) exist for scheduling real time tasks in an uniprocessor environment. Gillies [13] has studied the scheduling of tasks in AND OR graphs. However, the validity of most of these results is restricted to either uniprocessor systems or a sequential task model. Our interest is in managing parallel real time tasks in a parallel system: in this scenario, well known results such as EDF ....

D. Gillies and J. Liu. Scheduling tasks with and/or precedence constraints. In Proc. 2nd IEEE Conf. on Parallel and Distribute Processing, Dec. 1990.

....a fixed set of resources. Predictability for applications has been well studied in the real time systems literature. Several wellknown scheduling schemes, such as rate monotonic [12, 16] and earliest deadline first (EDF) exist for scheduling real time tasks in an uniprocessor environment. Gillies [8] has studied the scheduling of tasks in AND OR graphs, a problem very similar to our problem of scheduling task chains in tunable applications. However, the validity of most of these results is restricted to either uniprocessor systems or a sequential task model. Our interest is in managing ....

D. Gillies and J. Liu. Scheduling Tasks with AND/OR Precedence Constraints. Proc. of 2nd IEEE Conference on Parallel and Distributed Processing, Dec. 1990.

....fixed set of resources. Predictability for applications has been well studied in the real time systems literature. Several well known scheduling schemes, such as rate monotonic [12, 17] and earliest deadline first (EDF) exist for scheduling real time tasks in an uniprocessor environment. Gillies [8] has studied the scheduling of tasks in AND OR graphs. However, the validity of most of these results is restricted to either uniprocessor systems or a sequential task model. Our interest is in managing parallel real time tasks in a parallel system: in this scenario, well known results such as EDF ....

D. Gillies and J. Liu. Scheduling tasks with and/or precedence constraints. In Proc. of 2nd IEEE Conference on Parallel and Distribute Processing, Dec. 1990.

....rate, and the freshness of each consumed donut is bounded by the producer s cycle length. However, in our problem, because we allow many donuts in the shelf simultaneously, the freshness can not be simply bounded by the production rate. Our idea of constructing an AND XOR graph is partially from [Orl82, Bur86, GL90]. Assuming that every task has the same cycle time, i.e. T 1 = T 2 = Tn = K, Orlin [Orl82] models the resource minimization problem as a minimum cardinality chain covers problem in a periodic partial ordered set (periodic POSET) Burkard [Bur86] analyzed the optimal phasing to optimize ....

....the optimal phasing to optimize maximal waiting time, minimal safety interval, and average waiting time simultaneously with two periodic tasks with different cycle time. Unfortunately, no uniform results exist for when the number of tasks with different cycle length are greater than two. In [GL90], the AND OR precedence constraint scheduling problem is discussed with the goal to minimize the completion time of all the jobs. In this paper, we use the AND XOR graph instead of the AND OR graph in order to get a feasible MCF setting. 8 Remarks In this paper, we define and study a real time ....

Donald W. Gillies and Jane W.-S. Liu. Scheduling tasks with and/or precedence constraints. In 2nd IEEE Symposium on Parallel and Distributed Processing, pages 394--401, Dallas TX, December 1990.

....a fixed set of resources. Predictability for applications has been well studied in the real time systems literature. Several wellknown scheduling schemes, such as rate monotonic [12, 16] and earliest deadline first (EDF) exist for scheduling real time tasks in an uniprocessor environment. Gillies [8] has studied the scheduling of tasks in AND OR graphs, a problem very similar to our problem of scheduling task chains in tunable applications. However, the validity of most of these results is restricted to either uniprocessor systems or a sequential task model. Our interest is in managing ....

D. Gillies and J. Liu. Scheduling Tasks with AND/OR Precedence Constraints. Proc. of 2nd IEEE Conference on Parallel and Distributed Processing, Dec. 1990.

....the use of interactivity and simulation in the design flow. We conclude with Section 6. 2 Generalized Task Graph Model We model an embedded system using a directed graph called a generalized task graph. A more detailed discussion of this model and its relation to earlier models proposed in [3, 10, 11, 18, 19, 21, 23, 26, 30, 31] are given in [6] Each vertex corresponds to a task, and each arc corresponds to an asynchronous, unidirectional communication channel between its producer and consumer. Each task is a functional unit within the system, obtained after the functional decomposition of the system. Those tasks that ....

Gillies, D. W., and Liu, J. W.-S. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput. 24, 4 (Aug. 1995), 797--810.

....models cannot model this behavior, so they are extended to contain OR tasks. The extended dataflow model of Puchol and Mok [106] the Token Flow Model of Buck and Lee [11] the Task Flow Graph of Peng and Shin [105] the dataflow graph of Lent and Kurmann [89] the AND OR Model of Gillies and Liu [44], the timed AND OR automaton of Gunawardena [51] and the task graph of Silberman [113] are all AND OR models in that they have both AND and OR tasks. Some of these AND OR models such as the AND OR Model of Gillies and Liu are acyclic. Gillies and Liu also introduced the skipped and unskipped ....

D. W. Gillies and J. W.-S. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, August 1995.

....rather than an event graph. This section presents our task graph model. Notice that in this presentation, we expose the channel delay for each communicating task. Our task graph model is called the generalized task graph (GTG) model. It builds upon many previous models in the literature such as [2, 8, 9, 10, 12, 13, 14, 18, 20, 25]. Our model is an abstraction and combination of these models for the purposes of timing analysis. For example, we borrow both AND and OR causality types from [10] both acyclic and cyclic task dependencies from [2, 13, 14, 18] modeling of both control and data from [9, 20, 25] and skipped and ....

....and combination of these models for the purposes of timing analysis. For example, we borrow both AND and OR causality types from [10] both acyclic and cyclic task dependencies from [2, 13, 14, 18] modeling of both control and data from [9, 20, 25] and skipped and unskipped behaviors from [8]. We also kept our model as similar to these models as possible so that we can use the techniques developed for them as well as we can refer the readers to them for the concepts we borrowed from them. We now define the generalized task graph model. A GTGG = V; E) is a directed graph in which each ....

D. W. Gillies and J. W.-S. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, August 1995.

....provided for interprocessor communication. This is because the process implementing CPU related tasks is structured by the compiler optimizer such that blocking of the process only occurs, if this is called for by 4 The only exceptions known to us are the AND OR models used by Gillies and Liu [13] and the approach of Chou and Abraham [9] However, both approaches are too limited to handle our problems. AND OR models do not allow disjunctive output logic. Thus, they do not have to consider runtime information. The approach of Chou and Abraham is limited to a subset of series parallel ....

D.W. Gillies, J.W.S. Liu. Scheduling Tasks with AND/OR Precedence Constraints. Proc. 2nd IEEE Symposium on Parallel and Distributed Processing, Dallas, Texas, Dec. 1990.

....To carry out this approximation, negative task lengths may be input to the scheduling algorithms. This may be of use to developers of real time language timing tools. 1.2. Summary of Results Some of the work in this thesis has already appeared. In particular, Gillies90] Gillies91a] and [Gillies93b] contain some of the results in this thesis. We first show that our AND OR scheduling model subsumes some other models. Then it is shown that if the precedence constraints are arbitrary, the skipped problem subsumes the unskipped problem. We then describe why traditional AND only scheduling ....

Gillies, Donald W. and Jane W.-S. Liu. Scheduling Tasks With AND/OR Precedence Constraints. To appear, SIAM Journal on Computing (1993). 103

....on a fixed number of processors. To carry out this approximation, negative task lengths may be input to the scheduling algorithms. This may be of use to developers of real time language timing tools. 1.2. Summary of Results Some of the work in this thesis has already appeared. In particular, [Gillies90] [Gillies91a] and [Gillies93b] contain some of the results in this thesis. We first show that our AND OR scheduling model subsumes some other models. Then it is shown that if the precedence constraints are arbitrary, the skipped problem subsumes the unskipped problem. We then describe why ....

Gillies, Donald W. and Jane W.S. Liu. Scheduling tasks with AND/OR precedence constraints. Second Annual IEEE Symposium on Parallel Distributed Processing (December 1990), pp. 379-387.

....to be consistent with the notion of edges in a directed blocking graph. Often the meaning of the directed edge is reversed in scheduling literature. and C blocks A in a third. For this reason, we have made no a priori assumptions about the structure of the precedence relations. The work of Refs. [23,24] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many settings that are polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies and J. Liu, "Scheduling tasks with AND/OR precedence constraints," SIAM J. Comput. 24 (1995) 797--810.

....The precedence graph in general is a directed acyclic graph (DAG) An example of such a task is shown in Figure 9.1. If a subtask has two predecessors, such as subtask T 1;4 in Figure 9. 1, then the subtask can be either an AND subtask or an OR subtask, a case which was studied by Gillies and Liu [55]. If T 1;4 is an AND subtask, then an instance of T 1;4 cannot be released until both corresponding instances of T 1;2 and T 1;3 complete. Otherwise, if T 1;4 is an OR subtask, an instance of T 1;4 can be released when one of the corresponding instances of T 1;2 and T 1;3 completes. When a ....

D. Gillies and J. Liu. Scheduling tasks with and/or precedence constraints. SIAM Journal on Computing, 24(4):797--810, August 1995.

....the use of interactivity and simulation in the design flow. We conclude with Section 6. 2 Generalized Task Graph Model We model an embedded system using a directed graph called a generalized task graph. A more detailed discussion of this model and its relation to earlier models proposed in [3, 10, 11, 18, 19, 21, 23, 26, 30, 31] are given in [6] Each vertex corresponds to a task, and each arc corresponds to an asynchronous, unidirectional communication channel between its producer and consumer. Each task is a functional unit within the system, obtained after the functional decomposition of the system. Those tasks that ....

Gillies, D. W., and Liu, J. W.-S. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput. 24, 4 (Aug. 1995), 797--810.

....of the ability to treat each constraint independently depends on the ability to combine multiple constraints that affect the same operation. Although most work on real time systems suggests that such constraints are always combined by requiring all constraints to be met, it is sometimes necessary [GL91] to combine constraints by requiring that any one or more of these constraints be met. This concept of an oring constraint is quite new in the real time community. Suppose an event A 1 can be execute either before A 2 or after A 3 as shown in Figure 2. This can be mathematically expressed as A 1 ....

D. W. Gillibs and J. W. Liu. Scheduling tasks with and/or precedence constraints. Technical Report UIUC-ENG-1766, University of Illinois, October 1991.

....Section 3, we discuss related work. In Sections 4 and 5, we describe the LASM, SASM, and AIM models. In Section 6, we describe the BF model. In Section 7, we summarize this paper. 3. Related Work Several previous models attempt to capture application characteristics to enable resource management [1, 6, 7, 12, 9]. These models view an application as a set of subtasks, running on a set of homogeneous processors. The homogeneity assumption leads to simple models that unfortunately cannot be used in real distributed systems comprising heterogeneous resources. Our application model, on the other hand, is ....

D. Gilles and J. Liu, Scheduling Tasks with AND/OR Precedence Constraints, University of Illinois, UrbanaChampaign Technical Report UIUCDCS-R-90-1627, 20 March 1991.

....HW SW Interface synthesis 4 3 Figure 2: Hardware software codesign methodology using RADHA and RATAN. 3 Generalized Task Graph Model We model an embedded system using a hierarchical directed graph called a generalized task graph (the task graph for short) This model builds upon the models in [3, 8, 10, 12, 13, 15, 17, 18, 20, 21], as discussed in [6] Each node corresponds to a task, and each arc corresponds to an asynchronous, unidirectional communication channel between its producer and consumer. This model actually represents the system s data control flow diagram where each node is either a data or a control ....

Gillies, D. W., and Liu, J. W.-S. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput. 24, 4 (Aug. 1995), 797--810.

....for all tasks. It is worth noting that with classical and precedence constraints, this problem of minimizing the number of scheduled tasks can be solved exactly, in polynomially time by computing a depth order. A model for scheduling with and or precedence constraints has been studied earlier [15, 16], but with one key difference. The precedence constraints for an instance can be represented as a directed graph, with each node additionally tagged as either an and node or an or node; in this previous work, they assume that there is no cycle in this precedence graph, as that would make the ....

....the existence of a valid solution, and in fact cycles will often exist as it may be the case that part A blocks B in one direction, B blocks C in another, and C blocks A in a third. For this reason, we make no a priori assumptions about the structure of the precedence relations. Gillies and Lin [15, 16] prove the NP hardness of many variants of the problem, however they do not consider the approximability of the hard problems. Notation and Definitions. The input contains a set of tasks, T . Each task, t i 2 T , is labeled as either an and task or an or task, and has an associated set of tasks, P ....

[Article contains additional citation context not shown here]

D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, 1995.

....As a result, at higher levels of abstraction, we have a task graph rather than an event graph. This section presents our task graph model. Our task graph model is called the generalized task graph (GTG) model. It builds upon many previous models in the literature such as [Buck and Lee 1993; Gillies and Liu 1995; Gomaa 1993; Gunawardena 1993; Jeffay 1993; Karp and Miller 1966; Lee and Messerschmitt 1987; Mathur et al. 1998; Puchol and Mok 1994; Yen 1996] Our model is an abstraction and combination of these models for the purposes of timing analysis. For example, we borrow both AND and OR causality types ....

.... types from [Gunawardena 1993] both acyclic and cyclic task dependencies from [Buck and Lee 1993; Karp and Miller 1966; Lee and Messerschmitt 1987; Mathur et al. 1998] modeling of both control and data from [Gomaa 1993; Puchol and Mok 1994; Yen 1996] and skipped and unskipped behaviors from [Gillies and Liu 1995]. We also kept our model as similar to these models as possible so that we can use the techniques developed for them as well as we can refer the readers to them for the concepts we borrowed from them. Definition. A GTG G = V; E) is a directed graph in which each node in V is a task and each arc ....

Gillies, D. W. and Liu, J. W.-S. 1995. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput. 24, 4 (Aug.), 797--810.

....for all tasks. It is worth noting that with classical and precedence constraints, this problem of minimizing the number of scheduled tasks can be solved exactly, in polynomially time by computing a depth order. A model for scheduling with and or precedence constraints has been studied earlier [19, 20], but with one key difference. The precedence constraints for an instance can be represented as a directed graph 6 with each node additionally tagged as either an and node or an or node; in this previous work, they assume that there is no cycle in this precedence graph, as that would make the ....

....for the existence of a valid solution, and in fact cycles will often exist as it may be the case that part A blocks B in one direction, B blocks C in another, and C blocks A in a third. For this reason, we make no apriori assumptions about the structure of the precedence relations. Gillies and Lin [19, 20] prove the NPhardness of many variants of the problem, however they do not consider the approximability of the hard problems. It is important to note that in context of assembly sequencing, the precedence constraints for this scheduling problem could be more naturally modeled as DNF scheduling, ....

[Article contains additional citation context not shown here]

D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, 1995.

....notion of edges in a directed blocking graph. Often the meaning of the directed edge is reversed in scheduling literature. CHAPTER 5. AND OR SCHEDULING 39 precedence constraints. However, a series of papers by Gillies et al., study several variants of scheduling with and or precedence constraints [24, 25]. Our model for this problem was chosen to be similar to [24, 25] however with one key difference. As mentioned in Section 5.2, the precedence constraints for an instance can be represented as a directed graph. In this previous work, only the case of partial order precedence constraints is ....

....the directed edge is reversed in scheduling literature. CHAPTER 5. AND OR SCHEDULING 39 precedence constraints. However, a series of papers by Gillies et al., study several variants of scheduling with and or precedence constraints [24, 25] Our model for this problem was chosen to be similar to [24, 25], however with one key difference. As mentioned in Section 5.2, the precedence constraints for an instance can be represented as a directed graph. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence ....

[Article contains additional citation context not shown here]

D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, 1995.

....represented an AND model (in which every task is an AND task) e.g. see [16, 18, 19, 22, 31] Among these, only [18, 19, 22] allow cyclic dependencies between tasks. Since AND models do not model real life embedded systems properly [26] OR behavior is next introduced into the task graphs in [3, 8, 10, 12, 21, 25, 26, 30]. The skipped and unskipped concepts, and the joint and disjoint concepts are introduced in [10] and [30] respectively. Petri nets [23] can model both AND and OR behaviors but have some difficulty in handling all the different OR behaviors [30] Our earlier work [8] is the first generalization ....

....22] allow cyclic dependencies between tasks. Since AND models do not model real life embedded systems properly [26] OR behavior is next introduced into the task graphs in [3, 8, 10, 12, 21, 25, 26, 30] The skipped and unskipped concepts, and the joint and disjoint concepts are introduced in [10] and [30] respectively. Petri nets [23] can model both AND and OR behaviors but have some difficulty in handling all the different OR behaviors [30] Our earlier work [8] is the first generalization that combines all of the above concepts with AND and OR. It also discusses other important issues ....

Gillies, D. W., and Liu, J. W.-S. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput. 24, 4 (Aug. 1995), 797--810.

....of the ability to treat each constraint independently depends on the ability to combine multiple constraints that affect the same operation. Although most work on real time systems suggests that such constraints are always combined by requiring all constraints to be met, it is sometimes necessary [8] to combine constraints by requiring that any one or more of these constraints be met. This concept of an ORing constraint is quite new in the real time community. Suppose an event A 1 can be executed either before A 2 or after A 3 as shown in Figure 2. This can be mathematically expressed as A 1 ....

D. W. Gillibs and J. W. Liu. Scheduling tasks with and/or precedence constraints. Technical Report UIUC-ENG-1766, University of Illinois, October 1991.

....of the ability to treat each constraint independently depends on the ability to combine multiple constraints that affect the same operation. Although most work on real time systems suggests that such constraints are always combined by requiring all constraints to be met, it is sometimes necessary [GL91] to combine constraints by requiring that any one or more of these constraints be met. This concept of an oring constraint is quite new in the real time community. Suppose an event A 1 can be execute either before A 2 or after A 3 as shown in Figure 2. This can be mathematically expressed as A 1 ....

D. W. Gillibs and J. W. Liu. Scheduling tasks with and/or precedence constraints. Technical Report UIUC-ENG-1766, University of Illinois, October 1991.

....for this exact approximation problem. Minimizing the number of ones in satisfying a 3CNF formula is known to be n 0.5 # hard to approximate [29] and related minimization problems are studied in [34] Previous Work. A model for scheduling with and or precedence constraints was introduced in [14,15], however with one key di#erence. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence constraints, the existence of a cycle in the precedence constraints makes the scheduling problem infeasible, and ....

....drawn from assembly sequencing, as it may be the case that part A blocks part B in one direction, part B blocks part C in another, and part C blocks part A in a third direction. For this reason, we make no apriori assumptions about the structure of the precedence constraints. The work of [14,15] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many setting which were polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, 1995.

....this exact approximation problem. Minimizing the number of ones in satisfying a 3CNF formula is known to be n 0:5 Gammaffl hard to approximate [29] and related minimization problems are studied in [34] Previous Work. A model for scheduling with and or precedence constraints was introduced in [14,15], however with one key difference. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence constraints, the existence of a cycle in the precedence constraints makes the scheduling problem infeasible, and ....

....drawn from assembly sequencing, as it may be the case that part A blocks part B in one direction, part B blocks part C in another, and part C blocks part A in a third direction. For this reason, we make no apriori assumptions about the structure of the precedence constraints. The work of [14,15] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many setting which were polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies and J. Liu. Scheduling tasks with AND/OR precedence constraints. SIAM J. Comput., 24(4):797--810, 1995.

....words we must have U 1 fi Gamma O(1=fi 3 ) The non linear programming formulation that we used to find the bounds has not been used in previous papers on worst case scheduling analysis. The only similar work has been our previous work on linear programming models for aperiodic scheduling [5]. In previous work on static priority scheduling, all steps in the proof used ad hoc arguments. With this new approach, DRAFT Burst Processing 14 25 20 15 10 5 1 25 20 15 10 5 1 0 0.2 0.4 0.6 0.8 1 Burst Factor, b Tasks, n Fig. 2. Worst case utilization of bursty ....

D. W. Gillies, J. W.-S. Liu, Scheduling Tasks with AND/OR Precedence Constraints, SIAM J. on Computing, August 1995.

....number of processors. To carry out this approximation, negative task lengths may be input to the scheduling algorithms. This may be of use to developers of real time language timing tools. 1.2. Summary of Results Some of the work in this thesis has already appeared. In particular, Gillies90] [Gillies91a] and [Gillies93b] contain some of the results in this thesis. We first show that our AND OR scheduling model subsumes some other models. Then it is shown that if the precedence constraints are arbitrary, the skipped problem subsumes the unskipped problem. We then describe why traditional AND only ....

Gillies, D. W. and J. W. S. Liu. Scheduling Tasks with AND/OR Precedence Constraints. Report No. UIUCDCS-R-90-1627 (UIUC-ENG-1766), Department of Computer Science, University of Illinois at Urbana-Champaign (1991).

....and deadlines. Surprisingly, even this simplified AND OR scheduling problem is NP complete Theorem 3.4. The problem of AND OR unskipped scheduling to meet deadlines, where the task system is a simple in forest with identical processing times, is NPcomplete. Proof. The proof may be found in [11]. Theorems 3.1 3.3 allow us to arrive at the following conclusion. Every AND OR task graph with k OR tasks, each of which has l direct predecessors, corresponds to a set of l k different AND only task graphs. A feasible schedule of such a task system corresponds to an implicit selection of one ....

....first, least slack first, and some generalizations of the algorithms in [4] neglect to compare the deadlines among different in trees. In the worst case these algorithms may meet only p n deadlines when it is possible to meet n out of n 1 deadlines. For more information the reader is referred to [11] [12] 3.2. Scheduling to Minimize Completion Time. We now consider the problem of scheduling AND OR unskipped tasks with arbitrary processing times on m processors to meet a common deadline. This problem is equivalent to that of scheduling to minimize the overall completion time. Ullman has ....

[Article contains additional citation context not shown here]

D. W Gillies, and J. W.-S. Liu, Scheduling Tasks with AND/OR Precedence Constraints, Rep. No. UIUCDCS-R-90-1627 (UIUC-ENG-1766), Department of Computer Science, Univ. of Illinois, Urbana, 1991.

No context found.

D. Gillies and J. Liu, "Scheduling tasks with and/or precedence constraints," SIAM Journal on Computing, vol. 24, no. 4, pp. 797--810, Aug. 1995.

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