Louis E. Rosier

Preemptively scheduling hard-real-time sporadic tasks on one processor

Consideration is given to the preemptive scheduling of hard-real-time sporadic task systems on one processor. The authors first give necessary and sufficient conditions for a sporadic task system to be feasible (i.e., schedulable). The conditions cannot, in general, be tested efficiently (unless P=NP). They do, however, lead to a feasibility test that runs in efficient pseudo-polynomial time for a very large percentage of sporadic task systems.<<ETX>>

Algorithms and complexity concerning the preemptive scheduling of periodic, real-time tasks on one processor

We investigate the preemptive scheduling of periodic, real-time task systems on one processor. First, we show that when all parameters to the system are integers, we may assume without loss of generality that all preemptions occur at integer time values. We then assume, for the remainder of the paper, that all parameters are indeed integers. We then give, as our main lemma, both necessary and sufficient conditions for a task system to be feasible on one processor. Although these conditions cannot, in general, be tested efficiently (unless P=NP), they do allow us to give efficient algorithms for deciding feasibility on one processor for certain types of periodic task systems. For example, we give a pseudo-polynomial-time algorithm for synchronous systems whose densities are bounded by a fixed constant less than 1. This algorithm represents an exponential improvement over the previous best algorithm. We also give a polynomial-time algorithm for systems having a fixed number of distinct types of tasks. Furthermore, we are able to use our main lemma to show that the feasibility problem for task systems on one processor is co-NP-complete in the strong sence. In order to show this last result, we first show the Simultaneous Congruences Problem to be NP-complete in the strong sense. Both of these last two results answer questions that have been open for ten years. We conclude by showing that for incomplete task systems, that is, task systems in which the start times are not specified, the feasibility problem is ∑2p-complete.

On the competitiveness of on-line real-time task scheduling

With respect to on-line scheduling algorithms that must direct the service of sporadic task requests we quantify the benefit of clairvoyancy, i.e., the power of possessing knowledge of various task parameters of future events. Specifically, we consider the problem of preemptively sheduling sporadic task requests in both uni- and multi-processor environments. If a task request is successfuly scheduled to completion, a value equal to the tasks execution time is obtained; otherwise no value is obtained. We prove that no on-line scheduling algorithm can guarantee a cumulative value greater than 1/4th the value obtainable by a clairvoyant scheduler; i.e., we prove a 1/4th upper bound on the competitive factor of on-line real-time schedulers. We present an online uniprocessor scheduling algorithm TD1 that actually has a competitive factor of 1/4; this bound is thus shown to be tight. We further consider the effect of restricting the amount of overloading permitted (the loading factor), and quantify the relationship between the loading factor and the upper bound on the competitive factor. Other results of a similar nature deal with the effect of value densities (measuring the importance of type of a task). Generalizations to dual-processor on-line scheduling are also considered. For the dual-processor case, we prove an upper bound of 1/2 on the competitive factor. This bound is shown to be tight in the special case when all the tasks have the same density and zero laxity.

On-line scheduling in the presence of overload

The preemptive scheduling of sporadic tasks on a uniprocessor is considered. A task may arrive at any time, and is characterized by a value that reflects its importance, an execution time that is the amount of processor time needed to completely execute the task, and a deadline by which the task is to complete execution. The goal is to maximize the sum of the values of the completed tasks. An online scheduling algorithm that achieves optimal performance when the system is underloaded and provides a nontrivial performance guarantee when the system is overloaded is designed. The algorithm is implemented using simple data structures to run at a cost of O(log n) time per task, where n bounds the number of tasks in the system at any instant. Upper bounds on the best performance guarantee obtainable by an online algorithm in a variety of settings are derived.<<ETX>>

Feasibility problems for recurring tasks on one processor

Abstract We give a comprehensive summary of our recent research on the feasibility problems for various types of hard-real-time preemptive task systems on one processor. We include results on periodic, sporadic, and hybrid task systems. While many of the results herein have appeared elsewhere, this is the first paper presenting a holistic view of the entire problem.

Pinwheel scheduling with two distinct numbers

“The Pinwheel” is a real-time scheduling problem based on a problem in scheduling satellite ground stations but which also addresses scheduling preventive maintenance. Given a multiset of positive integers A = {a1, a2, ..., a n }, a schedule S for A is an infinite sequence over {1, 2, ..., n} such that any subsequence of length a i (1 ≤ i ≤ n) contains at least one i. Schedules can always be made cyclic; that is, a segment can be found that can be repeated indefinitely to form an infinite schedule. Interesting questions include determining whether schedules exist, determining the minimum cyclic schedule length, and creating an online scheduler. The “density” of an instance is defined as \(d = \sum\nolimits_{i = 1}^n {1/a} _i\). It has been shown that any instance with d > 1.0 cannot be scheduled. In the present paper we limit ourselves to instances in which A contains elements having only two distinct values. We prove that all such instances with d ≤ 1.0 can be scheduled, using a scheduling strategy based on balancing. The schedule so created is not always of minimum length, however. We use a related but more complicated method to create a minimum-length cyclic schedule, and prove its correctness. The former is computationally easier to obtain but not necessarily minimal. The latter, although still obtainable in polynomial time, requires significantly more computation. In addition, we show how to use either method to produce a fast online scheduler. Thus, we have solved completely the three major problems for this class of instances.

A taxonomy of fairness and temporal logic problems for Petri nets

In this paper, we define a temporal logic for reasoning about Petri nets. We show the model checking problem for this logic to be PTIME equivalent to the Petri net reachability problem. Using this logic and two refinements, we show the fair nontermination problem to be PTIME equivalent to reachability for several definitions of fairness. For other versions of fairness, this problem is shown to be either PTIME equivalent to the boundedness problem or highly undecidable. In all, 24 versions of fairness are examined.

Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states

Abstract In this paper, we analyse the complexity of the reachability, containment, and equivalence problems for two classes of vector addition systems with states (VASSs): finite VASSs and 2-dimensional VASSs. Both of these classes are known to have effectively computable semilinear reachibility sets (SLSs). By giving upper bounds on the sizes of the SLS representations, we achieve upper bounds on each of the aforementioned problems. In the case of the finite VASSs, the SLS representation is simply a listing of the reachability set; therefore, we derive a bound on the norm of any reachable vector based on the dimension, number of states, and amount of increment caused by any move in the VASS. The bound we derive shows an improvement of two levels in the primitive recursive hierarchy over results previously obtained by McAloon (1984), thus answering a question posed by Clote (1986). We then show this bound to be optimal. We feel that the techniques we use in deriving our upper bounds represent an original approach to the problem, and since they yield improvements over previous results, we feel these techniques may have applications to other problems. In the case of 2-dimensional VASSs, we analyse an algorithm given by Hopcroft and Pansiot (1979) that generates an SLS representation of the reachability set. Specifically, we show that the algorithm operates in 2 2 cln nondeterministic time, where l is the length of the binary representation of the largest integer in the VASS, n is the number of transitions, and c is some fixed constant. We also give examples for which this algorithm will take 2 2 dln nondeterministic time for some positive constant d . Finally, we give a method of determinizing the algorithm in such a way that it requires no more than 2 2 cln deterministic time. From this upper bound and special properties of the generated SLSs, we derive upper bounds of Dtime (2 2 cln ) for the three problems mentioned above.

Problems concerning fairness and temporal logic for conflict-free Petri nets

Abstract In this paper, we examine the complexity of the fair nontermination problem for conflict-free Petri nets under several definitions of fairness. For each definition of fairness, we are able to show the problem to be complete for either NP, PTIME, or NLOGSPACE. We then address the question of whether these results extend to the more general model checking problem with respect to the temporal logic for Petri nets introduced by Suzuki. Since many of the model checking problems concerning finite state systems can be reduced to a version of the fair nontermination problem, it would seem plausible that the model checking problem for conflict-free Petri nets would be decidable. However, it turns out that unless the logic is severely restricted, model checking is undecidable for conflict-free Petri nets. In particular, the problem is undecidable even when formulas are of the form Gƒ (“invariantly ƒ”) where ƒ contains no temporal logic operators. On the other hand, we show that model checking for conflict-free Petri nets is NP-complete for L(F, X)—the logic restricted to the operators F (eventually), X (next time), ∧, and ∨, with negations allowed only on the predicates.

Completeness results for conflict-free vector replacement systems

Abstract In this paper, we give completeness results for the reachability, containment, and equivalence problems for conflict-free vector replacement systems (VRSs). We first give an NP algorithm for deciding reachability, thus giving the first primitive recursive algorithm for this problem. Since Jones, Landweber, and Lien have shown this problem to be NP-hard, it follows that the problem is NP-complete. Next, we show as our main result that the containment and equivalence problems are ∏ 2 P -complete, where ∏ 2 P is the set of all languages whose complements are in the second level of the polynomial-time hierarchy. In showing the upper bound, we first show that the reachability set has a semilinear set (SLS) representation that is exponential in the size of the problem description, but which has a high degree of symmetry. We are then able to utilize in part a strategy introduced by Huynh (concerning SLSs) to complete our upper bound proof.