Pfair scheduling of generalized pinwheel task systems

[[abstract]]The scheduling of generalized pinwheel task systems is considered. It is shown that pinwheel scheduling is closely related to the fair scheduling of periodic task systems. This relationship is exploited to obtain new scheduling algorithms for generalized pinwheel task systems. When compared to traditional pinwheel scheduling algorithms, these new algorithms are both more efficient from a run-time complexity point of view, and have a higher density threshold, on a very large subclass of generalized pinwheel task systems.

On Asymptotic Optimality of Dual Scheduling Algorithm In A Generalized Switch

Generalized switch is a model of a queueing system where parallel servers are interdependent and have time-varying service capabilities. This paper considers the dual scheduling algorithm that uses rate control and queue-length based scheduling to allocate resources for a generalized switch. We consider a saturated system in which each user has infinite amount of data to be served. We prove the asymptotic optimality of the dual scheduling algorithm for such a system, which says that the vector of average service rates of the scheduling algorithm maximizes some aggregate concave utility functions. As the fairness objectives can be achieved by appropriately choosing utility functions, the asymptotic optimality establishes the fairness properties of the dual scheduling algorithm. The dual scheduling algorithm motivates a new architecture for scheduling, in which an additional queue is introduced to interface the user data queue and the time-varying server and to modulate the scheduling process, so as to achieve different performance objectives. Further research would include scheduling with Quality of Service guarantees with the dual scheduler, and its application and implementation in various versions of the generalized switch model

Fair Bandwidth Sharing Algorithms Based on Game Theory Frameworks for Wireless Ad-hoc Networks

This paper examines the theoretical aspects of bandwidth sharing in wireless, possibly mobile, ad-hoc networks (MANETs) through a game theoretic framework. It presents some applications to show how such a framework can be invoked to design efficient media access control protocols in a noncooperative, self-organized, topology-blind environment as well as in environments where the competing nodes share some basic information to guide their choice of channel access policies. For this purpose, contentions between concurrent links in a MANET are represented by a conflict graph, and each maximal clique in the graph defines a contention context which in turn imposes a constraint on the share of bandwidth that the links in the clique can obtain. Using this approach the fair bandwidth allocation problem is modeled as a general utility based constrained maximization problem, called the system problem, which is shown to admit a unique solution that can only be obtained when global coordination between all links is possible. By using Lagrange relaxation and duality theory, both a non-cooperative and a cooperative game formulation of the problem are derived. The corresponding mathematical algorithms to solve the two games are also provided where there is no need for global information. Implementation issues of the algorithms are also considered. Finally, simulation results are presented to illustrate the effectiveness of the algorithms

Guaranteeing Fair Service to Persistent Dependent Tasks

We introduce a new scheduling problem that is motivated by applications in the area of access and flow-control of high-speed and wireless networks. An instance of the problem consists of a set of persistent tasks that have to be scheduled repeatedly. Each task has a demand to be scheduled "as often as possible". There is no explicit limit on the number of tasks that can be scheduled concurrently. However, such limits gets imposed implicitly by the fact that some tasks are in conflict and cannot be scheduled simultaneously. The conflicts are presented in the form of a conflict graph. We define parameters that quantify the fairness and regularity of a given schedule. We then proceed to show lower bounds on these parameters, and present fair and efficient scheduling algorithms for the special case where the conflict graph is an interval graph. Some of the results presented here extend to the case of perfect graphs and circular-arc graphs as well

Fast algorithms for two scheduling problems

The thesis deals with problems from two distint areas of scheduling theory. In the first part we consider the preemptive Sum Multicoloring (pSMC) problem. In an instance of pSMC, pairwise conflicting jobs are represented by a conflict graph, and the time demands of jobs are given by integer weights on the nodes. The goal is to schedule the jobs in such a way that the sum of their finish times is minimized. We give the first polynomial algorithm for pSMC on paths and cycles, running in time O(min(n², n log p)), where n is the number of nodes and p is the largest time demand. This answers a question raised by Halldórsson et al. [51] about the hardness of this problem. Our result identifies a gap between binary-tree conflict graphs - where the question is NP-hard - and paths. In the second part of the thesis we consider the problem of scheduling n jobs on m machines of different speeds s.t. the makespan is minimized (Q||C_max). We provide a fast and simple, deterministic monotone 2.8-approximation algorithm for Q||C_max. Monotonicity is relevant in the context of truthful mechanisms: when each machine speed is only known to the machine itself, we need to motivate that machines "declare" their true speeds to the scheduling mechanism. So far the best deterministic truthful mechanism that is polynomial in n and m; was a 5-approximation by Andelman et al. [3]. A randomized 2-approximation method, satisfying a weaker definition of truthfulness, was given by Archer and Tardos [4, 5]. As a core result, we prove the conjecture of Auletta et al. [8], that the greedy list scheduling algorithm Lpt is monotone if machine speeds are all integer powers of two (2-divisible machines). Proving the worst case bound of 2.8 involves studying the approximation ratio of Lpt on 2-divisible machines. As a side result, we obtain a tight bound of (sqrt(3) + 1)/2 ~= 1.3660 for the "one fast machine" case, i.e., when m - 1 machine speeds are equal, and there is only one faster machine. In this special case the best previous lower and upper bounds were 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m), shown in a classic paper by Gonzalez et al. [42]. Moreover, the authors of [42] conjectured the bound 4/3 to be tight. Thus, the results of the thesis answer three open questions in scheduling theory.In dieser Arbeit befassen wir uns mit Problemen aus zwei verschiedenen Teilgebieten der Scheduling-Theorie. Im ersten Teil betrachten wir das sog. preemptive Sum Multicoloring (pSMC) Problem. In einer Eingabe für pSMC werden paarweise Konflikte zwischen Jobs durch einen Konfliktgraphen repräsentiert; der Zeitbedarf eines Jobs ist durch ein ganzzahliges, positives Gewicht in seinem jeweiligen Knoten gegeben. Die Aufgabe besteht darin, die Jobs so den Maschinen zuzuweisen, dass die Summe ihrer Maschinenlaufzeiten minimiert wird. Wir liefern den ersten Algorithmus für pSMC auf Pfaden und Kreisen mit polynomieller Laufzeit; er benötigt O(min(n², n log p)) Zeit, wobei n die Anzahl der Jobs und p die maximale Zeitanforderung darstellen. Dies liefert eine Antwort auf die von Halldórsson et al. [51] aufgeworfene Frage der Komplexitätsklasse von pSMC. Unser Resultat identifiziert eine Diskrepanz zwischen der Komplexität auf binären Bäumen - für diese ist das Problem NP-schwer - und Pfaden. Im zweiten Teil dieser Arbeit betrachten wir das Problem, n Jobs auf m Maschinen mit unterschiedlichen Geschwindigkeiten so zu verteilen, dass der Makespan minimiert wird (Q||C_max). Wir präsentieren einen einfachen deterministischen monotonen Algorithmus mit Approximationsgüte 2.8 für Q||C_max. Monotonie ist relevant im Zusammenhang mit truthful Mechanismen: wenn die Geschwindigkeiten der Maschinen nur diesen selbst bekannt sind, müssen sie motiviert werden, dem Scheduling Mechanismus ihre tatsächlichen Geschwindigkeiten offenzulegen. Der beste bisherige deterministische truthful Mechanismus mit polynomieller Laufzeit in n und m von Andelman et al. [3] erreicht Approximationsgüte fünf. Eine randomisierte Methode mit ApproximationsgÄute zwei, die jedoch nur eine schwächere Definition von truthful Mechanismen unterstützt, wurde von Archer und Tardos [4, 5] entwickelt. Als ein zentrales Ergebnis beweisen wir die Vermutung von Auletta et al. [8], dass der greedy list-scheduling Algorithmus Lpt monoton ist, falls alle Maschinengeschwindigkeiten ganze Potenzen von zwei sind (2-divisible Maschinen). Der Beweis der obigen Approximationsschranke von 2.8 benutzt die Approximationsgüte von Lpt auf 2-divisible Maschinen. Als Nebenresultat erhalten wir eine scharfe Schranke von (sqrt(3) + 1)/2 ~= 1.3660 für den Fall "einer schnellen Maschine", d.h. m - 1 Maschinen haben identische Geschwindigkeiten und es gibt nur eine schnellere Maschine. Die bisherigen besten unteren und oberen Schranken für diesen Spezialfall waren 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m). Letztere wurden 1977 von Gonzalez, Ibara und Sahni [42] bewiesen, die mutmaßten, dass die tatächliche obere Schranke bei 4=3 läge. Alles in allem, liefert diese Arbeit Antworten auf drei offene Fragen im Bereich der Scheduling-Theorie