On Centralized Smooth Scheduling

Abstract

This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a distributed environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation; namely, for ρ, ∆ ∈ R a set A of natural numbers is (ρ, ∆)-smooth if abs(|I| · ρ − |I ∩ A|) < ∆ for any interval I ⊂ N.The current paper studies scheduling persistent clients on a single slot-oriented resource in a flexible, predictable and distributed manner. Each client γ has a given rate ρ γ that defines the share of the resource he is entitled to receive and the goal is a smooth schedule in which, for some predefined ∆, each client γ is served in a (ρ γ , ∆)-smooth set of slots (natural numbers). The paper focuses on a distributed environment where each client by itself (without any inter-client communication) resolves (computes), slot after slot, whether or not it owns this slot. The paper presents extremely efficient schedules under which a client resolves each slot in a constant time.The paper considers two scheduling frameworks. The first one, the Flat Scheduling Framework, is the common problem where the rates of the clients are given a priori. In the second and novel framework, the Open-Market Scheduling Framework, fractions of the resource are bought and sold by dealers. Each dealer, upon receiving his fraction, may choose either to become a client and use his share, or to remain a dealer and sell fractions of his share to other dealers. In this framework, the allocation process is highly distributed; moreover, fractions of several resources can be combined into a single virtual resource of new capabilities.The paper presents two scheduling techniques. Both techniques, in both frameworks, produce smooth schedules with highly efficient distributed resolutions -a client resolves each slot in O(1) time on a RAM with a moderate number of memory words, all of a small size. Each technique has its pros and cons. For example, one technique utilizes 100% of the resource but its resolution algorithm requires a number of words which is linear in the number of clients; the other technique utilizes only 99% of the resource but its resolution algorithm requires just O(1) words.One of these techniques yields a solution to Tijdeman's Hierarchial Chairman Assignment Problem which outperforms prior solutions. The other technique naturally extends to the problem of scheduling multiple resources, under the restriction that a client may be served concurrently by at most one resource. The extension yields the first solution to this problem having efficient distributed resolution. Prior solutions produce a special type of smooth scheduling called P-fair scheduling, are centralized, and are less efficient than ours.

The Family Holiday Gathering Problem or Fair and Periodic Scheduling of Independent Sets

“…Each user or task has a weight and the goal is to schedule the users so that they obtain the resource proportionally to their weight. Sometime there are multiple identical resource, but each task can be assigned to one resource concurrently [6,20]. This is similar to our setting when the graph is a clique (or composed of components that are cliques) and all the weights are the same.…”

Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game

“…The classical p-processor cup game. The p processor cup game captures the general problem in which there are some number n of tasks competing for a smaller number p of processors [7,21,8,34,32,38,6,24,35,36,17,10,27,1,16,33,25]. A scheduler must assign tasks to processors over time in order to ensure that no individual task ever falls too far behind.…”

“…If one views the cup game as a scheduling problem, then the cups represent tasks, the water represents work, the filler represents an adversary that determines when work arrives, and the emptier represents a scheduler that can select p tasks to run on a given time step (we will use the terms "round" and "time step" interchangeably). Although we will primarily be interested in the cup game as a scheduling problem [7,21,8,34,32,38,6,24,35,36,1,33,17], it has also found applications to many other problems (e.g. deamortization of data structures [2,17,16,3,39,23,18,26,9], network-switch buffer management [22,4,40,20], quality-of-service guarantees [7,1,33], etc.…”

The Variable-Processor Cup Game

“…Starting with the seminal paper of Liu [30], work on the p processor cup game has spanned more than five decades [7,20,8,29,27,33,6,23,30,31,16,10,25,1,15,28]. In addition to processor scheduling [7,20,8,29,27,33,6,23,30,31,1,28,16], applications include network-switch buffer management [21,4,35,19], quality of service guarantees [7,1,28], and data structure deamortization [2,16,15,3,34,22,17,24,9]. The game has also been studied in many different forms.…”

“…The game has also been studied in many different forms. Researchers have studied the game with a fixed-filling-rate constraint [7,20,8,29,27,33,6,23,30,31], with various forms of resource augmentation [10,25,28,16], with both oblivious and adaptive adversaries [1,7,30,10,25], with smoothed analysis [25,10], with a semi-clairvoyant emptier [28], with competitive analysis [5,18,14], etc.…”