Smooth scheduling under variable rates or the analog-digital confinement game

Abstract

This work considers non-terminating scheduling problems in which a system of multiple resources serves clients having variable needs. The system has m identical resources and n clients; in each time slot each resource may serve at most one client; in each such slot t each client γ has a rate, a real number ργ(t), that specifies his needs in this slot. The rates satisfy the restriction È γ ργ(t) ≤ m for any slot t. Except of this restriction, the rates can vary in arbitrary fashion. (This contrasts most prior works in this area in which the rates of the clients are constant.) The schedule is required to be smooth as follows: a schedule is ∆-smooth if for all time intervals I the absolute difference between the amount of service received by each client γ to his nominal needs of È t∈I ργ(t) is less than ∆. Our objective are online schedulers that produce ∆-smooth schedules where ∆ is a small constant which is independent of m and n.Our paper constructs such schedulers; these are the first online ∆-smooth schedulers, with a constant ∆, for clients with arbitrarily variable rates in a single or multiple resource system. Furthermore, the paper also considers a non-concurrent environment in which there is an additional restriction that each client is served at most once in each time slot; it presents the first online smooth schedulers for variable rates under this restriction.The above non-concurrent restriction is crucial in some applications (e.g., CPU scheduling). It has been pointed out that this restriction "adds a surprising amount of difficulty" to the scheduling problem. However, this observation was never formalized and, of course, was never proved. This paper formalizes and proves some aspects of this observation.Another contribution of this paper is the introduction of a complete information, two player game called the analogdigital confinement game. In such a game pebbles are located on the real line; the two players, the analog player and the digital player, take alternating turns and each one, in his * The full paper is available in

Achieving Optimal Backlog in the Vanilla Multi-Processor Cup Game

“…This means that, even when a single cup contains far more water than any other cup, only a 1 p -fraction of the emptier's resources can be devoted to emptying that cup. As was first noted by Liu [28] in 1969, and later reiterated by other authors [7,9,23,26], this constraint adds a "surprising amount of difficulty" to the scheduling problem.…”

“…The p-processor cup game naturally arises in the study of processor scheduling [1,16,26]. The n cups represent tasks that are being scheduled to run in time slices on p processors.…”

“…The p-processor cup game (and its relaxations) has also appeared in a variety of other applications, ranging from deamortization [2, 3, 15-17, 22, 24, 31], to buffer management in network switches [4,19,21,32], to quality of service guarantees [1,7,26].…”

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.…”