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
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.
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