On distributed smooth scheduling

• Authors: Ami Litman;Shiri Moran-Schein
• Venue: Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
• Year: 2005

This paper studies evenly distributed sets of natural numbersand their applications to scheduling in a distributed environment.Such sets, called smooth sets, have the property that theirquantity within each interval is proportional to the size of theinterval, up to a bounded additive deviation; namely, forπ,Δ ∈ R a set A of natural numbers is(π, Δ)-smooth if abs(&vbar;I&vbar; ·π-- &vbar;I∩A&vbar;) 9 Δ for anyinterval I ⊂ N.The current paper studies scheduling persistentclients on a single slot-orientedresource in a flexible, predictable anddistributed manner. Each client γ has a givenrate ργ thatdefines the share of the resource he is entitled to receive and thegoal is a smooth schedule in which, for some predefined Δ,each client γ is served in aργ,Δ)-smooth set of slots(natural numbers). The paper focuses on a distributedenvironment where each client by itself (without anyinter-client communication) resolves (computes),slot after slot, whether or not it owns this slot. The paperpresents extremely efficient schedules under which a clientresolves each slot in a constant time.The paper considers two scheduling frameworks. The first one,the Flat Scheduling Framework, is the commonproblem where the rates of the clients are given a priori. In thesecond and novel framework, the Open-Market SchedulingFramework, fractions of the resource are bought and soldby dealers. Each dealer, upon receiving his setof slots, may choose either to become a client and use his share,or to remain a dealer and sell fractions of his share to otherdealers. In this framework, the allocation process is highlydistributed; moreover, fractions of several resources can becombined into a single virtual resource of new capabilities.The paper presents two scheduling techniques. Both techniques,in both frameworks, produce smooth schedules with highly efficientdistributed resolutions --- a client resolves each slot inO(1) time on a RAM with a moderate number ofmemory words, all of a small size. Each technique has its pros andcons. For example, one technique utilizes 100% of the resource butits resolution algorithm requires a number of words which is linearin the number of clients; the other technique utilizes only 99% ofthe resource but its resolution algorithm requires justO(1) words.One of these techniques yields a solution to Tijdeman'sHierarchial Chairman Assignment Problem which outperforms priorsolutions. The other technique naturally extends to the problem ofscheduling multiple resources, under the restriction that a clientmay be served concurrently by at most one resource. The extensionyields the first solution to this problem having efficientdistributed resolution. Prior solutions produce a special type ofsmooth scheduling called P-fair scheduling, arecentralized, and are less efficient than ours.