A Maiden Analysis of Longest Wait First Jeff Edmonds York University Kirk Pruhs University of Pittsburgh

Client-Server System Clients Server Requests for page transmission “pull”

Client-Server System Clients Server Transmit page Server scheduling problem: How does the server decide which requests to respond to first?

The Big Problem Movie Distribution Database Replication via Internet Harry Potter Book Download Software Download Olympics Pay-Per-View Movies Today’s Internet Audience Size Content Richness

The 1-1 communication is not scalable

Broadcast Common Pages From Newsweek magazine From

Time Requests Pages:

Time to minimize total “wait” (flow) time Scheduling Problem: NP-complete [EH, 2002] Given requests deciding when to broadcast

O(1)-Approximate Algorithms Online: ? Future Optimal: All Knowing All Powerful Time Requests Online Time Requests Optimal

O(1)-Approximate Algorithms Online: ? Future Optimal: All Knowing All Powerful no online O(1)-comp. Alg. [KPV ‘00, EP ‘02]

O(1)-Approximate Algorithms Online: ? Future Optimal: All Knowing All Powerful Time Requests Online Time Requests Optimal

Resource Augmentation Analysis Algorithm is s-speed c-competitive if max I Online s (I)/Opt 1 (I) < c Time Requests Time Requests Online Optimal

Classic Server QoS Curves Average response time Low load High load Fast processor Slow Processor Online Optimal Not O(1)-competitive O(1)-speed O(1)-competitive

Scheduling Algorithms 2-speed 2-competitive [KPV ‘01, EH ‘02, GKKW ‘02, GKPS ‘02] Difficult Off-line Linear Programming Algorithms

Time Requests Scheduling Algorithms First In First Out (FIFO) 2-competitive for Max-Wait but bad for Total-Wait.

Time Requests Most Requests First (MRF) Scheduling Algorithms not O(1)-speed O(1)-competitive. [KPV ‘00]

Time Requests Scheduling Algorithms Not 2-speed O(1)-competitive. (4+e)-speed O(1)-competitive for any page lengths [EP 2002] B-Equipoise Proportional to number of requests

Time Requests Scheduling Algorithms (8+e)-speed O(1)-competitive for unit sized files [EP 2002] B-Equipoise-EDF Non-preemptive ~ number of requests

Time Requests Longest Wait First(LWF) Scheduling Algorithms Best Experimentally [AM] Not 1.6-speed O(1)-competitive. New 6-speed O(1)-competitive. New Was hoped to be (1+  )-speed O(1)-competitive. Efficient implementation, [KTT ‘01]

LWF is not 1.6-speed O(1)-competitive.

With s=1.6 LWF catches up. LWF is competitive.

LWF is not 1.6-speed O(1)-competitive. 



LWF 6 (I) < Opt 1 (I) x c Time Requests Time Requests LWF Optimal xc LWF is 6-speed O(1)-competitive.

Time Requests Time Requests LWF Optimal LWF is 6-speed O(1)-competitive. xc LWF 6 (I) < Opt 1 (I) x c

LWF Optimal LWF is 6-speed O(1)-competitive.

LWF Optimal LWF is 6-speed O(1)-competitive. =

LWF Optimal LWF is 6-speed O(1)-competitive. xc

LWF is 6-speed O(1)-competitive. LWF Optimal

LWF is 6-speed O(1)-competitive. LWF

 LWF is 6-speed O(1)-competitive. LWF ?

LWF is 6-speed O(1)-competitive. LWF

LWF is 6-speed O(1)-competitive. Hall’s Theorem Needs to be paid Able to pay

LWF is 6-speed O(1)-competitive. LWF Optimal One of s.

LWF is 6-speed O(1)-competitive. LWF

LWF is 6-speed O(1)-competitive. LWF 

LWF is 6-speed O(1)-competitive. LWF 

LWF is 6-speed O(1)-competitive. LWF + 

Time Requests Time Requests LWF Optimal LWF is 6-speed O(1)-competitive. xc Everyone paid enough. No one pays to much. LWF 6 (I) < Opt 1 (I) x c

LWF is best experimentally [AM] LWF is not 1.6-speed O(1)-competitive. New LWF is 6-speed O(1)-competitive. New Was hoped to be (1+  )-speed O(1)-competitive. A Maiden Analysis of Longest Wait First Conclusion Efficient implementation, [KTT ‘01]

LWF is best experimentally [AM] LWF is not 1.6-speed O(1)-competitive. LWF is 6-speed O(1)-competitive. Was hoped to be (1+  )-speed O(1)-competitive. A Maiden Analysis of Longest Wait First Future (2+  ) for any file lengths Conclusion The End Efficient implementation, [KTT ‘01] No Online is ?

Multicast Pull Scheduling: When Fairness is Fine Jeff Edmonds York University Kirk Pruhs University of Pittsburgh

Time Requests Scheduling Algorithms Not 2-speed O(1)-competitive. (4+e)-speed O(1)-competitive for any page lengths [EP 2002] B-Equipoise Proportional to number of requests

The Power of the Adversary in Multicast Pull Basic idea of the proof that there is no O(1)-competitive online algorithm Immediately after the online algorithm broadcasts a document, the adversary requests that document  The adversary broadcasts the document after the second request to the document utilizing the power of broadcast After a while the online algorithm still has a lot of work left while the adversary has little work left Then a high load stream of work that requires the full processing power of the server arrives

More on the Power of the Adversary in Multicast Pull Hence, the adversary forces the online algorithm to labor on sequential work  Sequential work = increasing the processing power devoted to the work does not change the rate at which the remaining work decreases Parallel work = doubling the processing power devoted to work doubles that rate at which that work is completed IMHO, the main contribution of this paper is the insight that Multicast pull scheduling = scheduling of jobs with arbitrary speed-ups

Scheduling Jobs with Variable Speed-up Curves In the Context of Parallel Processing Equipoise (Round Robin) = Give each job equal processing time Equipoise is a 3-speed 6-competitive algorithm for jobs with arbitrary speed-up curves [E, 1999] Formally means that Equipose with a speed 3 processor has average flow time at most 6 times the optimal average flow time for a speed 1 processor Intuitively means that Equipoise will perform reasonably well at low loads

Proof by picture that Bequi is O(1)-speed O(1)- approximation algorithm

More proof by picture

Replace jobs by sequential and parallel work in such a way that Broadcast-Equipoise is unaffected, and optimal is not hurt. Then apply Equipoise analysis.

Why transformation doesn’t hurt optimal Each reverse L shaped region, which contains the parallel work that optimal must finish, is contained within two consecutive BEqui broadcasts

Time Requests Scheduling Algorithms (8+e)-speed O(1)-competitive for unit sized files [EP 2002] B-Equipoise-EDF Non-preemptive ~ number of requests

BEQUI-EDF Algorithm for Unit Sized Documents (no preemption) B-EQUI-EDF Algorithm: Simulate BEQUI to get deadlines for jobs Run EDF on the jobs using these deadlines B-EQUI-EDF is an O(1)-speed O(1)- competitive algorithm