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Competitive Management of Non-Preemptive Queues with Multiple Values. Nir Andelman Yishay Mansour Tel-Aviv University. Outline. Motivation Model Description Summary of Previous and New Results Smooth Selective Barrier Policy Policy definition and analysis Lower bound Open Questions.
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Competitive Management of Non-Preemptive Queues with Multiple Values Nir Andelman Yishay Mansour Tel-Aviv University
Outline • Motivation • Model Description • Summary of Previous and New Results • Smooth Selective Barrier Policy • Policy definition and analysis • Lower bound • Open Questions
Motivation • Quality of Service • Guaranteed performance • Limited resources • Premium Service
Motivation (cont.) • Assured Service • Relative (not guaranteed) performance • Different packet priorities (values) • High network utilization
Motivation (cont.) • Queue Management • Outgoing port • Limited queue size • Online packet scheduling 1
Our Model • Input: A stream of packets • Actions: Either accept or reject a packet • Send events: At integer times • Benefit = Total value of the packets sent • Main variations: • Non-Preemptive FIFO Queue • Preemptive FIFO Queue • Delay-Bounded Queue • Competitive Analysis: = max {offline/online}
Previous Results • Non-Preemptive Queue • (2-1)/ lower bound for 2 values (AMRR00) • (2-1)/ upper bound for 2 values (AMZ03) • ln()+1 general lower bound (AMZ03) • e ln() general upper bound (AMZ03) • Preemptive Queue • 1.28 lower bound for 2 values (Sviridenko01) • 1.30 upper bound for 2 values (LP02) • 2-o(1) competitive greedy algorithm (KLMPSS01) • 1.983 general upper bound (KMvS03) • 1.419 general lower bound (KMvS03)
Summary of Our Results Smooth-Selective-Barrier-Policy • Algorithm with = ln() + 2 + O(ln2()/B) • Better bounds for <5.558 • Lower bound of ln()+2-o(1) for similar policies
1 (1+)2 1+ 1 1 1 1 1 1+ (1+)2 1+ (1+)2 (1+)2 1+ 1+ (1+)2 (1+)2 (1+)2 (1+)2 (1+)2 (1+)2 (1+)2 1+ 1+ (1+)2 (1+)2 1 (1+)2 (1+)2 1+ 1+ 1+ 1+ 1 1 1 1 1 1 1 (1+)2 (1+)2 1+ 1+ 1+ 1+ 1 1 1 1 1 1 (1+)2 (1+)2 1+ 1+ 1+ 1+ Lower Bound of ln()+1 (AMZ03) • k bursts of B (queue’s size) packets • Packet values grow exponentially • Online accepts packets from all bursts • Offline accepts last burst (1+)2 1+ online offline
Smooth Selective Barrier • Accepting a packet depends on the packet value and the number of packets in the queue. • For each cell in the queue there is a minimal value for the packet that can occupy it. v 10 v 5 v 2 v 1
Upper Bound: sketch proof • Assume “worst case scenario” on input: the online accepts packets with minimal value • Calculate potential (t) How much the offline can gain, without changing the online • By induction: c on(t) off(t) + (t) • Show that c ln() + 2 + O(ln2()/B)
3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 1 3 Potential – Going up • A burst of packets is rejected by the online but accepted by the offline. 4 3 2 1 3 2 1 online offline
2 1 2 1 3 3 3 2 3 3 2 3 3 3 3 2 1 3 3 3 2 1 1 1 3 3 Potential – Going Down • Send one packet, then the offline accepts one packet that the online rejects. • Repeat until the online is willing to accept any packet. 4 3 2 1 1 2 1 2 online offline
Bound tightness • Going up: Due to the lower bound, for any similar policy: c.r. ln() + 1 • Going down: Inflicts a loss of approx. the queue’s contents • Up and Down: c.r. > ln() + 2 -
Open Questions • Non-Preemptive Queue • Gap between ln()+1 and ln()+2 (continuous case) • Preemptive Queue • Gap between 1.28 and 1.30 (2 values) • Gap between 1.419 and 1.983 (continuous case) • Delay-Bounded Queue • Few results for delay>2
