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This paper presents a comprehensive approach to competitive management of non-preemptive queues with multiple values, focusing on quality of service and guaranteed performance under limited resources. We define and analyze the Smooth Selective Barrier Policy, providing both lower and upper bounds for various queue types and showcasing new results alongside previous findings. The implications of packet prioritization with respect to network utilization and online packet scheduling are explored, culminating in robust solutions for managing finite queues efficiently. Open questions for future research are also discussed.
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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
