Competitive Buffer Management with Packet Dependencies Alex Kesselman, Google Boaz Patt-Shamir, Tel Aviv University Gabriel Scalosub, University of Toronto

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 2 Motivation: Video Streaming Smart encoding: – Suffices to recover many Every video frame is fragmented into packets Restoration depends on recovering all packets If packets are lost: – Affects other packets as well (become redundant) – Streaming: retransmission is not an option

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 3 Buffering Schematics buffer management incoming packets outgoing stream dropped packets finite link speed (“drain rate”)

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 4 Buffer Model Single FIFO queue of size Discrete time: – Delivery substep One packet delivered from head of queue – Arrival substep Packets arrive Some packets may be dropped Packets accommodated in the buffer

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 5 Traffic and Goal Traffic: frames consist of packets – E.g. Goal: Maximize number of whole frames delivered Methodology: Competitive analysis – Algorithm is -competitive if for every arrival sequence frame ‘s j-packet

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 6 Previous Work Buffer management for QoS – Multi-valued packets – Constant competitive ratio for finite values Some results for multiple buffers No results for co-dependent packets! [Kesselman et al., 2001], [Englert & Westerman, 2006], and many more [Mansour et al., 2000],

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 7 Our Results Frames consisting of k>1 parts Bad news: No finite competitive ratio in general Good news: If frame parts are consistently ordered – There exists an algorithm with c.r. – All deterministic algorithms have c.r.

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 8 Preliminaries Offline – Closely related to k-DM (as hard) – Simple greedy algorithm is a (k+1)-approximation Online (arbitrary traffic) – Not much you can do … time OPT ALG … packets ALG OPT

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 9 Restricted Traffic Problem: – Selective unbounded delay/burstiness Model requirement (solution): – Both ALG and OPT have to deal with same delay/burstiness Order-respecting traffic: – Frame order induced by j-packets is the same for every j OK: 2.1 (frame 2, part 1), 3.1, 2.2, 3.2 Not OK: 3.1, 2.1, 2.2, 3.2

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 10 All Is Well if Order-Respecting? Answer: Yes and No No: – Any deterministic algorithm has competitive ratio at least Yes: – A natural preemptive greedy approach – Conservative non-preemptive approaches

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 11 Greedy Algorithm (GA) Consider k=2 At every time t where overflow occurs, keep packets according to the following priority: 1.2-packets whose 1-packets were delivered 2.Complete frames (both 1-packets and 2-packets) 3.Remaining 1-packets Theorem: GA is constant-competitive for k=2 What if k>3?

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 12 Greedy Algorithm for k=3 GA OPT time OPT ALG ……… … 1-packets 2-packets 3-packets

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 13 Static-Partitioning Algorithm (SPA) Intuition – Think ahead: focus on admission control – Virtually partition the buffer into k levels of size Buffer is still FIFO!! – Level j only holds j-packets – Level j accepts j-packets that are “evenly” spaced in time Alternating accept/reject periods – Levels synchronize on frame index Ensures delivered packets correspond to the same frame Extra perk: non-preemptive

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 14 Example: SPA for k=2 Consider level 1, i.e., 1-packets 1-sync frame indices: Accepts first 1-packets after every 1-sync – Specifically, has sufficient buffer space time Accept packets Wait time units Accept packets Wait time units is the first 1-packet arriving after reject period ARRA is the first 1-packet

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 15 Example: SPA for k=2 Consider level 2, i.e., 2-packets 2-sync indices 1-sync indices Accepts first 2-packets after every 2-sync – Specifically, has sufficient buffer space time Accept packets Wait time units AR is the first 2-packet Accept packets Wait time units AR is the first 2-packet of a 1-sync arriving after reject period

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 16 Analysis of SPA for arbitrary k Consider two consecutive k-syncs : – SPA delivers out of frames – OPT delivers at most out of frames Intuition: at most out of every level Theorem: SPA is -competitive

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 17 Summary A new model in buffer management – Traffic has inter-packet dependencies Highly applicable to, e.g., video streaming First analytic results (still a lot to discover…) – Competitive algorithms (and lower bounds) – Complexity

Dagstuhl 2009Competitive Buffer Management with Packet Dependencies 18 Still Open Gap: vs. Randomization – Useful in the packet-weights models How does it work for real traffic? – Is greedy still an option? Using forward-error-correction (FEC) – Suffices to deliver m-of-k – Some preliminary results, but still a lot to discover High-order dependencies

Thank You!