competitive analysis in buffer management . Sergey I. Nikolenko - PowerPoint PPT Presentation

competitive analysis in buffer management . Sergey I. Nikolenko 1,2,3 Summer School on Operational Research and Applications Nizhny Novgorod, May 25, 2016 1 NRU Higher School of Economics, St. Petersburg 2 Steklov Institute of Mathematics at St. Petersburg 3 Deloitte Analytics Institute, Moscow

intro and problem setting .

problem setting . • A buffer B that handles a sequence of arriving packets. • Discrete time, each time slot contains: (1) arrival : new packets arrive, and the buffer management unit performs admission control and, possibly, push-out; (2) assignment and processing : a single packet is selected for processing by the scheduling module; (3) transmission : packets with zero required processing left are transmitted and leave the queue. • The goal is to transmit as many packets as possible (i.e., drop as little as possible). 3

packets, buffers, processing orders . • The structure of the buffer can be different: • single queue : all packets go to a single output port; • multiple queues : clearly separated queues leading to different output ports; • shared memory : different output ports but the memory is shared (and has to be balanced); • CIOQ (combined input-output queued) switches: several inputs, several outputs, one queue per output at every input; • crossbar switches: buffers at intersections. 4

packets, buffers, processing orders . • Packets can differ in various characteristics : • value v(p) ∈ {1, … , V} : how much the packet contributes to the objective function; • required processing cycles r(p) ∈ {1, … , k} : how long must a CPU work on the packet before transmission; • output port : where a packet is headed; • size in bytes (buffer slots). 4

packets, buffers, processing orders . • Finally, processing and transmission orders can be different too: • in FIFO order, packets should be processed and transmitted in the order they arrived; • in semi-FIFO order, processing is free but transmission should follow arrivals; • or the order does not matter, so we are free to construct priority queues. 4

competitiveness . • The goal is to transmit as many packets as possible (i.e., drop as little as possible). Definition An online algorithm A is said to be α -competitive (for some α ≥ 1 ) if for any arrival sequence σ the number of packets successfully transmitted by A is at least 1/α times the number of packets suc- cessfully transmitted by an optimal solution (denoted OPT ) ob- tained by an offline clairvoyant algorithm. • This is a worst-case definition, with guarantees over all traffic distributions. • Lower bounds are counterexamples, upper bounds are uniform guarantees (often interesting theorems). 5

plan . • Our plan: • try to study buffers, packets, processing orders in different combinations; • a lot of different combinations and works, we only look at some of the simplest and most interesting (best interest/technicality ratio); • start with uniform packets (shared memory); • then look at packets with heterogeneous processing; • and finally at packets with multiple characteristics. 6

uniform packets .

simplest setting . • Simplest setting: single queue, all packets are identical. • What is the competitive ratio? 8

simplest setting . • Simplest setting: single queue, all packets are identical. • What is the competitive ratio? • Naturally, 1 : the greedy algorithm is optimal. • Hasn’t been too hard, has it? Well, there’s more... • Based on this paper: • W. Aiello, A. Kesselman, Y. Mansour. Competitive buffer management for shared-memory switches. ACM Transactions on Algorithms, vol. 5, no. 1, 2008. 8

shared memory switch . • Let us now consider a shared memory switch. • N × N switch: each of N output ports has a queue, and the total • N input ports send in at most N packets per time slot (not necessary). • Non-preemptive (non-push-out) algorithms decide what to accept and cannot drop packets. • Preemptive (push-out) algorithms can push out already accepted packets. 9 number of packets in all queues is bounded by B .

push-out algorithms . • Push-out algorithms: there are N queues of total size B , each packet is labeled with its output port. • The queues can be assumed to be FIFO (doesn’t matter since packets are uniform). • A new packet comes in; if there is free space, we lose nothing by accepting. • If the buffer is congested (full), we have to decide: where do we push out from? • What policies can you propose? 10

push-out algorithms • What policies can you propose? • but first... • upper bound for LQD by matching (common technique); • lower bound for LQD by a specific counterexample; • We will demonstrate: • What is its competitive ratio? longest queue. • We’ll study LQD ( longest queue drop ): drop packet from the push out from? . • If the buffer is congested (full), we have to decide: where do we accepting. • A new packet comes in; if there is free space, we lose nothing by packets are uniform). • The queues can be assumed to be FIFO (doesn’t matter since packet is labeled with its output port. • Push-out algorithms: there are N queues of total size B , each 10

general lower bound . • There is one more flavor of results: general lower bounds . • How can we prove a lower bound for all online algorithms? 11

general lower bound . • There is one more flavor of results: general lower bounds . • How can we prove a lower bound for all online algorithms? • We have to show an adversarial bound, constructing a hard example for an online algorithm on the fly. 11

general lower bound queue. • But there are more interesting cases too... • This is how many lower bounds for many settings work. online algorithm . 3 . 4B • The sequence can be repeated, getting the ratio of 4B do the full . • Now ALG can transmit at most 4B • Next, for B time slots we send 1 packet per time slot to the other • At time • For 2B 12 2B • In this case: consider 2 active output ports with queues Q 1 and Q 2 (and N input ports). N−2 time slots, N 2 packets arrive for each port. N−2 , either Q 1 or Q 2 has ≤ B 2 packets ( B in total). N−2 + 3B 2 packets, and OPT can N−2 + 2B . 3B+2 → 4 • Hence, we have shown a general lower bound of 4 3 for any

lqd: lower bound . • LQD accepts more for overloaded ports... processing 2A packets per time slot; keeps all that comes for active ports (which amounts to exactly B ), • OPT accepts one of two packets for each overloaded port and busy in this way. nothing happens, then again; A input ports can keep A active port 13 • A idle: needed only to get enough inputs; • A overloaded: each receives 2 packets per time slot; • output ports: + A , L > A ; 2 • Proof by construction of a specific counterexample: • Lower bound for LQD : √2 . • consider a 3A × 3A switch with B = L(A−1) • A active: L packets arrive over L A time slots, then for L − L A slots

lqd: lower bound 2 2, 1 . ; • in overloaded ports, ≈ xAL ; • the total number of packets in active queues is • when a burst arrives to an active port, it stabilizes at xL ; • the max queue length is ≈ constant, say xL ; • LQD accepts more for overloaded ports: 14 xL + (xL − L A) + (xL − 2 L A) + … ≈ x 2 AL • and in total they should add up to B ≈ LA 2 , so we get 2x 2 + x ≈ 1 x ≈ √2 − 1; • each active port over L time slots transmits ≈ L A + xL ≈ xL , so the total throughput rate is ≈ xL L A + A = √2A , hence the bound.

lqd: upper bound . • Upper bound: 2 . • Proof: we construct a matching between the extra packets processed by OPT and LQD packets. • An extra packet is a packet transmitted by OPT when the corresponding LQD queue is idle. • A potential extra packet is a packet further in OPT’s queue than the entire corresponding LQD queue, pos t OPT (p) > L t LQD (q) . • Matching idea: 15

lqd: upper bound • else match p to an arbitrary unmatched packet in LQD buffer. • If so, we get a competitive ratio of 2 . packets). • the matching is feasible (there are always enough unmatched • all extra packets are matched before they are transmitted; • We have to prove that: they appear. • So the idea is to match all potential extra packets as soon as LQD , match p to itself; . • if p arrived at this time slot and was accepted by both OPT and LQD (q) as follows: OPT (p) > L t q for which pos t (2) at end of arrivals, match each unmatched OPT packet p in queue (1) if during arrival a matched LQD packet p is preempted by p ′ , • Matching routine: on each time slot t , 16 replace p by p ′ in the matching;

lqd: upper bound . • Sequence of lemmas: (1) all extra packets are matched (if matching works at all); then LQD is congested and some packets have been dropped in queue q ; pos t OPT (p) ≥ pos t LQD (p ′ ) (check all cases); (4) matching works (if there are unmatched packets in OPT then there are at least as many unmatched packets in LQD). 17 (2) if p ∈ OPT from queue q is matched to p ′ ∈ LQD and p ′ ≠ p , (3) if p ∈ OPT is matched to p ′ ∈ LQD at time t , then

single queue with heterogeneous processing .