1. Competitive Queueing Policies for QoS Switches Nir Andelman Yishay Mansour An Zhu TAUTAUStanford

2. Outline n Motivation n Model description n Summary of Previous and new results n Non-preemptive queue –Two packet types –Multiple packet types n Preemptive queue lower bound n Open Questions

3. Motivation n Quality of Service –Guaranteed performance –Limited resources n Premium Service

4. Motivation (cont.) n Assured service –Relative (not Guaranteed) Performance –Different packet priorities (values) –High Network Utilization

5. Motivation (cont.) n Queue management –Outgoing port –Limited queue space –Online packet scheduling 11

6. Our model n Input: a stream of valued packets. n Actions: either accept or reject a packet n Send events: at integer times n Benefit = Total value of the packets sent. n Main Variations: n Non-Preemptive FIFO Queue n Preemptive FIFO Queue n Delay-Bounded Queue n Competitive Analysis: ρ = max {offline/online}

7. Previous Results n Non-Preemptive Queue –(2  -1)/  lower bound for 2 values and Analyzes specific policies (AMRR00) n Preemptive Queue –2-o(1) competitive greedy algorithm (KLMPSS01) –1.28 lower bound for 2 values (Sviridenko01) –1.30 competitive algorithm for 2 values (LP02) n Delay-Bounded Queue (KLMPSS01) –2 competitive greedy algorithm –1.17 lower bound for  -uniform bounded delay –1.414  ρ  1.618 for 2-variable bounded delay –1.25  ρ  1.434 for 2-uniform bounded delay

8. Summary of Our Results n Non-preemptive queue –Algorithm with ρ = (2  -1)/  optimal for 2 values –tight(er) bounds for previous policies –ρ =  (ln(  )) for continuous values n Preemptive queue –General lower bound of 1.414 –Exact ρ =1.434 for queue size 2 n Delay-Bounded queue –1.366  ρ  1.414 for 2-uniform bounded delay –ρ = 1.618 for 2-variable bounded delay

9. Non-Preemptive Lower bound - 2 values 1 1 1 1 1 ONOFF 1 1 1 1 1 1 1 1 1 Online accepts xB packets. Offline accepts B packets. Ratio is x [From AMRR 2000]

10. Lower bound - 2 values (cont.) 1 1 1 1 1 ONOFF      1 1 1 1 Online accepts xB low and at most (1-x)B high. Offline accepts B high value packets. Ratio is [x+(1-x)  ]/   1      

11. Lower bound - 2 values (cont.) Optimize lower bound: x =  /(2  -1) Lower bound :   (2  -1)/ 

12. Ratio Partition (RP) Policy n Always accept high value packets. n Each high value packet marks  /(  -1) low value packets in the queue that arrived before it. n Accept a low packet if you can mark it by filling the queue with high value packets.

13. RP Example (1) 1 1 1 1 1 Let  = 2, Each high value marks 2 low values. 1 1 1 1     1 1 Lemma: When the queue is full, all packets in it are marked.  m m m m

14. RP Example (2) 1 1 1 1 1 1      1 1 1 Free slots left for (possible) future high values. 

15. RP Analysis n Full queue: –all low value packets are marked. n Online marked packets bound: – offline high value packets. n Marking parameter balances: –accepted low value packets –slots for future high value packets. n Optimizing the marking parameter gives ρ=(2  -1)/ . n Optimal competitive ratio.

16. Continuous Values n Create n= ln(  ) sub-queues n Sub-queue k accepts values [  k-1/n,  k/n ] n Sub-queues take turns in sending –Can be simulated by a FIFO queue. n Competitive ratio of e ln(  ) n Lower bound: ln(  ) +1

17. Lower bound (B=2) 1 T=1 11 22 T=2 22 T=3  k-2 T=k-1  k-1 T=k 11 33  k-1 kk kk T=k+1 kk At T=i packets  i-1 followed by  i arrive Scenario stops if online sends  i at T=i. Offline sends  1 +  2 + … +  k-2 +  k-1 +  k +  k +  k (or  1 +  2 + … +  i-2 +  i-1 +  i-1 +  i ) Online sends 1 +  1 +  2 + … +  k-2 +  k-1 +  k +  k (or 1 +  1 +  2 + … +  i-2 +  i ) Choosing appropriate  i the c.r. goes to 1.434

18. Preemptive Lower bound n Stage i includes: –A burst of B-1  i-1 packets followed by one  i –At the next Z times units, one  i packet each unit n End with B packets of value  k n Stop: if B-Z packets are preempted in a stage. n Optimize  i and Z=B/2 n the lower bound converges towards 1.414. n For B=2 the bound is 1.434.  i-1 ii ii ii ii ii B-1 Z

19. Open Problems n Non-Preemptive queue & continuous values –Close the constant gap between the upper (e ln(  )) and lower (ln(  )+1) bounds n Preemptive queue & continuous values –Is there a policy which has ρ ≤ 2-ε n Delay-Bounded queue: –Better than Greedy for delay > 2