1. 1 Input Queued Switches: Cell Switching vs. Packet Switching Abtin Keshavarzian Joint work with Yashar Ganjali, Devavrat Shah Stanford University

2. 2 Background Time is slotted Data units of fixed size  cells Buffers at input ports (Input-Queued Switch) To avoid HoL blocking, virtual output queues are used VOQ 11 VOQ 1N VOQ N1 VOQ NN Output 1 Output N Input 1 Input N Switching Fabric

3. 3 VOQ 11 VOQ 1N VOQ N1 VOQ NN Motivation Packets have different lengths –Splitter module –Combiner module (memory) Packet delays are more important than Cell delays Packet Based Scheduling algorithms Splitter Combiner Switch

4. 4 Outline Cell based algorithms review: –Stability concept –Maximum Weight Matching algorithm Packet based algorithms –Packet-Based Algorithms –PB-MWM and its stability –PB Algorithms Classification Work Conserving Waiting –Waiting PB Algorithms Conclusion

5. 5 Notation – Arrival rate : Number of cells arrived to VOQ ij up to time n : Number of cells departed from VOQ ij up to time n : Number of cells queued at VOQ ij at time n (SLLN) almost surely VOQ 11 VOQ 1N VOQ N1 VOQ NN Output 1 Output N Input 1 Input N Switching Fabric

6. 6 Admissibility and Rate Stability The arrival rate matrix is “admissible” iff A switch under a matching algorithm is “stable” (rate stable) if, almost surely,

7. 7 MWM algorithm A matching MWM: At each time slot, select the matching with maximum weight

8. 8 MWM Stability McKeown et al showed that MWM is stable under i.i.d. Bernoulli traffic Dai and Prabhakar using Fluid model technique showed MWM is stable for any admissible traffic J. G. Dai and B. Prabhakar, “The throughput of data switches with or without speedup,” INFOCOM 2000, pp. 556-564. N. McKeown,V. Ananthram, and J. Walrand, “Achieving 100% throughput in an input-queued switch,” INFOCOM 1996, pp. 296-302.

9. 9 Outline Cell based algorithms review: –Stability concept –Maximum Weight Matching algorithm Packet based algorithms –Packet-Based Algorithms –PB-MWM and its stability –Packet Based Algorithms Classification Work Conserving Waiting –Waiting Packet Based Algorithms Conclusion

10. 10 Packet-Based Switching Once the scheduler starts transmitting the first cell of a packet, it continues until the whole packet is received at output port

11. 11 Packet-Based Switching Once the scheduler starts transmitting the first cell of a packet, it continues until the whole packet is received at output port

12. 12 Packet-Based Switching Once the scheduler starts transmitting the first cell of a packet, it continues until the whole packet is received at output port.

13. 13 Cell-based to Packet-based Consider cell-based algorithm X At each time slot: –Busy ports : middle of sending a packet –Free ports : i/o ports can be assigned freely PB-X –Keep the assignments used by busy ports –Find a sub-matching for free ports using algorithm X.

14. 14 Stability of PB-MWM PB-MWM is stable under “regenerative admissible traffic” Traffic is called “regenerative” if on average it requires a finite time to reach the state where all ports are free if it keeps using any fixed matching. – Bernoulli i.i.d. is a regenerative traffic. M.A. Marsan, A. Bianco, P. Giaccone, E. Leonardi, and F. Nari, “Packet Scheduling in Input- Queued Cell-based switches,” INFOCOM 2001, pp. 1085-1094

15. 15 Proof Outline Matching m(n) is “k-imperfect” if For PB-MWM: Lemma: A scheduling algorithm is rate stable if the average value of its weight is larger than maximum weight matching minus a bounded constant.

16. 16 Question CB-MWM is stable under any admissible traffic PB-MWM is stable under any admissible regenerative traffic. Is the regenerative condition necessary?

17. 17 Counter-example

18. 18 Counter-example

19. 19 Counter-example

20. 20 Counter-example

21. 21 Counter-example

22. 22 Counter-example

23. 23 Counter-example

24. 24 Classification of PB algorithms Work Conserving (non-waiting): –No input is left unmatched when it has a packet for an unmatched output. Waiting : –Input ports may wait (do not start sending a packet) for infinite number of time slots. No work-conserving algorithm can be rate stable for all admissible traffic.

25. 25 PB-wMWM Segment #1 Segment #2 Switch runs at speedup Maximum packet length: L If use usual PB-MWM Else wait till all ports are free. PB-wMWM is rate stable for any admissible traffic with known max packet length

26. 26 Modified PB-wMWM Segment #1 Segment #2 The packet length is not known but has bounded expectation : the maximum length of packets left when waiting starts during lth segment Modified PB-wMWM is rate stable for any admissible traffic with bounded packet length

27. 27 Conclusion PB-MWM is rate stable under any admissible regenerative traffic. Work-conserving packet based algorithms can not be rate stable for all admissible traffics  Waiting is essential PB-wMWM and its modified version are stable under any admissible traffic (with bounded mean packet length) Future work: –Find simpler algorithms that are stable for any admissible traffic.

28. 28 Fluid model : number of time slots matching m being used up to time n