1. Algorithm Orals 2002 1 Algorithm Qualifying Examination Orals Achieving 100% Throughput in IQ/CIOQ Switches using Maximum Size and Maximal Matching Algorithms Sundar Iyer Stanford University sundaes@cs.stanford.edu www.stanford.edu/~sundaes

2. Algorithm Orals 2002 2 Outline  Introduction  Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch  Stability of critical MSM for any Bernoulli i.i.d. traffic  Stability of MSM for Bernoulli i.i.d. uniform traffic  Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch  A simple proof for stability

3. Algorithm Orals 2002 3 Simple Model of a Switch Port 1, inputPort 1, output Port 2, inputPort 2, output Port 3, inputPort 3, output Port 4, inputPort 4, output R R R R R R R R Example: Output Queued Switch

4. Algorithm Orals 2002 4 Input Queued Switch Model N N 1 1 R R Example: Input Queued Switch with virtual output queues (VOQs) Crossbar R R Port 1, input Port N, input Port 1, output Port 4, output VOQs

5. Algorithm Orals 2002 5 Relation to a Graph Matching 2 3 1 2 3 1 1 1 0 0 1 42 0 0 5 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1 2 3 1 VOQs

6. Algorithm Orals 2002 6 Classes of Scheduling Algorithms  Maximum Weight Matching (MWM)  Choose a matching which maximizes the weight of the matching  MWM gives 100% throughput  Maximum Size Matching (MSM)  Choose a matching which maximizes the size of the matching

7. Algorithm Orals 2002 7 Outline  Introduction  Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch  Stability of critical MSM for any Bernoulli i.i.d. traffic  Stability of MSM for Bernoulli i.i.d. uniform traffic  Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch  A simple proof for stability

8. Algorithm Orals 2002 8 MSM is Unstable N N 1 1 Request Graph N N 1 1 N N 1 1.. N N 1 1 Switch schedule based on MSM T=1 T=2 ……….

9. Algorithm Orals 2002 9 Questions  Are all MSMs unstable?  Is there a subclass of MSMs which are stable?  There is at least one MSM which is stable.  Are MSMs stable under uniform load?  Simulation seems to suggest this.  Can we prove this?

10. Algorithm Orals 2002 10 Non Pre-emptive Scheduling Batch Scheduling N N 1 1 R R Priority-2 Crossbar R R Port 1, input Port N, input Port 1, output Port N, output Priority-1 Batch- (k+1) Batch- (k)

11. Algorithm Orals 2002 11 Non Pre-emptive Scheduling Batch Scheduling N N 1 1 R R Priority-2 Crossbar R R Port 1, input Port N, input Port 1, output Port N, output Priority-1 Batch- (k+1) Batch- (k)

12. Algorithm Orals 2002 12 Degree of a Batch 1 2 3 0 1 0 2 1 0 0 0 1 1 2 3 Batch Request Graph Degree ( d v,k ):  The number of cells departing from (destined to) a vertex in batch k. Maximum Degree (D k )  The maximum degree amongst all inputs/outputs in batch k.

13. Algorithm Orals 2002 13 Critical Maximum Size Matching 2 3 1 2 3 1 1 2 3 2 3 1 2 3 1 2 3 1 2 3 1 0 1 0 2 1 0 0 0 1 1 2 3 Batch Request Graph degree =3

14. Algorithm Orals 2002 14 Outline  Introduction  Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch  Stability of Critical MSM for any Bernoulli i.i.d. traffic  Stability of MSM for Bernoulli i.i.d. uniform traffic  Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch  A Simple proof for stability

15. Algorithm Orals 2002 15 The Arrival Process

16. Algorithm Orals 2002 16 Stability of CMSM  Theorem 1:  CMSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform  Informal Arguments:  Let T k be the time to schedule batch k  Then for batch k+1 we buffer packets for time T k  We expect about  T k packets at every input/output  Hence, the maximum degree of batch k +1, i.e. D k+1   T k  Hence for a CMSM T k+1 = D k+1 =  T k < T k  Hence T k converges to a finite number

17. Algorithm Orals 2002 17 Formal Arguments … 1  We shall use the Chernoff bound to get  If we want to bound D k, we require that all the 2N vertices are bounded

18. Algorithm Orals 2002 18  We can choose (1 +  )  < 1 -  to get  Observe that  Q is now a function of T k only.  We can make Q as close to 1, by choosing a large T k  Also, T k+1  NT k  This gives Formal Arguments … 2

19. Algorithm Orals 2002 19 Formal Arguments …3  Hence, there is a constant T c which depends only on  (and hence only on  ), such that  Formally, using a linear Lyapunov function V(T k ) = T k, we can say that E(T k) is bounded.

20. Algorithm Orals 2002 20 Stability of CMSM  Theorem 2: CMSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d.

21. Algorithm Orals 2002 21 Outline  Introduction  Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch  Stability of Critical MSM for any Bernoulli i.i.d. traffic  Stability of MSM for Bernoulli i.i.d. uniform traffic  Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch  A Simple proof for stability

22. Algorithm Orals 2002 22 Example of a Uniform Graph 2 3 1 2 3 1 1 2 3 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1 1 1 1 1 1 1 1 2 3 Batch Request Graph degree =3

23. Algorithm Orals 2002 23 Properties of Uniform Graphs  Lemma-1:  If the request graph is uniform and the maximum degree is D, then any MSM can schedule the requests in exactly D time slots  Lemma-2:  Any request graph with maximum degree D, can be scheduled by any MSM within 2D time slots

24. Algorithm Orals 2002 24 Property of any Graph  Theorem:  Any request graph with maximum degree is D, and minimum VOQ length m, can be scheduled in less than 2D –Nm time slots  Proof:  Consider a request graph with minimum VOQ length m  The minimum degree of the graph is mN  Hence the original graph can be considered to be in two parts A uniform graph of degree mN Another graph of maximum degree D – mN  Hence the request graph can be scheduled in at most mN + 2(D-mN) = 2D - Nm

25. Algorithm Orals 2002 25 Stability of MSM..1  Theorem 3: MSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform  Informal Arguments  We can bound both the maximum degree D and the minimum VOQ length m  The rest of the proof is similar to the CMSM proof

26. Algorithm Orals 2002 26 Outline  Introduction  Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch  Stability of critical MSM for any Bernoulli i.i.d. traffic  Stability of MSM for Bernoulli i.i.d. uniform traffic  Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch  A simple proof for stability

27. Algorithm Orals 2002 27 Maximal Matching Algorithms  Maximal Matching (MXM)  Choose a matching such that no unmatched input or output has a packet meant for each other  They are easier to implement and have low complexity  They are known to be unstable and give low throughput for input queued switches

28. Algorithm Orals 2002 28 A Model for a CIOQ switch Combined Input-Output Queued Switch Bandwidth: 2NR 2R Port 1 Port 2 Port N 2R R R R Port 1 Port 2 Port N R R R  A CIOQ switch with a speedup of 2, gives 100% throughput for any MXM algorithm [Ref: Dai & Prabhakar, Leonardi. et. al.]

29. Algorithm Orals 2002 29  Let A j (t 1,t 2 ) denote the number of arrivals to output j in the interval between (t 1,t 2 )  A leaky bucket constrained traffic satisfies, the property that for each output j  Note that this means that for an ideal output queued switch no output has more than B packets in the switch  Let DT denote the departure time of a packet from this ‘ideal’ output queued switch Leaky Bucket Traffic

30. Algorithm Orals 2002 30 Stability of MXM  Theorem 4: A CIOQ switch with an MXM algorithm gives bounded delay and hence 100% throughput with a speedup greater than 2, under arrivals which satisfy the leaky bucket constraint

31. Algorithm Orals 2002 31 Constraint Set ‘Maximal’ Algorithm  The algorithm is greedy i.e. when a cell arrives, it immediately attempts to allot a time (in the future) when it should be transferred  Each input and output maintains a constraint set of the future times during which it is free to send/receive a packet  The algorithm attempts to bound the time of departure of a packet to within k time slots of its departure time DT, i.e each packet is transferred in the time (DT, DT+k)

32. Algorithm Orals 2002 32 Allocations as seen by the Output … DT + kDT- kDT c k  Packet has an OQ Departure Time = DT  Packet should leave in the interval (DT, DT + k)  In the interval (DT, DT + k)  There is one cell which tries to get allotted in that interval.  No more than k cells get delayed and are allotted to that interval  Number of Time Slots Available is more than

33. Algorithm Orals 2002 33 Allocations as seen by the Input … DT + kDT-B-kDT B + k DT-B  Packet has an OQ Departure Time = DT  Packet should leave during interval (DT, DT + k)  In the interval (DT, DT + k)  There is one cell which tries to get allotted in that interval  No cell which arrived before DT–B-k will be allotted to this interval  Number of Time Slots Available is more than c k

34. Algorithm Orals 2002 34 Sufficiency Conditions on Speedup  We are guaranteed a timeslot if  The above equation can be satisfied if  This means S > 2 is sufficient to guarantee that the delay is bounded  This implies 100% throughput

35. Algorithm Orals 2002 35 Stability of MXM  Theorem 5: A CIOQ switch with an MXM algorithm gives 100% throughput with a speedup greater than 2, under admissible arrivals which satisfy the strong law of large numbers

36. Algorithm Orals 2002 36 Summary  In an IQ switch with batch scheduling  A subclass of MSM called CMSM is stable, if the input traffic is admissible and Bernoulli i.i.d.  MSM is stable, if the input traffic is admissible and Bernoulli i.i.d. uniform  In a CIOQ switch with S>2,  MXM is stable under any traffic which satisfies the strong law of large numbers

37. Algorithm Orals 2002 37 Future Questions  We have seen that MSM is stable under the auspices of batch scheduling  Perhaps we could incorporate this (well known) idea into a number of other algorithms to prove stability?  It would be nice to nail down the stability of MSM with uniform load in the absence of batch scheduling  Other open questions remain

38. Algorithm Orals 2002 38 Backup

39. Algorithm Orals 2002 39 Stability of MSM …2  Informal Arguments:  Similar to the CMSM proof, derive P{D < (1 +  1 )  T k }  Use Chernoff bound, to derive P{mN > (1 -  2 )  T k }  We can now write the probability of using less than 2[(1 +  1 )  T k ] – (1 -  2 )  T k = (1 + 2  1 +  2 )  T k time slots  Then rest of the proof is similar to CMSM