Input Queued Switches: Cell Switching vs. Packet Switching

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

VOQ11 Output 1 Input 1 VOQ1N VOQN1 Output N Input N VOQNN Background Switching Fabric • 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

Motivation Switch • Packets have different lengths • Splitter module • Combiner module (memory) • Packet delays are more important than Cell delays Packet Based Scheduling algorithms VOQ11 Combiner VOQ1N Splitter VOQN1 VOQNN

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

VOQ11 Output 1 Input 1 VOQ1N VOQN1 Output N Input N VOQNN Notation – Arrival rate Switching Fabric • : Number of cells arrived to VOQij up to time n • : Number of cells departed from VOQij up to time n • : Number of cells queued at VOQij at time n • (SLLN) almost surely

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

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

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 N. McKeown,V. Ananthram, and J. Walrand, “Achieving 100% throughput in an input-queued switch,” INFOCOM 1996, pp. 296-302. J. G. Dai and B. Prabhakar, “The throughput of data switches with or without speedup,” INFOCOM 2000, pp. 556-564.

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

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

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.

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.

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

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.

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

Counter-example

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.

Segment #2 Segment #1 PB-wMWM • 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

Modified PB-wMWM • 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 Segment #2 Segment #1

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.

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

Input Queued Switches: Cell Switching vs. Packet Switching