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 1

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

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

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 4

Notation – Arrival rate Input 1 VOQ 11 Switching Fabric Output 1 VOQ 1 N Input N VOQN 1 Output N VOQNN • • • : 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 5

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

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

MWM Stability • Mc. Keown 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. Mc. Keown, 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. 8

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 9

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 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

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

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. 13

Stability of 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 14

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. 15

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

Counter-example 17

Counter-example 18

Counter-example 19

Counter-example 20

Counter-example 21

Counter-example 22

Counter-example 23

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. 24

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

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

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-w. MWM 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. 27

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