1. 048866: Packet Switch Architectures Dr. Isaac Keslassy Electrical Engineering, Technion isaac@ee.technion.ac.il http://comnet.technion.ac.il/~isaac/ Statistical Analysis of Output-Queued Switches

2. Spring 2006048866 – Packet Switch Architectures2 Where we are  We have studied output-queued switches from a deterministic point of view  We have obtained tools for statistical analysis  Now: statistical analysis of output- queued switches  What is the average output queue size?

3. Spring 2006048866 – Packet Switch Architectures3 Output-Queued Switch

4. Spring 2006048866 – Packet Switch Architectures4 Assumptions  Time is slotted  At each time-slot, at each of the N inputs: Bernoulli IID packet arrivals with probability   Each packet is destined for one of the N outputs uniformly at random

5. Spring 2006048866 – Packet Switch Architectures5 Notations  Consider some output.  Let A t and D t be the arrivals to (departures from) this output at time-slot t.  Let X t and Y t be the queue size at this output before (after) departures at time- slot t. time A1A1 A2A2 D1D1 D2D2 X1X1 Y1Y1 Y2Y2 X2X2 t=1t=2t=3

6. Spring 2006048866 – Packet Switch Architectures6 Recurrence Equation  Departures:  Recurrence equation:

7. Spring 2006048866 – Packet Switch Architectures7 Arrivals  A t is the total number of packet arrivals to a given output at time-slot t.  Packet arrivals from each input are Bernoulli I.I.D. with probability  /N.  Therefore A t is I.I.D. with

8. Spring 2006048866 – Packet Switch Architectures8 Poisson Limit  As N ! 1 and  is constant, binomial goes to Poisson:

9. Spring 2006048866 – Packet Switch Architectures9 Steady-State Equations  Taking expectations of (1):  Steady-state:  Using

10. Spring 2006048866 – Packet Switch Architectures10 Steady-State Equations  Taking expectations of (1) squared:  A t+1 independent of X t.  Steady-state:  Using (2):

11. Spring 2006048866 – Packet Switch Architectures11 Steady-State Equations  Finally:  Note:   = 0.8: EX=2.4 packets (and EY=1.6)   = 0.9: EX=4.95 packets (and EY=4.05)

12. Spring 2006048866 – Packet Switch Architectures12 Average Output Queue Size