Analysis of Input Queueing

1. Analysis of Input Queueing • More complex system to analyze than output queueing case. • In order to analyze it, we make a simplifying assumption of "heavy load", i.e. all queues are always full. • This is a worst-case assumption.

2. head-of-line (HOL) blocking Winning packet Input Queues Outputs Losing packet 3 1 1 2 1 Internally Nonblocking Switch 2 cannot access output 2 because it is blocked by the first packet 4 3 3 4

3. Internally Nonblocking Switch:Dropping packets 0= Pr[ carry a packet ] 1 2 3 2 Pr[ carry a packet ] =p for large N For p=1, 0= 0.632

4. the fictitious output queues used for analysis Output 1 Fictitious Output Queues formed by HOL packets (3,2) (1,2) Output 2 (2,3) Output 3 (4,4) Output 4 (input, output) Outputs (1,1) (1,2) 1 (2,1) (2,3) Internally Nonblocking Switch 2 (3,2) (3,2) 3 (4,1) (4,4) 4

5. N 2 3 4 5 * 0.75 0.68 0.66 0.64 • How about small N? * : the maximum throughput with input queueing • Simulation Results with Large N

6. Throughout of Input-Buffered Switch • Consider a fictitious queue i = # packets at start of time slot m. = # packets arriving at start of time slot m. • is Poisson and independent of as N 

7. e.g. Fictitious Queue i 1 i i 1 i i 1 2 2 2 3 i time slot m time slot m-1

8. – – under saturation –

9. Input Queue For finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped. Must keep p0 < p* Meaning of Saturation Throughput p0 =   p = throughput

10. Queuing scenario for the delay analysis of the input-buffered switch Fictitious Queues Output 1 Input Queue 1/N 1/N N 2 Output 2 HOL 1/N Time spent in HOL are independent for successive packets when N is large Output N Service times at different fictitious queues are independent

11. The busy periods and interpretations for delay analysis of an input queue U(t) X3  X0   X1  X2 X0 t Idle period Y Busy period Busy period Arrivals here are considered as arrivals in intervals i-2 Arrivals here are considered as arrivals in intervals i-1 Xi-1 Xi

12. Illustration of the meanings of random variables used in the delay analysis of an input queue mi =2 prior arrivals Arrival of the packet of focus. One simultaneous arrival to be served before the packet; L=1. Departure of packet of focus. (1) (1) (2) Xi Xi+1 Ri W -- Packet arrival in interval i. -- packet departure in interval i+1. (n) -- number of arrivals

13. Three Selection Policies • Random Selection Policy • If k packets are addressed to a particular output, one of the k packets is chosen at random, each selected with equal probability 1/k. • Longest Queue Selection Policy • The controller sends the packet from the longest queue • Fixed Priority Selection Policy • The N inputs have fixed priority levels and of the k packets, the controller send the one with highest priority

14. Different contention-resolution policies have different waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity. _ W p0

15. Conclusion • Mean queue length are always greater for queueing on inputs than on outputs • Output queues saturate only as the utilization approaches unity • Input queues saturate at a utilization that depends on N, but is approximately 0.586 when N is large