2. 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.

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

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

5. 1 2 3 4 Outputs Internally Nonblocking Switch (input, output) Fictitious Output Queues formed by HOL packets (1,2)(1,1) (2,3)(2,1) (3,2) (4,4)(4,1) (1,2)(3,2) (2,3) (4,4)Output 4 Output 3 Output 2 Output 1 the fictitious output queues used for analysis the fictitious output queues used for analysis

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

7. –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   – Throughout of Input-Buffered Switch

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

9. – under saturation – –

10. Meaning of Saturation Throughput p 0 =  p = throughput For finite buffer size, if p 0 > p * = 0.586 at least (p 0 - p * )/ p 0 fraction of packets are dropped. Must keep p 0 < p * Input Queue

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

12. X0X0 X3X3 X2X2 X 1  X 0  Busy period Idle period Busy period Y t U(t) Arrivals here are considered as arrivals in intervals i-2 Arrivals here are considered as arrivals in intervals i-1 X i-1 XiXi The busy periods and interpretations for delay analysis of an input queue

13. m i =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. XiXi X i+1 RiRi W -- Packet arrival in interval i. -- packet departure in interval i+1. -- number of arrivals(n) (1) (2) Illustration of the meanings of random variables used in the delay analysis of an input queue

14. 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

15. W _ p0p0 Different contention-resolution policies have different waiting time versus load relationships, but a common maximum load at which waiting time goes to infinity.

16. 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