02 - Chapter 2 Circuit Switch Design Principles Fig 2.1 An...

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Unformatted text preview: Chapter 2 Circuit Switch Design Principles Fig. 2.1. An N × N switch used to interconnect N inputs and N outputs 1 2 N 1 2 N . . . . . . Inputs Outputs Fig. 2.2. Bar and cross states of 2 × 2 switching elements Bar State Cross State Fig. 2.3. (a) Crossbar switch 1 2 3 4 1 2 3 4 Inputs Outputs Connections: Input 1 to Output 3 Input 2 to Output 4 Fig. 2.3. (b) banyan switch Blocking: Input 2 cannot be connected to output 2 if input 1 is already connected to output 1 1 2 3 4 1 2 3 4 Nonblocking Properties : SNB WSNB RNB RNB — Rearrangeably Nonblocking WSNB— Wide-sense Nonblocking SNB — Strictly Nonblocking Fig. 2.4. (a) A 4 × 4 rearrangeably nonblocking switch 1 2 3 4 1 2 3 4 Fig. 2.4. (b) a connection request from input 4 to output 1 is blocked Fig. 2.4. (c) Same connection request can be satisfied by rearranging the existing connection from input 2 to output 2 1 2 3 4 1 2 3 4 Connection cannot be set up between input 1 and output 4 Connection can now be set up between input 1 and output 4 1 2 3 4 1 2 3 4 Two states corresponding to the same mapping : 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 Output Input Complexity of nonblocking switches : How to build large switch from smaller switches? Problems with two-stage networks : 1 2 m 1 2 m . . . . . . (a) N = mn # lines = m 2 n = mN Bandwidth Expansion factor = m (b) 1 2 m 1 2 m . . . . . . n n An example of one-to-one mapping from input to output. In general, N! possible mappings (“macro states”) 1 2 2 2 3 2 4 2 2 1 2 2 2 3 2 4 Lower Bound on Complexity of Nonblocking Switches How many 2x2 crosspoints are needed to build an N x N nonblocking switch? Lower bound can be obtained by a non-constructive proof based on “number of states”. Number of crosspoints needed for nonblocking switch Argument based on number of micro states >= number of macro states 1 2 N 1 2 N . . . . . . N ! mappings M crosspoints N N N N M N m large for log ! log ! 2 mappings # states # 2 2 ≈ ≥ ≥ ≥ Lower Bound on Complexity of Nonblocking Switches Fig. 2.6. A three-stage Clos switch architecture n 1 × r 2 n 1 × r 2 n 1 × r 2 r 1 × r 3 r 1 × r 3 r 1 × r 3 r 2 × n 3 r 2 × n 3 r 2 × n 3 . . . . . . . . . . . . . . . . . . (1) (2) (r 1 ) (1) (2) (r 2 ) (1) (2) (r 3 ) . . . . . . n 1 r 1 = n 3 r 3 = N for N × N switch r i — # switch modules in column i n 1 — # inputs in column 1 module n 3 — # outputs in column 3 module Necessary condition for nonblocking: 3 1 2 , n n r ≥ Fig. 2.7. An example of blocking in a three-stage switch Key: Find a commonly accessible middle node from both input and output nodes A F G H B 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 A request for connection from input 9 to output 4 is blocked S A = set of middle-stage nodes used by A = { F, G } S B = set of middle-stage nodes used by B = { H } Fig. 2.8. The connection matrix of the three-stage network A B F G H F,G,H 2 1 A r 1 1 2 B r 2 Stage-1 module Stage-3 module Conditions of a Legitimate connection Matrix...
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This note was uploaded on 12/08/2010 for the course IEG IEG4020 taught by Professor Fengshen during the Spring '10 term at CUHK.

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02 - Chapter 2 Circuit Switch Design Principles Fig 2.1 An...

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