02 - Chapter 2 Circuit Switch Design Principles Inputs 1 2...

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Chapter 2 Circuit Switch Design Principles
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Fig. 2.1. An N × N switch used to interconnect N inputs and N outputs 1 2 N 1 2 N . . . . . . Inputs Outputs
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Fig. 2.2. Bar and cross states of 2 × 2 switching elements Bar State Cross State
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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
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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
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Nonblocking Properties : SNB WSNB RNB RNB — Rearrangeably Nonblocking WSNB— Wide-sense Nonblocking SNB — Strictly Nonblocking
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Fig. 2.4. (a) A 4 × 4 rearrangeably nonblocking switch 1 2 3 4 1 2 3 4
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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
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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
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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
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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”.
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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
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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
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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 }
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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
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