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The Transmission-Switching Duality of Communication Networks Tony T. Lee Shanghai Jiao Tong University The Chinese University of Hong Kong Xidian University, June 21, 2011

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A Mathematical Theory of Communication BSTJ, 1948 C. E. Shannon

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Contents Introduction Routing and Channel Coding Scheduling and Source Coding

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Reliable Communication Circuit switching network Reliable communication requires noise-tolerant transmission Packet switching network Reliable communication requires both noise-tolerant transmission and contention-tolerant switching

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Quantization of Communication Systems Transmissionfrom analog channel to digital channel Sampling Theorem of Bandlimited Signal (Whittakev 1915; Nyquist, 1928; Kotelnikou, 1933; Shannon, 1948) Switchingfrom circuit switching to packet switching Doubly Stochastic Traffic Matrix Decomposition (Hall 1935; Birkhoff-von Neumann, 1946)

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Noise vs. Contention Transmission channel with noise Source information is a function of time, errors corrected by providing more signal space Noise is tamed by error correcting code Packet switching with contention Source information f(i) is a function of space, errors corrected by providing more time Contention is tamed by delay, buffering or deflection Message=0101 Connection request f(i)= j Delay due to buffering or deflection

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Transmission vs. Switching SourceTransmitter Channel capacity C Receiver MessageSignal Received signal Shannons general communication system Temporal information source: function f(t) of time t Spatial information source: function f(i) of space i=0,1,…,N-1 Clos network C(m,n,k) 0 k-1 0 m-1 0 k-1 kxkmxnnxm o n-1 o N-n N-1 N-n N-1 Source Input moduleCentral moduleOutput module Internal contention Channel capacity = m Destination Noise source

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Noise Channel Coding Source Coding Contention Routing Scheduling Clos Network Communication Channel

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Apple vs. Orange 350mg Vitamin C 1.5g/100g Sugar 500mg Vitamin C 2.5g/100g Sugar

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Contents Introduction Routing and Channel Coding Scheduling and Source Coding Rate Allocation Boltzmann Principle of Networking

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Output Contention and Carried Load Nonblocking switch with uniformly distributed destination address ρ: offered load ρ: carried load The difference between offered load and carried load reflects the degree of contention 0 1 N-1 0 1

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Proposition on Signal Power of Switch (V. Benes 63) The energy of connecting network is the number of calls in progress ( carried load ) The signal power S p of an N×N crossbar switch is the number of packets carried by outputs, and noise power N p =N- S p Pseudo Signal-to-Noise Ratio (PSNR)

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Boltzmann Statistics a b c d n 0 = 5 n 1 = 2 n 2 = 1 ad b,c Micro State Packet: Energy Quantum Output Ports: Particles n i = number of outputs with energy level packets are distinguishable, the total number of states is, energy level of outputs = number of packets destined for an output. Number of Outputs 10r nnnN

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Boltzmann Statistics (contd) From Boltzmann Entropy Equation Maximizing the Entropy by Lagrange Multipliers Using Stirlings Approximation for Factorials Taking the derivatives with respect to n i, yields S: Entropy W: Number of States C: Boltzman Constant

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Boltzmann Statistics (contd) If offered load on each input is ρ, under uniform loading condition Probability that there are i packets destined for the output Carried load of output Poisson distribution

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Clos Network C(m,n,k) n(I+1)-1 D = nQ + R D is the destination address Q = D/n --- output module in the output stage R = [D] n --- output link in the output module G is the central module Routing Tag (G,Q,R) (n+1)Q-1 S n(k-1) 0 0 I k-1 m-1 Q 0 n-1 nI nk-1 Input stageMiddle stageOutput stage D k-1 n x m k x k m x n G 0 n m-1 0 G G 0 G 0 I k-1 m-1 0 G 0 G 0 I I k m-1 Q Q Q 0 0 n-1 n(k-1) nk-1 G R 0 n-1 nQ nQ+R 0 Slepian-Duguid condition m n

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Clos Network as a Noisy Channel Source state is a perfect matching Central modules are randomly assigned to input packets Offered load on each input link of central module Carried load on each output link of central module Pseudo signal-to-noise ratio (PSNR)

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Noisy Channel Capacity Theorem Capacity of the additive white Gaussian noise channel The maximum date rate C that can be sent through a channel subject to Gaussian noise is C: Channel capacity in bits per second W: Bandwidth of the channel in hertz S/N: Signal-to-noise ratio

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Planck's law can be written in terms of the spectral energy density per unit volume of thermodynamic equilibrium cavity radiation.energy density

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Clos Network with Deflection Routing Route the packets in C(n,n,k) and C(k,k,n) alternately Encoding output port addresses in C(n, n, k) Destination: D = nQ 1 + R 1 Output module number: Output port number: Encoding output port addresses in C(k, k, n) Destination: D = kQ 2 + R 2 Output module number: Output port number: Routing Tag = (Q 1,R 1, Q 2,R 2 ) 0 k-1 0 n-1 0 k-1 0 n-1 kxknxn kxk C(n, n, k) C(k, k, n)

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Loss Probability versus Network Length The loss probability of deflection Clos network is an exponential function of network length The loss probability of deflection Clos network is an exponential function of network length

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Shannons Noisy Channel Coding Theorem Given a noisy channel with information capacity C and information transmitted at rate R If R

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Binary Symmetric Channel The Binary Symmetric Channel(BSC) with cross probability q=1-p½ has capacity There exist encoding E and decoding D functions If the rate R=k/n=C-δ for some δ>0. The error probability is bounded by If R=k/n=C+ δ for some δ>0, the error probability is unbounded p p q q

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Parallels Between Noise and Contention Binary Symmetric Channel Deflection Clos Network Cross Probability q<½Deflection Probability q<½ Random CodingDeflection Routing RCRnRn Exponential Error ProbabilityExponential Loss Probability Complexity Increases with Code Length n Complexity Increases with Network Length L Typical Set DecodingEquivalent Set of Outputs

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Edge Coloring of Bipartite Graph A Regular bipartite graph G with vertex-degree m satisfies Halls condition Let A V I be a set of inputs, N A = {b | (a,b) E, a A}, since edges terminate on vertices in A must be terminated on N A at the other end.Then m|N A | m|A|, so |N A | |A|

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Route Assignment in Clos Network S=Input D=Output G=Central module Computation of routing tag (G,Q,R) 0 1 2

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Rearrangeabe Clos Network and Channel Coding Theorem (Slepian-Duguid) Every Clos network with mn is rearrangeably nonblocking (Slepian-Duguid) Every Clos network with mn is rearrangeably nonblocking The bipartite graph with degree n can be edge colored by m colors if mn There is a route assignment for any permutation Shannons noisy channel coding theorem Shannons noisy channel coding theorem It is possible to transmit information without error up to a limit C.

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LDPC Codes Low Density Parity Checking (Gallager 60) Bipartite Graph Representation (Tanner 81) Approaching Shannon Limit (Richardson 99) x0x0 x1x1 x2x2 x3x V L : n variablesV R : m constraints x 1 +x 3 +x 4 +x 7 =1 Unsatisfied x4x4 x5x5 x6x6 x7x x 0 +x 1 +x 2 +x 5 =0 Satisfied x 2 +x 5 +x 6 +x 7 =0 Satisfied x 0 +x 3 +x 4 +x 6 =1 Unsatisfied Closed Under (+) 2

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Benes Network Bipartite graph of call requests x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 G(V L X V R, E) x 1 + x 2 =1 x 3 + x 4 =1 x 5 + x 6 =1 x 7 + x 8 =1 x 1 + x 3 =1 x 6 + x 8 =1 x 4 + x 7 =1 x 2 + x 5 =1 Input Module Constraints Output Module Constraints Not closed under +

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Flip Algorithm Assign x 1 =0, x 2 =1, x 3 =0, x 4 =1…to satisfy all input module constraints initially Unsatisfied vertices divide each cycle into segments. Label them α and β alternately and flip values of all variables in α segments x 3 +x 4 = x 1 +x 2 =1 x 5 +x 6 =1 x 7 +x 8 =1 x 1 +x 3 =0 x 6 +x 8 =0 x 4 +x 7 =1 x 2 +x 5 =1 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 Input module constraints Output module constraints variables

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Bipartite Matching and Route Assignments Call requests Bipartite Matching and Edge Coloring

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Contents Introduction Routing and Channel Coding Scheduling and Source Coding

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Concept of Path Switching Traffic signal at cross-road Use predetermined conflict-free states in cyclic manner The duration of each state in a cycle is determined by traffic loading Distributed control N S WE Traffic loading: NS : 2ρ E W: ρ NS traffic EW traffic Cycle

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Call requests Connection Matrix

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Time slot 1 Time slot 2 Path Switching of Clos Network

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Capacity of Virtual Path Capacity equals average number of edges G1G1 G2G2 Time slot 0 Time slot 1 Virtual path G 1 U G

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Contention-free Clos Network 0 k-1 0 m-1 0 k-1 kxkmxnnxm Input bufferOutput buffer Predetermined connection pattern in every time slot Central module (nonblocking switch) Output module (output queued Switch) Input module (input queued switch) o n-1 o o o Buffer and scheduler Input module i Buffer and scheduler Input module j λ ij Source Destination Scheduling to combat channel noise Buffering to combat source noise Virtual path

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Complexity Reduction of Permutation Space Reduce the complexity of permutation space from N! to K Convex hull of doubly stochastic matrix Subspace spanned by K base states {P i } K min{F, N 2 -2N+2}, the base dimension of C

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BvN Capacity Decomposition and Sampling Theorems Packet switchingDigital transmission Network environment Time slotted switching systemTime slotted transmission system Bandwidth limitation Capacity limited traffic matrixBandwidth limited signal function Samples Complete matching, (0,1) Permutation matrixes Entropy, (0,1) Binary sequences Expansion Birkhoff decomposition (Halls matching theorem) Fourier series

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BvN Capacity Decomposition and Sampling Theorems Packet switchingDigital transmission Inversion by weighted sum by samples Reconstruction the capacity by running sum Reconstruction the signal by interpolation Complexity reduction Reduce number of permutation from N! to O(N 2 ). Reduce to O(N), if bandwidth is limited. Reduce to constant F if truncation error of order O( 1 / F ) is acceptable. Reduce infinite dimensional signal space to finite number 2tW in any duration t. QoS Buffering and scheduling, capacity guarantee, delay bound Pulse code modulation (PCM), error-correcting code, data compression, DSP

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Source Coding and Scheduling Source coding: A mapping from code book to source symbols to reduce redundancy Scheduling: A mapping from predetermined connection patterns to incoming packets to reduce delay jitter

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Scheduling of a set of permutation matrices generated by decomposition The sequence,,……, of inter-state distance of state P i within a period of F satisfies Smoothness of state P i Smoothness of Scheduling with frame size F PiPi PiPi PiPi PiPi PiPi F

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Entropy of Decomposition and Smoothness of Scheduling Any scheduling of capacity decomposition Entropy inequality The equality holds when (Krafts Inequality)

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Smoothness of Scheduling A Special Case If K=F, Ф i =1/F, and n i =1 for all i, then for all i=1, …,F Another Example Smoothness The Input Set The Expected Optimal Result P1P1 P2P2 P1P1 P3P3 P1P1 P2P2 P1P1 P4P4

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Optimal Smoothness of Scheduling Smoothness of random scheduling Kullback-Leibler distance reaches maximum when Always possible to device a scheduling within 1/2 of entropy

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(Krafts Inequality) Source Coding Theorem Necessary and Sufficient condition to prefix encode values x 1,x 2,…,x N of X with respective length n 1,n 2,…n N Any prefix code that assigns n i bits to x i Always possible to device a prefix code within 1 of entropy

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Huffman Round Robin (HuRR) Algorithm Initially set the root be temporary node P x, and S = P x …P x be temporary sequence. Apply the WFQ to the two successors of P x to produce a sequecne T, and substitute T for the subsequence P x …P x of S. If there is no intermediate node in the sequence S, then terminate the algorithm. Otherwise select an intermediate node P x appearing in S and go to step 2. Step1 Step2 Step3 PYPY PXPX 0.25 P1P1 0.5 P2P P3P3 P4P4 P5P5 PZPZ Huffman Code logarithm of interstate time = length of Huffman code

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Performance of Scheduling Algorithms P1P1 P2P2 P3P3 P4P4 RandomWFQWF 2 QHuRREntropy Better Performance

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Routing vs. Coding Noisy channel capacity theorem Noisy channel coding theorem Error-correcting code Sampling theorem Noiseless channel Noiseless coding theorem Random routing Deflection routing Route assignment BvN decomposition Path switching Scheduling Clos networkTransmission Channel

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Transmission-Switching Duality Boltzmann Equation S = k logW Permutation Matrix Entropy Clos Network Noisy Channel Route Assignment Channel Coding Halls Matching Theorem (BvN Decomposition) Bandlimited Sampling Theorem Scheduling and Buffering Source Coding Communication System

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Law of Probability Input signal to a transmission channel is a function of time The main theorem on noisy channel coding is proved by law of large number Input signal to a switch is a function of space Both theorems on deflection routing and smoothness of scheduling are proved by randomness

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