Chapter 9

1. IEG4020 Telecommunication Switching and Network Systems Chapter 9 Packet Switching and Information Transmission

2. Reliable Communication Circuit-switched network: • Reliable communication requires noise-tolerant transmission Packet-switched network: • Reliable communication requires both noise-tolerant transmission and contention-tolerant switching Quality of service QoS: • Determined by buffer delays and packet losses

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

4. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding

5. Duality of Switching and Transmission • 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 Connection request f(i) = j 0111 0001 Message=0101 0101 0100 1101 Delay due to buffering or deflection

6. Comparison of Transmission and Switching Shannon’s general communication system Received signal Message Signal Source Transmitter Channel capacity C Receiver Destination Temporal information source: function f(t) of time t Noise source Clos network C(m,n,k) Source Destination Input module Central module Output module nxm kxk mxn o o 0 0 0 n-1 n-1 Spatial information source: function f(i) of space i=0,1,…,N-1 N-n k-1 m-1 k-1 N-n N-1 N-1 Channel capacity = m Internal contention

7. Analogies between packet switching and information transmission Clos network Transmission channel • 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

8. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding

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

10. 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 Sp of an N ×N crossbar switch is the number of packets carried by outputs, and noise power Np = N - Sp • Pseudo Signal-to-Noise Ratio (PSNR)

11. Boltzmann Statistics 0 0 n0 = 5 1 3 4 6 7 a 1 1 b 2 2 0 5 n1 = 2 a d 3 3 c Micro State 2 n2 = 1 b,c 4 4 5 5 d Output Ports: Particles 6 6 Packet: Energy Quantum 7 7 energy level of outputs = number of packets destined for an output. ni = number of outputs with energy level packets are distinguishable, the total number of states is, = + + + N n n n Number of Outputs L 0 1 r

12. Boltzmann Statistics (cont.) • From Boltzmann Entropy Equation • Maximizing the Entropy by Lagrange Multipliers • Using Stirling’s Approximation for Factorials • Taking the derivatives with respect to ni, yields • S: Entropy • W: Number of States • C: Boltzman Constant

13. Boltzmann Statistics (cont.) • 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

14. Sum of i.i.d. random variables if output i is busy otherwise Contention as Pseudo Gaussian Noise • Noise power: • Signal power: • Sp and Np are Normal rv - Central limit theorem • Signal power Sp is normally distributed • Mean: E[Sp] = Nρ’, Variance: Var[Sp] = Nρ’(1- ρ’) • Noise power Npis also normally distributed • Mean: E[Np] = N(1-ρ’), Variance: Var[Np] = Nρ’(1- ρ’)

15. 0 0 n-1 n-1 Clos Network C(m,n,k) k x k n x m m x n 0 0 0 0 0 0 • 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) 0 0 0 G G I Q n-1 n-1 m-1 k-1 k-1 m-1 D 0 0 nI 0 0 I G S I Q G nQ k-1 0 0 k-1 n(I+1)-1 n-1 m-1 Q G R nQ+R m-1 n-1 (n+1)Q-1 n(k-1) 0 n(k-1) 0 0 0 0 0 m-1 k-1 k-1 I G G Q nk-1 nk-1 n-1 m-1 m-1 n-1 k-1 k-1 Input stage Middle stage Output stage Slepian-Duguid condition m ≥ n

16. 0 0 1 1 2 2 Connection Matrix 0 0 0 0 0 1 Call requests 1 2 2 3 3 1 1 1 4 4 5 5 6 6 2 2 2 7 7 8 8 0 1 2 0 1 2

17. 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)

18. 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

19. Tradeoff between Bandwidth and PSNR Circuit switching with nonblocking routing Packet switching with random routing

20. Transmission Channel Clos Network Noise Coding Contention Routing

21. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding

22. nxn kxk nxn kxk 0 0 0 0 k-1 n-1 k-1 n-1 C(n, n, k) C(k, k, n) Encoding output port addresses in C(k, k, n) Destination: D = kQ2 + R2 Output module number: Output port number: Encoding output port addresses in C(n, n, k) Destination: D = nQ1 + R1 Output module number: Output port number: Clos Network with Deflection Routing • Route the packets in C(n, n, k) and C(k, k, n) alternately Routing Tag = (Q1,R1, Q2,R2)

23. Example of Deflection Routing Deflection output output 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 0 2 2 1 2 3 0 0 3 1 1 0 4 1 4 2 1 2 5 5 2 C(3,3,2) C(2,2,3)

24. q q Q R O p p Input Analysis of Deflection Clos Network • Markov chain of deflection routing in C(n, n, n) network Output p Probability of success q = 1-p Probability of deflection

25. 1 0.8 0.6 0.4 0.2 0.0 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 Loss Probability Versus Network Length 0 -2 -4 -6 -8

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

27. Shannon’s Noisy Channel Coding Theorem • Given a noisy channel with information capacity C and information transmitted at rate R • If R < C, there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. • If R > C, the probability of error at the receiver increases without bound.

28. 0 0 q q 1 1 p Binary Symmetric Channel • The Binary Symmetric Channel(BSC) with cross probability q = 1-p < 1/2 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

29. Parallels Between Noise and Contention

30. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding

31. A A Hall’s Marriage Theorem Let G be a bipartite graph with input set VI, and edge set E. There exists a perfect matching f: VI → VO, if and only if for every subset A ⊂ VI, |NA| ≥ |A| where NA is the neighborhood of set A, NA = {b | (a,b) ∈ E, a∈A} ⊆ VO For subset A, |A|=2 |NA| = 4

32. Edge Coloring of Bipartite Graph • A Regular bipartite graph G with vertex-degree m satisfies Hall’s condition • Let A ⊆ VI be a set of inputs, NA = {b | (a,b) ∈ E, a∈A} , since edges terminate on vertices in A must be terminated on NA at the other end. Then m|NA| ≥ m|A|, so |NA| ≥ |A|

33. Route Assignment in Clos Network 0 0 0 0 1 0 1 2 2 1 1 3 3 1 4 4 2 2 5 5 6 6 2 3 3 7 7 Computation of routing tag (G,Q,R)

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

35. Gallager Codes • Low Density Parity Checking (Gallager 60) • Bipartite Graph Representation (Tanner 81) • Approaching Shannon Limit (Richardson 99) VL: n variables VR: m constraints 0 x0 x1+x3+x4+x7=1 + Unsatisfied x1 1 x2 0 x0+x1+x2+x5=0 + x3 0 Satisfied 1 x4 x2+x5+x6+x7=0 + x5 1 Satisfied Closed Under (+)2 x6 0 x0+x3+x4+x6=1 + x7 1 Unsatisfied

36. If for every S⊂VL Expander Codes • Expander Graph G(VL, VR, E) • VL: k-regular with |VL| = n • VR: 2k-regular with |VR|=n/2 • There exists α>0, such that for every S⊂VL • Distance (C(G))> αn • Decoding Algorithm (Sipser-Speilman 95) • The Algorithm can remove up to (αn)/2 errors |S| < αn |NS| > (k/2)|S| If there is a vertex v ∈ VL such that most of its neighbors (checks) are unsatisfied, flip the value of v. Repeat

37. Benes Network Bipartite graph of call requests 1 0 1 2 2 3 3 4 4 5 1 5 6 6 7 7 8 8 G(VLX VR, E) x1 + x1 + x2 =1 x2 x3 + x4 =1 + Input Module Constraints x3 x5 + x6 =1 + x4 x7 + x8 =1 + Not closed under + x5 x1 + x3 =1 + x6 x6 + x8 =1 Output Module Constraints + x7 x4 + x7 =1 + x8 x2 + x5 =1 +

38. Bipartite Matching and Route Assignments 1 1 2 2 Call requests 3 3 4 4 5 5 6 6 7 7 8 8 1 1 2 2 3 3 4 4 Bipartite Matching and Edge Coloring

39. Flip Algorithm • Assign x1= 0, x2 = 1, x3 = 0, x4 = 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 x1 0 x2 x1+x3=0 + + x1+x2=1 1 x3 0 x3+x4=1 x6+x8=0 + + x4 1 x5 0 x5+x6=1 x4+x7=1 + + x6 1 x7 x7+x8=1 x2+x5=1 0 + + x8 1 Input module constraints Output module constraints variables

40. Final Route Assignments Apply the algorithm iteratively to complete the route assignments 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

41. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding • Conclusion

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

43. Path Switching of Clos Network 0 0 0 0 0 1 1 2 2 3 3 1 1 1 4 4 5 5 6 6 2 2 2 7 7 8 8 0 1 2 0 1 2 0 0 1 1 2 2 Time slot 2 Time slot 1

44. Capacity of Virtual Path 1.5 0 0 0.5 1 • Capacity equals average number of edges 0.5 2 1 1 Time slot 0 0.5 Virtual path 0 0 1 0.5 2 2 1.5 1 1 2 2 G1 Time slot 1 G1 U G2 0 0 1 1 2 2 G2

45. nxm kxk mxn 0 0 0 k-1 m-1 k-1 Noiseless Virtual Path of Clos Network Input module (input queued switch) Central module (nonblocking switch) Output module (output queued Switch) o o n-1 n-1 o o n-1 n-1 Input buffer Predetermined connection pattern in every time slot Output buffer λij Source Buffer and scheduler Input module i Input module j Buffer and scheduler Destination Virtual path Scheduling to combat channel noise Buffering to combat source noise

46. Complexity Reduction of Permutation Space Subspace spanned by K base states {Pi} • Reduce the complexity of permutation space from N! to K Convex hull of doubly stochastic matrix • K ≤ min{F, N2-2N+2}, the base dimension of C • K ≤ F ≤(BN)/m, if C is bandlimitedcij ≤ B/F • K ≤ F can be a constant independent of N if round-off error of order • 1/F is acceptable

47. Parallels Between Sampling Theorems

48. Parallels Between Sampling Theorems

49. Contents • Duality of Noise and Contention • Parallel Characteristics of Contention and Noise • Clos Network with Deflection Routing • Route Assignments and Error-Correcting Codes • Clos Network as Noiseless Channel-Path Switching • Scheduling and Source Coding

50. 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