1. 89-850 Communication Networks Network Calculus: Deterministic Traffic Analysis Last updated: Thursday, May 07, 2015 Prof. Amir Herzberg Dept of Computer Science, Bar Ilan University http://AmirHerzberg.com

2. Network Calculus: Topics 1.Introduction Voice call processing example Network calculus Fixed-rate service and Reich’s equation Min-Plus Convolution 2.Arrival curves 3.Service curves  Backlog, delay bounds, IntServ routers (Latency- Rate servers), Shapers

3. Voice Call Processing & Delays Sound sampled and encoded  Constant Bit Rate (CBR), e.g. 64Kbps (8000*8b) Voice activity detection (and compression?)  Discard (long, e.g. >0.2s) inactive periods  On-off Variable Bit Rate (VBR) Question: what link rate C is needed?

4. Voice Call Processing & Delays (cont’) Sound sampled and CBR-encoded Voice activity detection, compression  VBR Shaping  Specifications of sent traffic  Allows efficient rates, buffers Packetization (and padding if necessary)

5. Voice Call Processing & Delays (cont’) Sender: Sampling and CBR-encoding  Voice Activity Detection, compression  VBR  Shaping  Packetization (and padding if necessary) Network (queues, transmission, propagation) Receiver  Immediate or delayed playout?

6. Voice Transport over Packet Network Deterministic vs. stochastic analysis

7. Playout Delay to `Hide` Jitter Lines: actual packet delays Is output Ok? What size of buffer (backlog)?

8. Playout Buffer Model Network system S:  CBR input: R(t)=rt  Delay variation (jitter) bounded by Δ  Namely for some D>0 holds: R(t) ≦ R*(t+D+Δ) and R*(t+D-Δ) ≦ R(t) Playout buffer S’:  Receive first bit at d(0), delay it by Δ  Then, serve at constant rate r, when not empty   if never empty after d(0), then S(t)=r(t-Δ-d(0)) +

9. Playout Buffer Analysis Claim: for every t>d(0) holds 0<R*(t)-S(t)<2Δr  Playout buffer always non- empty, produces at rate r  Buffer size of 2Δr is sufficient (to avoid overflow) Proof:  (omitted)

10. Network Calculus Deterministic analysis of scheduling/ queuing / flow Uses min, max as binary operators and integrals (min-plus. max-plus algebra)  Worst-case  wasteful [cf. probabilistic queuing analysis – later]  Applicable to many problems Voice, audio, data, …  Only `traffic flow`, no processing/control Discrete or continuous time (bit or fluid model, resp.)

11. Network Calculus: References and Sources Network Calculus by J. Le Boudec, P. Thiran, Springer, Aug 2001, LNCS 2050  In library  Online: http://ica1www.epfl.ch/PS_files/netCalBookv4.pdf Communication Networking – an analytical approach, Kumar, Manjunath and Kuri, Elsevier, 2004 (mainly ch. 4)  In (folder) in library

12. Network Calculus Network Calculus: Worst-case analysis of Arrivals, Service, Buffering and Departures Used in design of QoS protocols  Internet: mainly IntServ, DiffServ  ATM Arrivals A(t) Departures D(t)Service Buffer X(t)

13. Buffer Analysis: Fixed Rate Queue Bits sent (removed) at C bits/second (unless empty) a k : arrival of (complete) packet k L k  [0,∞) : bit-length of packet k A(t), D(t): total arrivals (deliveries) till time t; A(0 - )=0  t - : just before time t X(t): queued bits at time t X(t)

14. Reich’s Equation: Work- conserving schedule with output rate C t A(t) X(t) A(s) time v(t) this line has slopeC, and valueA(s) +C. (t  s) att s Proof:

15. Reich’s Equation: Let v be earliest time after which buffer was not empty: Hence:

16. Reich’s Equation: Illustration t A(t) X(t) A(s) time b this line has slopeC, and valueA(s) +C. (t  s) att s D(time)

17. Reich’s Equation: Departures

18. Departures and Convolutions A(s)=0 for s<0, and (t-s) + =0 for s≥t, hence… Definition: The convolution A*B of two non-decreasing functions A(t), B(t):    + is: Note: compare to linear systems theory convolution: Two equivalent notations: A*B, A ⊗ B

19. Convolution Operation - Visualization slopeρB(t) s t C t ρ τ t C A(t) (A * B)(t) A(τ ) + B(t  τ)τ) B(t-τ) A(t)=Ct +

20. Convolution Operator: Properties Convolution operator * is… Commutative: A*B=B*A Associative: (A*B)*C=A*(B*C) Distributive over min function: A*min{B,C} = = min {(A*B), (A*C)} The convolution A*B is: slopeρ τ t C A(t) (A * B)(t) A(τ ) + B(t  τ)τ)

21. Convolutions and the δ (delay) function The convolution A*B is: Define the delay function as: Since A(t) is non-decreasing: B(t)≥0 is causal if B(t)=0 for t<0; If A, B are causal then…

22. Network Calculus: Topics 1.Introduction 2.Arrival curves  Arrival curve and min-plus convolution  Leaky bucket  Good arrival curves are sub-additive  Minimal arrival curve and min-plus deconvolution 3.Service curves  Backlog, delay bounds, IntServ routers (Latency- Rate servers), Shapers

23. Cumulative flows Cumulative flow R(t)  F, t real (or integer) F = { x(t) | x(t) is non decreasing and x(t) = 0 for t < 0 } Models: time t bits 12567 time t bits 155.5 R 1 (t)R 2 (t) Fluid model (continuous)Packet model (left continuous)

24. Arrival Curves Flow R has arrival curve (envelope)  iff ∀ 0  s  t:  R(t) -R(s)   (t-s) and (x<0)(α(x)=0) Example 1: affine arrival curve  r,b  (t) =  r,b (t) = rt+b for t>0, r,b≥0 (Example 2, stair arrival curve  kv T,   : hidden next foil) time bits b slope r R(t)

25.  (t) = kv T,  (t) = k  (t+  )/T  with T = period,  = tolerance, k = constant packet size Characterizes flows that are periodic stream of packets of same size k (cells), that suffers a variable delay <=  time bits k kv T,  (t) = k  (t+  )/T   T-  2T-  3T-  2k 3k 4k Example 2: stair arrival curve  kv T,  o All packets of size k. Then R conforms to  =kv T,   R conforms to  =  r,b with r = k/T and b = k(  +T)/T

26. Leaky Bucket Conforming Arrivals Leaky Bucket (LB) conformance test: Parameters: rate r, bucket size b Initially: bucket  0 (empty) Upon packet arrival:  If bucket+|packet| ≦ b then {bucket+=|packet|; mark packet `conforming`} else mark packet `non-conforming’; Every 1/r seconds: if bucket>0 then bucket--; R(t) r x(t) b R(t)

27. Affine Arrivals are LB-Conforming Let R* be leaky-bucket conforming arrivals R* has affine arrival curve  (t) =  r,b (t) = rt+b for t>0, r,b≥0 If R conforms to (has envelope)  (t), then R*=R  All packets of R are conformant R(t)  r,b r x(t) b R(t) slope r b

28. GCRA (T,  ) All packets (cells) of flow R are of the same size k Arrival time of nth = A n Theoretical arrival just after nth arrival is  n = max(A n,  n-1 ) + T If A n+1 >=  n –  then cell is conformant, otherwise not Example: GCRA (10,2) n123345  n-1 01121213141 A n 11116202938 ccncccnc  Equivalences: R conforms to  GCRA (T,  )  R conforms to staircase arrival curve  = kv T,   R conforms to  leaky bucket (r = k/T, b = k(  +T)/T)  R conforms to  affine arrival curve  =  r,b

29. Min-Plus Convolution  Flow R has envelope (arrival curve)  iff ∀ 0  s  t:  R(t) -R(s)   (t-s) and (x<0)(α(x)=0) The min-plus convolution of R andαis: Flow R has arrival curve α,iff R ≦ R ⊗ α R has arrival curve    all 0  s  t, R(t) -R(s)   (t-s)  R(t)  R(s) +  (t-s) for all 0  s  t  R(t)  inf u { R(u) +  (t-u) }  R  R 

30. Min-plus convolution  (cont’) Definition (f  g) (t) = inf u { f(t-u) + g(u) } t f(t) g(t) (f  g)(t)

31. Enveloped Arrivals Process A(t) has envelope E if for every t, s (s  t) holds A(t)-A(s)  E(t-s)  A has envelope E if and only if A  A*E  Envelopes are non-decreasing, non-negative If A, E are causal (A(t)=0 for t<0, E  δ 0 ) then:  A*E  A* δ 0 = A  A*E   A=A*E If arrivals with envelope E pass via element with lower service curve S, define `max delay` as:

32. Some properties of min-plus convolution (f  g)  F  is associative  is commutative Neutral element:  0 : f  0 = f (  0 (t) = 0 for t = 0 and  0 (t) =  for t > 0)  is distributive with respect to min (   is isotone: f  f’ and g  g’  f  g  f’  g’ Functions passing through origin (f(0) = g(0) = 0): f  g  f  g Concave functions passing through the origin: f  g = f  g

33. Example: rate latency function  T (delay function)  R,T (rate-latency function) latency T  R,T (t) Slope (rate) R  delay T  T (t)  R (t)=Rt Rate R R (delay function)

34. Sub-Additive Functions Recall cumulative flow functions  A function f  F is sub-additive iff: If flow has arrival curve f  F, then it also has sub-additive arrival curve  In particular, the `sub-additive closure`  Focus on sub-additive arrival curves Mostly, `good` functions: sub-additive and f(0)=0

35. Affine Arrivals are Sub-Additive affine arrival curve:  r,b (t) = rt+b for t>0, r,b≥0 Is sub-additive (and `good`):  r,b (s+t) = r(s+t)+b ≦ ≦ rs+b+rt+b =  r,b (s)+  r,b (t) time bits b slope r

36. Sub-additivity and arrival curves T k 2T 3T 2k 3k 4k 4T T k 2T 3T 2k 3k 4k 4T Flow R has arrival curve    , the sub-additive closure, is arrival curve What is the sub-additive closure ?  Use min-plus convolution 

37. Sub-additive Closure The sub-additive closure of an arrival curve, is a (sub-additive) arrival curve f  F, f(0)=0  f ≥ f ⊗ f ≥ 0 Repeat… sequence of smaller and smaller functions: is sub-additive with (`good`) is sub-additive with f (0) = 0  = f  f = f  f

38. Examples time bits T  R,T (t)+K’ R K’ time bits T  R,T (t)+K’’ R K’’ 2T 2K’  R,T (t)+K’) (2) 2T  R,T (t)+K’ ’  R,T (t)+K’

39. Combining Two Leaky Buckets time bits b M slope r slope p time bits b slope r IntServ Standard Traffic Specifications:  T-Spec 4-tuple: (p,M, r, b)  Peak rate p (link speed), packet size M `Store and forward`: entire packet `received at once’  Burst size b, long-term average rate r  (u) = min (pu+M, ru+b) = (pu+M)  (ru+b) Exercise: show  (u) is sub-additive (and `good`)

40. Combining Two Arrival Curves (1) Generalize: Flow R has arrival curves  ’  ∀ 0  s  t:  R(t) -R(s)   (t-s), R(t) -R(s)   ’(t-s) Clearly: R has arrival curve  ’  E.g., two affine arrival curves (leaky bucket) But sometimes there is a tighter curve!

41. Combining Arrival Constraints (2) Flow R has arrival curves  ’ Clearly: R has arrival curve  ’ Example:  t)=  r,b (t),  ’(t)= v T,  (t)  Pick r=1, b=0, T=2,  =0…  Is  ’ sub-additive? time bits slope 1  1,0 (t) time bits 1 V 2,0 (t) [stair function] 2 3 4  46  ’  ’(2.5)=2  ’( 2)=1  ’( 0.5)=0.5

42. Combining with Sub-Additive Closure Flow R has arrival curves  ’  also  ’ Also arrival curves:  ⊗  ’ and  The sub-additive closure of  ’ If  ’ are `good`, then  ⊗  ’= time bits slope 1  1,0 (t) time bits 1 V 2,0 (t) [stair function] 2 3 4  46  ’

43. Minimal arrival curve Assume we only know arrival rate R (empirically) How can we compute the (minimal) arrival curve  ? The answer uses min-plus deconvolution R

44. Min-plus deconvolution Ø Definition (f Ø g) (t) = sup u { f(t+u) - g(u) } t f(t) g(t) (f  g)(t)

45. Properties of min-plus deconvolution (f  g)  F in general (f  f)  F (f  f) is sub-additive with (f  f) (0) = 0 (f  g)  h = f  (g  h) Duality with  : f  g  h  f  g  h

46. Minimal arrival curve The minimal arrival curve of flow R is  R Ø R . Proof:  It is an arrival curve because R(t) – R(s) = R((t-s)+s) - R(s)  sup u { R((t-s)+u) - R(u) } = (R Ø R) (t-s) l If  ’ is another arrival curve for flow R, then R  R  ’ [use duality rule: f  g  h  f  g  h ]   R Ø R  ’. ∎

47. Contents 1.Introduction 2.Arrival curves 3.Service curves, backlog, delay bounds  Service curve: definition  Backlog and delay bounds  Shapers

48. Goal of Service Curve and GR node define an abstract node model independent of a specific type of scheduler applies to real routers, which are not a single scheduler, but a complex interconnection of delay and scheduling elements applies to nodes that are not work-conserving

49. Service Curve System S has (minimal/lower) service curve , if  F, and for any arrival process A(t), the delivery process D(t) satisfies: D≥ A ⊗   Assuming system is full (some backlog) S A(t)D(t) t s A(s) A( ) D( )  (t)

50. Service Curve System S has (minimal/lower) service curve , if  F, and for any arrival process A(t), the delivery process D(t) satisfies: D≥ A ⊗   Assuming system is full (some backlog)  If  is continuous, for all t exists some s ≦ t such that D(t) - A(s)   (t-s) [see hidden foil]  (t) S A(t)D(t) t s A(s) A( ) D( )

51. Equivalence of Service Curve Defs Equivalence of min-plus convolution and rates definitions Exists some 0  s  t : y(t) - x(s)   (t-s)  y(t)  x(s) +  (t-s) for some 0  s  t  y(t)  inf u { x(u) +  (t-u) }  y  x 

52. Constant rate server  (t)=ct Proof: take s = beginning of busy period: y(t) – y(s) = c (t-s) and y(s) = x(s) -> y(t) – x(s) = c (t-s) buffer s t0 ct t

53. Rate-latency service curve Standard model for Internet (IntServ) router: (lower) service curve S(t)= Rt + ⊗ δ T E.g.: fixed rate, but (max) packet size L  T=L/R in this case T bits R seconds

54. WFQ Scheduler is a Latency Rate Server Notations:  Bound on length of packets  GPS departure time of k th packet (any queue)  WFQ departure time of same packet WFQ Delay Theorem [4.3 in KMK]: GPS deliveries for non-empty queue j : Hence GPS, WFQ have lower service curves: Latency Rate Server

55. Example 1: Constant-Rate Server Infinite buffer, output rate r `Fluid model` service curve S(t)=rt + (Lower) service curve S(t)=(rt + -L)=rt + * δ L / r A(t)D(t) S(t) S(t) r sloper sloper L r

56. Example 2: Coder and Constant-Rate Server Coder output bursts (can be < L bits), rate r Packetizer outputs packets: rate r, max delay L /r  D p (t)≥A(t-L /r), i.e. lower service curve is D(t)=(D p *(r t + )) ≥ (A*δ L/r )*(r t + ) Latency rate servers: rate r, delay (latency) d  Lower service curve S(t)=δ d *(rt + )=r(t-d) +  Typical `Per-Hop Behavior` service specifications A(t)D(t) D p (t) r Coder Packetizer L

57. The service curve of a Greedy shaper is its shaping curve x(t)  Shaper y(t) If  is sub-additive and  y(t) = (x  (t). The service curve of a shaper is thus 

58. Fixed-delay node: service curve  T seconds  T x y 0 T  T (t) Function  T t

59. Min-plus deconvolution Ø Definition (f Ø g) (t) = sup u { f(t+u) - g(u) } t f(t) g(t) (f  g)(t)

60. Tight Bound on Backlog If flow has arrival curve  and node offers service curve  then the backlog is at most: sup (  (s) -  (s)) = (  Ø  )(0) = v( ,  )  Proof: simple application of definitions of curves   δ(s) v(,)v(,) s Recall (f Ø g) (t) = sup u { f(t+u) - g(u) }

61. Tight Bound on Delay If flow has arrival curve  and node has service curve  then delay  h( ,  )=sup s { δ (s)}, δ (s)= inf{ τ≥0:  (s) ≦  (s+τ)}  Proof: omitted.   δ(s) v(,)v(,) s

62. Theorem: Delay of Enveloped Arrivals Given causal arrivals A with envelope E into element with lower service curve S(t), the departures D satisfy: Namely, delay is at most  Where: Proof: Since S is lower service curve: D≥A*S By definition of : Hence: A=A*E, since E is envelope of (causal) A Claim follows (from associativity of convolution *)

63. Leaky Bucket via IntServ Router Flow with LB-arrivals, A(t)=b+rT Latency-rate server (e.g. router): B(t)=R(t-T) + If r ≤ R: buffer ≦ b+rT, delay ≦ T+b/R  Burstiness of flow increased by rT. If r > R, the bounds are infinite. data Arrivals A(t)=b+rt b T b+rT T+b/R Latency-rate Service curve: B(t)=R(t-T) +

64. Leaky Bucket via WFQ Flow with LB-arrivals, A(t)=b+rt Latency-rate server (e.g. router): S(t)=R(t-T) + If r ≤ R: buffer ≦ b+rT, delay ≦ T+b/R  Burstiness of flow increased by rT. If r > R, the bounds are infinite. For WFQ: data Arrivals A(t)=b+rt b T B+rT T+b/R Latency-rate Service curve: S(t)=R(t-T) + Details: Network Calculus lecture (or [KMK/BT]

65. Service Curves: not just lower… Delay often bounded by service curves:  S(t) is a (lower/minimal) service curve if D≥A*S  S(t) is an upper service curve if D  A*S  S(t) is a service curve if D=A*S  S(t),S(t): non-negative, non-decreasing, causal

66. The composition theorem Theorem: the concatenation of two network elements each offering (lower) service curve  i offers the (lower) service curve  1   2       Holds for lower, upper service curvers

67. Cascade of Service Elements Traffic passes thru several `elements` Each has upper and lower service curves Composing: Composite lower service curve: Similarly, composite upper service curve: A(t) D 1 (t)D 2 (t) S 1 ( t ) D n (t) S 1 (t) S 2 (t) S 2 (t) S n (t) S n (t)

68. Cascade of Latency Rate Servers Latency rate servers: rate r, delay (latency) d  Lower service curve of k th server: S k (t)= r k (t-d k ) + A(t) D 1 (t)D 2 (t) S 1 ( t ) D n (t) S 1 (t) S 2 (t) S 2 (t) S n (t) S n (t)

69. R1R2 T2  = T1 Example: Cascade of Two Latency-Rate Servers (IntServ Routers) R1 T2T1+T2

70. Pay Bursts Only Once  D1D1 D2D2   D   D  b /R + T 1 + T 2 end to end delay bound is less!! D  b /R + T 1 + T 2 end to end delay bound is less!! D 1 +D 2  (2b + rT 1 )/ R + T 1 + T 2 Recall, LB via router: delay ≦ T+b/R b out =b in +RT

71. Regulators (aka Shapers) How to ensure envelope E?  Regulator: departures envelope is E (any arrivals) A regulator with envelope E satisfies:  Arrivals A(t), Departures D(t), and D(t)  A(t)  Departures envelope is E (any arrivals)  For all s  t holds D(t)-D(s)  E(t-s)  D  D*E

72. Regulators (Shapers) Lemma Def: A regulator with envelope E satisfies:  Arrivals A(t), Departures D(t), and D(t)  A(t)  Departures envelope is E (any arrivals)  For all s  t holds D(t)-D(s)  E(t-s)  D  D*E Lemma: Proof:

73. Leaky Bucket Regulator / Shaper D(t)-D(s)  σ+ρ(t-s)  Such process is (σ,ρ)-regulated, or (σ,ρ)-process  D has envelope E(t)= σ+ρt for t≥0, E(t)=0 for t<0  Or use notation: E(t)= (σ+ρt)I [t≥0]  D(0)=0 for t<0  D, E are causal   D=D*E Called leaky-bucket (LB) regulator / shaper σ ρ A(t) D(t) X 1 (t) X 2 (t) Source buffer Token bucket

74. Leaky Bucket Shaper Operation [fluid model - ignore packetization] LB becomes empty buffer becomes empty buffer nonempty amount of data in the buffer departure process time σ LB full LB nonempty amount of tokens in LB Bucket is full

75. LB Shaper: Properties, Operation Denote peak (input=output) rate by R D(t)-D(s)  σ+ρ(t-s) Bucket size σ, average rate ρ Max burst (at rate R ): σR/(R-ρ) LB becomes empty buffer becomes empty buffer nonempty amount of data in the buffer time σ LB nonempty σ

76. LB Shaper Theorem: D=A*E Lemma: Claim: Consider any time t.  If source buffer at t is empty then:  Source buffer non-empty at t  bucket empty at t  Let v(t) be last time before t bucket was full  At t all tokens were used for departures, so: A(t): arrivals to LB shaper with envelope E(t)= σ+ρt for t>0, E(t)=0 for t  0

77. Cascade of Leaky Buckets Shapers Theorem: D 2 =D 1 =A*E Associativity: D 2 =A*(E*E) E*E  E, since E is causal (E(t)=0 for t<0) E  E*E  Holds for any subadditive E [for all s  t holds E(t)  E(s)+E(t-s)]  E.g. for Leaky Bucket…  E=E*E D 2 =A*E=D 1 A(t) D 1 (t)D 2 (t) E(t)= =(σ+ρt)I [t≥0] E(t)