Background Concrete Greedy Shapers Min-Plus Algebra Abstract Greedy Shapers Acknowledgment. Greedy Shaper

Transcription

1 Greedy Shaper Modular Performance Analysis and Interface-Based Design for Embedded Real-Time Systems: Section 3.1 od Reading Group on Real-Time Calculus

2 Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

3 The Curves Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

4 The Curves Arrival Curves Number Of events Time Time

5 The Curves Bounds of Curves

6 Drawback Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

7 Drawback Some Drawbacks of Greedy

8 ig. 24: (a) A system architecture (a) a with internal re-shaping to reduce global buffer requirements. (b) A system architecture with external input-shaping to prevent Drawback Some Drawbacks of Greedy 3.1. Greedy Shapers 53 Burstiness of high-priority streams on shared BUS. (a) (b) R 1 R 2 CPU 1 CNI R 1 T 1 R1 σ R 1 1 CPU 2 CNI R 2 T 2 R2 σ R 2 2 Shared BUS R 1 CNI 3 R 2 CNI 4 R 1 R 2 R 3 σ1 σ2 σ3 R 1 R 2 R 3 MPSoC CPU 1 R T 1 R1 R T 2 R2 R T 3 R3

9 Drawback Some Drawbacks of Greedy. Greedy Shapers 53 R 1 R Greedy Shapers 53 Burstiness of high-priority streams on shared BUS. Buffer(a) requirements on subsequent network (b) nodes. (a) Shared (b) BUS Shared CPU 1 BUS MPSoC CPU 1 CNI R 1 MPSoC T 1 R1 CNI R σ R 1 R T 1 R1 σ R 1 CPU 1 1 CPU 1 R 1 R R R 1 R T 1 σ1 1 T 1 σ1 1 CPU R1 2 CPU R1 2 CNI 2 CNI R R 2 T 2 R 2 T 2 R2 σ R 2 R2 σ R R 2 R 2 σ2 R 2 σ2 R 2 R T 2 R2 R 1 R 1 CNI 3 CNI 3 R R 2 2 CNI CNI 4 4 R 3 R 3 σ3 σ3 R 3 R 3 T R2 R 2 T R T 3 R3 R3 ig. 24: (a) A system architecture (a) a with internal re-shaping to reduce (b) bglobal buffer requirements. architecture (b) A system with internal architecture re-shaping with external to reduce input-shaping globaltobuffer prevent ) A system re-

10 Playout Buffer Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

11 the first bit of data arrives, at time d r (0), where d r (0) = lim t 0,t>0 d(t) is the limit to the rig d 3, it is stored in the buffer until a fixed time Δ has elapsed. Then the buffer is served at a c whenever it is not empty. This gives us a second system S, with input R and output S. Cumulative functions are a powerful tool for studying delays and buffers. In order to illustrate the simple playout buffer problem that we describe now. Consider a packet switched netwo bits of information from a source with a constant bit rate r (Figure 1.2) as is the case for circuit emulation. We take a fluid model, as illustrated in Figure 1.2. We have a first system S with input function R(t) = rt. The network imposes some variable delay, because of q therefore the output R does not have a constant rate r. What can be done to recreate a cons? A standard mechanism is to smooth the delay variation in a playout buffer. It operates as f Playout Buffer Playout Buffer 4 J 4 J 5 J 5 5 \ > EJ I 4 J 4 J 5 J J @,, H H 4 Figure 1.2: A Simple Playout Buffer Example

12 the first bit of data arrives, at time d r (0), where d r (0) = lim t 0,t>0 d(t) is the limit to the rig d 3, it is stored in the buffer until a fixed time Δ has elapsed. Then the buffer is served at a c whenever it is not empty. This gives us a second system S, with input R and output S. Cumulative functions are a powerful tool for studying delays and buffers. In order to illustrate the simple playout buffer problem that we describe now. Consider a packet switched netwo bits of information from a source with a constant bit rate r (Figure 1.2) as is the case for circuit emulation. We take a fluid model, as illustrated in Figure 1.2. We have a first system S with input function R(t) = rt. The network imposes some variable delay, because of q therefore the output R does not have a constant rate r. What can be done to recreate a cons? A standard mechanism is to smooth the delay variation in a playout buffer. It operates as f Playout Buffer Playout Buffer 4 J 4 J 5 J 5 5 \ > EJ I 4 J 4 J 5 J J @,, H H 4 r(t d(0) ) R Figure (t) 1.2: r(ta Simple d(0) Playout + ) Buffer Example

13 Cumulative functions are a powerful tool for studying delays and buffers. In order to illustrate the simple playout buffer problem that we describe now. Consider a packet switched netwo bits of information from a source with a constant bit rate r (Figure 1.2) as is the case for circuit emulation. We take a fluid model, as illustrated in Figure 1.2. We have a first system S with input function R(t) = rt. The network imposes some variable delay, because of q therefore the output R does not have a constant rate r. What can be done to recreate a cons? A standard mechanism is to smooth the delay variation in a playout buffer. It operates as f Playout Buffer Playout Buffer 4 J 4 J 5 J 5 5 \ > EJ I 4 J 4 J 5 J J @,, H H 4 r(t d(0) ) R Figure (t) 1.2: r(ta Simple d(0) Playout + ) Buffer Example the first The bit of playout data arrives, buffer at time function d r (0), where σ(t) dis r (0) given = limby: t 0,t>0 d(t) is the limit to the rig d 3, it is σ(t) stored = in r(the buffer d(0) until ) a fixed time Δ has elapsed. Then the buffer is served at a c whenever it is not empty. This gives us a second system S, with input R and output S.

14 Leaky Bucket Shaper Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

15 The bucket has a hole and leaks at a rate of r units of fluid per second when it is not empty. Data from the flow R(t) has to pour into the bucket an amount of fluid equal to the amount of data. Data that Leaky Bucket wouldshaper cause the bucket to overflow is declared non-conformant, otherwise the data is declared conformant. Leaky Bucket Figure illustrates the definition. Fluid in the leaky bucket does not represent data, however, it is counted in the same unit as data. Data that is not able to pour fluid into the bucket is said to be non-conformant data. In ATM systems, non-conformant data is either discarded, tagged with a low priority for loss ( red cells), or can be put in a buffer (buffered leaky bucket controller). With the Integrated Services Internet, non-conformant data is in principle not marked, but simply passed as best effort traffic (namely, normal IP traffic). 4 J 4 J > N J # "! > EJI 4 J N J H! " # $ % & '! " Figure 1.4: A Leaky Bucket Controller. The second part of the figure shows (in grey) the level of the bucket x(t) for a sample input, with r =0.4 kbits per time unit and b =1.5 kbits. The packet arriving at time t =8.6 is not conformant, and no fluid is added to the bucket. If b would be equal to 2 kbits, then all packets would be conformant.

16 Leaky Bucket Shaper Shaping Curve Shaping Curve Shaping curve σ for a leaky bucket greedy shaper with bucket size b and filling rate r: σ( ) = b + r

17 Leaky Bucket Shaper Shaping Curve ǵr σ( ) = b + r b

18 In Figure 21(a) a greedy shaper component ingoing event stream R(t) with a shaping cur Leaky Bucket Shaper the events are emitted on the component s ou Upper Bound for Output Curve: σ outgoing event stream R (t). (a) Δ σ R(t) t σ R (t) t α(δ Fig. 21: (a) A concrete greedy shaper, shaping a concrete e greedy shaper, shaping an abstract event stream.

19 Cascade Leaky Bucket Shaper Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

20 Cascade Leaky Bucket Shaper Shaping Curve of Cascade Leaky Bucket Shaper 0 Chapter 3. Abstract Components R(t) σ b 1, r 1 b 2, r 2 R (t) r 1 r 2 σ 3 b 1 b ig. 22: A greedy shaper that is implemented by a cascade of two leaky bucket stages and the resulting total shaping curve.

21 Cascade Leaky Bucket Shaper Shaping Curve Shaping curve σ for a cascade leaky bucket greedy shaper with bucket sizes b i and filling rates r i : σ( ) = min i {(b i) + r i }

22 Cascade Leaky Bucket Shaper Properties Greedy Shapers Come for Free Shaping by a greedy shaper does not increase delay or buffer requirements. The output event trace R = R σ

23 Brief Introduction Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

24 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ).

25 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = 4 + 3

26 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5

27 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)...

28 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)... In min-plus algebra, we use (R,, +).

29 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)... In min-plus algebra, we use (R,, +). 3 4 = 4 3

30 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)... In min-plus algebra, we use (R,, +). 3 4 = (4 5) = (3 4) 5

31 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)... In min-plus algebra, we use (R,, +). 3 4 = (4 5) = (3 4) 5 (3 4) + 5 = (3 + 5) (4 + 5)...

32 Brief Introduction Brief Introduction fo Min-Plus Algebra Traditionally, we are used to work with (R, +, ) = (4 + 5) = (3 + 4) + 5 (3 + 4) 5 = (3 5) + (4 5)... In min-plus algebra, we use (R,, +). 3 4 = (4 5) = (3 4) 5 (3 4) + 5 = (3 + 5) (4 + 5)...

33 Brief Introduction A little Complex... Let f(t) be a real-valued function, Conventional integral: Min-plus integral: t 0 f(s)ds (1) inf {f(s)} (2) s R,0 s t

34 Brief Introduction More Complex... Conventional convolutions: (f g)(t) = + f(t s)g(s)ds (3)

35 Brief Introduction More Complex... Conventional convolutions: (f g)(t) = + f(t s)g(s)ds (3) if f(t) and g(t) are two functions that are zero for t < 0, (f g)(t) = t 0 f(t s)g(s)ds (4)

36 Brief Introduction More Complex... Conventional convolutions: (f g)(t) = + f(t s)g(s)ds (3) if f(t) and g(t) are two functions that are zero for t < 0, in min-plus algebra: (f g)(t) = t 0 f(t s)g(s)ds (4) (f g)(t) = inf {f(t s) + g(s)} (5) 0 s t

37 Brief Introduction More Complex... Conventional convolutions: (f g)(t) = + f(t s)g(s)ds (3) if f(t) and g(t) are two functions that are zero for t < 0, in min-plus algebra: (f g)(t) = t 0 f(t s)g(s)ds (4) (f g)(t) = inf {f(t s) + g(s)} (5) 0 s t Min-plus deconvolutions: (f g)(t) = sup{f(t + µ) g(µ)} (6) µ>0

38 Some Examples Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

39 Some Examples Some Examples For an arrival curve R that α l (t s) R[s, t) α u (t s), t s 0,

40 Some Examples Some Examples For an arrival curve R that α l (t s) R[s, t) α u (t s), t s 0, Upper bound α u ( ) = sup{r( + µ) R(µ)} = R R (7) µ 0

41 Some Examples Some Examples For an arrival curve R that α l (t s) R[s, t) α u (t s), t s 0, Upper bound α u ( ) = sup{r( + µ) R(µ)} = R R (7) µ 0 Lower bound α l ( ) = inf {R( + µ) R(µ)} = R R (8) µ 0

42 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ(9) σ R O t

43 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R σ(8) O t

44 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R σ(7) O t

45 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R σ(6) O t

46 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R σ(5) O t

47 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(5 ) R R(4 + ) O t

48 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(4) R R(5) O t

49 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(3) R R(6) O t

50 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(3 ) R R(6 + ) O t

51 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(2) R R(7) O t

52 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(2 ) R R(7 + ) O t

53 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ σ(1) R R(8) O t

54 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R O t

55 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R O t

56 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 σ R O t

57 Proof R = R σ R (t) = (R σ)(t) = inf {R(s) + σ(t s)} (9) 0 s t t = 9 R O t

58 Bounds Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

59 ding Greedy Shapers a Bounds greedy shaper component is depicted that shapes an ream R(t) with a shaping curve σ. After being shaped, itted Bounds on the component s output, resulting in a shaped tream R (t). a) (b) 8 Δ σ Δ σ σ R (t) t α(δ) Δ GS α (Δ) Δ edy shaper, shaping a concrete event stream. (b) An abstract ping an abstract α u = event α u stream. σ α l = α l (σ σ) alysis of systems with greedy shapers within the MPA eed to introduce a new abstract component that models, as depicted in Figure 21(b). Here, an abstract event

60 Proof Outline 1 Background The Curves Drawback 2 Concrete Greedy Shapers Playout Buffer Leaky Bucket Shaper Cascade Leaky Bucket Shaper 3 Min-Plus Algebra Brief Introduction Some Examples 4 Abstract Greedy Shapers Bounds Proof

61 Proof α u Use properties: then: (f g) h = f (g h) (f g) g f (g g) α u = R R = (R σ) (R σ) = ((R σ) R) σ = ((σ R) R) σ (σ (R R)) σ = (σ α u ) σ = (α u σ) σ α u σ (10) (11)

62 Proof Proof of The Property 1 CHAPTER 3. BASIC MIN-PLUS AND MAX-PLUS CAL ((f g) h)(t) = sup{(f g)(t + u) h(u)} u 0 { } = sup sup {f(t + u + v) g(v)} h(u) u 0 v 0 { } = sup u 0 { sup f(t + v ) g(v u) } h(u) v u = { sup sup f(t + v ) { g(v u)+h(u) }} u 0 v u { = sup sup f(t + v ) { g(v u)+h(u) }} v 0 0 u v { = sup f(t + v { ) inf g(v u)+h(u) } } v 0 0 u v { = sup f(t + v ) (g h)(v ) } =(f (g h))(t). v 0

63 Proof Proof of The Property = sup 2 3) One computes that { = sup f(t + v { ) inf g(v u)+h(u) } } v 0 0 u v { f(t + v ) (g h)(v ) } =(f (g h))(t). v 0 ((f g) g)(t) = sup{(f g)(t + u) g(u)} u 0 = sup inf {f(t + u s)+g(s) g(u)} u 0 0 s t+u = sup inf u 0 u s t sup inf u 0 0 s t sup inf u 0 0 s t = inf 0 s t { f(t s )+g(s + u) g(u) } { f(t s )+g(s + u) g(u) } { } f(t s )+sup{g(s + v) g(v)} v 0 } {g(s + v) g(v)} { f(t s )+sup v 0 = inf { f(t s )+(g g)(s ) } =(f (g g))(t). 0 s t

64 Proof α l Use properties (proved in A.3): then: (f g) (h j) (f h) (g( )j) (12) α l = R R = (R σ) (R σ) (R R) (σ σ) α l (σ σ) (13)

65 Background ã g' PPT. ã Úy²g5NETWORK CALCULUS6 JEAN-YVES LE BOUDEC and PATRICK THIRAN. Ì SNg5Modular Performance Analysis and Interface-Based Design for Embedded Real-Time Systems63.1!.