Delay bounds (Simon S. Lam) 1

Transcription

1 1 Pacet Scheduling: End-to-End E d Delay Bounds Delay bounds (Simon S. Lam) 1

2 2 Reerences Delay Guarantee o Virtual Cloc server Georey G. Xie and Simon S. Lam, Delay Guarantee o Virtual Cloc Server, IEEE/ACM Transactions on Networing, Vol. 3, No. 6, December End-to-End E Delay Guarantees and Bounds Simon S. Lam and Georey G. Xie, Group Priority Scheduling, IEEE/ACM Transactions on Networing, Vol. Vo. 5, No. 2, April Pawan Goyal, Simon S. Lam, and Harric Vin, Determining End-to-End E Delay Bounds in Heterogeneous Networs, ACM/Springer-Verlag Multimedia Systems, Vol. 5, No. 3, May Delay bounds (Simon S. Lam) 2

3 3 Virtual Cloc (VC) server - review Let r be the reserved rate in bits/s allocated to low. Let p denote a pacet in low, with length l(p) bits and arrival time, A(p) ( 0) Each low has a virtual cloc, priority(), which h is zero initially and updated d whenever a new pacet in low arrives: l( p) priority ( ) max{ priority ( ), A( p)} + (1) r The new value o priority() ) is assigned to pacet p as its virtual cloc value, denoted by P(p) Delay bounds (Simon S. Lam) 3

4 4 VC server review (cont.) Whenever the VC server is ready or another pacet, the pacet among all lows with the smallest virtual cloc value is selected FCFS within a low non-preemptive Meaning o virtual cloc value Consider a hypothetical server with rate r dedicated to low (processor sharing). Let F(p) denote the inishing time o pacet p. In this system, l ( p + 1) F ( p + 1) = max{ F( p), A( p + 1)} + r Virtual cloc value P(p) in previous slide is same as inishing time F(p) in processor sharing Delay bounds (Simon S. Lam) 4

5 Delay Guarantee o VC server VC by itsel does not provide a delay bound in the usual sense, i.e., L(p) A(p) cannot be bounded by the server alone, where L(p) denotes the departure time o pacet p Given admission control, VC can provide a delay guarantee to a low with no assumption that sources are low- controlled or well behaved It is a conditional guarantee based upon a low s reserved rate Delay bounds (Simon S. Lam) 5 5

6 Active lows VC server or a set F o lows, with reserved rate r allocated to low, or in F Deinition 1: A low is active at time t i priority () > t (2) That is, a low is active i its virtual cloc is running aster than real time. The virtual cloc o low is driven by the low s arrivals and pacet lengths only [see eq. (1)] Delay bounds (Simon S. Lam) 6 6

7 7 Active low interpretation At time t, the condition priority() > t holds at the hypothetical ti server (processor-sharing) o low i and only i the hypothetical server is baclogged i.e., there is at least one low pacet waiting or being served During a busy period o the VC server (pacet by pacet), low may be active when there is no low pacet in the system (queue + server); low may be inactive when the system has low pacets Delay bounds (Simon S. Lam) 7

8 8 Capacity constraint condition Deintion 2: Let C denote the capacity, in bits/s, o a VC server. The server s capacity is not exceeded at time t i the ollowing condition holds: a () t r C (3) where a(t), a subset o F, is the set o lows that are active at time t Above condition using a(t) allows more pacets to be admitted than using F but requires admission control or each pacet arrival Delay bounds (Simon S. Lam) 8

9 9 Delay guarantee theorem Theorem 1: I the capacity o a VC server has not been exceeded or a non-zero duration since the start o a busy period, then the ollowing holds or every pacet p served during the busy period: lmax L ( p ) P ( p ) + (4) C where l is the maximum pacet size max (Proo by contradiction) This bound is analogous to Pareh and Gallager s bound relating PGPS to GPS Delay bounds (Simon S. Lam) 9

10 10 Observations abouttheorem 1 The theorem holds without any assumption that sources are well-behaved or lowcontrolled. While each low, say, has a reserved rate r, its source can generate pacets at a much larger rate than r. Pacets can have arbitrary arrival times and pacet lengths The delay guarantee in (4) holds even i the lows in F generate pacets at an aggregate rate larger than C [assuming each low is allocated its own buers] It is a conditional guarantee based on the low s reserved rate Delay bounds (Simon S. Lam) 10

11 11 VC server needs to enorce the capacity constraint t The set o active lows, a(t), changes dynamically. At any time the server can determine whether low is active by comparing priority() with the current time t. Thus the server can determine the set a(t) In theory, the server may perorm admission control upon the arrival o each pacet to prevent violation o its capacity constraint Delay bounds (Simon S. Lam) 11

12 12 Practical admission controls Without a priori nowledge o source arrival characteristics: Flow sources are statically ti assigned reserved rates. Thus or all time t r r C a() t F On-demand assignment per session. A source can generate traic only ater it has been assigned a reserved rate on demand: r r C a() t D() t where D(t) is the set o sessions ss at time t with assigned reserved rates Delay bounds (Simon S. Lam) 12

13 13 Practical admission controls (more) On-demand assignment per application data unit (burst) All pacets o an application data unit are admitted or discarded together Examples: (i) a picture in a video, (ii) a chun o a video Given a priori nowledge o source traic statistics, a server can over-commit its capacity, maing use o the act that only some lows are active at any time Statistical rather than deterministic guarantee Delay bounds (Simon S. Lam) 13

14 14 Properties o VC delay guarantee A delay guarantee o the orm L(p) P(p) + β, where β is a constant, is a conditional guarantee A pacet s delay is bounded rom its expected arrival time based upon its reserved rate, that is, arriving early does not help On the other hand, pacets arriving late do not get better service or not using their low s reserved rate This encourages on-time arrivals, which are much better or buer management in a switch (router) Delay bounds (Simon S. Lam) 14

15 15 Delay Bounds Consider a low with reserved rate r and pacets indexed by n = 1, 2, in order o arrival. I P(n)-A(n) is bounded above, then the delay o pacet n, L(n)-A(n), is bounded above The goal o source control is to bound P(n) A(n) Method 1: Ensure that the inter-arrival time o two consecutive pacets is bounded below, ln ( ) ln ( ) An ( + 1) An ( ) Pn ( ) = An ( ) + r r max ( ) Ln ( ) Pn ( ) + l Thus, Ln ( ) An ( ) l n + l C r C max Delay bounds (Simon S. Lam) 15

16 16 Leay Bucet source control Let Arrival ( t1, t2) denote the number o low bits that arrive at server in interval [t 1, t 2 ] Bits o a pacet arrive when the last bit o the pacet arrives The arrival unction consists o a jump at each pacet arrival time and is right continuous, Arrival ( t, t) is the size o the pacet that arrives at time t Flow conorms to Leay Bucet with bucet size σ and rate ρ i and only i Arrival ( t, t ) σ + ρ ( t t ) or 0 t t Delay bounds (Simon S. Lam) 16

17 17 Delay Bounds (cont.) ( σ, ρ) Method 2: Source control by Leay Bucet with bucet size σ and toen arrival rate ρ equal to the reserved rate o the low, r It is proved [Goyal, Lam, Vin 1997] that or all n P ( n ) A ( n ) σ ρ σ Thus, we have Ln ( ) An ( ) + r For Spring semester 2017, sip ahead to slide 40 l max C Delay bounds (Simon S. Lam) 17

18 18 Generalizing Theorem 1 The delay guarantee theorem holds as long as the server capacity is not exceeded by the sum o the reserved rates o lows that t have had a pacet served during the busy period. We can change the active low deinition (to admit more pacets) Deinition 1 : A low is active at time t i (i) the system is not empty, (ii) a low pacet has arrived in the current busy period, (iii) priority() > t Delay bounds (Simon S. Lam) 18

19 19 Generalizing Theorem 1 by another way We can achieve the generalization, without changing the active low deinition, by resetting the virtual cloc o each low to zero at the end o a busy period. Speciically, when the system becomes empty upon a pacet departure, then or all in F, priority() <- 0 Side eect: Resetting means that the debt o low is orgotten (orgiven) Good or bad? Delay bounds (Simon S. Lam) 19

20 20 Deterministic End-to-End Delay Guarantees and Bounds Delay bounds (Simon S. Lam) 20

21 21 End-to-End Delay Guarantee Delay bounds (Simon S. Lam) 21

22 22 Notation or server Delay bounds (Simon S. Lam) 22

23 23 Guaranteed Deadline (GD) Server It provides the ollowing service to low e.g., a Virtual Cloc server Delay bounds (Simon S. Lam) 23

24 24 Traversing a path o GD servers = β + τ,+1 Delay bounds (Simon S. Lam) 24

25 25 Notation or low Delay bounds (Simon S. Lam) 25

26 26 Reerence cloc values or low v V () i () i V a time constant associated with pacet i (seconds) reerence cloc value o pacet i at server, to be interpreted as expected inishing time o pacet i at server (0) is 0 by deinition and or i 1 V ( i) = max{ V ( i 1), A ( i)}+ v ( i) (3) For the special case o VC servers v ( i) = s ( i) / λ ( i) where λ ( i) is reserved rate or pacet i in low The virtual cloc value o pacet i at server is its reerence cloc value and or all i P () i = V () i Delay bounds (Simon S. Lam) 26

27 27 Important relations Each server ensures that pacet i in low departs by its deadline which is P () i + β A ( i) depends on P ( i) as shown in (2) + 1 V ( i) depends on A ( i) as shown in (3) For a VC server,, we now that P () i = V () i What about other scheduling disciplines? Delay bounds (Simon S. Lam) 27

28 28 Putting the relations together Consider pacet i in low. Its end to end delay is A () i V () i 1 1 A () 1 i A () i A () i K σ ρ <- source control A () i () A + 1 i K () A i A () 1 i K K+1 source L () i L L + i () 1 () i + 1 LK () i P () i + β A () i = L () i + τ P () i + α + 1, + 1 β K + destination <- deadline guarantee by server is delay due to non-preemption Delay bounds (Simon S. Lam) 28

29 29 Dierence between a pacet s deadline and reerence cloc value For each GD server, suppose we now, or all i, P () i V () i Proo in [Lam and Xie 1997] by induction Recall that v s ( j) ( j) = and λ ( j) α = β + α +, 1 Delay bounds (Simon S. Lam) 29

30 30 End-to-end delay guarantee theorem Proo in [Lam and Xie 1997] by applying Lemma 2 recursively Delay bounds (Simon S. Lam) 30

31 31 End-to-end delay upper bound AK ( j ) A ( j ) The end-to-end delay o pacet j is K The end-to-end guarantee in (5) provides an upper bound on the end-to-end delay i a source control mechanism is used such that and V ( j) A ( j) 1 1 has a inite upper bound, server is a GD server, 1 K, such that the term P ( j ) V ( j ) has a inite upper bound Delay bounds (Simon S. Lam) 31

32 32 Example o source control I the source o low is controlled by a leay bucet with bucet depth σ and toen rate ρ = λ, which is the reserved rate o low, then or all pacet i in the low [Goyal, Lam, Vin 1997] To obtain an end-to-end delay upper bound or low, is instantiated to in the delay guarantee ormula o Theorem 1. Delay bounds (Simon S. Lam) 32

33 33 End-to-end delay upper bound (cont.) Note that dierent service disciplines may be chosen or dierent servers along a path also, the term P ( j) V ( j) may be positive or negative With Theorem 1, the problem o providing an upper bound on the end-to-end ddelay or a low along a path o heterogenous servers is reduced to a set o singlenode problems! Delay bounds (Simon S. Lam) 33

34 34 Examples o GD servers The GD class o servers is general including many pacet schedulers in the literature. They dier in their P (.) unctions, β constants, and admission control conditions We will show our: Virtual Cloc WFQ/PGPS Delay-EDD Leave-in-Time Delay bounds (Simon S. Lam) 34

35 35 1. Virtual Cloc (VC) as a GD server Delay bounds (Simon S. Lam) 35

36 36 2. WFQ/PGPS as a GD server The pacet deadline at server is, using P ( j) = L, GPS( j), s L ( j) L ( j ) rom Pareh and Gallager age [1993] max, PGPS j, GPS j + C where L ( j) is the departure time o being a GPS server GPS, { } with weights w g, g F At GPS server, consider the rate, λ where b ( t) is the set o baclogged lows, to be the reserved rate or low wc g w g b () t Set o lows must be inite or the reserved rate to be nonzero or all t Delay bounds (Simon S. Lam) 36

37 37 2..WFQ/PGPS as a GD server (cont.) From Goyal, Lam, and Vin [1997] eq. (7) s ( j ) LGPS, ( j) max{ LGPS, ( j 1), A ( j)} +. λ V ( j) = max{ V ( j 1), A ( j)} + v (j) rom eq. (3) s ( j) where v ( j)=. λ Thereore, P ( j) = L ( j) V ( j), GPS Finally, P ( j) V ( j) 0 Delay bounds (Simon S. Lam) 37

38 38 3. Delay-EDD as a GD server > 0 Delay bounds (Simon S. Lam) 38

39 39 4. Leave-in-Time as a GD server > 0 Delay bounds (Simon S. Lam) 39

40 40 End-to-end delay bound or a path o servers sthat tuse VC or WFQ/PGPS scheduling Using Eq. (5) o Theorem 1 together with leay bucet source control, we get σ A i A i K ( j) K K+ 1() 1 () + ( 1)max + 1 j i ρ ρ =1= 1 σ + ( K 1)max 1 j is ( j) K α ρ = 1 = +, + 1 s α A tight bound where ρ λ and α = β + τ, where β = s max C Delay bounds (Simon S. Lam) 40

41 41 The end-to-end delay bound o Pareh and Gallager A. Pareh and R. Gallager, A generalized processor-sharing approach to low control in integrated services networs: the multiple node case, IEEE/ACM Trans Networing, April 1994, equation (39). Previously nown bound or rate-proportional processor sharing (RPPS) rate assignment o PGPS networs when the sources conorm to Leay Bucet is σ + 2( K 1) s K max ρ = 1 max + α a loose bound where s is the maximum pacet size or low Delay bounds (Simon S. Lam) 41

42 42 Summary End-to-end delay bounds can be provided to lows by a variety o GD servers Delay guarantee theorem Decomposition o the end-to-end delay bound problem into single-node and source control problems Both VC and WFQ/PGPS servers provide the best delay bound (a tight bound) VC is easier to implement WFQ is slightly more air Delay bounds (Simon S. Lam) 42

43 43 The end Delay bounds (Simon S. Lam) 43