A Theory of Traffic Regulators for Deterministic Networks with Application to Interleaved Regulators

Transcription

1 1 A Theoy of Taffic Regulatos fo Deteministic Netwoks with Application to Inteleaved Regulatos Jean-Yves Le Boudec axiv: v1 [cs.ni] 25 Jan 2018 Abstact We define the minimal inteleaved egulato, which genealizes the Ugency Based Shape that was ecently poposed by Specht and Samii as a simple altenative to pe-flow eshaping in deteministic netwoks with aggegate scheduling. With this egulato, packets of multiple flows ae pocessed in one FIFO queue; the packet at the head of the queue is examined against the egulation constaints of its flow; it is eleased at the ealiest time at which this is possible without violating the constaints. Packets that ae not at the head of the queue ae not examined until they each the head of the queue. This egulato thus possibly delays the packet at the head of the queue but also all following packets, which typically belong to othe flows. Howeve, we show that, when it is placed afte an abitay FIFO system, the wost case delay of the combination is not inceased. This shaping-fo-fee popety is well-known with peflow shapes; supisingly, it continues to hold hee. To deive this popety, we intoduce a new definition of taffic egulato, the minimal Pi-egulato, which extends both the geedy shape of netwok calculus and Chang s max-plus egulato and also includes new types of egulatos such as packet ate limites. Incidentally, we povide a new insight on the equivalence between min-plus and max-plus fomulations of egulatos and shapes. Index Tems Netwok Calculus, FIFO Systems, Regulatos, Shapes I. INTRODUCTION We ae inteested in Fist-In-Fist-Out (FIFO o FIFOpe-class systems, as they ae found in Ethenet o outed netwoks with delay guaantees, see fo example the cuent time sensitive netwoking (TSN goup of IEEE [1] o the detnet goup of IETF [2]. In FIFO netwoks, the bustiness of a flow inceases at evey hop whee it shaes a queuing point with othe flows of its class, in diect elation to the bustiness of these flows [3]. The inceased bustiness of this flow may, in tun, incease the bustiness of othe flows in the same class. This ceates a positive feedback loop, which causes lage wost-case delays in FIFO pe-class netwoks [4]. Anothe consequence is that computing wostcase delays in FIFO netwoks is a vey difficult execise [5], [6]. An altenative appoach is to avoid cascades of inceased bustiness by e-shaping evey flow at evey hop. Howeve, this typically equies pe-flow queuing at evey hop, which defeats the pupose of FIFO netwoks. Specht and Samii intoduced in [7] a simple altenative, unde the name Ugency Based Schedule (UBS. With a UBS, the packet at the head of the queue is examined against the egulation constaints of its flow; it is eleased at the ealiest time whee this is possible without violating the constaints. Packets that ae not at the head of the queue ae not examined until they each the head of the queue. The egulation constaints ae eithe the Length Rate Quotient (LRQ ule, o a leaky bucket constaint (we explain the details in Section III. The motivation is to avoid pe-flow queuing, which is peceived as expensive, while keeping peflow state, which is inexpensive. Using tajectoy analyses, Specht and Samii compute a bound on the packet delay fo the combination of one FIFO node and one UBS. In this pape, we intoduce the concept of Minimal Inteleaved Regulato, which extends the concept of UBS to a vey lage class of egulation ules. Like the UBS, a minimal inteleaved egulato examines only the packet at the head of the queue; it possibly delays this packet but also, due to FIFO, all following packets, which typically belong to othe flows. Howeve, we show that, when a minimal inteleaved egulato is placed afte an abitay FIFO system, the wost case delay of the combination is the same as without the inteleaved egulato. This shaping-fo-fee popety is well-known with pe-flow shapes and pe-flow sevice cuve elements [8]; supisingly, it continues to hold with minimal inteleaved egulatos and FIFO systems that handle all flows as a single aggegate. UBS assumes that flows ae egulated eithe using LRQ o a leaky bucket constaint. Now the fome is an instance of Chang s g-egulato [9], which uses max-plus algeba (see Section III-B, wheeas the latte is an instance of an aival cuve constaint, which uses min-plus algeba (see Section III-C. It is known that a g-egulation constaint is not equivalent to an aival cuve constaint [10]. In contast, in [11] Liebehe shows that thee is an isomophism between the min-plus and max-plus epesentations of egulation constaints, and of the associated egulatos (called shapes. Theefoe the nonequivalence between Chang s g-egulato and aival cuve constaints is not explained by the choice a max-plus o min-plus algeba. Indeed, the diffeence is whethe taffic constaints ae expessed at an abitay packet aival time (as with Chang s g-egulation constaint o at an abitay point in time o space (as with aival cuves o the equivalent maxplus envelope of Liebehe. In Theoem 1 we give a new esult which claifies the elation between these viewpoints. This motivates the definition of a new fomalism fo egulatos that encompasses both g-egulation constaints and aival cuves. Moe pecisely, we intoduce in Section III the concept of Pi-egulaity and in Section IV of minimal Π egulato, adapted to the context of FIFO systems. We show that these concepts contain as special cases the classical geedy shape of netwok calculus [12] as well as Chang s minimal egulato [9]. We show that it also contains othe egulatos, which cannot be expessed in these classical famewoks, such as the TSN packet ate egulato. In Section V we intoduce and analyze inteleaved egulatos and show ou main esult (Theoem 4 on the maximum

2 2 latency induced by minimal inteleaved egulatos. Section II descibes the notation and povides some backgound esults. Poofs of lemmas and theoems ae in appendix. II. NOTATION AND GENERAL PRE-REQUISITES A. Packet Sequences and FIFO Systems We use a notation simila to Chang s maked point pocess notation in [9]. We conside a abitay FIFO systems that have packet sequences as input and output. In some cases the input and output packet sequences belong to one single flow, but in geneal we ae inteested in packets sequences whee packets may belong to diffeent flows. N = {0, 1, 2,...} and N + = {1, 2, 3,...}. R + = [0, is the set of non-negative eal numbes. F is the set of sequences such that A F wheneve A = (A 1, A 2,... with A n [, + fo n N +. F inc is the subset of F of wide-sense inceasing sequences, i.e. A F inc if and only if A F and A n A n+1 fo all n N +. G is the set of positive intege-valued sequences. i.e. L G wheneve L = (L 1, L 2,... with L n N + fo n N +. x y denotes the maximum of x and y, fo x, y [, + ]. Wheneve h R and A F, A + h and h + A denote the sequence A such that A n = A n + h fo all n N +. Fo A, A in F the notation A A means that A n A n fo all n N +. The supemum of an empty set is. The infimum of an empty set is. The summation of an empty set is 0. 1 {C} is equal to 1 when the condition C is tue and is equal to 0 othewise. Fo x R, x is the ceiling of x, namely the smallest intege x; x is the floo of x, namely the lagest intege x. A geneal packet sequence is a tiple such as (A, L, F whee: 1 The fist element A is the sequence of packet dates, i.e. the time instants A = (A 1, A 2,... at which the packets ae obseved. We assume that packet numbeing follows chonological ode; simultaneous packet obsevation times in the same sequence ae possible. Thus we equie that A F inc. Depending on the context, we may denote the sequence of packet dates with A (fo aivals o D (fo depatues etc. 2 The second element is a sequence of packet lengths L = (L 1, L 2,... with L G. Packet lengths ae counted in some abitay data unit, typically in bytes o wods of a fixed numbe of bytes. 3 The thid element is a sequence of flow numbes F = (F 1, F 2,... with F n N +, and n N +. In othe wods, F n = f means that packet n belongs to flow f. To avoid cumbesome notation, we assume without loss of geneality that the set of flow numbes is finite and that the subsequence of packets of flow f is infinite, fo evey f. With this notation, we can expess a FIFO system as a system that maps a given input packet sequence (A, L, F to an output packet sequence (D, L, F such that A D. When a packet sequence is fo a single flow, we will simply descibe it as a couple such as (A, L. B. Notation fo Flows Inside a Packet Sequence In Section V we need some specific notation fo flows inside a packet sequence. Given a sequence F of flow numbes, we define I( as the function that etuns the index of packet n in its flow. In othe wods: I(n = cad { m N + : m n and F m = F n } (1 We also define the function ind( such that ind(f, i is the index in the packet sequence of the i th packet of flow f; in othe wods: ind(f, i = n (F n = f and I(n = i (2 Note that the functions I( and ind( depend on the packet sequence F but we leave out the dependency on F fo the sake of simplicity in notation. When a flow f is pesent in a packet sequence (A, L, F we define A f [esp. L f ] as the subsequence extacted fom A [esp. L] by keeping only the packet dates [esp. lengths] coesponding to a packet of flow f, namely A f i = A ind(f,i, Lf i = L ind(f,i fo all i N (3 Fo example, assume the sequence of flow numbes is F = (3, 4, 1, 2, 1, 3..., i.e. the fist packet belongs to flow 3 (F 1 = 3, the second to flow 4 (F 2 = 4, etc. Packets 3 and 5 belong to flow 1, packet 5 is the second packet of flow 1 so ind(1, 2 = 5, I(5 = 2, A 1 = (A 3, A 5,... and A 3 = (A 1, A 6,... C. Pseudo-Inveses Let f( be a wide-sense inceasing function R + R +. Let f ( : R + [0, ] be its lowe pseudo-invese, defined by [9], [11] f (x = inf {s 0 such that f(s x} (4 = sup {s 0 such that f(s < x} (5 The lowe pseudo-invese is the same as the pseudo-invese in [12]. Similaly, let f ( : R + [0, ] be its uppe pseudoinvese, defined by [9], [11] f (x = sup {s 0 such that f(s x} (6 = inf {s 0 such that f(s > x} (7 Note that f (0 = 0. Futhemoe [12, Theoem 3.1.2]: and f(t x t f (x (8 t > f (x f(t x (9 but the convese may not hold. Howeve, we have: Lemma 1: If f is ight-continuous then fo all t, x R + : t f (x f(t x (10

3 3 It is known that f( is not necessaily continuous but (1 its set of discontinuities is countable and (2 f( has a limit to the ight and to the left at evey point. We denote with f + ( the ight-limit of f(, defined fo t R + by f + (t = lim s t,s>t f(s = inf s>t f(s. Note that f + ( is ight-continuous and wheneve 0 s < t: f(s f + (s f(t (11 Similaly, we denote with f ( the left-limit of f(, defined fo t R + by f + (t = lim s t,s<t f(s = sup s<t f(s. Last, we will use the following esults, which ae tue fo any wide-sense inceasing function f( : R + R + Lemma 2: f = (f + and f = (f. Lemma 3: (f = f and (f + = f. Poofs of the lemmas ae in appendix. III. PI-REGULARITY In this section and the next section we ae inteested in egulation of a single flow. We stat by intoducing a new concept, Pi-egulaity, which extends both aival cuve constaints and Chang s g-egulaity. This concept will pove to be essential in analyzing inteleaved egulatos in Section V. A. Definition of Pi-Regulaity Ou new definition of egulaity uses an opeato, say Π, which must satisfy the following conditions. C1 Π is a mapping F inc G F, i.e. Π takes as agument a single-flow packet sequence (A, L and tansfoms it into a sequence of time instants. The output sequence is in F, i.e. is not necessaily monotonic. C2 Π is causal: if Π(A, L = A then the value of A n may depend on A 1,...A n 1 and L 1,..., L n but not on A m fo m n no on L m fo m m + 1. C3 Π is homogeneous with espect to A: Π(A + h, L = Π(A, L + h fo any constant h R and any sequences A F inc, L G. C4 Π is isotone with espect to A: wheneve A, A F inc ae such that A A then also Π(A, L Π(A, L fo any sequence L G. Note that if an opeato Π is causal and homogeneous with espect to A then necessaily Π(A, L 1 = fo any input (A, L. This can be deived by obseving fist that Π(A, L 1 is independent of A by causality and theefoe has the fom c(l [,. Second, by homogeneity Π(A + h, L = Π(A, L + h fo any eal numbe h so c(l = c(l + h fo any h, which is possible only if c(l =. In the next thee subsections we give seveal examples of such opeatos. They all have the fom Π(A, L n = max {A m + H m,n (L} (12 fo some appopiate choice of the aay H m,n (L (with H m,n (L [, fo m, n N + and any L G. An opeato defined by an equation of the fom Eq.(12 is a maxplus-linea opeato and clealy satisfies C1-C4. Note that hee the identity Π(A 1 = follows fom the fact that the max of an empty set if. Definition 1 (Pi-Regulaity.: Given some opeato Π that satisfies C1-C4, we say that a single-flow packet sequence (A, L is Π egula if A Π(A, L. We next give some examples and show how this definition extends existing famewoks. B. Chang s g-regulaity Given some function (o sequence g( : N R, Chang [9] defines a single-flow packet sequence (A, L as g-egula if and only if fo all m, n N + such that m < n we have A n A m g (L m L n 1 (13 It is immediate to see that g-egulaity is a special case of Pi-egulaity with the opeato Π given by n 1 Π(A, L n = max A m + g L j (14 In the case whee g(x = x/ fo some, g-egulaity is called the Length Rate Quotient (LRQ constaint with ate in [7]. Any flow that is obseved on a physical communication link of ate satisfies the LRQ( constaint. Because in this case g is linea, it can easily be seen that, fo the LRQ( egulation constaint, Eq.(13 is equivalent to the simple condition A n A n 1 L n 1 (15 fo all n N +, n 2. Theefoe, LRQ( egulaity is an instance of Pi-egulaity with the opeato Π LRQ( given by Π LRQ( (A, L n = A n 1 + Ln 1 fo n 2 Π LRQ( (A, L 1 = C. Aival Cuve Constaint (16 This is a classical netwok calculus constaint, oiginally expessed with min-plus algeba. It uses some wide-sense inceasing function σ( : R + R +, called aival cuve o min-plus taffic envelope ; it can always be assumed without loss of geneality that σ is sub-additive and σ(0 = 0 (but we don t need such an assumption in the following theoems. The celebated leaky bucket constaint LB(, b [13] coesponds to σ(t = t + b fo t > 0 and σ(0 = 0, whee is the leaky bucket ate and b the bustiness. The aival cuve constaint is expessed in tems of the cumulative aival function R(t, defined fo t 0, and which can be deived fom the singleflow packet sequence (A, L by R(t = n N + L n 1 {An<t} (17 whee the indicato function 1 {An<t} has the value 1 when the condition {A n < t} is tue and 0 othewise. In this context, we assume that time is nonnegative and in paticula A n 0. The aival cuve constaint equies that R(t R(s σ(t s fo all 0 s t (18

4 4 Liebehe shows in [11] that the aival cuve constaint can be expessed in max-plus algeba. To this end, he intoduces the aival time function T ( = R (, which is equal to the uppe pseudo-invese of cumulative aival function R(, and is also given by T (x = inf n N + ( An 1 {L1+...+L n>x} fo x 0 (19 Note that R( can be ecoveed by T ( since R( = T (. Liebehe shows that the aival cuve constaint Eq.(18 is equivalent to the condition T (y T (x λ(y x fo all 0 x y (20 with λ( = σ (. In othe wods, the uppe pseudo-invese σ ( is a max-plus taffic envelope of T ( if and only if σ( is an aival cuve, o min-plus taffic envelope, of R(. If we enfoce that σ( be left continuous, then σ( is the lowe pseudo-invese of σ ( and thus thee is exact equivalence between min-plus and max-plus taffic envelopes. To make the link between aival cuve constaints and Piegulaity, we need to go one step futhe and undestand the elation between taffic constaints expessed at an abitay point in time (o space, as in Eq.(18 and Eq.(20 and constaints that ae expessed at packet aival times. This is povided by the following theoem. Theoem 1: Conside a single-flow packet sequence (A, L and the associated cumulative aival function R( given by Eq.(17. Let σ( be some wide-sense inceasing function, σ + ( its ight-limit and σ ( the lowe pseudo-invese of σ( (and hence, by Lemma 2 also of σ + (. The following thee conditions ae equivalent: 1 The aival cuve constaint in Eq.(18 is satisfied; 2 Fo any m, n with 1 m n: L j σ + (A n A m (21 3 Fo any m, n with 1 m n: A n A m σ L j (22 The poof is in appendix. We give above a vesion of the theoem using the aival cuve σ(. An equivalent fomulation can be given, assuming that we ae given not the aival cuve σ( but the max-plus taffic envelope λ(. Then define σ = λ so that Eq.(18 is equivalent to Eq.(20. Using Lemma 3 we have λ = σ + and λ = σ, so that Eq.(21 is equivalent to L j λ (A n A m (23 and Eq.(22 is equivalent to A n A m λ L j (24 which shows the duality between min-plus and max-plus epesentations. The most impotant outcome to the above theoem is that the condition in Eq.(22 o Eq.(24 has the fom in Eq.(12; this shows that an aival cuve constaint is a special fom of Pi-egulaity. The opeato Π that coesponds to an aival cuve σ( is given by Π(A, L n = max A m + σ L j (25 The theoem also claifies the elation between aival cuve constaints and Chang s g-egulaity. Notice that the aival cuve constaint in Eq.(18 implies the condition n 1 L j σ(a n A m, (26 fo any m n N +. This can be deived fom a diect application of Eq.(18 to s = A m, t = A n. Howeve, it can easily be seen that the convese is not tue, i.e. this last condition is not equivalent to the aival cuve constaint compae to Eq.(21. Obseve now that the definition of g-egulaity involves only equations such as Eq.(26. In paticula, it does not involve the length L n of the cuent packet, which explains why it cannot be equivalent to aival cuve constaints and why published elations between the two involve a bound with a tem in L max, the maximum packet size of the flow. In the est of this subsection we apply the above theoem to two classic aival cuve constaints. 1 Leaky Bucket Constaints: Fo the single leaky bucket constaint LB(, b, we can take σ(t = t+b fo some positive and b, so that σ + (t = σ(t and σ (x = 0 x b. The condition Eq.(22 is equivalent to A n A m L j b and the Pi-egulaity condition can be expessed by ( j=1 L n j b j=2 L j b A n A 1 A 2... A n 1 j=n 1 L j b (27 (28 Note that A n A m is always tue by constuction fo any 1 m n 1. Theefoe, the constaint in the pevious equation is equivalent to A n max A m + L j b (29 The opeato Π LB(,b that coesponds to the leaky bucket constaint LB(, b is theefoe given by Π LB(,b (A, L n = max A m + L j b (30

5 5 2 Staicase Aival Cuve: The staicase aival cuve SC(τ, b is defined fo τ > 0, b > 0 by t σ(t = b, t 0 (31 τ and is used to expess the constaint that at most b data units can be obseved ove any window of duation τ [12]. Hee σ( is left-continuous and staightfowad computations give: t σ + (t = b τ + 1, t 0 (32 x σ (x = τ b 1, x > 0 (33 σ (0 = 0 (34 The opeato Π SC(τ,b that coesponds to the staicase aival cuve SC(τ, b is theefoe given by Π SC(τ,b (A, L n = max D. Limits on Packet Rate Packet ate limitations ae used to put limits on the pocessing demand in netwoking boxes. Fo example, the IEEE TSN woking goup specifies a limit, say K N + on the numbe of packets sent by a flow ove a specified duation, say τ 0. If all packets ae of the same size, say l, this is the same as a staicase aival cuve constaint SC(τ, b with b = Kl. By Theoem 1 and Eq.(33, in this specific cases such a packet ate constaint is equivalent to n m + 1 K A n A m τ (36 K If packets ae not all of the same size, this cannot be specified exactly as an aival cuve constaint no as a g-egulation constaint. Howeve, it is still expessed by the constaint in Eq.(36. Theefoe, this TSN packet ate egulation is an instance of Pi-egulaity, with the opeato Π T SN(τ,K defined by { n m + 1 K Π T SN(τ,K (A, L n = max A m + τ K (37 Othe foms of packet ate limitations can be defined. Fo example, a packet spacing constaint PS(τ can be defined by A n A n 1 τ, fo all n = 2, 3... (38 whee τ 0 is the spacing inteval. Note the analogy with an LRQ egulation constaint. This egulation constaint, is obviously an instance of Pi-egulaity, with the opeato Π P S(τ given by Π P S(τ (A, L n = A n 1 + τ fo n 2 Π P S(τ (A, L 1 = (39 Last, simila to leaky bucket constaints, we can define a packet ate constaint by allowing a packet ate limit ρ } with some packet bustiness constaint K. In othe wods, the packet bustiness constaint PB(ρ, K specifies that the numbe of packets obseved ove an inteval of duation t must be uppe bounded by ρt + K. This can be expessed similaly to Eq.(29 by the condition A n max { A m + n m + 1 K ρ } (40 Theefoe, the packet butiness constaint is an instance of Piegulaity, with the opeato Π P B(ρ,K given by { Π P B(ρ,K (A, L n = max A m + n m + 1 K } ρ (41 Obviously, all of the constaints defined in this section can neithe be expessed with aival cuves no with g-egulaity. A m + τ L j b E. Combination of Regulation Constaints b Pi-egulation constaints can easily be combined, by taking the maximum of the opeatos. Indeed, it immediately follows (35 fom Definition 1 that a flow (A, L is both Π 1 and Π 2 -egula if and only if it is Π-egula, with Π being the maximum of Π 1 and Π 2, defined by Π(A, L n = Π 1 (A, L n Π 2 (A, L n (42 It is staightfowad to veify that if Π 1 and Π 2 both satisfy C1-C4, then so does Π. IV. PER-FLOW Π REGULATOR Afte defining Pi-egulaity we can now define a pe-flow Π-egulato as a FIFO system that may delay some o all of the packets of a flow in ode to make sue that the esulting output is Π-egula, fo some opeato Π. A minimal Π-egulato is one that outputs the packets of the flow as ealy as possible. Its existence is shown next. A. Minimal Pe-Flow Π Regulato Theoem 2: Conside a single-flow packet sequence (A, L and let Π be an opeato that satisfies C1-C4. The minimal Π egulato is defined as the FIFO system that tansfoms the input packet sequence (A, L into the output packet sequence (D, L such that D 1 = A 1 and D n = max {A n, D n 1, Π(D, L n } (43 1 The system defined in this way is a Π-egulato fo this flow. 2 (Minimality: Fo any othe Π-egulato that tansfoms (A, L into say (D, L we have D n D n fo all n N +. 3 The flow (A, L is Π-egula if and only if D = A. The poof is in appendix. Note that the definition in Eq.(43 may appea to be cicula, but it is not. This is because Π is assumed to be causal as in condition C2, which implies that Π(D, L n depends only on D 1,..., D n 1. Theefoe, the output sequence D is well defined by the initial condition D 1 = A 1 and Eq.(43.

6 6 To each of the egulatos descibed in the pevious section is thus associated a coesponding minimal pe-flow Pi-egulato, whose input output equation is given by Eq.(43. Sometimes the equation can be simplified; fo example, the packet spacing constaint in Eq.(39 gives, fo the packet-spacing egulato PS(τ, the elation D n = A n D n 1 (D n 1 + τ, which, because τ 0, can be witten moe simply as D n = A n (D n 1 + τ (44 In the est of this section we show that Chang s minimal egulato and the classic packetized geedy shape [14] ae all special cases of minimal pe-flow Π-egulatos. B. Chang s Minimal Regulato, LRQ(-Regulato. Chang s minimal g-egulato is the minimal FIFO system that delives an output that is g-egula [9]. We have seen in Section III-B that g-egulaity is equivalent to Pi-egulaity, with Π given by Eq.(14. Theefoe, the minimal g-egulato is the minimal g-egulato with Π given by Eq.(14. In paticula fo the minimal LRQ(-egulato, a simple fomulation of the opeato Π is given by Eq.(16, fom{ whee we deive the input-output } equations D n = max A n, D n 1, D n 1 + Ln 1. Obseve that Ln 1 0 so the tem D n 1 can be emoved fom the max. The input-output equations of the minimal LRQ(-egulato ae thus D 1 = A 1 and fo n 2: { D n = max C. Packetized Geedy Shape. A n, D n 1 + Ln 1 } (45 Given some aival cuve σ(, a packetized shape is a system that delives an output that is packetized and satisfies the aival cuve constaint σ(. Among all packetized shapes, the packetized geedy shape is the one that delives its output as ealy as possible [14]. The input and output of a packetized shape ae cumulative aival functions such as R( in Eq.(17. Given a sequence of packet lengths L and fo inputs and outputs that ae packetized, thee is a one-to-one mapping between the R( function and the sequence of packet aival times. Futhemoe, by Theoem 1, an aival cuve constaint is a fom of Pi-egulaity, with Π given by Eq.(25. It is then staightfowad to see that a packetized geedy shape fo the aival cuve σ( is the same as the minimal Π-egulato with Π given by Eq.(25. In the special case of a single leaky bucket constaint LB(, b, the packetized geedy shape is called the leaky bucket shape. Hee the Π opeato can take the simple fom in Eq.(30. The input-output equations of the leaky bucket shape LB(, b ae thus D 1 = A 1 and fo n 2: D n = A n D n 1 A L j b m +... A n 1 + j=n 1 L j b (46 The elation above is the max-plus equation of a leaky bucket shape. It is not the best fomula fo a pactical implementation (see [14] fo a discussion of the diffeent implementations of leaky bucket shapes and thei combinations but it can be used to deive fomal popeties of such shapes, as we do in Section V. V. INTERLEAVED REGULATOR In this section we conside a packet sequence (A, L, F of seveal multiplexed flows. Recall hee that L n is the length of the n th packet, which belongs to flow F n. Assume that fo evey flow f we have a Pi-egulation constaint, with opeato Π f fo flow f. The egulation opeatos Π f may be of any kind, as long as they satisfy the conditions C1-C4. Fo example, the egulation opeato may be of the LRQ type, leaky bucket o packet ate limit, o any combination of these opeatos. Futhemoe, the egulation opeatos of diffeent flows need not be of the same type. Recall that saying that flow f is Π f -egula means hee that A f Π f (A f, L f whee A f, L f ae the sequences extacted fom A and L by keeping only the indices coesponding to packets of flow f. In this context we define an inteleaved egulato as a system that is FIFO and may delay some o all of the packets of the input sequence so that evey flow f inside the output sequence is Π f -egula. Note that the FIFO condition imposes that a packet of a flow may not be deliveed befoe a packet of some othe flow that aived befoe it. Fomally, given a collection of egulation opeatos Π f, with one opeato pe flow that satisfies C1-C4, an inteleaved egulato is a FIFO system that tansfoms an input sequence (A, L, F into an output sequence (D, L, F such that (D f, L f is Π f egula fo evey flow f. A. Minimal Inteleaved Regulato The following esults establishes the existence of a minimal inteleaved egulato, i.e. one that delays the packets as little as possible. Theoem 3: Conside a packet sequence (A, L, F with, fo evey flow f, one egulation opeato Π f that satisfies C1-C4. The minimal inteleaved egulato is defined as the FIFO system that tansfoms the input packet sequence (A, L, F into the output packet sequence (D, L, F defined by D 1 = A 1 and { D n = max A n, D n 1, Π ( Fn D Fn, L Fn } (47 I(n Recall that, in the above fomula, I(n is the index of packet n in its flow (namely in flow f = F n. 1 The system defined in this way is an inteleaved egulato fo this packet sequence. 2 (Minimality: Fo any othe inteleaved egulato that tansfoms (A, L, F into say (D, L, F we have D n D n fo all n N +. 3 Evey flow f in (A, L, F is Π f -egula if and only if D = A. The poof is in appendix. Eq.(47 is the input-output chaacteization of the minimal inteleaved egulato. It has an

7 impotant consequence: it shows that the minimal inteleaved egulato can be implemented as a head of the line system, as in [7]. Moe pecisely, a possible implementation of the minimal inteleaved egulato is as follows. The Ugency Based Schedule of Specht and Samii in [7] is an instance of minimal inteleaved egulato, which coesponds to the case whee the egulation opeato Π f is eithe of the fom Π LRQ( f as in Eq.(16 o Π LB( f,b f as in Eq.(30. Since the minimal inteleaved egulato uses a FIFO queue, thee is no need fo pe-flow queuing. Note that thee is peflow state, but on one hand, this pe-fow state can be vey simple (a single numbe fo simple egulation ules such as leaky bucket, LRQ o packet spacing; on the othe hand, pe-flow state is typically pesent in switches and outes fo packet fowading and the pe-flow state of the inteleaved egulato can be placed thee. In contast, implementing one queue pe flow as would be equied by pe-flow egulatos has consideably lage complexity. It follows fom the stuctue of the minimal inteleaved egulato that, when the packet at the head of the queue is not eligible fo delivey, all packets behind it ae delayed. Theefoe a packet may be delayed eithe because it is too ealy with espect to the egulation imposed to its flow, o because a packet of some othe flow at the head of the queues is being delayed. B. Minimal Inteleaved Regulato Does Not Incease Wost Case Delay As discussed in the intoduction, inteleaved egulatos ae used in netwoks that handle multiple flows in the same queue, as a means to avoid bustiness cascades. Howeve, they also add some delay, which needs to be accounted fo in end-toend delay bounds. Supisingly, we show next a eshapingfo-fee popety, which is eminiscent of a simila popety of netwoks that use pe-flow queuing [8]. Moe pecisely, assume that a packet sequence (A, L, F is fed into a FIFO system (Figue 1. This FIFO system is typically a queuing point inside a switch, oute o end-system. Assume that evey flow f in the input packet sequence is Π f -egula. This popety is, geneally, lost by the output (D, L, F of the FIFO system. Assume that we place a minimal inteleaved shape just afte the output, in ode to eceate the egulaity that was lost by tavesing the FIFO node and let (E, L, F be the output sequence of the inteleaved egulato. The following theoem establishes that the minimal inteleaved egulato does not add anything to the wost-case delay of the FIFO system. Inteleaved Shaping Does Not Incease Wost Case Delay FIFO System Inteleaved Shape Packets of the multi-flow sequence ae queued in FIFO ode; The packet at the head of the queue is examined against aival time Fig. of 1: packet Configuation (numbeed fo Theoem in aival 4. ode the egulation constaints of its flow; it is eleased at the Evey flow is shaped befoe input to with paametes ealiest time whee this is possible without violating the Output of is fed to inteleaved shape with paametes fo constaints; flow Theoem 4: Assume the input to a FIFO system S is Π f Packets that ae not at the head of the queue ae not egula fo evey flow f and that fo evey f, the opeato Theoem: [Poof in appendix] examined until they each the head of the queue. Π f satisfies conditions C1-C4. The output is fed to a minimal inteleaved egulato with same opeato Π f fo flow f. The wost-case delay of the combination is the same as the wostcase delay of the FIFO system S alone. In othe wods, with the notation defined above: sup (D n A n = sup (E n A n (48 n N + n N + The poof is in appendix. This esult is quite impotant fo netwoks that do pe-class scheduling as it shows that it is possible to avoid the bustiness cascade while keeping only FIFO queues. Following [7], assume that we place one minimal inteleaved egulato pe switch input pot and pe taffic class, befoe the input to a queuing point, as illustated in Figue 2. The above theoem can then be applied, whee the FIFO system S is the upsteam node that feeds the inteleaved egulato. Note that in the uspteam node, thee ae othe flows that ae not fed to the inteleaved egulato. Thus, moe pecisely, the FIFO system S can be defined as the system that tansfoms the multi-flow packet sequence constituted of all packets of all flows that come fom the upsteam node. When computing end-to-end delay bounds, minimal inteleaved egulatos can then be ignoed, since, by the above theoem, thei delay can be absobed into the pevious node delay. Futhemoe, since the input flows to any queuing point is the output of an inteleaved egulato and thus is egulated, delay and backlog bounds can be computed using standad netwok calculus computations as in [12, Section 1.4.1]. TSN assumes that evey souce satisfies what we call hee the TSN packet ate egulation ule in Eq.(36. Theefoe, in a TSN netwok it would consistent to use an inteleaved shape whee the egulation ule fo evey flow is TSN packet ate egulation (athe than LRQ o leaky bucket, as is cuently poposed by UBS. On a side note, Theoem 4 can be paaphased by saying that inteleaved egulation comes fo fee, though thee may be a subtle diffeence. Indeed the theoem states that the wost case delay acoss all flows at system S is not inceased by the downsteam minimal inteleaved egulato. In some cases, the wost-case delay at system S may not be the same fo all flows (a typical case is when diffeent flows have diffeent maximum packet lengths and when the delay at S includes a tansmission delay popotional to packet length. Now, thee does not seem to be a pe-flow equivalent to Theoem 4; thus the only thing we can say at this stage is that the delay of the concatenation of system S and the inteleaved shape is uppe-bounded fo all flows by d, the wost case delay acoss 7 20

8 8 Minimal inteleaved egulato [13] Rene L Cuz. A calculus fo netwok delay. i. netwok elements in isolation. IEEE Tansactions on infomation theoy, 37(1: , [14] Jean-Yves Le Boudec. Some popeties of vaiable length packet shapes. IEEE/ACM Tansactions on Netwoking (TON, 10(3: , One minimal inteleaved egulato pe class and pe input Fig. 2: Use of Inteleaved Regulatos as poposed by [7]. all flows at S. Fo some flows, this may mean an inceased delay. Whethe a pe-flow equivalent of Theoem 4 can be found is fo futhe eseach. VI. CONCLUSION Motivated by the Ugency Based Schedule of [7], we have intoduced a new theoy of taffic egulatos that is able to explain the eshaping-fo fee popety of minimal inteleaved egulatos (Theoem 4. This theoy extends the existing, non compatible theoies of g-egulatos and aival cuve constaints, and also sheds some light on thei elationship. It also gives a pactical means to avoid bustiness cascades in pe-class FIFO netwoks. REFERENCES [1] Accessed: [2] Accessed: [3] Anna Chany and Jean-Yves Le Boudec. Delay bounds in a netwok with aggegate scheduling. In Quality of Futue Intenet Sevices, pages Spinge, [4] Jon CR Bennett, Kent Benson, Anna Chany, William F Coutney, and Jean-Yves Le Boudec. Delay jitte bounds and packet scale ate guaantee fo expedited fowading. IEEE/ACM Tansactions on Netwoking (TON, 10(4: , [5] Mac Boye and Chistian Faboul. Tightening end to end delay uppe bound fo afdx netwok calculus with ate latency fifo seves using netwok calculus. In Factoy Communication Systems, WFCS IEEE Intenational Wokshop on, pages IEEE, [6] Anne Bouillad and Giovanni Stea. Exact wost-case delay fo fifomultiplexing tandems. In Pefomance Evaluation Methodologies and Tools (VALUETOOLS, th Intenational Confeence on, pages IEEE, [7] Johannes Specht and Soheil Samii. Ugency-based schedule fo timesensitive switched ethenet netwoks. In Real-Time Systems (ECRTS, th Euomico Confeence on, pages IEEE, [8] L. Geogiadis, R. Guéin, V. Peis, and R. Rajan. Efficient suppot of delay and ate guaantees in an intenet. SIGCOMM Comput. Commun. Rev., 26(4: , August [9] Cheng-Shang Chang and Yih Hau Lin. A geneal famewok fo deteministic sevice guaantees in telecommunication netwoks with vaiable length packets. In Quality of Sevice, (IWQoS Sixth Intenational Wokshop on, pages 49 58, May [10] Cheng-Shang Chang. Pefomance Guaantees in Communication Netwoks. Spinge Science & Business Media, [11] Jög Liebehe et al. Duality of the max-plus and min-plus netwok calculus. Foundations and Tends R in Netwoking, 11(3-4: , [12] Jean-Yves Le Boudec and Patick Thian. Netwok calculus: a theoy of deteministic queuing systems fo the intenet, volume Spinge Science & Business Media, A. Poof of Lemma 1 APPENDIX is Eq.(8. : We have t f (x. If t > f (x, the conclusion follows fom Eq.(9. Else we have t = f (x. Fo any s > t, we have f(s x by Eq.(9. Theefoe lim s t,s>t f(s x. Since f( is ight-continuous, lim s t,s>t f(s = f(t. B. Poof of Lemma 2 We do the poof fo the statement f = (f +. The poof fo f = (f is exactly simila and is left to the eade. Note that f(t f + (t fo all t theefoe {t : f(t x} {t : f + (t x}, thus f (x (f + (x. Assume now, by contadiction, that f (x > (f + (x. Fo any t such that (f + (x < t < f (x, we have: 1 t < f (x and since f (x is the infimum of the set {t : f(t x}, it follows that t is not in this set, i.e. f(t < x 2 t > (f + (x and by Eq.(9 it follows that f + (t x Thus, fo any t ( (f + (x, f (x we have f + (t > f(t i.e. t is a point of discontinuity of f. But this is impossible because this inteval is not a countable set. Thus it is not possible that f (x > (f + (x, which poves that f (x = (f + (x. C. Poof of Lemma 3 We do the poof fo the statement (f = f. The poof fo (f + = f is exactly simila and is left to the eade. We can ewite the definitions in Eq.(5 and Eq.(6 as f (y = sup s R + ( s1{f(s>y} f (y = sup s R + ( s1{f(s y} Theefoe, using associativity of sup: (f (x (49 (50 def = sup f (y (51 0 y<x ( ( = sup sup s1{f(s y} (52 0 y<x s R ( + ( = sup sup s1{f(s y} (53 s R + 0 y<x = sup (s ϕ(s, x (54 s R + with ϕ(s, x def = sup 0 y<x ( 1{f(s y}. Now if x > f(s then ϕ(s, x = 1 and if x f(s then ϕ(s, x = 0. Theefoe ϕ(s, x = 1 {x>f(s}. Thus (f (x = sup s R + ( s 1{x>f(s} = f (x (55 whee the last equality is by Eq.(49.

9 9 D. Poof of Theoem 1 1 3: Conside some packet numbes 1 m n. If m = n then 3 tivially holds because σ (0 = 0. Assume now that m < n. Let T ( be the aival time function defined by Eq.(19. Take y = L L n ɛ with 0 < ɛ L n and x = L L m. Because L n is intege, we have T (y = A n and T (x = A m. By [11], the max-plus taffic envelope condition Eq.(20 also holds. Thus A n A m = T (y T (x σ (y x = σ (L m +...+L n ɛ (56 Take the limit of the above equation as ɛ 0 and obtain A n A m (σ ((L m L n (57 By Lemma 3, (σ = σ, which concludes this pat of the poof. 3 2: Conside some packet numbes 1 m n. If m = n then 2 tivially holds because σ(0 0 and σ + (0 0. Assume now that m < n. Eq.(21 follows fom Lemma 1 applied to f( = σ + (. 2 1: Pat 1: no simultaneous aivals. We fist pove this case assuming that thee cannot be simultaneous aivals, namely we assume A n < A n+1 fo all n N +. Conside s, t R + with 0 s t. If s = t then Eq.(18 tivially holds. Assume theefoe that 0 s < t. Let A = {A 1, A 2,...}. We conside seveal cases: Case 1: A [s, t] is empty. In this case. R(t R(s = 0 and Eq.(18 is tivially satisfied. Case 2: A [s, t] is nonempty, s A and t A. Let m be the smallest packet numbe such that s < A m and let n be the lagest packet numbe such that A n < t, so that A [s, A m and A (A n, t] ae empty. Theefoe and by Eq.(21 We must also have R(t R(s = L j (58 R(t R(s σ + (A n A m (59 and thus A n A m < t s; by Eq.(11 s < A m A n < t (60 σ + (A n A m σ(t s (61 This concludes the poof in this case. Case 3: s A and t A. Thus s = A m fo some m. Let n be the lagest packet numbe such that A n < t. We have theefoe R(t R(s = L j (62 and s = A m A n < t (63 The est of the poof in this case is as in Case 2. Case 4: s A and t A. Thus t = A n fo some n. Let m be the smallest packet numbe such that s < A m. We have theefoe and Thus R(t R(s R(t R(s = n 1 L j (64 s < A m A n = t (65 L j σ + (A n A m (66 whee the last inequality is by Eq.(21. Now A n A m < t s (67 R(t R(s σ(a n A m (68 Now A n A m t s and σ( is wide-sense inceasing, thus σ(a n A m σ(t s, which concludes the poof in this case. Case 5: s A and t A. Thus s = A m and t = A n fo some m n. We have theefoe R(t R(s = n 1 L j (69 If m = n then R(t R(s = 0 and Eq.(18 is tivially veified. We can theefoe assume m < n. It follows that 0 A n 1 A m < A n A m theefoe By Eq.(21 σ + (A n 1 A m σ(a n A m (70 R(t R(s σ + (A n 1 A m σ(a n A m (71 Pat 2: with simultaneous aivals. We now allow simultaneous aivals in the flow (A, L. We assume that thee is a finite numbe of packet aivals in evey bounded inteval. Indeed, if this does not hold, the conditions in the theoem ae false and the equivalence holds. We deive fom (A, L anothe packet sequence, (A, L obtained by aggegating all packets that aive at the same time unde (A, L. Fomally, (A, L is defined by: A 1 = min {A m, m N + } A n = min { } A m, m N +, A m > A n 1 L n = j N L j1 + {Aj=A n }

10 10 Note that L n is finite fo evey n by ou assumption and L n is the sum of all packet sizes of all packets that aive at the same instant. Note that A F inc and thee ae no simultaneous aivals in the flow (A, L. We next show that, fo i = 1, 2, condition i of the theoem holds fo (A, L if and only if it holds fo (A, L, which will conclude the poof. Condition 1: The cumulative aival function R( is the same fo (A, L and (A, L so Condition 1 of the theoem holds fo (A, L if and only if it holds fo (A, L. Condition 2: Assume fist that Condition 2 holds fo (A, L. Conside some fixed m, n N + and let m be the index of the fist packet such that A m = A m and n the index of the last packet such that A n = A n. We have n L j = L i (72 i=m and A n A m = A n A m. Since Eq.(21 holds fo (A, L, it follows that it also holds fo (A, L. Convesely, assume that Condition 2 holds fo (A, L and conside some fixed m, n N +. Let Define and m, n by A m = A m and A n = A n. We have L j n i=m L i (73 and A n A m = A n A m. Since Eq.(21 holds fo (A, L, it follows that it also holds fo (A, L. This concludes the poof in this case. E. Poof of Theoem 2 1 We fist pove that the system defined by D 1 = A 1 and Eq.(43 is a Π-egulato. We obviously have D n D n 1 i.e. D F inc and D n A n fo all n N thus this is a FIFO system. Also D n Π(D, L n by constuction. 2 Next, we show by induction that D n D n. Base Step: We have D 1 = A 1 D 1 because the Π- egulato is a FIFO system. Induction Step: Assume D m D m fo 1 m n 1. Let D be the sequence defined by Dm = D m fo 1 m n 1 and D m = fo m n. We have D D by induction hypothesis and by Condition C4, Π(D, L Π( D, L. By Condition C2, Π( D, L n = Π(D, L n. Theefoe Π(D, L n Π(D, L n. But since D is Π-egula, we also have D n Π(D, L n. Thus Now D n Π(D, L n (74 D n D n 1 D n 1 and D n A n (75 because the Π-egulato is a FIFO system. Combining the last two inequalities gives D n max {A n, D n 1, Π(D, L n } = D n (76 3 If D = A then since D is Π-egula by 1 obviously A is Π-egula. Convesely, if (A, L is Π-egula then the identity system, which maps (A, L into itself, is a Π-egulato fo this flow. By item 2, we have D A. But since D A by constuction it follows that D = A. F. Poof of Theoem 3 1 We fist pove that the system defined by D 1 = A 1 and Eq.(47 is an inteleaved egulato. We obviously have D n D n 1 i.e. D F inc and D n A n fo all n N thus this is a FIFO system. Also by constuction which is the same as D n Π Fn ( D Fn, L Fn I(n (77 D Fn I(n ΠFn ( D Fn, L Fn I(n (78 which shows that D f Π ( f D f, L f fo evey flow f, i.e. evey flow at the output is Π f -egula. 2 Next, we show by induction that D n D n. Base Step: We have D 1 = A 1 D 1 because the inteleaved egulato is a FIFO system. Induction Step: Assume D m D m fo 1 m n 1. Let D be the sequence defined by Dm = D m fo 1 m n 1 and D m = fo m n. We have D D by induction hypothesis and thus D Fn D Fn. By Condition C4, Π Fn (D Fn, L Fn Π Fn ( D Fn, L Fn. By Condition C2, Π Fn ( D Fn, L Fn I(n = Π Fn (D Fn, L Fn I(n. Theefoe Π Fn (D Fn, L Fn I(n Π Fn (D Fn, L Fn I(n. But since D is Π Fn -egula, we also have D n = (D Fn I(n Π Fn (D Fn, L Fn I(n. Thus Now D n Π Fn (D Fn, L Fn I(n (79 D n D n 1 D n 1 and D n A n (80 because the inteleaved egulato is a FIFO system. Combining the last two inequalities gives D n max { A n, D n 1, Π Fn (D Fn, L Fn I(n } = Dn (81 3 Since the system is an inteleaved egulato by item 1, evey flow f in the output sequence is Π f egula. Thus, if D = A, the input is equal to the output and evey flow f in the input sequence is also Π f egula. Convesely, if evey flow in (A, L, F is Π f -egula then the identity system, which maps (A, L, F into itself, is an inteleaved egulato fo this packet sequence. By item 2, we have D A. But since D A by constuction it follows that D = A. G. Poof of Theoem 4 Note that, hee, the equation of the minimal inteleaved egulato becomes E 1 = D 1 and { E n = max D n, E n 1, Π ( Fn E Fn, L Fn } (82 I(n 1 We pove by induction that fo any bound d on the delay at S, we have E n A n + d (83

11 11 Base Step: E 1 = D 1 theefoe Eq.(83 tivially holds by definition of d. Induction Step: Assume that E m A m + d fo 1 m n 1. Let n be fixed and wite F n = f, I(n = i, so that A f i = A n. Eq.(82 can now be witten moe simply as E n = max { D n, E n 1, Π f ( E f, L f } (84 i We have, by hypothesis on d D n A n + d (85 By the induction hypothesis E n 1 A n 1 + d and futhemoe A n 1 A n thus E n 1 A n + d (86 Next, if i = 1, packet n is the fist packet of its flow and Π ( f E f, L f =. In this case we tivially have 1 Π ( f E f, L f i A n + d. Othewise, by induction hypothesis, E f j Af j + d fo all 1 j i 1 (87 Because Π f is causal and isotone (Conditions C2 and C4, it follows that Π f (E f, L f i Π f (A f + d, L f i (88 By homogeneity (condition C3: Π f (E f, L f i Π f (A f, L f i + d (89 Now flow f is Π f -egula in the input sequence, theefoe Combining the two last equations gives Π f (A f, L f i A f i (90 Π f (E f, L f i A f i + d = A n + d (91 Combining Eq.(84, Eq.(85, Eq.(86 and Eq.(91 gives E n A n + d (92 which concludes the induction step. 2 Let s = sup n N + (D n A n and s = sup n N + (E n A n. We have shown that if d is an uppe bound on (D n A n then it also an uppe bound on (E n A n and theefoe s d. Since this is tue fo any uppe bound d, it is also tue fo the supemum, namely s s. Convesely, since E n D n fo any n N +, we have s s which finally shows that s = s.