Packetization and Aggregate Scheduling

We present a new formalism for data packetization in Network Calculus. Packet curves are introduced to model constraints on the packet lengths of data flows. Indeed, a more precise knowledge of the packet characteristics can be efficiently exploited to get tighter performance bounds, specially when dealing with scheduling policies based on packet count, such as round-robin. A second use of packet curves is the packetization of a superposition of periodic flows. Finally, we show that packet curves can be used to compute a global service curve for the aggregate of several flows, with different service curves, sharing a unique queue. Key-words: Network Calculus, packetization, scheduling, quality of service guarantees This work has been supported by ANR project PEGASE, 2009-SEGI-009. ∗ ENS/INRIA TREC [email protected] † Université Paris-Est, IFSTTAR [email protected] ‡ INRIA/LIG MESCAL [email protected] in ria -0 06 08 85 2, v er si on 1 15 J ul 2 01 1 Paquetisation et ordonnancement agrégé Résumé : Nous présentons dans ce rapport un nouveu formalisme de paquétisation de donnée en Network Calculus. La notion de courbe de paquets est introduite pour modéliser les contraintes sur les longueurs des paquets de flux de données. En effet, une connaissance plus précise des caractéristiques de paquets peuvent être exploitées efficacement pour obtenir de meilleurs bornes de performances, spécialement dans le cas de politiques de service basées sur le comptage de paquets, comme la politique round-robin. Deuxièmement, les courbes de paquets permettent de calculer des caractéristiques (courbes d’arrivées) plus précises de flux périodiques superposés. Enfin, nous montrons comment ces courbes peuvent être utilisées pour calculer une courbe de service globale pour des flux partageant différemment une même ressource. Mots-clés : Network calculus, paquétisation, ordonnancement, garanties de service. in ria -0 06 08 85 2, v er si on 1 15 J ul 2 01 1 Packetization and Aggregate Scheduling 3 Introduction The purpose of this article is to present a new data packetization approach in network calculus. Network calculus [1, 2, 3] is a theory based on min-plus algebra [4] and developed for the calculus of performance bounds in computer and communication networks. Remarkably, this theory is almost only based on two objects: arrival curves and service curves, that are used to express constraints on arrival flows and service capacities. Performance bounds are then derived by cleverly handling arrival and service curves, and by taking into account the service policies. Although several alternative approaches of the network calculus exist, such as trajectory methods [5] or model checking [6], the network calculus approach is applied on a range of fields, e.g. internet Quality of Service (QoS) [7], wireless sensor networks [8], with several advantages on other approaches. Network calculus has recently received a lot of attention because its algebraic framework provides an efficient and elegant way to compose elementary network elements into more complex systems in order to get worst-case performances upper bounds. Unfortunately, those bounds are often over-pessimistic. Indeed, as soon as several flows and servers are composed together, tight bounds cannot be obtained from purely algebraic methods. This phenomenon has been observed under several assumptions (blind multiplexing [9] or FIFO [10]). Some exact methods have been derived using linear programming [11] in general acyclic networks, but are algorithmically costly and there are no general results for networks with cyclic dependencies. While network composition and flow aggregation has received a lot of attention, few works in network calculus concern the packet nature of flows. The main technique to deal with packets so far is called packetization and only uses the maximal and minimal sizes of the packets. This is rather unsatisfactory because most actual flows in communicating embedded systems are made of packets (often of different sizes) and the interaction between the flows inside a node of the system is also often packet-based (for example when no preemption is possible). In this paper, we propose a more refined modeling of packet flows, that may have different packet lengths. We propose a new object that we call packet curve, that captures information about the distribution of packets in a flow the same way as arrival time constraints of data are captured by arrival curves, in network calculus theory. As mentioned before, when packets may have different lengths, only the minimum and the maximum lengths have been taken into account in the calculation of performance bounds. We show here that the whole available information on the packet lengths, given by a packet curve, can be taken into account in that calculation. In Section 3 we provide closed form formulas for the packets curves in one important case, the superposition of several periodic flows. In Section 4, we apply the approach using packet curves to calculate residual services of arrival flows routed under the round-robin policy, where packets of each flow may have different lengths, and where information on the sequence of packet lengths of a given flow is given in packet curves. Although this approach is quite efficient for the round-robin service discipline, we will also see that the approach is not as good to other service policies such as packet-based fixed priority or packet-based FIFO. In Section 5, we treat the problem of determining a global minimum strict service curve for the aggregation of flows that guarantee some given minimum services for each flow. This problem only has some meaning when data is set in packets, in which case, the server is supposed to be reinitiated each time it starts to serve a new packet flow. Under the general RR n° 7685 in ria -0 06 08 85 2, v er si on 1 15 J ul 2 01 1 Packetization and Aggregate Scheduling 4 case where packets of one flow may have different lengths, we show the role of packet curves on the calculus of global minimum service curves. 1 Network Calculus Preliminaries Network calculus is based on (min,plus) algebra [4]. Data arrivals and services are modeled by (min,plus) functions and (min,plus) operators such as (min,plus) convolution and deconvolution, and used to express and handle constraints on data arrivals and service. More precisely, the set of functions considered is F def = {f : R+ → R+ ∪ {+∞} | f(0) = 0 and f is non-decreasing}, where R+ is the set of non-negative reals, and the two operators are defined as follows: let f, g ∈ F , then, ∀t ∈ R+, • (min,plus)-convolution: f ∗ g(t) = inf0≤s≤t f(s) + g(t− s); • (min,plus)-deconvolution: f g(t) = sups≥0 f(t+ s)− g(s). The set F equipped with the minimum and the (min,plus)-convolution is a semi-ring with zero element : t 7→ +∞ and unit element e : 0 7→ 0; t 7→ +∞. We also define the power of a function as f0 = e and ∀n ∈ N \ {0}, fn = f ∗ fn−1. Two important notions in network calculus theory are arrival curves and service curves. One of the main objectives of this theory is to calculate upper bounds of end-to-end delays and data backlogs on servers. This section provides a brief review of the basic results of network calculus. A more detailed presentation can be found in [1, 2]. Consider a data flow arriving at a server. For t ∈ R+, the cumulative amount of data between times 0 and t is denoted by A(t) ∈ F . The function A is then non-decreasing, and A(0) = 0. Definition 1 ((Maximum) arrival curve). A function α (resp. γ) is a maximal (resp. minimal) arrival curve for A if ∀s, t ∈ R+, s ≤ t, A(t)−A(s) ≤ α(t− s) ( resp. A(t)−A(s) ≥ γ(t− s) ) . Let A be an arrival flow at a given network server. We denote the output flow from this server by Ā. Definition 2 (Minimum simple service curve). The node offers a minimum simple service curve β if Ā ≥ A ∗ β. A backlogged period of a server is an interval (s, t] such that ∀u ∈ (s, t], A(u) > Ā(u). Definition 3 (Minimum strict service curve). A minimum service curve β is strict if during any backlogged period (s, t] of the server, Ā(t)− Ā(s) ≥ β(t− s). Basic results of network calculus give upper bounds of the worst-case backlog, the worstcase delay and the output burstiness of a server. Those bounds are computed using a maximum arrival curve α for the input flow A and a minimum service curve β for the server. RR n° 7685 in ria -0 06 08 85 2, v er si on 1 15 J ul 2 01 1 Packetization and Aggregate Scheduling 5 • The backlog at time t is defined by B(t) = A(t)− Ā(t). The maximum backlog Bmax = supt∈R+ B(t) is bounded as follows: Bmax ≤ sup s≥0 [α(s)− β(s)] = α β(0). • The virtual delay at time t is defined by dv(t) = inf{d ≥ 0 | Ā(t + d) ≥ A(t)}. The maximum virtual delay dmax = supt∈R+ dv(t) satisfies: dmax ≤ sup t≥0 {inf{d ≥ 0 | β(t+ d) ≥ α(t)}}. • Output burstiness: the curve α β is a maximum arrival curve for the output flow Ā. The difference between simple and strict service curves is important when dealing with residual service curves: when several flows share the same server, it may be necessary (in the case of arbitrary multiplexing or fixed priorities, for example) to have strict service curves to compute a service curve for a single flow (basically removing the arrival curve of the crosstraffic from the service curve). Unfortunately, in the case of arbitrary multiplexing, the curve obtained is not a strict service curve [12]. The next theorem provides a residual strict service curve when there already exist individual strict service curves. Doing this, those individual services are improved. Theorem 1. Let A1, A2, . . . , An be n arrival flows to a given server that offers a minimum strict service curve β. Let α1, α2, . . . , αn be maximum arrival curves for A1, A2, . . . , An respectively. Let β1, β2, . . . , βn be minimum strict service curves offered by this server to A1A2, . . . , An respectively. Then, for all i = 1, . . . , n, max (β −∑ j 6=i min(αj βj , αj (β − ∑