Min-plus Algebra Research Articles

Perturbation of eigenvalues of matrix pencils and the optimal assignment problem

We extend the perturbation theory of Višik, Ljusternik and Lidskiı̆ to the case of eigenvalues of matrix pencils. This extension allows us to solve certain degenerate cases of this theory. We show that the first order asymptotics of the eigenvalues of a perturbed matrix pencil can be computed generically by methods of min-plus algebra and optimal assignment algorithms. We illustrate this result by discussing a singular perturbation problem considered by Najman. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Service curve based routing subject to deterministic QoS constraints

The problem of traffic engineered routing subject to deterministic QoS constraints with a minimum cost objective is considered. A novel blend of generic service curves as abstraction of network links of the associated network calculus under min-plus algebra and of existing heuristics for the restricted shortest path problem, is presented as a generic approach applicable in arbitrary network infrastructures. In particular, we define load-dependent link cost and present in pseudo-code notation the details of the different steps of the developed algorithm, which comprises of a pre-computation and an on-demand phase. A tabu-search is deployed as a local search heuristic to exploit the neighborhood of a pre-computed minimum weight path. The presented approach provides a generic technical solution for minimum-cost routing subject to deterministic QoS constraints, which includes only a reasonably limited number of tuning knobs and thus facilitates the network operator to concentrate on the rather economical task of link cost definition.

A general framework for deterministic service guarantees in telecommunication networks with variable length packets

By extending the recently developed filtering theory under the min-plus algebra to the max-plus algebra, we develop a general framework for providing deterministic quality-of-service guarantees in telecommunication networks with variable length packets. The traffic in such networks is modeled by marked point processes that consist of two sequences of variables: the arrival times and the packet lengths. By modifying Cruz's characterization for a counting process, we propose a traffic characterization, called g-regularity, to characterize a marked point process. Based on the traffic characterization, we introduce two basic network elements: traffic regulators that generate g-regular marked point processes; and g-servers that provides quality-of-service for marked point processes. As in the filtering theory under the min-plus algebra, network elements can be joined by concatenation, filter bank summation, and feedback to form a composite network element. We illustrate the use of the framework by various examples that include G/G/1 queues, VirtualClock, guaranteed rate servers in tandem, segmentation and reassembly, jitter control, dampers, and window flow control.

THE USE OF CONVENTIONAL AND MINPLUS ALGEBRA FOR THE MODELING OF HYBRID PETRI NETS

In this paper we are interested in a mathematical description of the class of hybrid systems, which can be modeled by hybrid Petri nets HPNs . A state space model using conventional algebra for the continuous subsystem and minplus algebra for the discrete subsystem is given. The interface between the discrete and continuous part appears in our model. This model is illustrated through an example. We introduce the concept of discrete control of HPNs.

Application of network calculus to guaranteed service networks

We use previous network calculus results to study some properties of lossless multiplexing as it may be used in guaranteed service networks. We call network calculus a set of results that apply min-plus algebra to packet networks. We provide a simple proof that shaping a traffic stream to conform to a burstiness constraint preserves the original constraints satisfied by the traffic stream. We show how all rate-based packet schedulers can be modeled with a simple rate latency service curve. Then we define a general form of deterministic effective bandwidth and equivalent capacity. We find that call acceptance regions based on deterministic criteria (loss or delay) are convex, in contrast to statistical cases where it is the complement of the region which is convex. We thus find that, in general, the limit of the call acceptance region based on statistical multiplexing when the loss probability target tends to 0 may be strictly larger than the call acceptance region based on lossless multiplexing. Finally, we consider the problem of determining the optimal parameters of a variable bit rate (VBR) connection when it is used as a trunk, or tunnel, given that the input traffic is known. We find that there is an optimal peak rate for the VBR trunk, essentially insensitive to the optimization criteria. For a linear cost function, we find an explicit algorithm for the optimal remaining parameters of the VBR trunk.

Timed-event graphs with multipliers and homogeneous min-plus systems

The authors study fluid analogues of a subclass of Petri nets, called fluid timed event graphs with multipliers, which are a timed extension of weighted T systems studied in the Petri net literature. These event graphs can be studied naturally, with a new algebra, analogous to the min-plus algebra, but defined on piecewise linear concave increasing functions, endowed with the pointwise minimum as addition and the composition of functions as multiplication. A subclass of dynamical systems in this algebra, which have a property of homogeneity, can be reduced to standard min-plus linear systems after a change of counting units. The authors give a necessary and sufficient condition under which a fluid timed-event graph with multipliers can be reduced to a fluid timed event graph without multipliers. In the fluid case, this class corresponds to the so-called expansible timed-event graphs with multipliers of Munier (1993), or to conservative weighted T-systems. The change of variable is called here a potential. Its restriction to the transition nodes of the event graph is a T-semiflow.

Analyse de systèmes min-max

We will study extensions of known results of max-plus (or min-plus) systems to systems where the underlying algebra consists of the three operations max, min and addition simultaneously (such systems are called min-max systems). Such systems are nonlinear in both the max-plus and the min-plus algebra. The notion of a pulsative circuit is introduced, which characterizes the speed of a stationary solution of the system. In general, the cycle mean of a pulsative circuit is neither maximal (as is the case for max-plus systems) nor minimal (as is the case for min-plus systems). Subject to certain conditions, the solutions of min-max systems show a periodic behaviour, which is directly related to the pulsative circuits. Solutions starting from arbitrary initial conditions converge in a finite number of steps to such a periodic behaviour. Game-theoretic notions are helpful for the construction of pulsative circuits. Both analytic and graph-theoretic reasoning is used in the derivation of the various results.