Generalized matrix period in max-plus algebra

Molnárová, Monika

doi:10.1016/j.laa.2005.02.033

Cited by 20 publications

(14 citation statements)

References 6 publications

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“…Lemma 7 (Molnárová (2005)) Consider two almost linear periodic sequences a * and b * , where a * has period p a and factor q, and b * has period p b and the same factor q. Then the sequence max(a * , b * ) is almost linear periodic with period p as an integer divisor of lcm( p a , p b ), and factor q.…”

Section: Definition 10mentioning

confidence: 99%

“…Molnárová (2005)) to prove their linear periodicity Baccelli et al (1992), i.e., they can be represented by a finite aperiodic part and a periodic part repeated infinitely often. The required proof technique is significantly different from the previous work Zeng and Di Natale (2012) (which provides initial results on the periodicity of execution matrix for synchronous state machines), including the consideration of the sequence of maximum elements of matrix power and the required task graph transformation.…”

Section: Our Contributionsmentioning

confidence: 99%

“…However, even when the matrix is not linear periodic, it is still possible to avoid computing the rbf function over a long time interval by leveraging the concept of general periodicity (Molnárová (2005)), a generalization of linear periodicity.…”

Section: Definition 7 the Matrixmentioning

confidence: 99%

“…The problem of computing the general period gper(A) or general factor matrix gfac(A) for a given square matrix A is shown to be NP-hard Molnárová (2005), where the complexity is expressed in terms of the size of the matrix (or the corresponding graph). However, it is asymptotically independent from the power of the matrix (or the time interval for the rbf and dbf functions), which is not part of the input to the algorithm.…”

Section: Definition 7 the Matrixmentioning

confidence: 99%

“…Two other lemmas in Molnárová (2005) provide sufficient conditions for the maximum of two almost linear periodic sequences to be linear periodic.…”

Section: Definition 10mentioning

confidence: 99%

See 4 more Smart Citations

Computing periodic request functions to speed-up the analysis of non-cyclic task models

Zeng

Natale

2014

Real-Time Syst

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Tasks are units of sequential code implementing the system actions and executed concurrently by an operating system. Techniques have been developed to determine, at design time, whether a set of tasks can safely complete before their deadlines. Several models have been proposed to represent conditional executions and dependencies among concurrent tasks for the purpose of schedulability analysis. Among them, task graphs with cyclic recurrent behavior (i.e., those modeled with a single source vertex and a period parameter specifying the minimum amount of time that must elapse between successive activations of the source job) allow for efficient schedulability analysis based on the periodicity of the request and demand bound functions (rbf and dbf). In this paper, we leverage results from max-plus algebra to identify a recurrent term in rbf and dbf of general task graph models, even when the execution is neither recurrent nor controlled by a period parameter. As such, the asymptotic complexity of calculating rbf and dbf is independent from the length of the time interval. Experimental results demonstrate significant improvements on the runtime for system schedulability analysis.

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Section: Definition 10mentioning

confidence: 99%

Section: Our Contributionsmentioning

confidence: 99%

Section: Definition 7 the Matrixmentioning

confidence: 99%

Section: Definition 7 the Matrixmentioning

confidence: 99%

“…Two other lemmas in Molnárová (2005) provide sufficient conditions for the maximum of two almost linear periodic sequences to be linear periodic.…”

Section: Definition 10mentioning

confidence: 99%

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Computing periodic request functions to speed-up the analysis of non-cyclic task models

Zeng

Natale

2014

Real-Time Syst

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Bounded Model Checking of Max-Plus Linear Systems via Predicate Abstractions

Mufid

Adzkiya

Abate

2019

Lecture Notes in Computer Science

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This paper introduces the abstraction of max-plus linear (MPL) systems via predicates. Predicates are automatically selected from system matrix, as well as from the specifications under consideration. We focus on verifying time-difference specifications, which encompass the relation between successive events in MPL systems. We implement a bounded model checking (BMC) procedure over a predicate abstraction of the given MPL system, to verify the satisfaction of time-difference specifications. Our predicate abstractions are experimentally shown to improve on existing MPL abstractions algorithms. Furthermore, with focus on the BMC algorithm, we can provide an explicit upper bound on the completeness threshold by means of the transient and the cyclicity of the underlying MPL system.

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A comparison of schedulability analysis methods using state and digraph models for the schedulability analysis of synchronous FSMs

2019

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Generalized matrix period in max-plus algebra

Abstract: The matrix power sequences and their periodic properties in max-plus algebra are studied. The notion of generalized periodicity is introduced for which both periodicity and linear periodicity are special cases. Every matrix over a max-plus algebra is shown to be almost generally periodic.

Cited by 20 publications

References 6 publications

Computing periodic request functions to speed-up the analysis of non-cyclic task models

Computing periodic request functions to speed-up the analysis of non-cyclic task models

Bounded Model Checking of Max-Plus Linear Systems via Predicate Abstractions

A comparison of schedulability analysis methods using state and digraph models for the schedulability analysis of synchronous FSMs

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