An Extension of Norton's Theorem for Queueing Networks

Balsamo, Simonetta; Iazeolla, Giuseppe

doi:10.1109/tse.1982.235424

Cited by 37 publications

(13 citation statements)

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“…The results of [10] can easily be generalised to subnetworks consisting of several stations such that customers enter the subnetwork through a single input node and leave the subnetwork through a single output node. Balsamo and Iazeolla [1], Kritzinger et al [19], and Vantilborgh [23] extend Norton's theorem to BCMP-networks consisting of two arbitrary subnetworks. A further extension is given by Towsley [22], where elementary state-dependent routing is incorporated.…”

Section: Aggregationmentioning

confidence: 96%

Decomposition and Aggregation in Queueing Networks

Huisman¹,

Boucherie

2010

International Series in Operations Research &Amp; Management Science

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This chapter considers the decomposition and aggregation of multiclass queueing networks with state-dependent routing. Combining state-dependent generalisations of quasi-reversibility and biased local balance, sufficient conditions are obtained under which the stationary distribution of the network is of product-form. This product-form factorises into one part that describes the nodes of the network in isolation, and one part that describes the routing and the global network state. It is shown that a decomposition holds for general nodes if the input-output behaviour of these nodes is suitably compensated by the state-dependent routing function. When only a subset of the nodes is of interest, it is shown that the other nodes may be aggregated into nodes that only capture their global behaviour. The results both unify and extend existing classes of product-form networks, as is illustrated by several cases and an example of an assembly network. IntroductionIn the analysis of queueing networks, two at first sight different techniques have been used to derive product form results: quasi-reversibility and local balance. Quasi-reversibility is a property of the nodes of the network, roughly stating that they should preserve input and output flows when they are considered in isolation and fed by a Poisson process. If such nodes are coupled into a network by Markov

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Section: Aggregationmentioning

confidence: 96%

Decomposition and Aggregation in Queueing Networks

Huisman¹,

Boucherie

2010

International Series in Operations Research &Amp; Management Science

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“…As far as we know this is the first work in which a connection between the notion of lumpability and product-form is shown. The application of aggregation and lumpability techniques has been proposed to cope with the solution of models with a large state space, and it has been widely applied for various formalisms, e.g., exact and approximate aggregation in queueing networks (Balsamo and Iazeolla 1982;Buchholz 2010), decomposability and lumpability for Markov chains (Kemeny and Snell 1960;Stewart 1994), aggregation of stochastic Petri nets (Buchholz 2004), stochastic automata or Markovian process algebras (Hillston 1996;Gilmore et al 2001), where the references should be considered just as examples of remarkable work in the corresponding field.…”

Section: Related Workmentioning

confidence: 99%

“…However, some exact aggregation methods have been defined directly in terms of model components at a higher level of abstraction. Moreover, under special constraints, conditions for exact aggregation have been defined for various classes of Markov models and for product-form models, such as product-form queueing networks Balsamo and Iazeolla 1982). In (Chiola et al 1993) the high level nets are restricted to class called well-formed nets in order to allow for a more efficient analysis based on aggregations of states.…”

Section: Related Workmentioning

confidence: 99%

Lumping and reversed processes in cooperating automata

2014

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Performance evaluation of computer software or hardware architectures may rely on the analysis of a complex stochastic model whose specification is usually given in terms of a high level formalism such as queueing networks, stochastic Petri nets, stochastic automata or Markovian process algebras. Compositionality is a key-feature of many of these formalisms and allows the modeller to combine several (simple) components to form a complex architecture. However, although these formalisms lead to relative compact specifications of possibly complex models, the derivation of the performance indices may be computationally very time and space consuming since the set of possible states of the model tends to grow exponentially (or even faster) with the number of components. In this paper we focus on models with underlying continuous time Markov chains (CTMCs) and we introduce a notion of typed lumpability, which gives sufficient conditions under which a lumping of the process can be derived, allowing the exact computation of marginal stationary probabilities of the cooperating components. The peculiarity of our method relies on the fact that lumping is applied at the component-level rather than to the CTMC underlying the joint process, thus reducing both the memory requirements and the computational cost of the subsequent solution of the model. Moreover, we investigate the properties of the lumping of reversed automata and we prove that, if these are reversible, a conditional product-form solution of their cooperation with other non-blocking automata can be derived. Although conditional product-forms have been previously investigated by other authors with the aim of approximating non-product-form models, the contribution of this paper consists in giving sufficient conditions for this approach to yield exact results and providing examples to support the modeller’s intuition

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“…4c is obtained by substituting the subnetwork with the aggregated node C. The aggregated network and the original one have the same marginal queue length distribution and average performance indices. Exact aggregation in queueing networks holds for any subnetwork, i. e. for subnetworks with multiple entry and exit points and for which we have also to define a new routing matrix for the aggregated network [3], and for multichain networks [39].…”

Section: Propertiesmentioning

confidence: 99%

Product Form Queueing Networks

Balsamo

2000

Performance Evaluation: Origins and Directions

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An Extension of Norton's Theorem for Queueing Networks

Cited by 37 publications

References 1 publication

Decomposition and Aggregation in Queueing Networks

Decomposition and Aggregation in Queueing Networks

Lumping and reversed processes in cooperating automata

Product Form Queueing Networks

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