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Exact Analysis of Discrete Part Production Lines: The Markovian Queueing Network and the Stochastic Automata Networks Formalisms

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Handbook of Stochastic Models and Analysis of Manufacturing System Operations

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 192))

Abstract

Manufacturing systems are quite complex and many research works have been devoted to their analysis, modeling, design and operation. This work is concerned with the exact analysis of discrete part production lines. More specifically, two formalisms are provided: (a) the Markovian Queueing Network and (b) the Stochastic Automata Network (SAN). These two methods are described explicitly via the use of an example of a production line consisting of three stations. SAN methodology utilizes both classical and generalized tensor algebra. The tensor or Kronecker representation of the SAN three-station example is given and comparisons are made between these two exact methods. Their limitations are also examined regarding the numerical results concerned with throughput of discrete part production lines. It is seen that with the SAN formalism one may solve exactly much larger production line configurations than those traditional Markovian formalism can handle.

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Notes

  1. 1.

    In general, the service rates need not be identical, i.e., μ i ≠μ j for i≠j.

  2. 2.

    http://purl.oclc.org/NET/prodline.

  3. 3.

    The choice of order of the states is arbitrary. It affects the numerical values of the matrices, but it does not affect the complexity of the derivation of the model solution, which, as expected, is not dependent on the ordering of the states.

  4. 4.

    The choice of the automaton to carry the rates of a synchronizing event is arbitrary and may change the descriptor numeric values. However, it does not affect the solution of the model.

  5. 5.

    We consider two matrices to be stochasticly equivalent when they represent the same stochastic process, i.e., they deliver the same steady state and transient solution.

  6. 6.

    A functional element can be a rate (\(\tau ({\mathcal{S}}^{(\omega )})\)) or a probability (\(\pi ({\mathcal{S}}^{(\omega )})\)), and it can be also expressed in the evaluated form: \(\tau (\tilde{{x}}^{(\omega )})\) and \(\pi (\tilde{{x}}^{(\omega )})\) respectively.

  7. 7.

    The master/slave semantic is used to the formal definition of synchronizing events. However, any semantics can be used without loss of generality.

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Authors and Affiliations

  1. Department of Computer Science, PUCRS-PPGCC, Av. Ipiranga, 6681, Porto Alegre, Brazil

    P. Fernandes & A. Sales

  2. Department of Economics, Aristotle University of Thessaloniki, Thessaloniki, 541 24, Greece

    C. T. Papadopoulos

  3. Waterford Institute of Technology, Waterford, Ireland

    M. E. J. O’Kelly

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  1. P. Fernandes
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  2. M. E. J. O’Kelly
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  3. C. T. Papadopoulos
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  4. A. Sales
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Correspondence to C. T. Papadopoulos .

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Editors and Affiliations

  1. , MIE Department, University of Massachusetts Amherst, Governors Drive 160, Amherst, 01003, Massachusetts, USA

    J. MacGregor Smith

  2. College of Admin. Sciences & Econ., Koç University, Rumelifeneri Yolu, Sarıyer, 34450, Istanbul, Turkey

    Barış Tan

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Fernandes, P., O’Kelly, M.E.J., Papadopoulos, C.T., Sales, A. (2013). Exact Analysis of Discrete Part Production Lines: The Markovian Queueing Network and the Stochastic Automata Networks Formalisms. In: Smith, J., Tan, B. (eds) Handbook of Stochastic Models and Analysis of Manufacturing System Operations. International Series in Operations Research & Management Science, vol 192. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6777-9_3

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