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Analysis of Markov-modulated fluid polling systems with gated discipline

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  • In this paper we present two different analytical descriptions of the fluid polling model with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain (the special case when the modulating Markov chains are independent is also included). The fluid is removed at the stations during the service period by a station dependent constant rate.

    The first analytical description is based on the relationships of steady-state fluid levels at embedded server arrival and departure epochs. We derive the steady-state vector Laplace transform of the fluid levels at the stations at arbitrary epoch and its moments. The second analytical description applies the method of supplementary variables and results in differential equations, from which the joint density function of the fluid levels can be obtained.

    We also propose computational methods for both analytical descriptions and provide numerical examples to illustrate the numeric computations.

    Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 60J25.


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  • Figure 1.  The joint distribution of the fluid level and the one-dimensional marginals

    Figure 2.  The joint distribution of the fluid level at polling and at departure epochs

    Table 1.  Steady-state vector moments of the fluid levels at polling epochs

    1st moment 1st moment 2nd moment 2nd moment
    element 0 element 1 element 0 element 1
    Station 1: 1.0614 0.7386 2.1640 1.7821
    Station 2: 2.1759 0.7170 8.3775 2.2387
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    Table 2.  Mean fluid levels of the queue in different phases of the server

    St. 1. busy St. 1. vacation St. 2. busy St. 2. vacation
    Station 1: 7.559 5.827 7.861 9.418
    Station 2: 3.915 5.932 4.362 2.194
     | Show Table
    DownLoad: CSV
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