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Marked point processes in discrete time

Author

Listed:
  • Karl Sigman

    (Columbia University)

  • Ward Whitt

    (Columbia University)

Abstract

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ { ( t j , k j ) : j ∈ Z } of times $$t_j\in {\mathbb {Z}}$$ t j ∈ Z and marks $$k_j\in {\mathbb {K}}$$ k j ∈ K , with batch arrivals (i.e., $$t_j=t_{j+1}$$ t j = t j + 1 ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

Suggested Citation

  • Karl Sigman & Ward Whitt, 2019. "Marked point processes in discrete time," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 47-81, June.
  • Handle: RePEc:spr:queues:v:92:y:2019:i:1:d:10.1007_s11134-019-09612-3
    DOI: 10.1007/s11134-019-09612-3
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    References listed on IDEAS

    as
    1. Glynn, Peter & Sigman, Karl, 1992. "Uniform Cesaro limit theorems for synchronous processes with applications to queues," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 29-43, February.
    2. Ward Whitt & Xiaopei Zhang, 2019. "Periodic Little’s Law," Operations Research, INFORMS, vol. 67(1), pages 267-280, January.
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    Cited by:

    1. Xiaoting Li & Christian Genest & Jonathan Jalbert, 2021. "A self‐exciting marked point process model for drought analysis," Environmetrics, John Wiley & Sons, Ltd., vol. 32(8), December.

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