Marked point processes in discrete time

Karl Sigman () and Ward Whitt
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Karl Sigman: Columbia University
Ward Whitt: Columbia University

Queueing Systems: Theory and Applications, 2019, vol. 92, issue 1, No 3, 47-81

Abstract: 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.

Keywords: Marked point processes; Discrete-time stochastic processes; Batch arrival processes; Queueing theory; Periodic stationarity; Primary: 60G10; 60G55; Secondary: 60K25; 90B22 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11134-019-09612-3

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