Marked point processes in discrete time

  • Karl SigmanEmail author
  • Ward Whitt


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 \(\{(t_j,k_j): j\in {\mathbb {Z}}\}\) of times \(t_j\in {\mathbb {Z}}\) and marks \(k_j\in {\mathbb {K}}\), with batch arrivals (i.e., \(t_j=t_{j+1}\)) 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.


Marked point processes Discrete-time stochastic processes Batch arrival processes Queueing theory Periodic stationarity 

AMS 2010 Subject Classification

Primary: 60G10 60G55 Secondary: 60K25 90B22 



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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of IEORColumbia UniversityNew YorkUSA

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