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 \(\{(t_j,k_j): j\in {\mathbb {Z}}\}\) 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.
Similar content being viewed by others
References
Zulfiker Ali, M., Misic, J., Misic, V.B.: Uplink access protocol in IEEE 802.11ac. IEEE Trans. Wirel. Commun. 17, 5535–5551 (2018)
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Baccelli, F., Bremaud, P.: Elements of Queueing Theory. Springer, Berlin (1994)
Billingsley, L.: Probability. Addison-Wesley, New York (1968)
Brandt, A., Franken, P., Lisek, B.: Stationary Stochastic Models. Wiley, New York (1990)
Breiman, L.: Probability. Addison-Wesley, New York (1968). Reprinted by SIAM
Bruneel, H., Kim, B.G.: Discrete-Time Models for Communication Systems Including ATM. Springer, New York (2012)
Durrett, R.: Probability: Theory and Examples. Wadsworth and Brooks/Cole, Pacific Grove (1991)
Franken, P., König, D., Arndt, U., Schmidt, V.: Queues and Point Processes. Akademie, Berlin (1981)
Glynn, P., Sigman, K.: Uniform Cesaro limit theorems for synchronous processes with applications to queues. Stoch. Process. Appl. 40, 29–43 (1992)
Halfin, H.: Batch delays versus customer delays. Bell Syst. Tech. J. 62, 2011–2015 (1983)
Kallenberg, O.: Random Measures: Theory and Applications. Springer, New York (2017)
Khintchine, A. Ya.: Mathematical Methods in the theory of Queueing (in Russian, English translation by Griffin, London, 1960) (1955)
Loève, M.: Probability Theory II, 4th edn. Springer, New York (1978)
Loynes, R.M.: The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520 (1962)
Ma, S., Xu, X., Wang, H.: A discrete queue with double thresholds policy and its application to SVC systems. J. Appl. Math. Comput. 56, 351–366 (2018)
Matthes, K.: Zur theorie des Bedienungsprocesses. Trans. Third Prague Onf. Information Theory, Prague, pp. 513–528 (1962)
Miyazawa, M.: Derivation of Little’s and related formulas by rate conversation law with multiplicity. Science University of Tokyo, Preprint (1990)
Miyazawa, M., Takahashi, Y.: Rate conservation principle for discrete-time queues. Queueing Syst. 12, 215–230 (1992)
Munkres, J.: Topology: A First Course. Princeton-Hall, New Jersey (1975)
Nieuwenhuis, G.: Equivalence of functional limit theorems for stationary point processes and their Palm distributions. Probability Theory and Related Fields 81, 593–608 (1989)
Palm, C.: Intensity variations in telephone traffic. Ericsson Technics, 44, pp. 1–189. (in German, English translation by North Holland, Amsterdam, 1988) (1943)
Rolski, T.: Relationships between characteristics in periodic Poisson queues. Queueing Syst. 4, 17–26 (1989)
Sigman, K.: Stationary Marked Point Processes: An Intuitive Approach. Chapman and Hall, New York (1995)
Takagi, H.: Queueing Analysis, Vol 3: Discrete-Time Systems, North-Holland, Amsterdam (1993)
Whitt, W.: Comparing batch delays and customer delays. Bell Syst. Tech. J. 62(7), 2001–2009 (1983)
Whitt, W.: The renewal-process stationary-excess operator. J. Appl. Probab. 22, 156–167 (1985)
Whitt, W.: \(H=\lambda G\) and the Palm transformaton. Adv. Appl. Probab. 24, 755–758 (1992)
Whitt, W.: Stochastic Process Limits. Springer, New York (2002)
Whitt, W., Zhang, X.: A data-driven model of an emergency department. Oper. Res. Health Care 12, 1–15 (2017)
Whitt, W., Zhang, X.: Periodic Little’s law. Oper. Res. 67(1), 267–280 (2019)
Wolff, R.W.: Stochastic Modeling and the Theory of Queues. Prentice Hall, New Jersey (1989)
Zhong, J., Yates, R. D., Soljanin, E.: Timely lossless source coding for randomly arriving symbols. In: Proceedings IEEE Information Theory Workshop, Guangzhou, China, pp. 25–29 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sigman, K., Whitt, W. Marked point processes in discrete time. Queueing Syst 92, 47–81 (2019). https://doi.org/10.1007/s11134-019-09612-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-019-09612-3
Keywords
- Marked point processes
- Discrete-time stochastic processes
- Batch arrival processes
- Queueing theory
- Periodic stationarity