Abstract
We consider a storage model where the input and demand are modulated by an underlying Markov chain. Such models arise in data communication systems. The input is a Markov-compound Poisson process and the demand is a Markov linear process. The demand is satisfied if physically possible. We study the properties of the demand and its inverse, which may be viewed as transformed time clocks. We show that the unsatisfied demand is related to the infimum of the net input and that, under suitable conditions, it is an additive functional of the input process. The study of the storage level is based on a detailed analysis of the busy period, using techniques based on infinitesimal generators. The transform of the busy period is the unique solution of a certain matrix-functional equation. Steady state results are also obtained; these are not obvious generalizations of the results for simple storage models. In particular, a generalization of the Pollaczek-Khinchin formula brings new insight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data-handling system with multiple sources, Bell Syst. Tech. J. 61 (1982) 1871–1894.
R.M. Blumenthal and R.K. Getoor,Markov Processes and Potential Theory (Academic Press, New York, 1968).
R.M. Blumenthal,Excursions of Markov Processes (Birkäuser, Boston, 1992).
S. Browne and P. Zipkin, Inventory models with continuous, stochastic demands, Ann. Appl. Prob. 1 (1991) 419–435.
A.I. Elwalid and D. Mitra, Analysis and design of rate-based congestion control of high speed networks, I: Stochastic fluid models, access regulation, Queueing Systems 9 (1991) 29–64.
A. Friedman,Foundations of Modern Analysis (Dover, New York, 1982).
D.P. Gaver and J.P. Lehoczky, Channels that cooperatively service a data stream and voice messages, IEEE Trans. Commun. 30 (1982) 1153–1162.
J.A. Goldstein,Semigroups of Linear Operators and Applications (Oxford University Press, New York, 1985).
I. Karatzas and S.E. Shreve,Brownian Motion and Stochastic Calculus (Springer, New York, 1988).
D. Mitra, Stochastic theory of a fluid model of producers and consumers coupled by a buffer, Adv. Appl. Prob. 20 (1988) 646–676.
N.U. Prabhu,Stochastic Storage Processes (Springer, New York, 1980).
N.U. Prabhu and Y. Zhu, Markov-modulated queueing systems, Queueing Systems 5 (1989) 215–245.
N.U. Prabhu, Markov renewal and Markov-additive processes — A review and some new results,Proc. KAIST Mathematics Workshop, Vol. 6, eds. B.D. Choi and J.W. Yim, Korea Advanced Institute of Science and Technology, Taejon (1991) pp. 57–94.
B.A. Sevast'yanov, Influence of a storage bin capacity on the average standstill time of a production line, Theory Prob. Appl. 7 (1962) 429–438.
N.U. Prabhu and L.C. Tang, Markov-modulated single server queueing systems, to appear in J. Appl. Prob.
E.A. Van Doorn and G.J.K. Regterschot, Conditional pasta, Oper. Res. Lett. 7 (1988) 229–232.
E.A. Van Doorn, A.A. Jqaers and J.S.J. de Wit, A fluid reservoir regulated by a birth-death process, Stoch. Models 4 (1988) 457–472.
Author information
Authors and Affiliations
Additional information
Research supported by Grant BD/645/90-RM from Junta Nacional de Investigação Cientifica e Tecnológica.
Rights and permissions
About this article
Cite this article
Prabhu, N.U., Pacheco, A. A storage model for data communication systems. Queueing Syst 19, 1–40 (1995). https://doi.org/10.1007/BF01148938
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01148938