Abstract
We prove a functional central limit theorem for Markov additive arrival processes where the modulating Markov process has the transition rate matrix scaled up by \(n^{\alpha }\) (\(\alpha >0\)) and the mean and variance of the arrival process are scaled up by n. It is applied to an infinite-server queue and a fork–join network with a non-exchangeable synchronization constraint, where in both systems both the arrival and service processes are modulated by a Markov process. We prove functional central limit theorems for the queue length processes in these systems joint with the arrival and departure processes, and characterize the transient and stationary distributions of the limit processes. We also observe that the limit processes possess a stochastic decomposition property.
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Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H., de Turck, K.: A functional central limit theorem for a Markov-modulated infinite-server queue. Methodol. Comput. Appl. Probab. 18(1), 153–168 (2016)
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, Berlin (2003)
Baykal-Gursoy, M., Xiao, W.: Stochastic decomposition in \(M/M/\infty \) queues with Markov modulated service rates. Queueing Syst. 48(1), 75–88 (2004)
Blom, J., Kella, O., Mandjes, M., Thorsdottir, H.: Markov-modulated infinite-server queues with general service times. Queueing Syst. 76(4), 403–424 (2014)
Blom, J., Mandjes, M., Thorsdottir, H.: Time-scaling limits for Markov-modulated infinite-server queues. Stoch. Models 29(1), 112–127 (2013)
Blom, J., de Turck, K., Mandjes, M.: Analysis of Markov-modulated infinite-server queues in the central-limit regime. Probab. Eng. Inf. Sci. 29(3), 433–459 (2015)
Blom, J., de Turck, K., Mandjes, M.: Functional central limit theorems for Markov-modulated infinite-server systems. Math. Methods Oper. Res. 83(3), 351–372 (2016)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (2009)
D’Auria, B.: Stochastic decomposition of the \(M/G/\infty \) queue in a random environment. Oper. Res. Lett. 35(6), 805–812 (2007)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (2009)
Falin, G.: The \(M/M/\infty \) queue in a random environment. Queueing Syst. 58, 65–76 (2008)
Keilson, J., Servi, L.: The matrix \(M/M/\infty \) system: retrial models and Markov modulated sources. Adv. Appl. Probab. 25, 453–471 (1993)
Krichagina, E.V., Puhalskii, A.A.: A heavy-traffic analysis of a closed queueing system with a \(GI/\infty \) service center. Queueing Syst. 25(1–4), 235–280 (1997)
Lu, H., Pang, G.: Gaussian limits for a fork-join network with non-exchangeable synchronization in heavy traffic. Math. Oper. Res. 41(2), 560–595 (2015a)
Lu, H., Pang, G.: Heavy-traffic limits for an infinite-server fork-join network with dependent and disruptive services. Submitted (2015b)
Lu, H., Pang, G.: Heavy-traffic limits for a fork-join network in the Halfin-Whitt regime. Submitted (2015c)
Nazarov, A., Baymeeva, G.: The \(M/G/\infty \) queue in a random environment. In: Dudlin, A. et al. (eds.) ITMM 2014, CCIS 487, pp. 312–324 (2014)
Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat. 42(4), 1285–1295 (1971)
O’Cinneide, C., Purdue, P.: The \(M/M/\infty \) queue in a random environment. J. Appl. Probab. 23(1), 175–184 (1986)
Pang, G., Whitt, W.: Two-parameter heavy-traffic limits for infinite-server queues. Queueing Syst. 65(4), 325–364 (2010)
Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, New York (1996)
Skorohod, A.V.: Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2, 138–171 (1957)
Steichen, J.L.: A functional central limit theorem for Markov additive processes with an application to the closed Lu-Kumar network. Stoch. Models 17(4), 459–489 (2001)
Straf, M.L.: Weak convergence of stochastic processes with several parameters. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 2, pp. 187–221 (1972)
Whitt, W.: Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Applications to Queues. Springer, Berlin (2002)
Whitt, W.: Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Applications to Queues, Online Supplement (2002)
Acknowledgments
Hongyuan Lu and Guodong Pang acknowledge the support from the NSF Grant CMMI-1538149. Michel Mandjes acknowledges the support from Gravitation project NETWORKS, Grant number 024.002.003, funded by the Netherlands Organization for Scientific Research (NWO).
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Lu, H., Pang, G. & Mandjes, M. A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems. Queueing Syst 84, 381–406 (2016). https://doi.org/10.1007/s11134-016-9496-8
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DOI: https://doi.org/10.1007/s11134-016-9496-8
Keywords
- Markov additive arrival process
- Functional central limit theorem
- Infinite-server queues
- Fork–join networks with non-exchangeable synchronization
- Gaussian limits
- Stochastic decomposition