Far Eastern Mathematical Journal

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Jackson network in a random environment: strong approximation

Elena Bashtova, Elena Lenena

2020, issue 2, Ñ. 144-149
DOI: https://doi.org/10.47910/FEMJ202015

We consider a Jackson network with regenerative input flows in which every server is subject to a random environment influence generating breakdowns and repairs. They occur in accordance with two independent sequences of i.i.d. random variables. We establish a theorem on the strong approximation of the vector of queue lengths by a reflected Brownian motion in positive orthant.

Jackson network, Strong approximation, Unreliable systems

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[1] J. M. Harrison, “The heavy tra?c approximation for single server queues in series”, J. Appl. Probab., 10:3, (1973), 613–629.
[2] W. Whitt, “Heavy tra?c limit theorems for queues: a survey”, In Mathematical Methods in Queueing Theory, 98, (1974), 307–350.
[3] A. J. Lemoine, “State of the art – networks of queues: a survey of weak convergence results”, Manag. Sci., 24:11, (1978), 1175–1193.
[4] M. I. Reiman, “Open queueing networks in heavy tra?c”, Mathematics of Operations Re-search, 9:3, (1984), 441–458.
[5] H. Chen, D. D. Yao, Fundamentals of Queueing Networks, Springer, New York, 2001.
[6] N. V. Djellab, “On the M |G|1 retrial queue subjected to breakdowns”, RAIRO - Oper. Res., 36, (2002), 299-310.
[7] D. P. Gaver, “A waiting line with interrupted service, including priorities”, J. R. Stat. Soc. (B), 24:1, (1962), 73–90.
[8] E. Kalimulina, Analysis of unreliable Jackson-type queueing networks with dynamic routing, SSRN Working Paper, https://ssrn.com/abstract=2881956.
[9] N. Sherman, J. Kharoufen, M. Abramson, “An M |G|1 retrial queue with unreliable server for streaming multimedia applications”, Prob. Eng. Inf. Sci., 23, (2009), 281–304.
[10] L. G. Afanasyeva, E. E. Bashtova, “Coupling method for asymptotic analysis of queues with regenerative input and unreliable server”, Queueing Systems, 76, (2014), 125–147.
[11] J. M. Harrison, M. I. Reiman, “Re?ected Brownian motion on an orthant”, Ann. Probab., 9:2, (1981), 302–308.
[12] W. L. Smith, “Regenerative stochastic processes”, Proc. Royal Soc. London Ser. A, 232:1188, (1955), 6–31.
[13] E. Bashtova, A. Shashkin, “Strong Gaussian approximation for cumulative processes with heavy tails”, arXiv: 2007.15481.
[14] M. Cs?org?o, L. Horv?ath, J. Steinebach, “Invariance principles for renewal processes”, Ann. Probab., 15:4, (1987), 1441–1460.

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