Abstract
This paper develops a Gaussian model for an open network of queues having a path-dependent net-input process, whose evolution depends on its early history, and satisfies a non-ergodic law of large numbers. We show that the Gaussian model arises as the heavy-traffic limit for a sequence of open queueing networks, each with a multivariate generalization of a Polya arrival process. We show that the net-input and queue-length processes for the Gaussian model satisfy non-ergodic laws of large numbers with tractable distributions.
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References
Cha, J.H.: Characterization of the generalized Polya process and its applications. Adv. Appl. Probab. 46(4), 1148–1171 (2014)
Cha, J.H., Badia, F.G.: On a multivariate generalized Polya process without regularity property. Probab. Eng. Inf. Sci. 34(4), 484–506 (2020)
Cha, J.H., Finkelstein, M.: Point Processes for Reliability Analysis. Springer, New York (2018)
Cha, J.H., Mercier, S.: Poisson generalized gamma process and its properties. Stochastics 93(8), 1123–1140 (2021)
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem, Society for Industrial and Applied Mathematics (1992)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer, New York (2003)
Dai, J., Nguyen, V., Reiman, M.I.: Sequential bottleneck decomposition: an approximation method for generalized Jackson networks. Oper. Res. 42(1), 119–136 (1994)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn. John Wiley, New York (1968)
Fendick, K.W.: Brownian motion minus the independent increments: representation and queueing application. Probab. Eng. Inf. Sci. 36(1), 144–168 (2020)
Fendick, K.W., Whitt, W.: Queues with path-dependent arrival processes. J. Appl. Probab. 58, 484–504 (2021)
Fendick, K.W., Whitt, W.: Heavy traffic limits for queues with non-stationary path-dependent arrival processes. Queueing Syst. 101, 113–135 (2022)
Fendick, K.W., Whitt, W.: On Gaussian Markov processes and Polya processes. unpublished paper submitted for publication, available on the authors’ web pages (2023)
Glynn, P.W.: Diffusion Approximations. Handbooks in Operations Research and Management Science, vol. 2, pp. 145–198. Elsevier, New York (1990)
Grandell, J.: Mixed Poisson Processes. Chapman and Hall, London (1997)
Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1985)
Harrison, J.M., Nguyen, V.: The QNET method for two-moment analysis of open queueing networks. Queueing Syst. 101(1), 1–32 (1990)
Harrison, J.M., Reiman, M.I.: Reflected Brownian motion on an orthant. Ann. Probab. 9(2), 302–308 (1981)
Kleinrock, K.: Queueing Systems, vol. 2. WIley, New York (1976)
Konno, T.H.: On the exact solution of a generalized Polya process. Adv. Math. Phys. 2010, 504267 (2010)
Mandjes, M.: Large Deviations for Gaussian Queues: Modelling Communication Networks. John Wiley, New York (2007)
Ross, S.: Stochastic Processes, 2nd edn. John Wiley, New York (1996)
Simeu-Abazi, Z., Di Mascolo, M., Gascard, E.: Performance evaluation of centralized maintenance workshop by using Queueing Networks. IFAC Proc. Vol. 45(31), 175–180 (2012)
Whitt, W.: Stochastic Process Limits. Springer, New York (2002)
Whitt, W., You, W.: Time-varying robust queueing. Oper. Res. 67(6), 1766–1782 (2019)
Whitt, W., You, W.: A robust queueing network analyzer based on indices of dispersion. Nav. Res. Logist. 69(1), 36–56 (2022)
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Fendick, K., Whitt, W. Queueing networks with path-dependent arrival processes. Queueing Syst 105, 17–46 (2023). https://doi.org/10.1007/s11134-023-09885-9
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DOI: https://doi.org/10.1007/s11134-023-09885-9
Keywords
- Path-dependent stochastic processes
- Generalized Polya process
- Gaussian Markov process
- Diffusion approximations
- Queues
- Heavy-traffic limit