Асимптотический анализ и оценивание качества обслуживания систем с гауссовским входным потоком
Диссертация
Столь существенное отличие в свойствах сетевого трафика потребовало разработки новых моделей и методов их анализа. В частности, наличие долговременной зависимости между данными сетевого трафика сделало весьма популярными модели, основанные на гауссовских процессах. Самым известным и изученным самоподобным гауссовским процессом с долговременной зависимостью является ДБ Д. Так, например, данный… Читать ещё >
Список литературы
- Биллингсли П. Сходимость вероятностных мер. М.: Наука, 1977.
- Боровков А. А. Асимптотические методы в теории массового обслуживания. М.: Наука, 1980.
- Жакод Ж., Ширяев А. Н. Предельные теоремы для случайных процессов. М.: Физматлит, 1994.
- Кельтон В., Лоу А. Имитационное моделирование. Спб.: Питер, 2004.
- Лидбеттер М., Линдгрен Г., Ротсен X. Экстремумы случайных последовательностей и процессов. Москва: Мир, 1989.
- Лифшиц М. А. Гауссовские случайные функции. Киев: ТВиМС, 1995.
- Лукашенко О. В., Морозов Е. В. Асимптотика максимума процесса нагрузки для некоторого класса гауссовских очередей // Информатика и ее применения. 2012. Т. 6, № 3. С. 81−89.
- Лукашенко О. В., Морозов Е. В., Пагано М. Применение гауссовских процессов в моделировании сетевого трафика // Труды Карельского научного центра Российской академии наук. 2010. № 3. С. 51−58.
- Лукашенко О. В., Морозов Е. В., Пагано М. Статистическое моделирование гауссовской очереди // Труды Карельского научного центра Российской академии наук. 2011. № 5. С. 55−62.
- Льюис Д., Фистер К. Термодинамическая теория вероятностей: некоторые аспекты больших уклонений // Успехи математических наук. 1995. Т. 50. С. 47−88.
- Питербарг В. И. Лекции по теории гауссовских процессов. М.: Изд-во МГУ, 1986.
- Питербарг В. И. Асимптотические методы в теории гауссовских случайных процессов и полей. М.: Изд-во МГУ, 1988.
- Сенета Е. Правильно меняющиеся функции. Москва: Наука, 1985.
- Такач Л. Комбинаторные методы в теории случайных процессов. М.: Мир, 1971.
- Addie R., Mannersalo P., Norros I. Performance formulae for queues with Gaussian input // Proceedings of ITC 16. 1999. Pp. 1169−1178.
- Addie R., Mannersalo P., Norros I. Most probable paths and performance formulae for buffers with Gaussian input traffic // European Transactions in Telecommunications. 2002. Vol. 13. Pp. 183−196.
- Adler R. J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics, Hayward, CA, 1990.
- Asmussen S. Applied Probability and Queues. Springer, 2002.
- Asmussen S., Glynn P. Stochastic Simulation: algorithms and analysis. Springer, 2007.
- Baldi P., Pacchiarotti B. Importance Sampling for the Ruin Problem for General Gaussian Processes. 2004.
- Bingham N. H., Goldie C. M., Teugels J. L. Regular Variation. Cambridge University Press, 1987.
- Botvich D., Duffield N. Large deviations, the shape of the loss curve, and economies of scale in large multiplexers // Queueing Systems. 1995. Vol. 20. Pp. 293−320.
- Bucklew J. A. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, 1990.
- Burnecki Z., Michna Z. Simulation of Pickands constants // Probab. Math. Stat. 2002. Vol. 22. Pp. 193−199.
- Chang C. S. Performance Guarantees in Communication Networks. Springer, 2000.
- Choe J., Shroff N. A central-limit-theorem-based approach for analyzing queue behavior in high-speed networks // IEEE/ACM Trans. Netw. 1998. Vol. 6. Pp. 659−671.
- Choe J., Shroff N. On the supremum distribution of integrated stationary Gaussian processes with negative linear drift // Adv. Appl. Probab. 1999. Vol. 31. Pp. 135−157.
- Choe J., Shroff N. Use of the supremum distribution of Gaussian processes in queueing analysis with long-range dependence and selfsimilarity // Stoch. Models. 2000. Vol. 16. Pp. 209−231.
- Coeurjolly J. F. Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study // Journal of Stat. Software. Vol. 5. 2000.
- Crovella M. E., Bestavros A. Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes // IEEE/ACM Transactions on Networking. 1997. Vol. 5, no. 6. Pp. 835−846.
- Debicki K. A note on LDP for supremum of Gaussian processes over infinite horizon // Stat. Probab. Lett. 1999. Vol. 44. Pp. 211−220.
- Debicki K. Ruin probabilities for Gaussian integrated processes // Stoch. Process. Appl. 2002. Vol. 98. Pp. 151−174.
- Debicki K. Gaussian Processes // Encyclopedia of Actuarial Sciences. 2004. Vol. 2. Pp. 752−757.
- Debicki K. Some properties of generalized Pickands constants // Prob. Th. Appl. 2006. Vol. 50. Pp. 290−298.
- Debicki K., Es-Saghouani A., Mandjes M. Transient characteristic of Gaussian fluid queues // Queueing Syst. 2009. Vol. 62. Pp. 383−409.
- Debicki K., Kisowski P. A note on upper estimates for Pickands constants // Stat. Probab. Lett. 2008. Vol. 78. Pp. 2046−2051.
- Debicki K., Mandjes M. Exact overflow asymptotics for queues with many Gaussian inputs //J. Appl. Probab. 2003. Vol. 40. Pp. 702−720.
- Debicki K., Mandjes M. Traffic with an FBM limit: convergence of the stationary workload process // Queueing Syst. 2003. Vol. 46. Pp. 113−127.
- Debicki K., Michna Z., Rolski T. Simulation of the asymptotic constant in some fluid models // Stoch. Models. 2003. Vol. 19. Pp. 407−423.
- Debicki K., Palmowski Z. Heavy traffic Gaussian asymptotics of on-off fluid model // Queueing Syst. 1999. Vol. 33. Pp. 327−338.
- Debicki K., Rolski T. A Gaussian fluid model // Queueing Syst. 1995. Vol. 20. Pp. 433−452.
- Debicki K., Rolski T. Gaussian fluid models- a survey // Symposium on Performance Models for Information Communication Networks, Sendai, Japan. 2000.
- Debicki K., Rolski T. A note on transient Gaussian fluid models // Queueing Syst. 2002. Vol. 41. Pp. 321−342.
- Dembo A., Zeitouni O. Large Deviations Techniques and Applications. Springer, 1998.
- Dieker A. B. Simulation of fractional Brownian motion. Master’s thesis, the Vrije Universiteit, Amsterdam, 2002.
- Dieker A. B. Extremes of Gaussian processes over an infinite horizon // Stoch. Process. Appl. 2005. Vol. 115. Pp. 207−248.
- Dieker A. B. Extremes and fluid queues: Ph. D. thesis / the Vrije Universiteit, Amsterdam. 2006.
- Dieker A. B., Mandjes M. On asymptotically efficient simulation of large deviation probabilities // Adv. in Appl. Probab. 2005. Vol. 37. Pp. 539−552.
- Dieker A. B., Mandjes M. Fast simulation of overflow probabilities in a queue with Gaussian input // ACM Trans. Model. Comput. Simul. 2006. Vol. 16, no. 2. Pp. 119−151.
- Duffield N., O’Connell N. Large deviations and overflow probabilities for the general single server queue, with applications // Proceedings of the Cambridge Philosophical Society. 1995. Vol. 118. Pp. 363−374.
- Duffy K., Lewis J. T., Sullivan W. G. Logarithmic asymptotics for the supre-mum of a stochastic process // Ann. Appl. Probab. 2003. Vol. 13, no. 2. Pp. 430−445.
- Erramilli A., Narayan O., Willinger W. Experimental queueing analysis with long-range dependent packet traffic // IEEE/ACM Trans. Netw. 1996. Vol. 4. Pp. 209−223.
- Erramilli A., Roughan M., Veitch D., Willinger W. Self-similar traffic and network dynamics // Proc. IEEE 90. 2002. Pp. 800−819.
- Es-Saghouani A., Mandjes M. On the correlation structure of Gaussian queues. // Stoch. Models. 2009. Vol. 25. Pp. 221−247.
- Fraleigh C., Tobagi F., Diot C. Provisioning IP backbone networks to support latency sensitive traffic // Proc. IEEE Infocom. 2003.
- Ganesh A., O’Connell N., Wischik D. Big Queues. Springer, 2004.
- Giordano S., Gubinelli M., Pagano M. Bridge Monte-Carlo: a novel approach to rare events of Gaussian processes // Proc. of the 5th St. Petersburg Workshop on Simulation. St. Petersburg, Russia: 2005. Pp. 281−286.
- Glynn P., Whitt W. Logarithmic asymptotics for steady-state tail probabilities in a single-server queue //J. Appl. Probab. 1994. Vol. 31 A. Pp. 413−430.
- Heidelberger P. Fast simulation of rare events in queueing and reliability models // ACM Transactions on Modeling and Computer Simulation. 1995. Vol. 5. Pp. 43−85.
- Hollander F. Large Devations. Fields Institute Monographs, AMS, 2000.
- Hiisler J., Piterbarg V. I. Extremes of a certain class of Gaussian processes // Stochastic Processes and their Applications. 1999. Vol. 83. Pp. 257−271.
- Hiisler J., Piterbarg V. I. Limit theorem for maximum of the storage process with fractional Brownian as input // Stochastic Processes and their Applications. 2004. Vol. 114. Pp. 231−250.
- Kaj I., Taqqu M. Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach: Tech. rep.: Department of Mathematics, Uppsala University, U.U.D.M., 2004.
- Kilpi J., Norros I. Testing the Gaussian approximation of aggregate traffic // In Proceedings of the 2nd Internet Measurement Workshop. 2002. Pp. 49−61.
- Kim H. S., Shroff N. B. Loss Probability Calculations and Asymptotic Analysisfor Finite Buffer Multiplexers // IEEE/ACM Transactions on Networking. 2001. Vol. 9. Pp. 755−768.
- Kim H. S., Shroff N. B. On the Asymptotic Relationship between the Overflow Probability and the Loss Ratio // Advances in Applied Probability. 2001. Vol. 33. Pp. 836−863.
- Konstantopoulos T., Zazanis M., Veciana G. D. Conservation laws and reflection mappings with application to multiclass mean value analysis for stochastic fluid queues // Stochastic Processes and their Applications. 1996. Vol. 65. Pp. 139−146.
- Kroese D. P., Taimre T., Botev Z. I. Handbook of Monte Carlo Methods. Wiley, 2011.
- Kulkarni V., Rolski T. Fluid model driven by an Ornstein-Uhlenbeck process // Probability in the Engineering and Informational Sciences. 1994. Vol. 8. Pp. 403−417.
- Lau W., Erramilli A., Wang J. L., Willinger W. Self-similar traffic generation: the random midpoint displacement algorithm and its properties // Proceedings of ICC '95. 1995.
- Leland W. E., Taqqu M. S., Willinger W., Wilson D. V. On the self-similar nature of ethernet traffic (extended version) // IEEE/ACM Transactions of Networking. 1994. Vol. 2. Pp. 1−15.
- Lewis J. T., Russell R. An Introduction to Large Deviations for Teletraffic Engineers //In Performance '96, October 1996. 1996.
- Likhanov N., Mazumdar R. Cell loss asymptotics in buffers fed with a largenumber of independent stationary sources // Journal of Applied Probability. 1999. Vol. 36. Pp. 86−96.
- Lukashenko 0. V. Gaussian queues in communication networks // Third Northern Triangular seminar. Programe and abstract. 2011. P. 14.
- Lukashenko 0. V., Morozov E. V. Gaussian Processes in Communication Networks // Proceedings of AMICT'2009. 2009. Pp. 112−118.
- Lukashenko O. V., Morozov E. V. On the maximum workload for a class of Gaussian queues // International conference «Probability theory and its applications» in Commemoration of the Centennial of B. V. Gnedenko. 2012. Pp. 231−232.
- Mandelbrot B. B., van Ness J. W. Fractional Brownian motions, fractional noises and applications // SIAM Review. 1968. Vol. 10. Pp. 422−437.
- Mandjes M. Large Deviations of Gaussian Queues. Chichester: Wiley, 2007.
- Marcus M. B., Shepp L. A. Sample behaviour of Gaussian processes // Proc. Sixth Berkeley Symp. Math. Stat. Prob. 1971. Vol. 2. Pp. 423−442.
- Massouli6 L., Simonian A. Large buffer asymptotics for the queue with FBM input // J. Appl. Probab. 1999. Vol. 36. Pp. 894−906.
- Michna Z. On tail probabilities and first passage times for fractional Brownian motion // Math. Methods Oper. Res. 1999. Vol. 49. Pp. 335−354.
- Mikosch T., Resnick S., Rootzen H., Stegeman A. Is Network Traffic Apprixi-mated by Stable Levy Motion or Fractional Brownian Motion? // Ann. Appl. Probab. 2002. Vol. 12, no. 1. Pp. 23−68.
- Morozov E. Communications Systems: Rare Events and Effective Bandwidths. Public University of Navarre, 2004.
- Narayan O. Exact asymptotic queue length distribution for fractional Brownian traffic // Advances in Performance Analysis. 1998. Vol. 1. Pp. 39−63.
- Norros I. Studies on a model for connectionless traffic, based on fractional Brownian motion // Conference on Applied Probability in Engineering, Computer and Communication Sciences INRIA/ORSA/TIMS/SMAI, Paris. 1993.
- Norros I. A storage model with self-similar input // Queueing Syst. 1994. Vol. 16. Pp. 387−396.
- Norros I. On the use of fractional Brownian motion in the theory of connectionless networks // IEEE J. Sel. Areas Comm. 1995. Vol. 13. Pp. 953−962.
- Norros I., Mannersalo P., Wang J. Simulation of fractional Brownian motion with conditionalized random midpoint displacement // Advances in Performance Analysis. 1999. Vol. 2. Pp. 77−101.
- Pickands J. Asymptotic properties of the maximum in a stationary Gaussian process // Trans. Amer. Math. Soc. 1969. Vol. 145. Pp. 75−86.
- Piterbarg V. Large deviations of a storage process with fractional Brownian motion as input // Extremes. 2001. Vol. 4. Pp. 147−164.
- Reich E. On the On the integrodifferential equation of Takacs I. // Ann. Math. Stat. 1958. Vol. 29. Pp. 563−570.
- Ross S. M. Simulation. Elsevier, 2006.
- Rubino G., Tuffin B. Rare Event Simulation using Monte Carlo Methods. Wiley, 2009.
- Samorodnitsky G., Taqqu M. S. Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Chapman & Hall, 1994.
- Shao Q. M. Bounds and estimators of a basic constant in extreme value theory of Gaussian processes // Stat. Sin. 1996. Vol. 6. Pp. 245−257.
- Simonian A., Virtamo J. Transient and stationary distributions for fluid queue and input processes with density // SIAM Journal of Applied Mathematics. 1991. Vol. 51. Pp. 1731−1739.
- Taqqu M. S., Willinger W., Sherman R. Proof of a fundamental result in self-similar traffic modeling // Computer communication review. 1997. Vol. 27. Pp. 5−23.
- Van de Meent R., Mandjes M., Pras A. Gaussian traffic everywhere? //In Proceedings of the IEEE International Conference on Communications. 2006.
- Weiss A. A new technique for analyzing large traffic systems // Adv. Appl. Probab. 1986. Vol. 18. Pp. 506−532.
- Whitt W. Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and their Application to Queues. Springer, New York, NY, USA, 2002.
- Willinger W., Taqqu M. S., Leland W. E., Wilson D. V. Self-similarity in highspeed packet traffic: analysis and modeling of Ethernet traffic measurements // Statistical Sciences. 1995. Vol. 10, no. 1. Pp. 67−85.
- Стабилизация оценки 7Гю (1) с ростом числа наблюдений.71
- Сравнение двух методов оценки стационарной вероятности переполнения 7ГП (1).72
- Зависимость вероятности переполнения 7ГП (1) от количества источников п.72
- Точность аппроксимации (2.40).74
- Выборочные траектории процессов нагрузки (конечный и бесконечный буфер).75
- Доверительный интервал для Р^(4).78
- Зависимость длины доверительного интервала от размера буфераЪ. 78
- Относительная ошибка оценки (ДБД).87
- Оценка 7Г1 (6) в сравнении с аппроксимацией (2.9).88г)
- Выборочные реализации У .89
- Гистограмма распределения У (п = 50).89
- Гистограмма распределения У (п = 100) .90
- Гистограмма распределения У (п — 500) .90
- Гистограмма распределения У (п = 2000).91г)
- Выборочные реализации У .91