More Web Proxy on the site http://driver.im/

research-article

Copula Analysis of Temporal Dependence Structure in Markov Modulated Poisson Process and Its Applications

Authors:

Venkatesh SrinivasanAuthors Info & Claims

ACM Transactions on Modeling and Performance Evaluation of Computing Systems (TOMPECS), Volume 2, Issue 3

Article No.: 14, Pages 1 - 28

https://doi.org/10.1145/3089254

Published: 29 June 2017 Publication History

Abstract

The Markov Modulated Poisson Process (MMPP) has been extensively studied in random process theory and widely applied in various applications involving Poisson arrivals whose rate varies following a Markov process. Despite the rich literature on MMPP, very little is known on its intricate temporal dependence structure. No exact solution is available so far to capture the functional temporal dependence of MMPP at the stationary state over slotted times.

This article tackles the above challenges with copula analysis. It not only presents a novel analytical framework to capture the temporal dependence of MMPP but also provides the exact copula-based solutions for single MMPP as well as the aggregate of independent MMPP. This theoretical contribution discloses functional dependence structure of MMPP. It also lays the foundation for many applications that rely on the temporal dependence of MMPP for adaptive control or predictive resource provisioning. We demonstrate case studies, with real-world trace data as well as simulation, to illustrate the practical significance of our analytical results.

Supplementary Material

a14-dong-apndx.pdf (dong.zip)

Supplemental movie, appendix, image and software files for, Copula Analysis of Temporal Dependence Structure in Markov Modulated Poisson Process and Its Applications

Download
193.09 KB

References

[1]

Kjersti Aas. 2004. Modelling the dependence structure of financial assets: A survey of four copulas. Technical Report SAMBA/22/04, Norsk Regnesentral (2004).

[2]

Allan T. Andersen and Bo Friis Nielsen. 1997. An application of superpositions of two state markovian source to the modelling of self-similar behaviour. In Proceedings of INFOCOM’97. IEEE, Kobe, Japan, 196--204.

Digital Library

[3]

Allan T. Andersen and Bo Friis Nielsen. 1998. A Markovian approach for modeling packet traffic with long-range dependence. IEEE J. Select. Areas Commun. 16, 5 (1998), 719--732.

Digital Library

[4]

Konstantin Avrachenkov, Natalia M. Markovich, and Jithin K. Sreedharan. 2015. Distribution and dependence of extremes in network sampling processes. Computat. Soc. Netw. 2, 1 (2015), 1.

[5]

Kazim Azam and Michael K. Pitt. 2014. Bayesian inference for a semi-parametric copula-based Markov chain. Working Paper, University of Warwick. (2014).

[6]

Andrea Baiocchi and Nicola Blefari-Melazzi. 1993. Steady-state analysis of the MMPP/G/1/K queue. IEEE Trans. Commun. 41, 4 (1993), 531--534.

[7]

Soshant Bali and Victor S. Frost. 2007. An algorithm for fitting MMPP to IP traffic traces. IEEE Commun. Lett. 11, 2 (2007), 207--209.

[8]

Khalid Begain, Gunter Bolch, and Helmut Herold. 2012. Practical Performance Modeling: Application of the MOSEL Language. Springer Science 8 Business Media, U.S.

[9]

Martina Beil. 2013. Modeling Dependencies Among Financial Asset Returns Using Copulas. Ph.D. Dissertation. Technische Universität München.

[10]

Vladislav Bína and Radim Jiroušek. 2013. A short note on multivariate dependence modeling. Kybernetika 49, 3 (2013), 420--432.

[11]

Eric Bouyé, Valdo Durrleman, Ashkan Nikeghbali, Gaël Riboulet, and Thierry Roncalli. 2000. Copulas for finance-a reading guide and some applications. Available at SSRN 1032533 (2000).

[12]

Lothar Breuer and Alfred Kume. 2010. An EM algorithm for Markovian arrival processes observed at discrete times. In Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance. Springer, Berlin, 242--258.

Digital Library

[13]

Yulia Burkatovskaya, Tatiana Kabanova, and Sergey Vorobeychikov. 2015. CUSUM algorithms for parameter estimation in queueing systems with jump intensity of the arrival process. In Information Technologies and Mathematical Modelling-Queueing Theory and Applications. Springer, Switzerland, 275--288.

[14]

Xiaohong Chen, Wei Biao Wu, and Yanping Yi. 2009. Efficient estimation of copula-based semiparametric Markov models. Ann. Stat. 37, 6B (2009), 4214--4253.

[15]

Tiberiu Chis and Peter G. Harrison. 2015. Adapting hidden Markov models for online learning. Electron. Notes Theoret. Comput. Sci. 318 (2015), 109--127.

Digital Library

[16]

Doo Il Choi, Tae-Sung Kim, and Sangmin Lee. 2008. Analysis of an MMPP/G/1/K queue with queue length dependent arrival rates, and its application to preventive congestion control in telecommunication networks. Eur. J. Operat. Res. 187, 2 (2008), 652--659.

[17]

Tibor Csóka and Jaroslav Polec. 2015. Modeling poisson error process on wireless channels. Int. J. Commun. Netw. Inform. Secur. 7, 1 (2015), 1--7.

[18]

Fang Dong, Kui Wu, Venkatesh Srinivasan, and Jianping Wang. 2016. Copula analysis of latent dependency structure for collaborative auto-scaling of cloud services. In Proceedings of ICCCN. IEEE, 1--8.

[19]

Fang Dong, Kui Wu, and Srinivasan Venkatesh. 2015. Copula analysis for statistical network calculus. In Proceedings of INFOCOM’15. IEEE, Hong Kong, 1535--1543.

[20]

Qing Du. 1995. A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process. J. Appl. Probabil. 32, 4 (1995), 1103--1111.

[21]

Valdo Durrleman, Ashkan Nikeghbali, Thierry Roncalli, and others. 2000. Which copula is the right one? Working paper, Crédit Lyonnais. (2000).

[22]

Robert J. Elliott and W. Paul Malcolm. 2008. Discrete-time expectation maximization algorithms for Markov-modulated Poisson processes. IEEE Trans. Automat. Control 53, 1 (2008), 247--256.

[23]

Paul Embrechts, Filip Lindskog, and Alexander McNeil. 2003. Modelling dependence with copulas and applications to risk management. Handbook Heavy Tail. Distrib. Fin. 8, 1 (2003), 329--384.

[24]

Wolfgang Fischer and Kathleen Meier-Hellstern. 1993. The Markov-modulated Poisson process (MMPP) cookbook. Perform. Eval. 18, 2 (1993), 149--171.

Digital Library

[25]

Christian Genest, Bruno Rémillard, and David Beaudoin. 2009. Goodness-of-fit tests for copulas: A review and a power study. Insur.: Math. Econ. 44, 2 (2009), 199--213.

[26]

Mahmood Mollaei Gharehajlu, Saadan Zokaei, and Yousef Darmani. 2015. Statistical analysis of different traffic types effect on QoS of wireless ad hoc networks. J. Inform. Syst. Telecommun. 3, 1(9) (2015), 7--15.

[27]

Paolo Giacomazzi. 2007. Statistical traffic envelopes for Markov-modulated Poisson packet sources. In Proceedings of IEEE GLOBECOM’07. IEEE, Washington, D.C., 2628--2633.

[28]

Paolo Giacomazzi. 2009. Closed-form analysis of end-to-end network delay with Markov-modulated Poisson and fluid traffic. Comput. Commun. 32, 4 (2009), 640--648.

Digital Library

[29]

Daniel P. Heyman and David Lucantoni. 2003. Modeling multiple IP traffic streams with rate limits. IEEE/ACM Trans. Netw. 11, 6 (2003), 948--958.

Digital Library

[30]

Alexander Ihler, Jon Hutchins, and Padhraic Smyth. 2007. Learning to detect events with markov-modulated poisson processes. ACM Trans.n Knowl. Discov. Data 1, 3 (2007), 13.

Digital Library

[31]

Shoji Kasahara. 2001. Internet traffic modeling: Markovian approach to self-similar traffic and prediction of loss probability for finite queues. IEICE Trans. Commun. 84, 8 (2001), 2134--2141.

[32]

Natalia Markovich. 2008. Nonparametric Analysis of Univariate Heavy-tailed Data: Research and Practice. Vol. 753. John Wiley 8 Sons, England.

[33]

Natalia M. Markovich. 2010. Modeling of dependence in a peer-to-peer video application. In Proceedings of the 6th International Wireless Communications and Mobile Computing Conference. ACM, 316--320.

Digital Library

[34]

Luca Muscariello, Marco Mellia, Michela Meo, M. Ajmone Marsan, and R. Lo Cigno. 2005. Markov models of internet traffic and a new hierarchical MMPP model. Comput. Commun. 28, 16 (2005), 1835--1851.

Digital Library

[35]

Roger B. Nelson. 2006. An Introduction to Copulas. Springer, New York.

Digital Library

[36]

David Neuhäuser, Christian Hirsch, Catherine Gloaguen, and Volker Schmidt. 2013. A parametric copula approach for modelling shortest-path trees in telecommunication networks. In Analytical and Stochastic Modeling Techniques and Applications. Springer, Berlin, 324--336.

[37]

Marcel F. Neuts. 1989. Structured Stochastic Matrices of M/G/1 Type and Their Applications. Taylor 8 Francis, New York.

[38]

António Nogueira, Paulo Salvador, Rui Valadas, and António Pacheco. 2003. Modeling self-similar traffic through markov modulated poisson processes over multiple time scales. In High-Speed Networks and Multimedia Communications. Springer, Berlin, 550--560.

[39]

Hiroyuki Okamura, Tadashi Dohi, and Kishor S. Trivedi. 2009. Markovian arrival process parameter estimation with group data. IEEE/ACM Trans. Netw. 17, 4 (2009), 1326--1339.

Digital Library

[40]

Sergio Pacheco-Sanchez, Giuliano Casale, Bryan Scotney, Sally McClean, Gerard Parr, and Stephen Dawson. 2011. Markovian workload characterization for qos prediction in the cloud. In Proceedings of the 2011 IEEE International Conference on CLOUD. IEEE, 147--154.

Digital Library

[41]

Andrew Patton. 2012. Copula methods for forecasting multivariate time series. Handbook Econ. Forecast. 2 (2012), 899--960.

[42]

Ali Rajabi and Johnny W. Wong. 2012. MMPP characterization of web application traffic. In Proceedings of the 2012 IEEE 20th International Symposium on MASCOTS. IEEE, 107--114.

Digital Library

[43]

Ali Rajabi and Johnny W. Wong. 2014. Provisioning of computing resources for web applications under time-varying traffic. In Proceedings of the 2014 IEEE 22nd International Symposium on MASCOTS. IEEE, 152--157.

Digital Library

[44]

Bruno Rémillard, Nicolas Papageorgiou, and Frédéric Soustra. 2012. Copula-based semiparametric models for multivariate time series. J. Multivar. Anal. 110 (2012), 30--42.

Digital Library

[45]

William J. J. Roberts, Yariv Ephraim, and Elvis Dieguez. 2006. On Rydén’s EM algorithm for estimating MMPPs. IEEE Signal Process. Lett. 13, 6 (2006), 373--376.

[46]

Sheldon M. Ross. 2003. Introduction to Probability Models. Academic Press, Burlington.

[47]

Tobias Rydén. 1994. Parameter estimation for Markov modulated Poisson processes. Stoch. Models 10, 4 (1994), 795--829.

[48]

Tobias Rydén. 1996. An EM algorithm for estimation in Markov-modulated Poisson processes. Computat. Stat. Data Anal. 21, 4 (1996), 431--447.

Digital Library

[49]

Paulo Salvador, Rui Valadas, and António Pacheco. 2003. Multiscale fitting procedure using Markov modulated Poisson processes. Telecommun. Syst. 23, 1--2 (2003), 123--148.

Digital Library

[50]

Steven L. Scott. 2000. Detecting network intrusion using a Markov modulated nonhomogeneous poisson process. Available Online (2000). https://astro.temple.edu/&sim;msobel/courses_files/scott-smythe.pdf.

[51]

Shou-Kuo Shao, Malla Reddy Perati, Meng-Guang Tsai, Hen-Wai Tsao, and Jingshown Wu. 2005. Generalized variance-based markovian fitting for self-similar traffic modelling. IEICE Trans. Commun. 88, 4 (2005), 1493--1502.

[52]

Pravin K. Trivedi and David M. Zimmer. 2007. Copula Modeling: An Introduction for Practitioners. Now Publishers Inc., Boston.

[53]

Maarten R. C. Van Oordt and Chen Zhou. 2012. The simple econometrics of tail dependence. Econ. Lett. 116, 3 (2012), 371--373.

[54]

Ury Yechiali and Pinhas Naor. 1971. Queuing problems with heterogeneous arrivals and service. Operat. Res. 19, 3 (1971), 722--734.

Digital Library

[55]

Tadafumi Yoshihara, Shoji Kasahara, and Yutaka Takahashi. 2001. Practical time-scale fitting of self-similar traffic with Markov-modulated Poisson process. Telecommun. Syst. 17, 1--2 (2001), 185--211.

Digital Library

[56]

Ming Yu and Mengchu Zhou. 2006. A model reduction method for traffic described by MMPP with unknown rate limit. IEEE Commun. Lett. 10, 4 (2006), 302--304.

Cited By

Lei LHu JZhu C(2021)Discrete-Event Stochastic Systems with Copula Correlated Input ProcessesIISE Transactions10.1080/24725854.2021.1943571(1-30)Online publication date: 18-Jun-2021
https://doi.org/10.1080/24725854.2021.1943571
Saha AGanguly NChakraborty SDe A(2019)Learning Network Traffic Dynamics Using Temporal Point ProcessIEEE INFOCOM 2019 - IEEE Conference on Computer Communications10.1109/INFOCOM.2019.8737622(1927-1935)Online publication date: Apr-2019
https://doi.org/10.1109/INFOCOM.2019.8737622
Dong FWu KSrinivasan V(2018)A New Dependence Model for Heterogeneous Markov Modulated Poisson Processes2018 IFIP Networking Conference (IFIP Networking) and Workshops10.23919/IFIPNetworking.2018.8696419(397-405)Online publication date: May-2018
https://doi.org/10.23919/IFIPNetworking.2018.8696419

Index Terms

Copula Analysis of Temporal Dependence Structure in Markov Modulated Poisson Process and Its Applications
1. Computing methodologies
  1. Modeling and simulation
    1. Model development and analysis
2. Theory of computation
  1. Theory and algorithms for application domains

Recommendations

A Mathematical Programming Approach to Optimal Markovian Switching of Poisson Arrival Streams to Queueing Systems

Motivated by certain situations in manufacturing systems and communication networks, we look into the problem of maximizing the profit in a queueing system with linear reward and cost structure and having a choice of selecting the streams of Poisson ...
Single-Server Queues with Markov-Modulated Arrivals and Service Speed

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is ...
A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞queue with binomial catastrophes. When a binomial catastrophe occurs, each customer is either removed with a probability or is retained with the complementary ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Transactions on Modeling and Performance Evaluation of Computing Systems

ACM Transactions on Modeling and Performance Evaluation of Computing Systems Volume 2, Issue 3

September 2017

135 pages

ISSN:2376-3639

EISSN:2376-3647

DOI:10.1145/3119902

Editors:
Sem Borst
Nokia Bell Labs / Eindhoven University of Technology, Netherlands
,
Carey Williamson
University of Calgary, Canada

Issue’s Table of Contents

Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 29 June 2017

Accepted: 01 April 2017

Revised: 01 April 2017

Received: 01 April 2016

Published in TOMPECS Volume 2, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Natural Sciences and Engineering Research Council of Canada

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
197
Total Downloads

Downloads (Last 12 months)16
Downloads (Last 6 weeks)3

Reflects downloads up to 18 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Lei LHu JZhu C(2021)Discrete-Event Stochastic Systems with Copula Correlated Input ProcessesIISE Transactions10.1080/24725854.2021.1943571(1-30)Online publication date: 18-Jun-2021
https://doi.org/10.1080/24725854.2021.1943571
Saha AGanguly NChakraborty SDe A(2019)Learning Network Traffic Dynamics Using Temporal Point ProcessIEEE INFOCOM 2019 - IEEE Conference on Computer Communications10.1109/INFOCOM.2019.8737622(1927-1935)Online publication date: Apr-2019
https://doi.org/10.1109/INFOCOM.2019.8737622
Dong FWu KSrinivasan V(2018)A New Dependence Model for Heterogeneous Markov Modulated Poisson Processes2018 IFIP Networking Conference (IFIP Networking) and Workshops10.23919/IFIPNetworking.2018.8696419(397-405)Online publication date: May-2018
https://doi.org/10.23919/IFIPNetworking.2018.8696419

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents