Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T22:44:44.069Z Has data issue: false hasContentIssue false
  • English
  • Français

FINITE TWO LAYERED QUEUEING SYSTEMS

Published online by Cambridge University Press:  18 May 2016

Efrat Perel  and
Uri Yechiali
Efrat Perel
Affiliation:
Afeka College of Engineering, Tel-Aviv, Israel Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel E-mail: [email protected]
Uri Yechiali
Affiliation:
Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.

These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 3: Erol Gelenbe's 70th Birthday , July 2016 , pp. 492 - 513
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2016

References

1.Androutsellis-Theotokis, S. & Spinellis, D. (2004). A survey of peer-to-peer content distribution technologies. ACM Computing Surveys 36(4): 335–371.Google Scholar
2.Arazi, A., Ben-Jacob, E., & Yechiali, U. (2005). Controlling an oscillating Jackson-type network having state-dependant service rates. Mathematical Methods of Operations Research 62: 453–466.Google Scholar
3.Bright, L.W. & Taylor, P.G. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stochastic Models 11(3): 497–526.Google Scholar
4.Cooper, R.B. (1981). Introduction to queueing theory. 2nd ed.New York: North Holland.Google Scholar
5.De Nitto Personè, V. & Grassi, V. (1996). Solution of finite QBD processes. Journal of Applied Probability 33(4): 1003–1010.Google Scholar
6.Elhafsi, E.H. & Molle, M. (2007). The solution to QBD processes with finite state space. Stochastic Analysis and Applications 25: 763–779.Google Scholar
7.Franks, G., Al-Omari, T., Woodside, M., Das, O., & Derisavi, S. (2009). Enhanced modeling and solution of layered queueing networks. IEEE Transactions on Software Engineering 35: 148–161.CrossRefGoogle Scholar
8.Hajek, B. (1982). Birth-and-death processes on the integers with phases and general boundaries. Journal of Applied Probability 19: 488–499.Google Scholar
9.Latouche, G. & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia: SIAM and ASA.Google Scholar
10.Neuts, M.F. (1981). Matrix geometric solutions in stochastic models—an algorithmic approach. Baltimore and London: The Johns Hopkins University Press.Google Scholar
11.Perel, E. & Yechiali, U. (2008). Queues where customers of one queue act as servers of the other queue. Queueing Systems 60: 271–288.Google Scholar
12.Perel, E. & Yechiali, U. (2013). On customers acting as servers. Asia-Pacific Journal of Operational Research 30(5): Article 1350019, 23 pages. DOI: 10.1142/S021759591350019X.Google Scholar
13.Wu, D., Liu, Y., & Ross, K.W. (2009). Queuing network models for multi-channel live streaming systems. Proceedings of IEEE INFOCOM 2009, pp. 73–81.Google Scholar