Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-03T19:50:30.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Jackson networks in nonautonomous random environments

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  10 June 2016

Ruslan Krenzler ,
Hans Daduna  and
Sonja Otten
Ruslan Krenzler*
Affiliation:
University of Hamburg
Hans Daduna*
Affiliation:
University of Hamburg
Sonja Otten*
Affiliation:
University of Hamburg
*
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
* Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.

Keywords

Randomised random walkJackson networkskippingprocesses in random environmentreflectionproduct form steady statebreakdown of nodesdegrading servicespeed-up of service

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60K25: Queueing theory 90B22: Queues and service 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 315 - 331
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Balsamo, S. and Marin, A. (2013).Separable solutions for Markov processes in random environments.Europ. J. Operat. Res. 229,391–403.Google Scholar
[2]Bell, C. E. and StidhamS., Jr. S., Jr. (1983).Individual versus social optimization in the allocation of customers to alternative servers.Manag. Sci. 29,831–839.CrossRefGoogle Scholar
[3]Boucherie, R. J. and van Dijk, N. M. (eds) (2011).Queueing Networks: A Fundamental Approach (Internat. Ser. Operat. Res. Manag. Sci.154).Springer,New York.CrossRefGoogle Scholar
[4]Cogburn, R. (1980).Markov chains in random environments: the case of Markovian environments.Ann. Prob. 8,908–916.Google Scholar
[5]Cogburn, R. and Torrez, W. C. (1981).Birth and death processes with random environments in continuous time.J. Appl. Prob. 18,19–30.Google Scholar
[6]Cornez, R. (1987).Birth and death processes in random environments with feedback.J. Appl. Prob. 24,25–34.Google Scholar
[7]Economou, A. (2005).Generalized product-form stationary distributions for Markov chains in random environments with queueing applications.Adv. Appl. Prob. 37,185–211.Google Scholar
[8]Economou, A. (2014). Personal communication.Google Scholar
[9]Economou, A. and Fakinos, D. (1998).Product form stationary distributions for queueing networks with blocking and rerouteing.Queueing Systems Theory Appl. 30,251–260.Google Scholar
[10]Falin, G. (1996).A heterogeneous blocking system in a random environment.J. Appl. Prob. 33,211–216.Google Scholar
[11]Gannon, M.,Pechersky, E.,Suhov, Y. and Yambartsev, V. (2016).Random walks in a queueing network environment. To appear in J. Appl. Prob.CrossRefGoogle Scholar
[12]Gibbens, R. J.,Kelly, F. P. and Key, P. B. (1995).Dynamic alternative routing. In Routing in Communications Networks, ed. M. E. Steenstrup,Prentice Hall,Englewood Cliffs, NJ, pp.13–47.Google Scholar
[13]Jackson, J. R. (1957).Networks of waiting lines.Operat. Res. 5,518–521.Google Scholar
[14]Kleinrock, L. (1976).Queueing Systems, Vol. II.John Wiley,New York.Google Scholar
[15]Krenzler, R. and Daduna, H. (2014).Modeling and performance analysis of a node in fault tolerant wireless sensor networks. In Measurement, Modeling, and Evaluation of Computing Systems and Dependability and Fault Tolerance, eds K. Fischbach and U. R. Krieger,Springer,Heidelberg, pp.73–78.Google Scholar
[16]Krenzler, R. and Daduna, H. (2015).Loss systems in a random environment: steady state analysis.Queueing Systems 80,127–153.Google Scholar
[17]Krenzler, R. and Daduna, H. (2015).Performability analysis of an unreliable M/M/1-type queue. In Leistungs-, Zuverlässigkeits- und Verlässlichkeitsbewertung von Kommunikationsnetzen und verteilten Systemen: 8. GI/ITG-Workshop MMBnet 2015,Berichte des Fachbereichs Informatik der Universität Hamburg,Universität Hamburg, pp.90–95.Google Scholar
[18]Krenzler, R.,Daduna, H. and Otten, S. (2014).Randomization for Markov chains with applications to networks in a random environment. Preprint 2014--02, Deptarment of Mathematics, University of Hamburg.Google Scholar
[19]Krishnamoorthy, A.,Pramod, P. K. and Chakravarthy, S. R. (2014).Queues with interruptions: a survey.TOP 22,290–320.Google Scholar
[20]Krishnamoorthy, A. and Viswanath, N. C. (2013).Stochastic decomposition in production inventory with service time.Europ. J. Operat. Res. 228,358–366.CrossRefGoogle Scholar
[21]Kulkarni, V. and Yan, K. (2012).Production-inventory systems in stochastic environment and stochastic lead times.Queueing Systems 70,207–231.CrossRefGoogle Scholar
[22]Nucci, A.,et al. (2003).IGP link weight assignment for transient link failures. In Proc. 18th Internat. Teletraffic Congress,Elsevier,Amsterdam, pp.321–330.Google Scholar
[23]Otten, S.,Krenzler, R. and Daduna, H. (2016).Models for integrated production-inventory systems: Steady state and cost analysis. To appear in Internat. J. Production Res.Google Scholar
[24]Ramaswami, V. and Taylor, P. G. (1996).An operator-analytic approach to product-form networks.Commun. Statist. Stoch. Models 12,121–142.CrossRefGoogle Scholar
[25]Saffari, M.,Asmussen, S. and Haji, R. (2013).The M/M/1 queue with inventory, lost sale, and general lead times.Queueing Systems 75,65–77.Google Scholar
[26]Sauer, C. and Daduna, H. (2003).Availability formulas and performance measures for separable degradable networks.Econom. Quality Control 18,165–194.Google Scholar
[27]Schassberger, R. (1984).Decomposable stochastic networks: some observations. In Modelling and Performance Evaluation Methodology (Lecture Notes Control Inf. Sci.60),Springer,Berlin, pp.137–150.Google Scholar
[28]Schwarz, M.et al. (2006).M/M/1 queueing systems with inventory.Queueing Systems 54,55–78.CrossRefGoogle Scholar
[29]Serfozo, R. F. (1999).Introduction to Stochastic Networks.Springer,New York.CrossRefGoogle Scholar
[30]Shah, D. and Shin, J. (2012).Randomized scheduling algorithm for queueing networks.Ann. Appl. Prob. 22,128–171.Google Scholar
[31]StidhamS., Jr. S., Jr. (2009).Optimal Design of Queueing Systems.CRC,Boca Raton, FL.CrossRefGoogle Scholar
[32]Tsitsiashvili, G. S.,Osipova, M. A.,Koliev, N. V. and Baum, D. (2002).A product theorem for Markov chains with application to PF-queueing networks.Ann. Operat. Res. 113,141–154.Google Scholar
[33]Van Dijk, N. M. (1988).On Jackson's product form with `jump-over' blocking.Operat. Res. Lett. 7,233–235.Google Scholar
[34]Van Dijk, N. M. (1993).Queueing Networks and Product Forms: A Systems Approach.John Wiley,Chichester.Google Scholar
[35]Whittle, P. (1985).Scheduling and characterization problems for stochastic networks.J. R. Statist. Soc. B 47,407–415.Google Scholar
[36]Yamazaki, G. and Miyazawa, M. (1995).Decomposability in queues with background states.Queueing Systems Theory Appl. 20,453–469.Google Scholar
[37]Yechiali, U. (1973).A queueing-type birth-and-death process defined on a continuous-time Markov chain.Operat. Res. 21,604–609.CrossRefGoogle Scholar
[38]Zhu, Y. (1994).Markovian queueing networks in a random environment.Operat. Res. Lett. 15,11–17.Google Scholar
[39]Zolotarev, V. M. (1966).Distribution of queue length and number of operating lines in a system of Erlang type with random breakage and restoration of lines. In Selected Translations in Mathematical Statistics and Probability, Vol. 6,American Mathematical Society,Providence, RI, pp.89–99.Google Scholar