Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-07T22:29:33.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady-state analysis of a multiclass MAP/PH/c queue with acyclic PH retrials

Part of: Markov processes Computer system organization

Published online by Cambridge University Press:  09 December 2016

Tuǧrul Dayar  and
M. Can Orhan
Tuǧrul Dayar*
Affiliation:
Bilkent University
M. Can Orhan*
Affiliation:
Bilkent University
*
* Postal address: Department of Computer Engineering, Bilkent University, TR‒06800 Bilkent, Ankara, Turkey.
* Postal address: Department of Computer Engineering, Bilkent University, TR‒06800 Bilkent, Ankara, Turkey.
Get access Rights & Permissions [Opens in a new window]

Abstract

A multiclass c-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.

Keywords

Markovian arrival processphase-type service time distributionacyclic phase-type retrial time distributionKronecker productMarkov chain

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1098 - 1110
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] Artalejo, J. R. (1999).Accessible bibliography on retrial queues.Math. Comput. Modelling 30,1–6.Google Scholar
[2] Artalejo, J. R. (2010).Accessible bibliography on retrial queues: progress in 2000‒2009.Math. Comput. Modelling 51,1071–1081.Google Scholar
[3] Artalejo, J. R. and Gómez-Corral, A. (2007).Modelling communication systems with phase type service and retrial times.IEEE Commun. Lett. 11,955–957.CrossRefGoogle Scholar
[4] Artalejo, J. R. and Gómez-Corral, A. (2008).Retrial Queueing Systems: A Computational Approach.Springer,Berlin.CrossRefGoogle Scholar
[5] Artalejo, J. R. and Phung-Duc, T. (2012).Markovian retrial queues with two way communication.J. Ind. Manag. Optimization 8,781–806.CrossRefGoogle Scholar
[6] Asmussen, S. (2003).Applied Probability and Queues(Appl. Math. (New York) 51).Springer,New York.Google Scholar
[7] Avrachenkov, K.,Morozov, E. and Steyaert, B. (2016).Sufficient stability conditions for multi-class constant retrial rate systems.Queueing Systems 82,149–171.Google Scholar
[8] Baumann, H.,Dayar, T.,Orhan, M. C. and Sandmann, W. (2013).On the numerical solution of Kronecker-based infinite level-dependent QBD processes.Performance Evaluation 70,663–681.Google Scholar
[9] Bause, F.,Buchholz, P. and Kemper, P. (1998).A toolbox for functional and quantitative analysis of DEDS. In Computer Preformance Evaluation (Lecture Notes Comput. Sci. 1469),Springer,Berlin,pp. 356–359.Google Scholar
[10] Breuer, L.,Dudin, A. and Klimenok, V. (2002).A retrial BMAP/PH/N system.Queueing Systems 40,433–457.Google Scholar
[11] Bright, L. and Taylor, P. G. (1995).Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes.Commun. Statist. Stoch. Models 11,497–525.Google Scholar
[12] Buchholz, P. The Nsolve package. Available at http://ls4-www.cs.tu-dortmund.de/download/buchholz/struct-matrix-market.html.Google Scholar
[13] Buchholz, P.,Kriege, J. and Felko, I. (2014).Input Modeling with Phase-Type Distributions and Markov Models: Theory and Applications.Springer,Cham.Google Scholar
[14] Chakravarthy, S. R. (2013).Analysis of MAP/PH/c retrial queue with phase type retrials ‒ simulation approach. In Modern Probabilistic Methods for Analysis of Telecommunication Networks (Commun. Comput. Inf. Sci. 356),Springer Berlin,pp. 37–49.Google Scholar
[15] Chiola, G.,Dutheillet, C.,Franceschinis, G. and Haddad, S. (1993).Stochastic well-formed colored nets and symmetric modeling applications.IEEE Trans. Comput. 42,1343–1360.Google Scholar
[16] Choi, B. D. and Chang, Y. (1999).MAP1, MAP2/M/c retrial queue with the retrial group of finite capacity and geometric loss.Math. Comput. Modelling 30,99–113.Google Scholar
[17] Choi, B. D.,Chang, Y. and Kim, B. (1999).MAP1, MAP2/M/c retrial queue with guard channels and its application to cellular networks.Top 7,231–248.Google Scholar
[18] Dayar, T. and Orhan, M. C. RetrialQueueSolver. Available at http://www.cs.bilkent.edu.tr/∼tugrul/software.html.Google Scholar
[19] Dayar, T. and Orhan, M. C. (2012).Kronecker-based infinite level-dependent QBD processes.J. Appl. Prob. 49,1166–1187.CrossRefGoogle Scholar
[20] Dayar, T.,Hermanns, H.,Spieler, D. and Wolf, V. (2011).Bounding the equilibrium distribution of Markov population models.Numer. Linear Algebra Appl. 18,931–946.Google Scholar
[21] Dayar, T.,Sandmann, W.,Spieler, D. and Wolf, V. (2011).Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics.Adv. Appl. Prob. 43,1005–1026.CrossRefGoogle Scholar
[22] Diamond, J. E. and Alfa, A. S. (1998).The MAP/PH/1 retrial queue.Commun. Statist. Stoch. Models 14,1151–1177.Google Scholar
[23] Diamond, J. E. and Alfa, A. S. (1999).Approximation method for M/PH/1 retrial queues with phase type inter-retrial times.Europ. J. Operat. Res. 113,620–631.Google Scholar
[24] Dudin, A. and Klimenok, V. (2000).A retrial BMAP/SM/1 system with linear repeated requests.Queueing Systems Theory Appl. 34,47–66.Google Scholar
[25] Falin, G. I. (1988).On a multiclass batch arrival retrial queue.Adv. Appl. Prob. 20,483–487.CrossRefGoogle Scholar
[26] Falin, G. I. and Templeton, J. G. C. (1997).Retrial Queues.Chapman & Hall,London.Google Scholar
[27] Fayolle, G.,Malyshev, V. A. and Menshikov, M. V. (1995).Topics in the Constructive Theory of Countable Markov Chains.Cambridge University Press.Google Scholar
[28] Gharbi, N.,Dutheillet, C. and Ioualalen, M. (2009).Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems.Math. Comput. Modelling 49,1436–1448.Google Scholar
[29] Gómez-Corral, A. (2006).A bibliographical guide to the analysis of retrial queues through matrix analytic techniques.Ann. Operat. Res. 141,163–191.Google Scholar
[30] He, Q.-M.,Li, H. and Zhao, Y. Q. (2000).Ergodicity of the BMAP/PH/s/s+K retrial queue with PH-retrial times.Queueing Systems Theory Appl. 35,323–347.CrossRefGoogle Scholar
[31] Kim, B. (2011).Stability of a retrial queueing network with different classes of customers and restricted resource pooling.J. Ind. Manag. Optimization 7,753–765.Google Scholar
[32] Kim, C. S.,Mushko, V. and Dudin, A. N. (2012).Computation of the steady state distribution for multi-server retrial queues with phase type service process.Ann. Operat. Res. 201,307–323.Google Scholar
[33] Kim, J. and Kim, B. (2016).A survey of retrial queueing systems. To appear in Ann. Operat. Res. Google Scholar
[34] Kulkarni, V. G. (1986).Expected waiting times in a multiclass batch arrival retrial queue.J. Appl. Prob. 23,144–154.Google Scholar
[35] Kumar, M. S.,Sohraby, K. and Kiseon, K. (2013).Delay analysis of orderly reattempts in retrial queueing system with phase type retrial time.IEEE Commun. Lett. 17,822–825.Google Scholar
[36] Meyer, C. (2000).Matrix Analysis and Applied Linear Algebra.SIAM,Philadelphia, PA.Google Scholar
[37] Neuts, M. F. (1979).A versatile Markovian point process.J. Appl. Prob. 16,764–779.Google Scholar
[38] Neuts, M. F. (1981).Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach.Johns Hopkins University Press,Baltimore, MD.Google Scholar
[39] Neuts, M. F. and Rao, B. M. (1990).Numerical investigation of a multiserver retrial model.Queueing Systems 7,169–189.Google Scholar
[40] Phung-Duc, T. and Kawanishi, K. (2011).Multiserver retrial queues with after-call work.Numer. Algebra Control Optimization 1,639–656.Google Scholar
[41] Phung-Duc, T. and Kawanishi, K. (2014).Performance analysis of call centers with abandonment, retrial and after-call work.Performance Evaluation 80,43–62.Google Scholar
[42] Phung-Duc, T.,Masuyama, H.,Kasahara, S. and Takahashi, Y. (2010).A simple algorithm for the rate matrices of level-dependent QBD processes. In Proceedings of the 5th International Conference on Queueing Theory and Network Applications,ACM,New York, pp. 46–52.CrossRefGoogle Scholar
[43] Ramaswami, V. and Lucantoni, D. M. (1985).Algorithms for the multiserver queue with phase-type service.Commun. Statist. Stoch. Models 1,393–417.Google Scholar
[44] Sakurai, H. and Phung-Duc, T. (2015).Two-way communication retrial queues with multiple types of outgoing calls.TOP 23,466–492.Google Scholar
[45] Shin, Y. W. (2011).Algorithmic solution for M/M/c retrial queue with PH2-retrial times.J. Appl. Math. Inform. 29,803–811.Google Scholar
[46] Shin, Y. W. and Moon, D. H. (2011).Approximation of M/M/c retrial queue with PH-retrial times.Europ. J. Operat. Res. 213,205–209.Google Scholar
[47] Shin, Y. W. and Moon, D. H. (2014).M/M/c retrial queue with multiclass of customers.Methodol. Comput. Appl. Prob. 16,931–949.Google Scholar
[48] Tweedie, R. L. (1975).Sufficient conditions for regularity, recurrence and ergodicity of Markov processes.Math. Proc. Camb. Phil. Soc. 78,125–136.Google Scholar
[49] Uysal, E. and Dayar, T. (1998).Iterative methods based on splittings for stochastic automata networks.Europ. J. Operat. Res. 110,166–186.Google Scholar