Skip to main content

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

  • Tuǧrul Dayar (a1) and M. Can Orhan (a1)

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.

Corresponding author
* Postal address: Department of Computer Engineering, Bilkent University, TR‒06800 Bilkent, Ankara, Turkey.
** Email address:
Hide All
[1] Artalejo, J. R. (1999).Accessible bibliography on retrial queues.Math. Comput. Modelling 30,16.
[2] Artalejo, J. R. (2010).Accessible bibliography on retrial queues: progress in 2000‒2009.Math. Comput. Modelling 51,10711081.
[3] Artalejo, J. R. and Gómez-Corral, A. (2007).Modelling communication systems with phase type service and retrial times.IEEE Commun. Lett. 11,955957.
[4] Artalejo, J. R. and Gómez-Corral, A. (2008).Retrial Queueing Systems: A Computational Approach.Springer,Berlin.
[5] Artalejo, J. R. and Phung-Duc, T. (2012).Markovian retrial queues with two way communication.J. Ind. Manag. Optimization 8,781806.
[6] Asmussen, S. (2003).Applied Probability and Queues(Appl. Math. (New York) 51).Springer,New York.
[7] Avrachenkov, K.,Morozov, E. and Steyaert, B. (2016).Sufficient stability conditions for multi-class constant retrial rate systems.Queueing Systems 82,149171.
[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 70,663681.
[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. 356359.
[10] Breuer, L.,Dudin, A. and Klimenok, V. (2002).A retrial BMAP/PH/N system.Queueing Systems 40,433457.
[11] Bright, L. and Taylor, P. G. (1995).Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes.Commun. Statist. Stoch. Models 11,497525.
[12] Buchholz, P. The Nsolve package. Available at
[13] Buchholz, P.,Kriege, J. and Felko, I. (2014).Input Modeling with Phase-Type Distributions and Markov Models: Theory and Applications.Springer,Cham.
[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. 3749.
[15] Chiola, G.,Dutheillet, C.,Franceschinis, G. and Haddad, S. (1993).Stochastic well-formed colored nets and symmetric modeling applications.IEEE Trans. Comput. 42,13431360.
[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 30,99113.
[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 7,231248.
[18] Dayar, T. and Orhan, M. C. RetrialQueueSolver. Available at∼tugrul/software.html.
[19] Dayar, T. and Orhan, M. C. (2012).Kronecker-based infinite level-dependent QBD processes.J. Appl. Prob. 49,11661187.
[20] Dayar, T.,Hermanns, H.,Spieler, D. and Wolf, V. (2011).Bounding the equilibrium distribution of Markov population models.Numer. Linear Algebra Appl. 18,931946.
[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. 43,10051026.
[22] Diamond, J. E. and Alfa, A. S. (1998).The MAP/PH/1 retrial queue.Commun. Statist. Stoch. Models 14,11511177.
[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. 113,620631.
[24] Dudin, A. and Klimenok, V. (2000).A retrial BMAP/SM/1 system with linear repeated requests.Queueing Systems Theory Appl. 34,4766.
[25] Falin, G. I. (1988).On a multiclass batch arrival retrial queue.Adv. Appl. Prob. 20,483487.
[26] Falin, G. I. and Templeton, J. G. C. (1997).Retrial Queues.Chapman & Hall,London.
[27] Fayolle, G.,Malyshev, V. A. and Menshikov, M. V. (1995).Topics in the Constructive Theory of Countable Markov Chains.Cambridge University Press.
[28] Gharbi, N.,Dutheillet, C. and Ioualalen, M. (2009).Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems.Math. Comput. Modelling 49,14361448.
[29] Gómez-Corral, A. (2006).A bibliographical guide to the analysis of retrial queues through matrix analytic techniques.Ann. Operat. Res. 141,163191.
[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. 35,323347.
[31] Kim, B. (2011).Stability of a retrial queueing network with different classes of customers and restricted resource pooling.J. Ind. Manag. Optimization 7,753765.
[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. 201,307323.
[33] Kim, J. and Kim, B. (2016).A survey of retrial queueing systems. To appear in Ann. Operat. Res.
[34] Kulkarni, V. G. (1986).Expected waiting times in a multiclass batch arrival retrial queue.J. Appl. Prob. 23,144154.
[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. 17,822825.
[36] Meyer, C. (2000).Matrix Analysis and Applied Linear Algebra.SIAM,Philadelphia, PA.
[37] Neuts, M. F. (1979).A versatile Markovian point process.J. Appl. Prob. 16,764779.
[38] Neuts, M. F. (1981).Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach.Johns Hopkins University Press,Baltimore, MD.
[39] Neuts, M. F. and Rao, B. M. (1990).Numerical investigation of a multiserver retrial model.Queueing Systems 7,169189.
[40] Phung-Duc, T. and Kawanishi, K. (2011).Multiserver retrial queues with after-call work.Numer. Algebra Control Optimization 1,639656.
[41] Phung-Duc, T. and Kawanishi, K. (2014).Performance analysis of call centers with abandonment, retrial and after-call work.Performance Evaluation 80,4362.
[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. 4652.
[43] Ramaswami, V. and Lucantoni, D. M. (1985).Algorithms for the multiserver queue with phase-type service.Commun. Statist. Stoch. Models 1,393417.
[44] Sakurai, H. and Phung-Duc, T. (2015).Two-way communication retrial queues with multiple types of outgoing calls.TOP 23,466492.
[45] Shin, Y. W. (2011).Algorithmic solution for M/M/c retrial queue with PH2-retrial times.J. Appl. Math. Inform. 29,803811.
[46] Shin, Y. W. and Moon, D. H. (2011).Approximation of M/M/c retrial queue with PH-retrial times.Europ. J. Operat. Res. 213,205209.
[47] Shin, Y. W. and Moon, D. H. (2014).M/M/c retrial queue with multiclass of customers.Methodol. Comput. Appl. Prob. 16,931949.
[48] Tweedie, R. L. (1975).Sufficient conditions for regularity, recurrence and ergodicity of Markov processes.Math. Proc. Camb. Phil. Soc. 78,125136.
[49] Uysal, E. and Dayar, T. (1998).Iterative methods based on splittings for stochastic automata networks.Europ. J. Operat. Res. 110,166186.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 31 *
Loading metrics...

Abstract views

Total abstract views: 199 *
Loading metrics...

* Views captured on Cambridge Core between 9th December 2016 - 16th August 2018. This data will be updated every 24 hours.