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Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems. (English) Zbl 1165.90361
Summary: Most retrial models assume that customers and servers are homogeneous. However, multiclass (or heterogeneous) retrial systems arise in various practical areas such as telecommunications and cellular mobile networks. Multiclass models are far more difficult for mathematical analysis than single class ones. So, explicit results are available only in few special cases. Actually, so far multiclass retrial systems have been analyzed only by means of queueing theory and almost all studies consider models with several customer’s classes and a service station consisting in one single server or multiple homogeneous (identic) servers and an infinite population size. In this paper, we propose an approach for modelling and analyzing finite-source retrial systems with several customer’s classes and server’s classes using the Colored Generalized Stochastic Petri Nets (CGSPNs). This high-level mathematical model is appropriate for describing and analyzing the performance of systems exhibiting concurrency and synchronization, possibly with heterogeneous components. Using a high-level formalism makes the description of the system easier, while preserving the possibility of obtaining exact performance results. We show how the main steady-state performance indices can be derived and we analyze the behaviour of heterogeneous retrial systems under two service disciplines. The numerical results are graphically displayed to illustrate the effect of system parameters and service discipline on the mean response time.

MSC:
90B15 Stochastic network models in operations research
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
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[1] Kulkarni, V.G.; Liang, H.M., Retrial queues revisited, () · Zbl 0871.60074
[2] Artalejo, J.R.; Falin, G.I., Standard and retrial queueing systems: A comparative analysis, Revista matematica complutense, XV, 101-129, (2002) · Zbl 1009.60079
[3] Falin, G.I.; Templeton, J.G.C., Retrial queues, (1997), Chapman and Hall London · Zbl 0944.60005
[4] Artalejo, J.R., A classified bibliography of research in retrial queues: progress in 1990-1999, Top, 7, 187-211, (1999) · Zbl 1009.90001
[5] Artalejo, J.R., Accessible bibliography on retrial queues, Mathematical and computer modelling, 30, 1-6, (1999) · Zbl 1009.90001
[6] Choi, B.D.; Chang, Y., Single server retrial queues with priority calls, Mathematical and computer modelling, 30, 7-32, (1999) · Zbl 1042.60533
[7] Choi, B.D.; Chang, Y.; Kim, B., MAP1, MAP2/M/c retrial queue with guard channels and its application to cellular networks, Top, 7, 231-248, (1999) · Zbl 0949.60097
[8] Choi, B.D.; Chang, Y., MAP1, MAP2/M/c retrial queue with the retrial group of finite capacity and geometric loss, Mathematical and computer modelling, 30, 99-113, (1999) · Zbl 1042.60534
[9] Tran-Gia, P.; Mandjes, M., Modeling of customer retrial phenomenon in cellular mobile networks, IEEE journal on selected areas in communications, 15, 1406-1414, (1997)
[10] Falin, G.I., On a multiclass batch arrival retrial queue, Advances in applied probability, 20, 483-487, (1988) · Zbl 0672.60089
[11] Grishechkin, S.A., Multiclass batch arrival retrial queues analyzed as branching processes with immigration, Queueing systems, 11, 395-418, (1992) · Zbl 0770.60084
[12] Langaris, C.; Moutzoukis, E., Non-pre-emptive priorities and vacations in a multiclass retrial queueing system, Communications in statistics stochastic models, 12, 455-472, (1996) · Zbl 0858.60086
[13] Bolch, G.; Almasi, B.; Sztrik, J., Heterogeneous finite-source retrial queues, Journal of mathematical sciences, 121, 2590-2596, (2004) · Zbl 1070.60078
[14] J. Sztrik, J. Roszik, Retrial queues for performance modelling and evaluation of heterogeneous networks, in: Proc. of Conf. on Performance Modelling and Evaluation of Heterogeneous Networks, HET-NET’04, Ilkey, England, 2004
[15] G. Bolch, J. Roszik, J. Sztrik, Heterogeneous finite-source retrial queues in the analysis of communication systems with CSMA/CD protocols, in: Proc. of the Inter. Conf. on Modern Mathematical Methods of Analysis and Optimization of Telecommunication Networks, Gomel, Belarus, 2003, pp. 39-45
[16] J. Sztrik, J. Roszik, B. Almasi, Multiserver retrial queues with finite number of heterogeneous sources, in: 6th Inter. Conf. on Applied Informatics, Eger, Hungary, 2004 · Zbl 1411.60139
[17] Efrosinin, D.; Breuer, L., Threshold policies for controlled retrial queues with heterogeneous servers, Annals of operations research, 141, 139-162, (2006) · Zbl 1114.90020
[18] J. Sztrik, G. Bolch, H.de. Meer, J. Roszik, P. Wuechner, Modeling finite-source retrial queueing systems with unreliable heterogeneous servers and different service policies using MOSEL, in: Proc. of 14th Inter. Conf. on Analytical and Stochastic Modelling Techniques and Applications, ASMTA’07, Pargue, Czech Republic, 2007, pp. 75-80
[19] Sztrik, J.; Roszik, J., Performance analysis of finite-source retrial queues with non-reliable heterogeneous servers, Journal of mathematical sciences, 146, 6033-6038, (2007)
[20] Artalejo, J.R., Retrial queues with a finite number of sources, Journal of Korean mathematical society, 35, 503-525, (1998) · Zbl 0930.60079
[21] Houck, D.J.; Lai, W.S., Traffic modeling and analysis of hybrid fiber-coax systems, Computer networks and ISDN systems, 30, 821-834, (1998)
[22] Sztrik, J.; Roszik, J.; Kim, C.S., Retrial queues in the performance modeling of cellular mobile networks using MOSEL, International journal of simulation: systems science & technology, 6, 38-47, (2005)
[23] Janssens, G.K., The quasi-random input queueing system with repeated attempts as a model for collision-avoidance star local area network, IEEE transactions on communications, 45, 360-364, (1997)
[24] Chiola, G.; Dutheillet, D.; Franceschinis, G.; Haddad, S., Stochastic well-formed colored nets and symmetric modeling applications, IEEE transactions on computers, 42, 1343-1360, (1993)
[25] Dutheillet, D.; Haddad, S., Aggregation and disaggregation of states in colored stochastic Petri nets: application to a multiprocessor architecture, ()
[26] Gharbi, N.; Ioualalen, M., Performance analysis of retrial queueing systems using generalized stochastic Petri nets, Electronic notes in theoretical computer science, 65, (2002) · Zbl 1270.68051
[27] Gharbi, N.; Ioualalen, M., GSPN analysis of retrial systems with servers breakdowns and repairs, Applied mathematics and computation, 174, 1151-1168, (2006) · Zbl 1156.68319
[28] Ajmone Marsan, M.; Balbo, G.; Conte, G.; Donatelli, S.; Franceschinis, G., Modelling with generalized stochastic Petri nets, (1995), John Wiley and Sons New York · Zbl 0843.68080
[29] Jensen, K., An introduction to the practical use of coloured Petri nets, (), 237-292
[30] Chiola, G.; Franceschinis, C.; Gaeta, R.; Ribaudo, M., Greatspn 1.7: graphical editor and analyzer for timed and stochastic Petri nets, Performance evaluation, 24, 47-68, (1995) · Zbl 0875.68663
[31] Falin, G.I., A multiserver retrial queue with a finite number of sources of primary calls, Mathematical and computer modelling, 30, 33-49, (1999) · Zbl 1042.60537
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