Strong stability in a Jackson queueing network

Lekadir, O.; Aissani, D.

doi:10.1090/S0094-9000-09-00750-9

Strong stability in a Jackson queueing network

Authors: O. Lekadir and D. Aissani
Journal: Theor. Probability and Math. Statist. 77 (2008), 107-119
MSC (2000): Primary 60K20, 60K25
DOI: https://doi.org/10.1090/S0094-9000-09-00750-9
Published electronically: January 16, 2009
MathSciNet review: 2432775
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Non-product networks are extremely difficult to analyze, so they are often solved by approximate methods. However, it is indispensable to delimit the domain wherever these approximations are justified. Our goal in this paper is to prove the applicability of the strong stability method to the queueing networks in order to be able to approximate non-product form networks by product ones. Therefore, we established the strong stability of a Jackson network $M/M/1\to M/M/1$ (ideal model) under perturbations of the service time distribution in the first station of a non-product network $M/GI/1\to GI/M/1$ (real model).

References

D. Aïssani, Application of the operator methods to obtain inequalities of stability in an $M_2/G_2/1$ system with a relative priority, Annales Maghrébines de l’ Ingénieur, Numéro Hors série 2 (1991), 790–795.

D. Aĭssani and N. V. Kartashov, Strong stability of an imbedded Markov chain in an $M/G/1$ system, Teor. Veroyatnost. i Mat. Statist. 29 (1983), 3–7 (Russian). MR 727097

Dzh. Aĭssani and N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topologies in a space of transition kernels, Dokl. Akad. Nauk Ukrain. SSR Ser. A 11 (1983), 3–5 (Russian, with English summary). MR 728475

Forest Baskett, K. Mani Chandy, Richard R. Muntz, and Fernando G. Palacios, Open, closed, and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach. 22 (1975), 248–260. MR 365749, DOI https://doi.org/10.1145/321879.321887

B. Baynat, Réseaux de files d’attente: Des Chaînes de Markov aux Réseaux à Forme Produit, Eyrolles Edition, 2000.

B. Baynat and Y. Dallery, A Unified View of Product-form Approximation Techniques for General Closed Queueing Networks, Technical Report 90.48, Institut Blaise Pascal, Paris, Octobre 1990.

Mustapha Benaouicha and Djamil Aissani, Strong stability in a $G/M/1$ queueing system, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 22–32; English transl., Theory Probab. Math. Statist. 71 (2005), 25–36. MR 2144318, DOI https://doi.org/10.1090/S0094-9000-05-00645-9

Louisa Berdjoudj and Djamil Aissani, Strong stability in retrial queues, Teor. Ĭmovīr. Mat. Stat. 68 (2003), 11–17; English transl., Theory Probab. Math. Statist. 68 (2004), 11–17. MR 2000390, DOI https://doi.org/10.1090/S0094-9000-04-00595-2

Louiza Bouallouche-Medjkoune and Djamil Aissani, Performance analysis approximation in a queueing system of type $M/G/1$, Math. Methods Oper. Res. 63 (2006), no. 2, 341–356. MR 2264753, DOI https://doi.org/10.1007/s00186-005-0022-8

Robert B. Cooper, Fundamentals of Queueing Theory, 4th ed., Ceep Press Books, 1990.

J. G. Dai, On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models, Ann. Appl. Probab. 5 (1995), no. 1, 49–77. MR 1325041

G. Fayolle, V. A. Malyshev, M. V. Menshikov, and A. F. Sidorenko, Lyaponov Functions for Jackson Networks, Rapport de Recherche 1380, INRIA, Domaine de Voluceau, LeChenay, Janvier 1991.

Erol Gelenbe, Product-form queueing networks with negative and positive customers, J. Appl. Probab. 28 (1991), no. 3, 656–663. MR 1123837, DOI https://doi.org/10.2307/3214499

W. J. Gordon and F. Newell, Closed queueing systems with exponential servers, Operations Research 15 (1967).

Donald Gross and Carl M. Harris, Fundamentals of queueing theory, 3rd ed., Wiley Series in Probability and Statistics: Texts and References Section, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1600527

Ilse C. F. Ipsen and Carl D. Meyer, Uniform stability of Markov chains, SIAM J. Matrix Anal. Appl. 15 (1994), no. 4, 1061–1074. MR 1293904, DOI https://doi.org/10.1137/S0895479892237562

James R. Jackson, Networks of waiting lines, Operations Res. 5 (1957), 518–521. MR 93061, DOI https://doi.org/10.1287/opre.5.4.518

N. V. Kartashov, Strongly stable Markov chains, Problems of stability of stochastic models (Panevezhis, 1980) Vsesoyuz. Nauch.-Issled. Inst. Sistem. Issled., Moscow, 1981, pp. 54–59 (Russian). MR 668559

N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375

Hisashi Kobayashi, Application of the diffusion approximation to queueing networks. I. Equilibrium queue distributions, J. Assoc. Comput. Mach. 21 (1974), 316–328. MR 350899, DOI https://doi.org/10.1145/321812.321827

P. R. Kumar, A tutorial on some new methods for performance evaluation of queueing networks, IEEE Journal on Selected Areas in Communications 13 (1995), no. 6, 970–980.

J. Labetoulle and G. Pujolle, Isolation method in a network of queues, IEEE Transaction on Software Engineering 6 (1980), 373–381.

Marcel F. Neuts, Matrix-geometric solutions in stochastic models, Johns Hopkins Series in the Mathematical Sciences, vol. 2, Johns Hopkins University Press, Baltimore, Md., 1981. An algorithmic approach. MR 618123

Boualem Rabta and Djamil Aïssani, Stability analysis in an inventory model, Theory Stoch. Process. 10 (2004), no. 3-4, 129–135. MR 2329786

B. Rabta and D. Aïssani, Strong stability in an $(R,s,S)$ inventory model, International Journal of Production Economics 97 (2005), 159–171.

S. T. Rachev, The problem of stability in queueing theory, Queueing Systems Theory Appl. 4 (1989), no. 4, 287–317. MR 1018523, DOI https://doi.org/10.1007/BF01159470

P. J. Schweitzer, A Survey of Mean Value Analysis, its Generalizations, and Applications for Networks of Queues, Technical report, University of Rochester, Rochester, NY, 1990.

K. Sigman, Correction: “Notes on the stability of closed queueing networks” [J. Appl. Probab. 26 (1989), no. 3, 678–682; MR1010957 (90j:60104)], J. Appl. Probab. 27 (1990), no. 3, 735. MR 1067041

Karl Sigman, The stability of open queueing networks, Stochastic Process. Appl. 35 (1990), no. 1, 11–25. MR 1062580, DOI https://doi.org/10.1016/0304-4149%2890%2990119-D

Nico M. van Dijk, Queueing networks and product forms, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1993. A systems approach. MR 1266845

I. Y. Wang and T. G. Robertazzi, Recursive computation of steady-state probabilities of nonproduct form queueing networks associated with computer network models, IEEE Transactions on Communications 38 (January 1990), no. 1.

W. Whitt, The queueing network analyzer, The Bell System Technical Journal 62 (November 1983), no. 9.