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
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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.
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. Aïssani and N. V. Kartashov, Strong stability of the imbedded Markov chains in an $M/G/1$ system, Theor. Probability and Math. Statist. 29 (1984), 1–5. MR 727097 (85d:60167)
- D. Aïssani and N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels, Dokl. Akad. Nauk Ukr. SSR, ser. A 11 (1983), 1–5. MR 728475 (85c:60110)
- F. Baskett, K. M. Chandy, R. Muntz, and G. F. Palacios, Open, closed and mixed networks of queueing with different classes of customers, J. ACM 22 (1975), 248–260. MR 0365749 (51:2001)
- 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.
- M. Benaouicha and D. Aïssani, Estimate of the strong stability in a $G/M/1$ queueing system, Theory of Probab. and Math. Statist. 71 (2005), 22–32. MR 2144318 (2006a:60171)
- L. Berdjoudj and D. Aïssani, Strong stability in retrial queues, Theor. Probability and Math. Statist. 68 (2004), 11–17. MR 2000390 (2004f:60188)
- L. Bouallouche and D. Aïssani, Performance analysis approximation in a queueing system of type $M/G/1$, Mathematical Methods of Operations Research 63 (2006), no. 2. MR 2264753 (2007i:90017)
- 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 approch via fluid limit models, Annals of Applied Probability 5 (1993), no. 1, 49–77. MR 1325041 (96c:60113)
- 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.
- E. Gelenbe, Product form queueing networks with positive and negative customers, J. Appl. Probab. 28 (1991), 656–663. MR 1123837 (92k:60210)
- W. J. Gordon and F. Newell, Closed queueing systems with exponential servers, Operations Research 15 (1967).
- D. Gross and C. M. Harris, Fundamentals of Queueing Theory, 3rd ed., Wiley–InterScience, 1998. MR 1600527 (98m:60144)
- C. F. Ipsen and C. D. Meyer, Uniform stability of Markov chains, Siam. J. Matrix Anal. Appl. 15 (1994), no. 4, 1061–1074. MR 1293904 (95h:65110)
- J. R. Jackson, Networks of waiting lines, Operations Research 5 (1957), 518–521. MR 0093061 (19:1203c)
- N. V. Kartashov, Strong stability of Markov chains, Vsesoyuzn. Seminar on Stability Problems for Stochastic Models, VNIISI, Moscow, 1981, pp. 54–59; see also Journal of Soviet Mathematics 34 (1986), 1493–1498. MR 668559 (84b:60089)
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
- H. Kobayashi, Application of the diffusion approximation to queueing network, J. Assoc. Comput. Mach. 21 (1974), 316–328. MR 0350899 (50:3391)
- 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.
- M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, 1981. MR 618123 (82j:60177)
- B. Rabta and D. Aïssani, Stability analysis in an inventory model, Theory of Stochastic Processes 10(26) (2004), no. 3–4, 129–135. MR 2329786 (2008g:60219)
- 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 4 (1989), 287–318. MR 1018523 (91c:60132)
- 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, Notes on the stability of closed queueing networks, J. Appl. Probab. 27 (1990), 735. MR 1067041 (91j:60151)
- K. Sigman, The stability of open queueing networks, Stochastic Processes and their Applications 35 (1990), 11–25. MR 1062580 (91m:60178)
- N. M. van Dijk, Queueing Networks and Product Forms—A Systems Approach, Wiley, 1993. MR 1266845 (95b:90001)
- 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.
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Additional Information
O. Lekadir
Affiliation:
LAMOS Laboratory, University of Bejaia 06000, Algeria
Email:
ouiza_lekadir@yahoo.fr
D. Aissani
Affiliation:
LAMOS Laboratory, University of Bejaia 06000, Algeria
Email:
lamos_bejaia@hotmail.com
Keywords:
Queueing networks,
strong stability,
product form,
Jackson networks,
Markov chain,
perturbation
Received by editor(s):
February 10, 2006
Published electronically:
January 16, 2009
Article copyright:
© Copyright 2009
American Mathematical Society