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**Strong stability in a Jackson queueing network.**
*(English)*
Zbl 1199.60321

Teor. Jmovirn. Mat. Stat. 77, 96-108 (2007) and Theory Probab. Math. Stat. 77, 107-119 (2008).

Queueing networks have been extensively applied in the last decades as a powerful tool for modelling, performance evaluation and prediction of systems as well as production and manufacturing systems, communication networks, computer systems, see D. Gross, John F. Shortle, James M. Thompson and C. M. Harris [Fundamentals of queueing theory. 4th ed., Hoboken, NJ: John Wiley & Sons (2008; Zbl 1151.60001)]. The important contribution of queueing networks theory is that it allows obtaining a simple exact solution for the joint queue length distribution as the product of the distributions of the single queues. This famous product form was introduced by J. R. Jackson [Oper. Res. 5, 518–521 (1957)] for opening exponential networks and W. J. Gordon and F. Newell [Oper. Res. 15, 254–265 (1967; Zbl 0168.16603)] for closed exponential networks.

Non-product form networks are extremely difficult to analyze. Therefore great deal of effort has been devoted to establishing approximate methods for these networks, such as decomposition methods, mean value methods, isolation methods, diffusion methods, aggregation methods, and many other numerical methods.

In this paper, the authors deal with the method which consists in substituting a non-product form network (real model) by a product one (ideal model). When this substitution is performed, the stability problem arises. Stability analysis of queueing networks have received a great attention recently.

The goal of this paper is to prove the applicability of the strong stability method (also called method of operators, see Dzh. Ajssani and N. V. Kartashov [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1983, No. 11, 3–5 (1983; Zbl 0528.60067)]) to the queueing networks in order to be able to approximate non-product form networks by product ones. The strong stability of a Jackson network (ideal model) under perturbations of the service time distribution in the first station of a non-product network (real model) is established.

Non-product form networks are extremely difficult to analyze. Therefore great deal of effort has been devoted to establishing approximate methods for these networks, such as decomposition methods, mean value methods, isolation methods, diffusion methods, aggregation methods, and many other numerical methods.

In this paper, the authors deal with the method which consists in substituting a non-product form network (real model) by a product one (ideal model). When this substitution is performed, the stability problem arises. Stability analysis of queueing networks have received a great attention recently.

The goal of this paper is to prove the applicability of the strong stability method (also called method of operators, see Dzh. Ajssani and N. V. Kartashov [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1983, No. 11, 3–5 (1983; Zbl 0528.60067)]) to the queueing networks in order to be able to approximate non-product form networks by product ones. The strong stability of a Jackson network (ideal model) under perturbations of the service time distribution in the first station of a non-product network (real model) is established.

Reviewer: Mikhail P. Moklyachuk (Kyïv)