## The Annals of Applied Probability

### Condensation in large closed Jackson networks

#### Abstract

We consider finite closed Jackson networks with N first come, first serve nodes and M customers. In the limit $M \to \infty, N \to \infty, M/N \to \lambda > 0$, we get conditions when mean queue lengths are uniformly bounded and when there exists a node where the mean queue length tends to $\infty$ under the above limit (condensation phenomena, traffic jams), in terms of the limit distribution of the relative utilizations of the nodes. In the same terms, we also derive asymptotics of the partition function and of correlation functions.

#### Article information

Source
Ann. Appl. Probab., Volume 6, Number 1 (1996), 92-115.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968067

Digital Object Identifier
doi:10.1214/aoap/1034968067

Mathematical Reviews number (MathSciNet)
MR1389833

Zentralblatt MATH identifier
0863.60100

#### Citation

Malyshev, Vadim A.; Yakovlev, Andrei V. Condensation in large closed Jackson networks. Ann. Appl. Probab. 6 (1996), no. 1, 92--115. doi:10.1214/aoap/1034968067. https://projecteuclid.org/euclid.aoap/1034968067

#### References

• 1 BASKETT, F., CHANDY, K. M., MUNTZ, R. R. and PALACIOS, F. G. 1975. Open, closed, and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22 248 260.
• 2 BENDER, C. M. and ORSZAG, S. A. 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.
• 3 BIRMAN, A. and KOGAN, Y. 1991. Asy mptotic analysis of closed queueing networks with bottlenecks. In Proceedings of the International Conference on the Performance of Z Distributed Sy stems and Integrated Communication Networks T. Hasegawa, H. Tagaki. and Y. Takahashi, eds. 237 252. North-Holland, Amsterdam.
• 4 BIRMAN, A. and KOGAN, Y. 1992. Asy mptotic evaluation of closed queueing networks with many stations. Comm. Statist. Stochastic Models 8 543 563.
• 5 BUZEN, J. P. 1973. Computational algorithms for closed queueing networks with exponential servers. Comm. ACM 16 527 531.
• 6 BUZEN, J. P. and DENNING, P. J. 1978. The operational analysis of queueing network models. Comput. Survey s 10 225 261.
• 7 COPSON, E. T. 1965. Asy mptotic Expansions. Cambridge Univ. Press.
• 8 GORDON, J. J. 1990. The evaluation of normalizing constants in closed queueing networks. Oper. Res. 38 863 869.
• 9 GORDON, W. J. and NEWELL, G. F. 1967. Closed queueing sy stems with exponential servers. Oper. Res. 15 254 265.
• 10 HARRISON, P. G. 1985. On normalizing constants in queueing networks. Oper. Res. 33 464 468.
• 11 JACKSON, J. R. 1963. Jobshop-like queueing sy stems. Management Sci. 10 131 142.
• 12 KELLER, J. B. 1978. Ray s, waves, and asy mptotics. Bull. Amer. Math. Soc. 84 727 750.
• 13 KELLY, F. P. 1980. Reversibility and Stochastic Networks. Wiley, New York.
• 14 KNESSL, C. and TIER, C. 1990. Asy mptotic expansions for large closed queueing networks. J. Assoc. Comput. Mach. 37 144 174.
• 15 KNESSL, C. and TIER, C. 1992. Asy mptotic expansions for large closed queueing networks with multiple job classes. IEEE Trans. Comput. 41 480 488.
• 16 KOBAy ASHI, H. and REISER, M. 1975. Queueing networks with multiple closed chains: theory and computational algorithms. IBM J. Res. Develop. 19 283 294.
• 17 KOGAN, Y. 1992. Another approach to asy mptotic expansions for large closed queueing networks. Oper. Res. Lett. 11 317 321.
• 18 LAVENBERG, S. S., ed. 1983. Computer Performance Modeling Handbook. Academic Press, New York.
• 19 LAVENBERG, S. S. and REISER, R. 1980. Stationary state probabilities at arrival instants for closed queueing networks with multiple ty pes of customers. J. Appl. Probab. 17 1048 1061.
• 20 MARKUSHEVICH, A. I. 1965. Theory of Functions of a Complex Variable 1, 2. Prentice-Hall, Englewood Cliffs, NJ.
• 21 MASSEY, W. A. 1987. Stochastic orderings for Markov processes on partially ordered spaces. Math. Oper. Res. 12 350 367.
• 22 MCKENNA, J. 1987. Asy mptotic expansions of the sojourn time distribution functions of jobs in closed, product-form queueing networks. J. Assoc. Comput. Mach. 34 985 1003.
• 23 MCKENNA, J., MITRA, D. and RAMAKRISHAN, K. G. 1981. A class of closed Markovian queueing networks: integral representations, asy mptotic expansions, and generalizations. Bell Sy stem Tech. J. 60 599 641.
• 24 MCKENNA, J. and MITRA, D. 1982. Integral representations and asy mptotic expansions for closed Markovian queuing networks: normal usage. Bell Sy stem Tech. J. 61 661 683.
• 25 MCKENNA, J. and MITRA, D. 1984. Asy mptotic expansions and integral representations of moments of queue lengths in closed Markovian networks. J. Assoc. Comput. Mach. 31 346 360.
• 26 MEI, J.-D. and TIER C. 1994. Asy mptotic approximations for a queueing network with multiple classes. SIAM J. Appl. Math. 54 1147 1180.
• ROCQUENCOURT, B.P.105 U.F.R. FACULTE DES SCIENCES, B.P. 6759 ´ 78153 LE CHESNAY CEDEX 45067 ORLEANS CEDEX 2 FRANCE FRANCE E-mail: Vadim.Maly shev@inria.fr