Ergodicity of Jackson-type queueing networks

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMPs. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.

[1]  Jean Walrand,et al.  On the ergodicity of networks of ·/ GI /1/ N queues , 1990 .

[2]  S. Foss,et al.  Two ergodicity criteria for stochastically recursive sequences , 1994 .

[3]  T. B. Boffey,et al.  Applied Graph Theory , 1973 .

[4]  A. Borovkov.  Limit Theorems for Queueing Networks. I , 1987 .

[5]  D. Yao,et al.  Stochastic monotonicity in general queueing networks , 1989, Journal of Applied Probability.

[6]  Cheng-Shang Chang,et al.  On the stability of open networks: A unified approach by stochastic dominance , 1994, Queueing Syst. Theory Appl..

[7]  M. Bramson.  Instability of FIFO Queueing Networks , 1994 .

[8]  V. Schmidt,et al.  Queues and Point Processes , 1983 .

[9]  R. M. Loynes,et al.  The stability of a queue with non-independent inter-arrival and service times , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  A. A. Borovkov,et al.  ERGODICITY OF A JACKSON NETWORK WITH BATCH ARRIVALS , 1994 .

[11]  Iosif Ilitch Gikhman,et al.  Introduction to the theory of random processes , 1969 .

[12]  A. A. Borovkov,et al.  Asymptotic Methods in Queueing Theory. , 1986 .

[13]  F. Baccelli,et al.  On a Class of Stochastic Recursive Sequences Arising in Queueing Theory , 1992 .

[14]  Ward Whitt,et al.  Networks of infinite-server queues with nonstationary Poisson input , 1993, Queueing Syst. Theory Appl..

[15]  F. Baccelli,et al.  Palm Probabilities and Stationary Queues , 1987 .

[16]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[17]  François Baccelli,et al.  Elements Of Queueing Theory , 1994 .

[18]  K. Sigman.  The stability of open queueing networks , 1990 .

[19]  P. Franken,et al.  Stationary Stochastic Models. , 1992 .

[20]  A. A. Borovkov,et al.  STOCHASTICALLY RECURSIVE SEQUENCES AND THEIR GENERALIZATIONS , 1992 .

[21]  Sean P. Meyn,et al.  Stability of queueing networks and scheduling policies , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[22]  S. Foss,et al.  Ergodicity of queuing networks , 1991 .

[23]  F. Baccelli,et al.  On the saturation rule for the stability of queues , 1995, Journal of Applied Probability.

[24]  Aleksandr Alekseevich Borovkov,et al.  Stochastic processes in queueing theory , 1976 .

[25]  L. G. Afanas’eva.  On the Ergodicity of an Open Queueing Network , 1982 .

[26]  Sean P. Meyn,et al.  Stability of Generalized Jackson Networks , 1994 .

[27]  François Baccelli,et al.  Recursive equations and basic properties of timed Petri nets , 1991 .

[28]  A. Mandelbaum,et al.  Regenerative closed queueing networks , 1992 .