Limit non-stationary behavior of large closed queueing networks with bottlenecks

Yaakov Kogan, Robert Sh Liptser

Research output: Contribution to journalArticlepeer-review


In this paper martingales methods are applied for analyzing limit non-stationary behavior of the queue length processes in closed Jackson queueing networks with a single class consisting of a large number of customers, a single infinite server queue, and a fixed number of single server queues with large state independent service rates. It is assumed that one of the single server nodes forms a bottleneck. For the non-bottleneck nodes we show that the queue length distribution at time t converges in generalized sense to the stationary distribution of the M/M/1 queue whose parameters explicitly depend on t. For the bottleneck node a diffusion approximation with reflection is proved in the moderate usage regime while fluid and Gaussian diffusion approximations are established for the heavy usage regime.

Original languageEnglish
Pages (from-to)33-55
Number of pages23
JournalQueueing Systems
Issue number1-2
StatePublished - Mar 1993
Externally publishedYes


  • Queueing networks
  • bottlenecks
  • limit theorems


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