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.
- Queueing networks
- limit theorems