Stability criteria for controlled queueing networks

Abstract

We give criteria for the stability of a very general queueing model under different levels of control. A complete classification of stability (or positive recurrence), transience and null-recurrence is presented for the two queue model. The stability and instability results are extended for models with N > 3 queues. We look at a broad class of models which can have the following features: Customers arrive at one, several or all of the queues from the outside with exponential inter arrival times. We often have the case where a arrival stream can be routed so that under different routing schemes each queue can have external arrivals, i.e. we assume we have some control over the routing of the arrivals. We also consider models where the arrival streams are fixed. We view the service in a more abstract way, in that we allow a number к of different service configurations. Under every such service configuration service is provided to some or all of the queues, length of service time can change from one service configuration to another and we can change from one configuration to another according two some control policy. The service times are assumed to be exponentially distributed. The queueing models we consider are networks where, after completion at one queue, a customer might be fed back into another queue where it will be served another time often under with a different service time. These feedback probabilities change with the service configurations. Our interest is in different types of control policies which allow us to change the routing of arrivals and configurations of the service from time to time so that the controlled queue length process (which in most cases is Markov) is stable. The semi-martingale or Lyapunov function methods we use give necessary and sufficient conditions for the stability classification. We will look at some two queue models with different inter arrival and service times where the queueing process is still Markov.