Stability, queue length and delay. I. Deterministic queueing networks

Conditions for deterministic queueing networks that render bounded delays for customers are identified. These conditions are referred to as stability conditions for deterministic queues. Parallel to the three classical stability conditions, stationarity, ergodicity, and the traffic condition, for queues with random inputs, the notions of envelope process, subadditivity, and the traffic condition are used as the stability conditions for deterministic queues. It is shown that the delay in a single queue is bounded under a work-conservative scheduling policy if the minimum envelope rate, which is the average rate of the smallest envelope process, is less than the capacity. Moreover, the delay cannot be bounded if the minimum envelope rate is larger than the capacity. Similar results are extended to multiclass networks with feedforward routing. For networks with nonfeedforward routing, it is shown that the stability result holds for networks with a single class of customers under the first come first served (FCFS) policy. Various scheduling policies that stabilize multiclass networks with nonfeedforward routing are discussed, and a sufficient condition for the stability under the FCFS policy is given