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.<<ETX>>

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

[2]  P. R. Kumar,et al.  Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[3]  J. Walrand,et al.  Monotonicity of Throughput in Non-Markovian Networks. , 1989 .

[4]  Prabandham M. Gopal,et al.  Network Delay Considerations for Packetized Voice , 1989, Perform. Evaluation.

[5]  J. R. Perkins,et al.  Stable, distributed, real-time scheduling of flexible manufacturing/assembly/diassembly systems , 1989 .

[6]  H. Chen,et al.  Optimal scheduling control in a multi-class fluid network , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[7]  S. Jamaloddin Golestani.  Congestion-free transmission of real-time traffic in packet networks , 1990, Proceedings. IEEE INFOCOM '90: Ninth Annual Joint Conference of the IEEE Computer and Communications Societies@m_The Multiple Facets of Integration.

[8]  P. R. Kumar,et al.  Distributed scheduling based on due dates and buffer priorities , 1991 .

[9]  Rene L. Cruz,et al.  A calculus for network delay, Part I: Network elements in isolation , 1991, IEEE Trans. Inf. Theory.

[10]  Rene L. Cruz,et al.  A calculus for network delay, Part II: Network analysis , 1991, IEEE Trans. Inf. Theory.

[11]  James F. Kurose,et al.  On computing per-session performance bounds in high-speed multi-hop computer networks , 1992, SIGMETRICS '92/PERFORMANCE '92.

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

[13]  Cheng-Shang Chang.  Stability, Queue Length and Delay, Part 11: Stochastic Queueing Networks , 1992 .

[14]  Roch Guérin,et al.  Buffer Size Requirements Under Longest Queue First , 1993, Perform. Evaluation.