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Note---Comments on a Queueing Inequality

Author

Listed:
  • Daniel P. Heyman

    (Bell Laboratories, Holmdel, New Jersey)

Abstract

In a G/G/c/N system (a queue with general distributions of inter-arrival and service time, c servers and N - c \ge 0 queueing positions), let B be the steady-state probability that an arriving customer finds all queue positions filled and p be the time average probability that all queue positions are filled. By assuming p = B, Matthew Sobel proved where p is the traffic intensity. He also showed, by numerical examples, that the lower bound is a good approximation when p \ge 1.5 and c \ge 2. In this paper, we show that the lower bound does not require the assumption p = B to hold, and that it follows from the conservation of load. This derivation also explains why the bound does not depend on N and is a good approximation in heavy traffic. We also show that the upper bound depends critically on the assumption that p = B.

Suggested Citation

  • Daniel P. Heyman, 1980. "Note---Comments on a Queueing Inequality," Management Science, INFORMS, vol. 26(9), pages 956-959, September.
    • Handle: RePEc:inm:ormnsc:v:26:y:1980:i:9:p:956-959
    DOI: 10.1287/mnsc.26.9.956
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