Little's law when the average waiting time is infinite

  • Authors:

  • Ronald W. Wolff;Yi-Ching Yao

  • Affiliations:

  • Department of Industrial Engineering and Operations Research, University of California, Berkeley, USA 94720;Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC 11529

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 2014

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Abstract

One version of Little's law, written as $$L = \lambda w$$, is a relation between averages along a sample path. There are two others in a stochastic setting; they readily extend to the case where the average waiting time $$w$$ is infinite. We investigate conditions for the sample-path version of this case to hold. Published proofs assume (our) Eq. (3) holds. It is only sufficient. We present examples of what may happen when (3) does not hold, including one that may be new where $$w$$ is infinite and $$L$$ is finite. We obtain a sufficient condition called "weakly FIFO" that is weaker than (3), and through truncation, a necessary and sufficient condition. We show that (3) is sufficient but not necessary for the departure rate to be equal to the arrival rate.