Little’s law when the average waiting time is infinite

One version of Little’s law, written as , 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 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 is infinite and 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.