Embedded renewal processes in theGI/G/squeue

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Abstract
The stableGI/G/squeue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for theGI/G/1 queue, but a simpleGI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in theGI/G/1 andGI/G/squeues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class ofGI/G/squeues in light traffic are implied.

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