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Operational analysis replaces certain classical queueing theory assumptions with the conditions of "homogeneous service times" and "on-line= off-line behavior." In this paper we explore the relationship between the operational and classical concepts for the sample paths of an M/G/1 queueing system. The primary results are that the sample paths can have these operational properties with nonzero probability if and only if the service time is exponential. We also state dual results for interarrival times in G/M/l. Additionally, we show that open, feedforward networks of single server queues can have product form solutions valid across a range of system arrival rates if and only if all of the service times are exponential. Finally, we consider the relationship between the operational quantities S(n) and the mean service time in M/G/1. This relationship is shown to depend on the form of the service time distribution. It follows that using operational analysis to predict the performance of an M/G/1 queueing system will be most successful when the service time is exponential. Simulation evidence is presented which supports this claim.