ABSTRACT
A “piecewise cyclic trajectory” is a “single sample path” of a regenerative process, with regenerative cycles. We assume that the cycles may derive from probabilistic models or from deterministic models. Many stochastic models have sample paths with this property. Examples are: the virtual wait in M/M/1 queues; the “content” in an M/M/r https://www.w3.org/1998/Math/MathML"> · https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math9_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> dam; the “surplus” in a C-L Insurance risk model; the “net inventory” in an https://www.w3.org/1998/Math/MathML"> s , S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math9_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> inventory model; the serum “concentration of a drug” in a pharmacokinetic model with multiple dosing; the position of a point moving in simple harmonic motion (“SHM”), with examples: the position of the weight at the end of a swinging pendulum about a center point, as in a grandfather clock; the position of a weight at the bottom end of a vertical spring, which is fixed in position at the top end–like a bungee jump). Note: these positions are not the same as time-dependent (transient) results.
