ABSTRACT
Consider 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/math6_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> inventory, having “inter-demand” times https://www.w3.org/1998/Math/MathML"> = d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Exp λ ; “demand sizes” https://www.w3.org/1998/Math/MathML"> = d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Exp μ ; no product decay; and no lead time. The SP of the corresponding sample path of net inventory describes its motion in the state space https://www.w3.org/1998/Math/MathML"> − ∞ , S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> over time. Let https://www.w3.org/1998/Math/MathML"> I ( t ) t ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denote the net inventory at time https://www.w3.org/1998/Math/MathML"> t ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Note that https://www.w3.org/1998/Math/MathML"> I ( t ) t ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a “regenerative process”, as we next show. Once the SP “enters a fixed level” in 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/math6_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the SP remains at that level for a time https://www.w3.org/1998/Math/MathML"> = d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Exp λ , until the next “demand” instant (see Figure 6.1). The SP returns to level S at ends of a sequence of “ordering cycles”. These cycles are iid r.v.s due to the identical probabilistic structure of each cycle. Therefore, the stationary mixed pdf https://www.w3.org/1998/Math/MathML"> Π S , f ( x ) s < x < S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_11.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> exists (see, e.g., [55] (1993) or [57] (2003)). The term “ https://www.w3.org/1998/Math/MathML"> Π S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_12.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ” is an “atom” at level S (positive probability). The term https://www.w3.org/1998/Math/MathML"> f ( x ) , 0 < x < S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_13.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , is the continuous part of the pdf. Every level https://www.w3.org/1998/Math/MathML"> x ∈ ( 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/math6_14.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a continuous point (not an atom) of https://www.w3.org/1998/Math/MathML"> f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_15.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , because the probability of entering any https://www.w3.org/1998/Math/MathML"> x ∈ 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/math6_16.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is zero due to continuous demand sizes https://www.w3.org/1998/Math/MathML"> = d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math6_17.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Exp μ..
