ABSTRACT
In the standard G/M/1 queue, assume that arrivals occur according to a renewal process (for definition of a renewal process, see Section 1.8.1 in Chapter 1). For the common inter-arrival time denote the cdf, complementary cdf (ccdf), and pdf, respectively, by https://www.w3.org/1998/Math/MathML"> A ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> x > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> A ¯ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> = https://www.w3.org/1998/Math/MathML"> 1 − A ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> x ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> a ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> = https://www.w3.org/1998/Math/MathML"> d A ( x ) / d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math5_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> wherever the derivative exists. It can be shown that the stationary pdf of the virtual wait has an “exponential form” (see Section 5.1.5). We derive the stationary pdf and cdf of the virtual wait, the stationary pdf and cdf of the actual (arrival-point) wait just before arrival instants, the expected busy period, expected idle period, and related results.
