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- Title
Growth-collapse effects applied to cash management and queues.
- Authors
Stadje, W.; Perry, D.
- Abstract
When the proportion is a function HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><mo stretchy="false">)</mo></mrow></math> ht of a uniformly distributed random variable, we obtain the I Pollaczek-Khinchine i balance equation 2 HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><msubsup><mo> </mo><mrow><mi>x</mi></mrow><mi> </mi></msubsup><mi> </mi><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><mi>h</mi><mfenced close=")" open="("><mfrac><mi>x</mi><mi>w</mi></mfrac></mfenced><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>w</mi><mo>=</mo><mi> </mi><msubsup><mo> </mo><mrow><mn>0</mn></mrow><mi>x</mi></msubsup><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>-</mo><mi>G</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></mrow><mi mathvariant="normal">d</mi><mi>F</mi><mrow><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> ht Graph The left-hand side of the balance equation (2) is the rate of downcrossings of level HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo> </mo><mn>0</mn></mrow></math> ht . This means that I V i (0) is stochastically equal to HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>U</mi><mi>V</mi><mo stretchy="false">(</mo><mi> </mi><mo stretchy="false">)</mo></mrow></math> ht where I U i is a uniform random variable (0, 1).
- Subjects
CASH management; POISSON processes; RANDOM variables; DISTRIBUTION (Probability theory); STOCHASTIC systems
- Publication
Queueing Systems, 2022, Vol 100, Issue 3/4, p257
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-022-09820-4