Queueing models for addictive tasks.

It is easy to see that I f i ( I k i ) is of order HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>log</mo><mi>k</mi></mrow></math> ht if HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi></math> ht 's are light-tailed. Assume that HT <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi> </mi><mi>k</mi></msub></math> ht new jobs arrive at the beginning of time slot I k i , and HT <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mo stretchy="false">{</mo><msub><mi> </mi><mi>k</mi></msub><mo stretchy="false">}</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi> </mi></msubsup></math> ht is an i.i.d. sequence. Denote by HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>M</mi><mi>k</mi></msub><mo>=</mo><msub><mo movablelimits="true">max</mo><mrow><mi>i</mi><mo><=</mo><mi>k</mi></mrow></msub><msub><mi> </mi><mi>i</mi></msub></mrow></math> ht and let I f i ( I k i ) be such that HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="double-struck">P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">-></mo><mn>1</mn></mrow></math> ht for HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mrow><mo stretchy="false">{</mo><msub><mi>M</mi><mi>k</mi></msub><mo><=</mo><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">}</mo></mrow></mrow></math> ht . [Extracted from the article]