\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\newcommand {\tcbset }[1]{}\)
\(\newcommand {\tcbsetforeverylayer }[1]{}\)
\(\newcommand {\tcbox }[2][]{\boxed {\text {#2}}}\)
\(\newcommand {\tcboxfit }[2][]{\boxed {#2}}\)
\(\newcommand {\tcblower }{}\)
\(\newcommand {\tcbline }{}\)
\(\newcommand {\tcbtitle }{}\)
\(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\)
\(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\)
\(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\)
\(\newcommand {\Batched }{b\text {-}{\rm B{\small ATCHED}}}\)
\(\newcommand {\OneChoice }{{\rm O{\small NE}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\TwoChoice }{{\rm T{\small WO}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\GBounded }{g\text {-}{\rm B{\small OUNDED}}}\)
\(\newcommand {\GMyopic }{g\text {-}{\rm M{\small YOPIC}}}\)
\(\newcommand {\RelativeThreshold }{{\rm R{\small ELATIVE}\text {-}T{\small RHESHOLD}}}\)
\(\newcommand {\MeanThinning }{{\rm M{\small EAN}\text {-}T{\small HINNING}}}\)
\(\newcommand {\TwoThinning }{{\rm T{\small WO}\text {-}T{\small HINNING}}}\)
\(\newcommand {\Twinning }{{\rm T{\small WINNING}}}\)
\(\newcommand {\Packing }{{\rm P{\small ACKING}}}\)
\(\newcommand {\Quantile }{{\rm Q{\small UANTILE}}}\)
\(\newcommand {\Memory }{{\rm M{\small EMORY}}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\newcommand {\Oh }{\mathcal {O}}\)
\(\newcommand {\AdvTauDelay }{\tau \text {-}{\rm A{\small DV}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\RandomTauDelay }{\tau \text {-}{\rm R{\small AND}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\TauDelay }{\tau \text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\SigmaNoisyLoad }{\sigma \text {-}{\rm N{\small OISY}}\text {-}{\rm L{\small OAD}}}\)
The Twinning Process
The \(\Twinning \) process allocates \(w_+\) balls when sampling an overloaded bin (i.e., a bin \(i\) with \(x_i^t \geq t/n\)) and \(w_-\) balls when sampling an underloaded bin.
The \(\Twinning \) process:
Iteration: At each step \(t = 1, 2, \ldots \),
-
• Sample a bin \(i = i^t\) uniformly at random.
-
• Update its load:
\[ x_i^{t+1} = \begin {cases} x_i^t + 2 & \text {if }x_i^t < W^t/n \\ x_i^t + 1 & \text {if }x_i^t \geq W^t/n. \end {cases} \]
.