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\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
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\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
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\(\let \Tilde \tilde \)
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\(\newcommand {\Batched }{b\text {-}{\rm B{\small ATCHED}}}\)
\(\newcommand {\OneChoice }{{\rm O{\small NE}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\TwoChoice }{{\rm T{\small WO}}\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\DSample }{d\text {-}{\rm S{\small AMPLE}}}\)
\(\newcommand {\TwoSample }{{\rm T{\small WO}}\text {-}{\rm S{\small AMPLE}}}\)
\(\newcommand {\Graphical }{{\rm G{\small RAPHICAL}}}\)
\(\newcommand {\DChoice }{d\text {-}{\rm C{\small HOICE}}}\)
\(\newcommand {\GBounded }{g\text {-}{\rm B{\small OUNDED}}}\)
\(\newcommand {\GMyopic }{g\text {-}{\rm M{\small YOPIC}}}\)
\(\newcommand {\RelativeThreshold }{{\rm R{\small ELATIVE}\text {-}T{\small RHESHOLD}}}\)
\(\newcommand {\MeanThinning }{{\rm M{\small EAN}\text {-}T{\small HINNING}}}\)
\(\newcommand {\TwoThinning }{{\rm T{\small WO}\text {-}T{\small HINNING}}}\)
\(\newcommand {\DThinning }{d\text {-}{\rm T{\small HINNING}}}\)
\(\newcommand {\Twinning }{{\rm T{\small WINNING}}}\)
\(\newcommand {\Threshold }{{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\Thinning }{{\rm T{\small HINNING}}}\)
\(\newcommand {\KThreshold }{k\text {-}{\rm T{\small HRESHOLD}}}\)
\(\newcommand {\OnePlusBeta }{(1+\beta )}\)
\(\newcommand {\Packing }{{\rm P{\small ACKING}}}\)
\(\newcommand {\Quantile }{{\rm Q{\small UANTILE}}}\)
\(\newcommand {\Memory }{{\rm M{\small EMORY}}}\)
\(\newcommand {\N }{\mathbb {N}}\)
\(\newcommand {\R }{\mathbb {R}}\)
\(\newcommand {\Oh }{\mathcal {O}}\)
\(\newcommand {\AdvTauDelay }{\tau \text {-}{\rm A{\small DV}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\RandomTauDelay }{\tau \text {-}{\rm R{\small AND}}\text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\TauDelay }{\tau \text {-}{\rm D{\small ELAY}}}\)
\(\newcommand {\SigmaNoisyLoad }{\sigma \text {-}{\rm N{\small OISY}}\text {-}{\rm L{\small OAD}}}\)
\(\newcommand {\Heterogeneous }{{\rm H{\small ETEROGENEOUS}}}\)
\(\newcommand {\Gap }{\mathrm {Gap}}\)
\(\newcommand {\eps }{\epsilon }\)
The σ-Noisy-Load Process
The \(\SigmaNoisyLoad \) process is a noisy version of \(\TwoChoice \) where the load estimates are perurbed by a Gaussian distribution \(\mathcal {N}(0, \sigma ^2)\).
The \(\SigmaNoisyLoad \) process:
Iteration: At each step \(t = 0, 1, 2, \ldots \),
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• Sample two bins \(i_1 = i_1^t\) and \(i_2 = i_2^t\) uniformly at random.
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• Compute the load estimates \(\tilde {x}_{i_1}^t = x_{i_1}^{t} + \mathcal {N}(0, \sigma ^2)\) and \(\tilde {x}_{i_2}^t = x_{i_2}^{t} + \mathcal {N}(0, \sigma ^2)\).
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• Allocate to bin \(i \in \{ i_1, i_2 \}\) satisfying \(\tilde {x}_i^t = \min \{ \tilde {x}_{i_1}^t, \tilde {x}_{i_2}^t \}\).
.