TY - JOUR

T1 - The loss of serving in the dark

AU - Azar, Yossi

AU - Cohen, Ilan Reuven

AU - Gamzu, Iftah

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2023/2

Y1 - 2023/2

N2 - We study the following balls and bins stochastic process: There is a buffer with B bins, and there is a stream of balls X=〈X1,X2,…,XT〉 such that Xi is the number of balls that arrive before time i but after time i−1. Once a ball arrives, it is stored in one of the unoccupied bins. If all the bins are occupied then the ball is thrown away. In each time step, we select a bin uniformly at random, clear it, and gain its content. Once the stream of balls ends, all the remaining balls in the buffer are cleared and added to our gain. We are interested in analyzing the expected gain of this randomized process with respect to that of an optimal gain-maximizing strategy, which gets the same online stream of balls, and clears a ball from a bin, if exists, at any step. We name this gain ratio the loss of serving in the dark. In this paper, we determine the exact loss of serving in the dark. We prove that the expected gain of the randomized process is worse by a factor of ρ+ϵ from that of the optimal gain-maximizing strategy where [Formula presented] and ρ=maxα>1αeα/((α−1)eα+e−1)≈1.69996. We also demonstrate that this bound is essentially tight as there are specific ball streams for which the above-mentioned gain ratio tends to ρ. Our stochastic process occurs naturally in packets scheduling and mechanisms design applications.

AB - We study the following balls and bins stochastic process: There is a buffer with B bins, and there is a stream of balls X=〈X1,X2,…,XT〉 such that Xi is the number of balls that arrive before time i but after time i−1. Once a ball arrives, it is stored in one of the unoccupied bins. If all the bins are occupied then the ball is thrown away. In each time step, we select a bin uniformly at random, clear it, and gain its content. Once the stream of balls ends, all the remaining balls in the buffer are cleared and added to our gain. We are interested in analyzing the expected gain of this randomized process with respect to that of an optimal gain-maximizing strategy, which gets the same online stream of balls, and clears a ball from a bin, if exists, at any step. We name this gain ratio the loss of serving in the dark. In this paper, we determine the exact loss of serving in the dark. We prove that the expected gain of the randomized process is worse by a factor of ρ+ϵ from that of the optimal gain-maximizing strategy where [Formula presented] and ρ=maxα>1αeα/((α−1)eα+e−1)≈1.69996. We also demonstrate that this bound is essentially tight as there are specific ball streams for which the above-mentioned gain ratio tends to ρ. Our stochastic process occurs naturally in packets scheduling and mechanisms design applications.

KW - Oblivious algorithms

KW - Online algorithms

KW - Prompt mechanisms

KW - Randomized algorithms

KW - Scheduling

UR - http://www.scopus.com/inward/record.url?scp=85140059086&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2022.106334

DO - 10.1016/j.ipl.2022.106334

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AN - SCOPUS:85140059086

SN - 0020-0190

VL - 180

JO - Information Processing Letters

JF - Information Processing Letters

M1 - 106334

ER -