We consider variants of the online stochastic bipartite matching problem motivated by Internet advertising display applications, as introduced in Feldman et al. [6]. In this setting, advertisers express specific interests into requests for impressions of different types. Advertisers are fixed and known in advance while requests for impressions come online. The task is to assign each request to an interested advertiser (or to discard it) immediately upon its arrival.In the adversarial online model, the ranking algorithm of Karp et al.[11] provides a best possible randomized algorithm with competitive ratio 1−1/e ≈ 0.632. In the stochastic i.i.d. model, when requests are drawn repeatedly and independently from a known probability distribution over the different impression types, Feldman et al. [6] prove that one can do better than 1 − 1/e. Under the restriction that the expected number of request of each impression type is an integer, they provide a 0.670-competitive algorithm, later improved by Bahmani and Kapralov [3] to 0.699, and by Manshadi et al.[13] to 0.705. Without this integrality restriction, Manshadi et al. [13] are able to provide a 0.702-competitive algorithm.In this paper we consider a general class of online algorithms for the i.i.d. model which improve on all these bounds and which use computationally efficient offline procedures (based on the solution of simple linear programs of maximum flow types). Under the integrality restriction on the expected number of impression types, we get a 1 − 2e −2 (≈ 0.729)-competitive algorithm. Without this restriction, we get a 0.706-competitive algorithm.Our techniques can also be applied to other related problems such as the online stochastic vertex-weighted bipartite matching problem as defined in Aggarwal et al. [1]. For this problem, we obtain a 0.725-competitive algorithm under the stochastic i.i.d. model with integral arrival rate.Finally we show the validity of all our results under a Poisson arrival model, removing the need to assume that the total number of arrivals is fixed and known in advance, as is required for the analysis of the stochastic i.i.d. models described above.

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Commercially available friction modifiers are used in many different countries that have widely different atmospheric conditions. These variations in atmospheric conditions lead to varying levels of railhead oxidation and debris build-up. Friction modifiers can be applied to the rail without any prior cleaning of the rail and this can lead to varying friction modifier/iron oxide ratios potentially affecting the performance of the friction modifier. This paper reports the results of an investigation that was performed to determine the effects of varying atmospheric and oxide conditions on the performance of friction modifiers. A pin-on-disk test rig with an attached environmental chamber was used for the study. Results show that relative humidity has a pronounced effect on the way in which the friction modifier affects friction levels, and also the amount of time it remains on the disk. This also depends on the concentration of oxide in the friction modifier. Glow discharge optical emission spectroscopy analysis was also carried out to assess the effect of the friction modifier and atmospheric conditions on the chemical composition of the surface of the disk. Results show that the depth of surface modification is vastly different depending on the conditions and level of railhead debris.

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