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The parallel batch Sort is an important model of the modern sort in parallel batches Sort , a capacity for b machine can b workpiece as the number of simultaneous machining the workpiece in the same batch have the same starting time and completion time. processing time per batch and the batch processing time of the longest workpiece . under in LKβ model at time t, the online algorithm can foresee in the time interval ( t, t β ] arrive workpiece not compatible workpiece artifacts belonging to different groups can not be processed in the same batch , in this paper , we study the four able to forward-looking information under the workpiece within a time interval of unit length of the workpiece parallel sub- batch machine online scheduling problems , including stand-alone and parallel machine , batch capacity is unbounded . objective function is to make the maximum completion time of all artifacts minimum . using Graham et al ( 1979) provide three-parameter representation , these problems . that is the the Pm | on - line , the p -batch , b = ∞ , pj = 1 , LKβ | Cmax, 1 | on-line , p -batch , b = ∞ pj = 1 , LKβ , two families | Cmax 1 | on-line, p-batch, b = ∞, pj = 1, LKβ, families | Cmax, Pm | on-line, p-batch, b = ∞, pj = 1, LKβ, f = m | Cmax. below we specifically the main results of this paper is given in the second chapter , for the first scheduling problem Pm | on-line , the p -batch , b = ∞ , pj = 1 , LKβ | Cmax when β ≥ 1 / m when we are given an optimal online algorithm H ∞ ( β ≥ 1 / m ) when 0 ≤ β lt; 1 / m , we first prove that the problem online algorithm competition than the lower bound 1 αm, wherein 0 LT ; α LT ; equation ( 1 αm ) (M 1 ) = αm 2- βΣi = 1m ( 1 αm ) i a root , and then provide a competitive than 1 αm best possible online algorithm H ∞ (β lt; 1 / m). third chapter , after three online scheduling problems with non - compatible work group , we were proved that the competition of these problems than the Lower Bound least 1 α2, 1 af and 1 a , wherein α2, af and α are equations 2α22 ( β 1 ) α2 of β- 2 = 0 , the F · αf2 ( β 1 ) of αf β - f = 0 and α2 (β 1 ) α and β - 1 = 0 of the one positive root and then sequentially provides the best possible online algorithm H2 (β), Hf (β) and HM (β ) .
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