Average-case analysis of the Modified Harmonic algorithm

In this paper we analyze the average-case performance of the Modified Harmonic algorithm for on-line bin packing. We first analyze the average-case performance for arbitrary distribution of item sizes over (0,1]. This analysis is based on the following result. Letf ₁ andf ₂ be two linear combinations of random variables {N _i } _i=1 ^k where theN _i 's have a joint multinomial distribution for eachn=σ _i=1 ^k ,N _i . LetE(f ₁) ≠ O andE(f ₂)≠ 0. Then lim_{n →}∞E(max(_{f 1},f ₂ ))/n = lim _n →∞ max(E(f ₁),E(f ₂))/n. We then consider the special case when the item sizes are uniformly distributed over (0,1]. For specific values of the parameters, the Modified Harmonic algorithm turns out to be better than the other two linear-time on-line algorithms—Next Fit and Harmonic—in both the worst case as well as the average case. We also obtain optimal values for the parameters of the algorithm from the average-case standpoint. For these values of the parameters, the average-case performance ratio is less than 1.19. This compares well with the performance ratios 1.333. and 1.2865. of the Next Fit algorithm and the Harmonic algorithm, respectively.