We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate $r$ and depart after an independent exponentially distributed unit mean service time. The arriving item sizes (lengths) are assumed to be independent and identically distributed (i.i.d.) from a common distribution $H$. A widely employed algorithm for allocating the items is the "first-fit" discipline, namely, each arriving item is placed in the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp ([6]) proved that in the special case of unit length items (i.e. degenerate $H$), as $r$ tends towards infinity, the first-fit algorithm is asymptotically optimal in the following sense: the steady-state ratio of expected "empty space" (gaps between items) to expected occupied space tends towards $0$. In a sequel to [6], Coffman, Kadota, and Shepp ([5]) conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate $H$. In this paper we provide the first proof of first-fit asymptotic optimality for non-degenerate distributions $H$ of item sizes. Our main result is for the case when $H$ is concentrated on countably many positive real sizes forming an increasing sequence that is either finite or goes to infinity, with the average item size being finite. We prove that under the first-fit discipline, as $r$ tends towards infinity, the steady-state packing configuration (scaled down by $r$) converges in distribution to the limiting packing configuration with smaller items on the left, larger items on the right, and with no gaps between. In particular, this proves asymptotic optimality of first-fit in the sense that in steady-state the empty space (scaled down by $r$) vanishes.

翻译：我们重新审视动态存储分配中的一个经典问题。物品以泊松速率 $r$ 到达线性存储介质（建模为半轴），并在独立指数分布的单位均值服务时间后离开。到达物品的大小（长度）假设为独立同分布（i.i.d.），服从共同分布 $H$。广泛用于分配物品的一种算法是“首次适应”规则，即每个到达的物品被放置在足够容纳它的最左边空闲区间。在1985年的一篇开创性论文中，Coffman、Kadota 和 Shepp ([6]) 证明了在单位长度物品（即退化 $H$）的特殊情况下，当 $r$ 趋于无穷时，首次适应算法在以下意义上是渐近最优的：稳态下期望“空空间”（物品之间的间隙）与期望占用空间之比趋于0。在 [6] 的后续研究中，Coffman、Kadota 和 Shepp ([5]) 推测首次适应规则对于非退化 $H$ 也是渐近最优的。本文首次证明了物品大小分布 $H$ 为非退化时首次适应的渐近最优性。我们的主要结果适用于 $H$ 集中在可数多个正实数大小上，这些大小构成一个递增序列（有限或趋于无穷），且平均物品大小有限的情况。我们证明，在首次适应规则下，当 $r$ 趋于无穷时，稳态装箱配置（按 $r$ 缩放）依分布收敛到极限装箱配置，其中较小的物品在左侧，较大的物品在右侧，且之间无间隙。特别地，这证明了首次适应在稳态下空空间（按 $r$ 缩放）消失的意义上的渐近最优性。