Asymptotic optimality of dynamic first-fit packing on the half-axis

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 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 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), the first-fit algorithm is asymptotically optimal in the following sense: the ratio of expected empty space to expected occupied space tends towards 0 as the occupied space tends towards infinity. In a sequel to [6], the authors of [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 a non-degenerate distribution H, namely the case when items can be of sizes 1 and 2. Specifically, we prove that, under first-fit, the steady-state packing configuration, scaled down by r, converges in distribution to the optimal limiting packing configuration, i.e. the one with smaller items on the left, larger items on the right, and with no gaps between.