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We give some asymptotic results for the following storage allocation model: There are m primary spaces, ranked {1, 2, ... , m}, and R secondary spaces, ranked {m + 1, m + 2, ..., m + R} . Customers arrive according to a Poisson process with rate parameter λ, a new arrival takes the lowest ranked space, and occupies it for an exponentially distributed amount of time, with mean 1/ μ Letting N1 and N2 be, respectively, the numbers of occupied primary and secondary spaces, we recently [15] obtained an explicit, albeit complicated, expression for the joint steady-state distribution, π (k, r) = Prob[N1 = k, N2 = r]. We also obtained various asymptotic results [14] for this distribution, in the limit where ρ ≡ λ /μ→∞, but with m, R = ϴ(ρ). Here we consider a different limit, where ρ → ∞ withm = ρ + O( √ρ ) and R = O(√ρ). Then we obtain a limiting two-dimensional density as a leading order approximation to π (k, r). We also examine various asymptotic properties of this density, including tail behaviors, and give numerical comparisons to the discrete model. Finally we show that the density satisfies a certain two-dimensional parabolic partial differential equation with appropriate boundary/corner conditions. [ABSTRACT FROM AUTHOR]

We consider a processor-sharing storage allocation model, which has m primary holding spaces and infinitely many secondary ones, and a single processor servicing the stored items. All of the spaces are numbered and ordered. An arriving customer takes the lowest available space. Dynamic storage allocation and the fragmentation of computer memory are well-known applications of this model. We define the traffic intensity ρ to be λ/μ, where λ is the customers' arrival rate and μ is the service rate of the processor. We study the joint probability distribution of the numbers of occupied primary and secondary spaces. We study the problem in two asymptotic limits: (1) m → ∞ with a fixed ρ < 1, and (2) ρ ↑ 1, m → ∞ with m(1-ρ) = O(1). The asymptotics yield insight into how many secondary spaces tend to be needed, and into the sample paths leading to the occupation of the two types of spaces. We show that the asymptotics lead to accurate numerical approximations. [ABSTRACT FROM AUTHOR]