Some Exact and Asymptotic Solutions to Single Server Models of Dynamic Storage

Abstract

We consider models of queue storage, where items arrive accordingly to a Poisson process of rate λ and each item takes up one cell in a linear array of cells, which are numbered {1, 2, 3,…}. The arriving item is placed in the lowest numbered available cell. The total service rate provided to the items is the constant μ (with ρ = λ/μ), but service may be provided simultaneously to more than one item. If there are 𝒩 items stored and each is serviced at the rate μ/𝒩, this corresponds to processor-sharing (PS). We analyze several models of this type, which have been shown to provide bounds on the PS model. We shall assume that (1) the server works only on the leftmost item, (2) on the two rightmost items, or (3) on all items by the PS discipline, but with a departure of the rightmost item causing a rearrangement of the items within the cells, so that the wasted space remains unchanged. The set of occupied cells at any time is {i ₁, i ₂,…, i _𝒩} where i ₁ < i ₂ < … <i _𝒩 and we are interested in the wasted space (W = i _𝒩 − 𝒩), the maximum occupied cell (i _𝒩), and the joint distribution of W and 𝒩. We study these exactly and asymptotically, especially in the heavy traffic limit where ρ ↑ 1.

Keywords:

• Asymptotics
• Process-sharing
• Storage

Mathematics Subject Classification:

• 60K25
• 90B05
• 34E10