Advances in Operations Research

Volume 2016 (2016), Article ID 1925827, 55 pages

http://dx.doi.org/10.1155/2016/1925827

## Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

^{1}Department of Science and Mathematics, Columbia College Chicago, 623 South Wabash Avenue, Chicago, IL 60605, USA^{2}Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA

Received 7 August 2015; Accepted 26 January 2016

Academic Editor: Hsien-Chung Wu

Copyright © 2016 Eunju Sohn and Charles Knessl. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a storage allocation model with a finite number of storage spaces. There are primary spaces and secondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We let and denote the numbers of occupied primary and secondary spaces and study the joint distribution in the steady state. The joint process behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate becomes large, and the storage capacities and are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.

#### 1. Introduction

We consider the following storage allocation model. There are primary and secondary storage spaces. The primary spaces are numbered and the secondary ones are numbered . Customers arrive according to a Poisson process of rate , and each customer occupies a storage space for an exponentially distributed amount of time, with the mean occupation time . A new arrival takes the lowest ranked available space. If all spaces are filled, then a new arrival is turned away and lost. The policy of taking the lowest ranked space is called “first-fit allocation.”

We can consider the storage spaces as parking spaces of a restaurant. The primary spaces are in a lot right next to the restaurant, and the secondary spaces are located somewhere further away from the restaurant. Lower ranked spaces will be closer to the restaurant so it is natural for a customer to use the first-fit policy. Since spaces are occupied and emptied at random times, this model is called a dynamic storage allocation model. Design and analysis of algorithms for dynamic storage allocation are a fundamental part of computer science [1]. In such applications we can consider the customers as records, files, or lists and the storage device as a memory device. As time evolves, items are inserted and deleted, and the storage device, which is a linear array of “cells,” will have regions of occupied cells alternating with interior holes. This is referred to as memory fragmentation in computers, and collapsing the holes corresponds to running a defragmentation program.

In the language of queueing theory, the model with finite secondary storage spaces can be called the queue (the Erlang loss model) with ranked servers. The main contribution here is to study the effects of the finite storage capacity, for systems with a large number of both primary and secondary storage spaces and a commensurably large traffic intensity, which we denote by . Thus we study the model asymptotically for with .

We let and be the numbers of occupied primary and secondary spaces, and we will focus on the joint distribution of and , in the steady state. The distributions of both and are readily computed, as these processes behave as Erlang loss models, with and servers, respectively. Thus their steady state distributions are truncated Poisson distributions. However, the distribution of the number of occupied secondary spaces is much more complicated, as is the joint distribution .

We focus here on only the steady state distribution but comment that the transient behavior of the standard Erlang loss model can be analyzed by singular perturbation methods of the type employed here (see [2]). Thus we believe that, with significant additional effort, the transient behavior of the joint process (, ) could also be ultimately analyzed.

There has been much past work on the model with an infinite (secondary) storage capacity () since Kosten [3]. Various aspects of the solution were also studied in [4–7], but the solutions are in a complicated form, which is difficult to evaluate asymptotically for , due to the presence of an alternating sum. We derived the joint steady state distribution of the process in [8] using a discrete version of the classic method of separation of variables. We obtained the solution as a contour integral that involves certain polynomials related to hypergeometric functions. Such representations enabled us to obtain a complete set of asymptotic results including the joint distribution , for [9–12].

The solution of the finite capacity model with seems more complicated than the solution of the model with . But we will show here that a singular perturbation analysis is again fruitful, and we will obtain a complete set of asymptotic results for , which depends also parametrically on , and the numbers and of primary and secondary storage spaces. Most of the time we will scale all of , , , and to be of the same order as the traffic intensity . We will focus on understanding the effects of the finiteness of the secondary storage capacity .

The remainder of the paper is organized as follows. In Section 2 we state the basic equations and briefly describe their forthcoming analysis. In Section 3 we summarize all of the main results, and the joint distribution will have different asymptotic expansions in three main regions of the state space, which is the lattice rectangle . Moreover, there are also various boundary, corner, and transition curves where different expansions will be needed. In Section 4 we derive the asymptotics of the joint distribution in the three main regions, while in Sections 5–7 we treat the boundary, corner, and transition ranges. In Section 8 we will do some numerical comparisons to test the accuracy and robustness of our asymptotic results. Some discussion of our results also appears in Section 8. Since the analysis is quite technical, we have written this paper so that the derivations in Sections 4–7 can be omitted upon a first (and perhaps even later) reading(s).

#### 2. Statement of the Problem

We consider a system with primary and secondary storage spaces (or servers). The primary spaces are ranked and numbered while the secondary spaces are numbered . Customers arrive according to a Poisson process with rate parameter and a new arrival takes the lowest ranked available space, if possible a primary one. If all spaces are occupied further arrivals are turned away and lost. All of the storage spaces are identical and a customer occupies a space for an exponentially distributed amount of time, with the mean occupation (or service) time being . We then let and be the numbers of occupied primary and secondary spaces, respectively. We also introduce a dimensionless parameterto denote the traffic intensity.

The joint process corresponds to a continuous time random walk in a lattice rectangle. Figure 1 indicates transition rates. The steady state distributionis independent of the initial values and and satisfies the following balance equations:The main balance equation (3) applies in the interior of the lattice rectangle and along the boundary , (4)–(6) correspond to boundary conditions along three of the four boundaries of the rectangle, and (7)–(9) are corner conditions. Also, (6) applies at so the corner condition at is . We also have the normalization condition