Research Article | Open Access
Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution
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.
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 . 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 ). 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 . 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  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
The process by itself behaves precisely as the Erlang loss model ( queue with servers). This is well known to have, in the steady state, a truncated Poisson distribution; hence
The total number, , of occupied servers also follows a truncated Poisson distribution. Therefore,
We recently obtained in  explicit expressions for the joint distribution , but they are not very insightful due to their complexity. Thus we study the problem asymptotically, for with . This means that there are many arrivals but the numbers of storage spaces, both primary and secondary ones, are commensurately large. Note that if with then the probability distribution would concentrate on a single lattice point, with as . Here , , and , . Thus this limit would not be particularly interesting. There are, however, certain cases where either or is large but , that should lead to interesting results, but we do not analyze them here.
We next introduce the parametersand in the present limit we have , , where we view and as fixed as . We may then view the process on a “coarse” spatial scale, withOn the scale the random walk takes small steps () and the state space may be approximately viewed as the continuous rectangle Setting the main balance equation (3) becomeswhich is a difference equation with small differences, of order . The boundary condition along in (4) may be replaced by the “artificial boundary condition”The above is obtained by requiring that (3) holds also at and comparing this to (4). Introducing simplifies some of the calculations, but this quantity has no physical meaning.
The asymptotic structure of the joint distribution will be very different for four main regions in the parameter space. We call these regions and they are sketched in Figure 2. They are defined by the inequalitiesIt will also prove useful to define as follows the curves that separate these four regions:Note that the union of all the sets in (19)–(25) is the entire open quarter plane in parameter space. We purposefully exclude the coordinate axes and , as they would require entirely different asymptotic analyses. The separating curves in (23)–(25) will also require separate analyses, and we will obtain results that apply not only along the curves but also in small neighborhoods of these curves, which will be defined precisely later. This will produce results that asymptotically match to those in the main regions.
The presence of the different regions can be explained intuitively. If () there are enough primary spaces to service all storage requests and the secondary spaces will generally not be needed. If but () the primary spaces are insufficient but the total number of spaces is adequate. Then we might expect that typically all primary spaces and about () secondary spaces will be occupied. If then typically all primary and secondary spaces will be occupied. Then we might expect to be concentrated near , . The further splitting of into the regions and is difficult to explain intuitively in terms of the basic model, but we will explain this dichotomy via our asymptotic analysis. We also note that the asymptotic behavior of the distribution in (11) undergoes a transition when passes through 1, while (12) undergoes an analogous transition when passes through 1. However, neither (11) nor (12) undergoes a transition along . In the analysis that follows we will also need to, for each region of parameter space, separately analyze several different regions of the state space, which corresponds to the rectangle , on the coarse spatial scale. It will sometimes prove necessary to analyze boundary and corner regions where the discrete nature of the model must be considered.
3. Summary of Results
In the analysis it proves sometimes useful to use the variables whereso that (resp., ) measures the number of unoccupied primary (resp., secondary) spaces. Then we also letso that corresponds to the probability that all of the storage spaces are full. In (27) we did not indicate the dependence of and on the parameters , , and .
We begin by giving asymptotic results for .
Proposition 1. For and fixed one has
Here and throughout the paper, we use the convention that corresponds to the union of the open sets and and also the separating curve (cf. (23)). Similar comments apply for and . We refer to the asymptotic limit in (29) as corresponding to , where we now give the precise scaling, , that applies near the separating curve in (24). The results in (28)–(30) will follow from our asymptotic analysis of the joint distribution, but we note that these also follow easily from (12), by setting and expanding the result for and different ranges of (thus ).
It will prove convenient to express some of our results in terms of the three constants , , and ; these depend only on the parameters , , and . We summarize below the leading order asymptotics of these constants.
Proposition 2. Define the constants , , and by the relationswhere is the Airy function and its maximal root
We note that the relation holds for all cases of the parameters. In (38) we have thus defined the precise scaling near the separating curve , as . Note that is not defined for region while is not defined for , as then the corresponding constant will play no role in the analysis. We conclude by giving the precise scaling for (near ), which will beNote that (46) can be predicted from the marginal distribution in (11), as the sum in the denominator undergoes a transition for , which is the same scaling as in (46).
3.1. Joint Distribution and Its Limits
Now we consider the joint distribution; for . We recall that and are the scaled numbers of primary and secondary spaces. The state space of the random walk is the lattice rectangle in Figure 1, and on the coarse spatial scale this can be viewed as the continuous rectangle . Our goal is to give a complete asymptotic description of the joint distribution for , including ranges of the state space where there is appreciable mass and also ranges where is asymptotically small. This corresponds to the tails of the distribution and in such ranges is typically exponentially small for large . We first discuss the ranges where there is significant mass, and this will lead to certain limiting distributions, which will be very different for regions of parameter space in (19)–(25).
Proposition 3. For one has the following limiting distributions:(i) (thus ) which can be recast as the limit(ii) (thus ) and this applies for and (i.e., ).(iii) (thus ) which holds for and .
When we have so the secondary storage spaces will be rarely needed, and then approximately follows the Poisson distribution in (47), which has also the Gaussian limit in (48). The results in (47) and (48) provide no information on for , but later we will estimate precisely these probabilities. We also note that when , (47) ceases to be valid for , for then if almost all primary spaces are full there may well be some secondary spaces also occupied, and thus may become comparable to for , for this range of . If and the primary storage spaces are insufficient to meet the demand, but the total number of spaces does suffice. Then (49) shows that primary spaces and secondary spaces will tend to be occupied, with the joint distribution being a product of a geometric and a Gaussian. This also shows that, to leading order for large , the processes and decouple. When we have and the totality of storage spaces is not enough to meet the demand. Then typically all but a few spaces, both primary and secondary, will tend to be occupied, with the numbers and of available spaces following the discrete joint distribution in (50). From (50) we can easily show that so that the total number of empty spaces is geometrically distributed; this result also follows easily from the exact expression in (12). We will later see that the tail behavior of (50), for and/or , is quite different according as or , which again will indicate that the triangle in parameter space needs to be split into the two regions and .
We next study the transitions between the three limiting results in Proposition 4.
Proposition 4. For one has the limiting distributions: (i), , and (thus and ) where is the parabolic cylinder function of order and argument . When the above simplifies to(ii), , and
As the truncated Gaussian distribution in (55) approaches the free space Gaussian in (49), which applies for . For , (55) asymptotically matches to (50), when the latter is expanded for and simultaneously , with the product held fixed.
The complicated distribution in (53) is a necessary intermediate result since (47) and (49) do not asymptotically match. The right-hand side of (53) is of the form (density in ), with the density having support in the quarter plane , . Thus if there will tend to be empty primary spaces and full secondary spaces, with now an intricate coupling between the processes and . Finally, we note that the results in items (i) and (ii) in Proposition 3 and in item (i) of Proposition 4 are independent of the secondary storage capacity , while item (iii) in Proposition 3 and item (ii) in Proposition 4 do depend upon .
3.2. Joint Distribution: Main Regions of State Space
The asymptotic expansion of will be different for the four parameter ranges indicated in Figure 2 and also for three main regions of the state space, which we call , , and , and we define/discuss these below.
First consider region of parameter space, so that , and define the curveThis curve depends on both and and thus on both of the total numbers, and , of primary and secondary storage spaces. The curve is defined for and we haveFor region (and indeed also for and ) we have so that (56) connects the point to the corner point in the scaled state space. The curve divides the state space into the two regions and , withHere we defined as an open set, while is bounded by the four curves , , , and and we include only the third of these as a part of . This is because the asymptotic expansion that will apply in the interior of will remain valid near , but not near the other three bounding curves. The expansion valid in will break down if either , , or . We sketch in Figure 3 the curve and we recall that if most of the mass in is concentrated in the range and (see Proposition 3), and this corresponds to the lower bounding curve for .
Proposition 5. For () the asymptotic expansions of are as follows: (i) where are related to via the mapping, for , where is the Jacobian associated with (62); that is,(ii) where is given by (45) for region where are related to by where with with corresponding to , and where is the Jacobian associated with (68), so that
We can view (68) as representing a family of curves in the plane, with indexing the family and increasing along a curve. When the curves in (68) meet at the corner point and we also note that the Jacobian in (71) vanishes when , indicating a singularity in the transformation in (68). When the curve becomes the horizontal segment . But then (69) shows that so that in (67). Thus near the expansion in (68) becomes invalid. When the curve in (68) becomesand eliminating we see that (72) is precisely the curve in (56) that separates from . For the curves in (68) fill a portion of , but then the leading term for is given by (60), and (65) corresponds to only an exponentially small correction to (60). When the curve in (68) is tangent to the line at the point , which will have significance for the parameter region . We also note that for when the curves in (68) hit the -axis (then by (68)) and then the first factor in (70) becomes singular, which indicates that the asymptotics become invalid. Along corresponding to , in (70) is again singular. Thus (70) is singular when , , and , corresponding to the three curves (, , and ) that bound the region . We will give the appropriate expansions near these bounding curves in Section 3.3.
For the curves in (62) fill the entire region , with corresponding to the line segment , , and corresponding to the curve (then (62) coincides with (72)). When we have and the curves in (62) hit the line at finite and nonzero slopes, for all . As increases each curve will hit first either the -axis or the -axis. When the -axis is hit first. For this occurs at a finite value of , when , but if in order to approach the -axis we must let and in such a way that is held fixed. In this limit (62) may be approximated by and . We discuss in more detail the behavior of (60) as later, when we give the asymptotic expansion(s) for that apply for and . When the curves in (62) hit the -axis when , for then . In particular if the corresponding curve hits the -axis at in (57). Near both the - and -axes, (60) will have singular behaviors and other expansions must be constructed. Note, however, that (60) is not singular along the curve , whereas (65) is singular. We can simplify (60) near , and then we obtain the more explicit form which holds for and . However, we note that since we have and thus in (73) is exponentially small in . This is true for the entire domains and , as there is very little probability mass in these ranges if .
We next consider regions and in parameter space, where it will become necessary to break up the state space into the three regions , , and . These regions are sketched in Figures 4 and 5. The curve that separates from is again given by (56), while the curve separating from will beand thus, since now ,
We thus define asand, for regions and , now becomeswith still defined by (59).
Proposition 6. For the asymptotic expansions of are as follows: (i) where