Minimum cost modeling for hierarchical storage reallocation using dynamic usage frequencies.

Cope, Rachelle F. ; Cope, Robert F., III ; Baldwin, Yvette B. 等

ABSTRACT

In computer information systems, some programs are used more frequently than others producing skewed distributions of program usages. We investigate the claim that static views of program usage frequencies are insufficient when they are used for storage allocation decisions making it necessary to study the implications of the use of dynamic frequencies in storage allocation.

The use of dynamic frequencies provides a natural extension to previously presented static cost model literature for hierarchical storage allocation. In our work, we present the value of incorporating dynamic usage frequencies into program usage cost models. Thus, an optimization-based cost modeling methodology using Simon's Model for dynamic hierarchical storage allocation is presented.

To illustrate, a simple example of program storage allocation is presented in both static and dynamic form. Cost-saving comparisons are then discussed.

INTRODUCTION

One problem faced by organizations in today's rapidly changing field of computer information systems is the consumption of storage capacity due to the growing base of software assets. Research has shown that very few firms effectively monitor program usage (Willet, 1994). Storage management issues arise when many of the programs occupying valuable storage space are used infrequently. Many IS managers simply buy more primary storage when they feel it is necessary instead of performing maintenance.

Many application programs in a firm are subject to decreases as well as increases in usage rates. The problem is that the usage rate of an individual software item is dynamic. There is no feasible way to predict the behavior of each individual program in an organization. Instead, it is more practical to provide a methodology for focusing on those programs whose changing usage is critical to storage allocation and cost minimization.

With a better understanding of normal program usage behavior, organizations can develop policies based on easily understood criteria. Simon's Model has been well studied in literature concerning information usage (Simon, 1991). Program usage can also be studied using the model, making it the tool of choice to develop such criteria. In turn, we will show that it is possible to create organizational policy based on program usage observations.

The true value in deriving a policy using Simon's Model is the relative ease that an organization can assess its own model parameters. Once model parameters are evaluated, a firm would be able to determine the category into which their software falls. Each category would provide a recommendation for the intervals at which the "significant few" should be assessed for usage. The cost of storage allocation could then be minimized.

SUPPORTING LITERATURE

While investigating various algorithms developed for the purpose of optimizing storage in multi-level storage systems, it was found that the models developed by P. Chen (1973) and Ramamoorthy & Chandy (1970) assumed a static, or constant, usage frequency. Both models are optimization models that serve to either minimize storage cost or minimize average access time. The focus of Chen's model was on the placement of files using various types of storage devices that minimize cost. In the work of Ramamoorthy & Chandy, their model involved the placement of programs and associated data in a storage hierarchy to minimize access time.

General observations from the study of Simon's Model indicate that it is possible to isolate dynamic usage scenarios prevalent in organizational program usage behavior (Simon, 1968). Simon's model provides us with a powerful mechanism for studying program usage scenarios. Therefore, we adapt the model in a dynamic way by expressing a (alpha--an increasing usage rate) and ? (gamma--a decreasing usage rate) as functional values. This adaptation provides us new insight into the interrelation of program entry and program aging (Simon, 1991).

By incorporating Simon's Model, we have the ability to categorize usage patterns. For example, we can determine which software programs fall into the category of the "trivial many." Research thus far indicates that in many firms at least 80% of programs fall into this category (Willet, 1994). Thus, in most organizations, a very large portion of programs can be eliminated from the group requiring maintenance attention. Additionally in many organizations, there is usually a very small portion of programs whose usage consistently remains in a high usage ranking position. It would also be reasonable to eliminate these programs from the group requiring maintenance since they show little to no decay in usage.

Once the group of programs of concern in the "significant few" is identified, it then becomes valuable to examine the changes in their usage. In order for organizations to have the ability to establish a policy for better management of their storage assets, we extend the research of P. Chen and Ramamoorthy & Chandy. We develop and present the methodology: Dynamic Storage Reallocation with Cost Minimization, a cost-oriented optimization model for the assignment of programs to various levels of storage.

THE OPTIMAL STORAGE ALLOCATION HIERARCHY MODEL

The model developed here is targeted toward the placement of programs in different storage media, but it is also applicable to the placement of data. We have chosen to concentrate on program storage in order to accommodate programs consisting of a variable number of memory blocks that are assigned to a single storage medium. If data files are considered in the storage hierarchy, then one can assume that the data can be divided into equal size blocks, and blocks of the same file may be stored on different devices in the memory hierarchy.

Model Assumptions

1. Let M denote the total number of devices in the storage hierarchy.

2. For each device j, the cost per block is Cj for devices j = 1, 2, ... , M. It is assumed that Cj is a known, constant value. Also, it is proportional to the efficiency with which the program can be retrieved from storage. That is, a program stored in primary storage will have a higher associated cost per block than a program that is stored in some other secondary storage device. Cost figures for the use of primary and secondary storage devices are generally expressed in dollars per megabyte. Ghandeharizadeh, Ierardi and Zimmerman (1994) make reference to costs associated with such storage options.

3. Let L denote the total number of programs, where each is stored in a file consisting of Ni blocks for i = 1, 2, ..., L. Assume that the blocks of a program cannot be separated from the program file on different media. Namely, blocks of a program file cannot be divided between several storage devices.

4. Suppose that decision variable Xij = 1 if program i is assigned to device j, or 0 if program i is not assigned to device j. Thus:

[M.summation over (j=1)] [X.sub.ij] = 1

5. Let fi be the reference frequency for program i in per unit time. Here, the time unit is expressed in terms of cycles, where a cycle is defined as the number of total usages over an observed distribution of frequencies. Thus, the total request rate (?j) for device j is found below. We assume here that a single program is allocated to only one storage device, and any usage frequency profile (fi) will reflect only the usage history since the previous allocation.

[[lambda[.sub.j] = [L.summation over (i=1)] [X.sub.ij] = 1

6. It is assumed that one program is transferred per input/output request. The service time for each device is presumed to be a random variable that varies accordingly for input/output requests due to the electromechanical nature of storage devices. Thus, request service time is assumed to be exponentially distributed with a mean of 1/[[mu].sub.j] for [[mu].sub.1],? [[mu].sub.2],? ..., [[mu].sub.M] > 0, where :j is the service rate of device j. In order to prevent the queue length for requests from growing without bound, it is required that ?j < [[mu].sub.j]. Namely, the overall request rate to a device must be less than the service rate.

7. It is also necessary to define BSj as the maximum allowable number of blocks that can be stored on device j. This is a constant value that is assigned as blocks of storage are added.

Minimum Cost Storage Allocation Model Formulation

From the assumptions stated above, a cost minimization model for the allocation of programs to various levels of storage can be stated as follows:

Minimize Cost

[M.summation over (j=1)] [C.sub.j] [L.summation over (i=1)] [N.sub.i] [X.sub.ij] (1)

Subject To

[M.summation over (j=1)] [X.sub.ij] = 1 for i = 1,...,L (2)

[L.summation over (i=1)] [N.sub.i][X.sub.ij] [less than or equal to] B[S.sub.j] for j = 1,...,M (3)

[L.summation over (i=1)] [f.sub.i][X.sub.ij] [less than or equal to] [[mu].sub.j] for j = 1,...,M (4)

A PROGRAM STORAGE ALLOCATION EXAMPLE

Suppose we have two storage devices. In addition, assume we have five program files, and we want to allocate them between the two devices. Further, assume that the cost of the two storage devices are C1 = $4/block and C2 = $2/block. Also for the example, assume that the usage frequencies of the programs are known. (Note: The assumption of known, or static, frequency is a limitation to storage allocation cost modeling.) Finally, suppose for this example that the service rates for each storage device are [[mu].sub.1] = 30 and [[mu].sub.2] = 20. Table 1 below summarizes the example computer information system where i is the index of programs, Ni is the block size of each program, and fi denotes the usage frequency of program i.

Thus, the example can be modeled as a storage allocation linear programming problem with L = 5 and M = 2.

Capacity constraints for the two storage devices are defined by total block sizes BS1 and BS2. For the example, we let BS1 = 200 and BS2 = 500. Appendix 1 contains the example problem's storage allocation cost formulation. To solve this optimization problem LINDO/PC, Release 6.1, was employed. Initial results are presented in Table 2 below.

The initial results indicate that programs 1 and 5 were allocated to device 1, and programs 2, 3 and 4 were allocated to device 2. Implementation of this algorithm would cause the programs to be moved to their appropriate locations in order to optimize storage cost. Intuitively, one might think that files will always be allocated to the cheaper storage device in order to minimize cost. In this model, program access is constrained by the service time requirements that would ensure that some files are stored on the faster and more expensive storage medium.

At this point we have solved the storage allocation problem using a single usage frequency profile. The contention of this paper is that program usage frequencies are not static over time. There are many occurrences such as changes in technology, changing organizational needs, and new program production that cause program usage behavior to be dynamic. We know that this dynamic behavior can be captured using Simon's Model to incorporate the dynamic parameter values alpha and gamma.

These parameters are described by Simon and Van Wormer (1963) as the entrance rate of new items and the decay, or aging rate, of old items, respectively. Many studies have been performed that demonstrate the interaction between alpha and gamma. It is reasonable, in the context of this study to assume that alpha is increasing, while gamma is decreasing. In our use of alpha and gamma, both parameters are functions of the total number of usages and maximum number of programs.

INCORPORATING SIMON'S MODEL FOR DYNAMIC STORAGE REALLOCATION

With insight gained through the study of dynamic usage frequencies of Simon's Model, we extend the static cost optimization storage allocation model presented earlier. We call this modeling methodology: Dynamic Storage Reallocation with Cost Minimization. The process for implementation is shown in Figure 1. Also, for the current example, we show the effect of using dynamic usage frequencies on the model.

[FIGURE 1 OMITTED]

In the static example, we started out with 5 programs totaling 49 usages (sum of all usage frequencies). Thus, we estimate alpha to be 0.102 (total number of programs/sum of all frequencies). We also assume that alpha is increasing and gamma is decreasing. That is, we are observing the changing usage distribution under the assumption that new programs are "arriving" at a rate of alpha, which in turn causes a "decay" at a rate gamma in the usage of the existing programs. Using Simon's Autoregressive Model with increasing alpha and decreasing gamma, we observe the frequency distribution when alpha is estimated to be 0.20 (Chen, Chong and Tong, 1993). Table 3 below presents frequency data for the initial allocation of programs made according to the original optimal solution. Appendix 2 contains the example problem's storage allocation cost formulation.

Note that in the dynamic example, the increasing value of alpha has caused the addition of 3 new programs. We then observe the changes in allocation to the appropriate storage medium that occur when the model is adjusted for the entry of these new programs in order to minimize cost.

Results of the model indicate an increased objective function value (cost) of $770. This implies that the addition of the 3 new programs increased the cost by $130 (770--640). It is interesting to compare these results to those that would have been incurred if the model did not account for all dynamic changes. If that were the case, the three new programs would have simply been stored on device 1, the more expensive storage medium. We make this assumption since most organizations choose to store new programs on the fastest medium. With the addition of these new programs, and no change made to the allocation of existing programs, the total cost incurred would have been $1000. This cost was computed by adding the cost of the initial allocation to the cost of allocating 3 new programs to device 1 [(3*30*4)+640]. Thus, for the static example, the savings were $1000--$770 = $230.

EXAMPLE CONSIDERING FURTHER REALLOCATION

We now consider a second dynamic example that considers a different value for alpha. In this example we set alpha at 0.40. Our goal here is to show that there will still be a cost improvement as alpha increases and gamma decreases. Again using Simon's Model, a frequency distribution was generated according to these new parameters. Table 4 below summarizes the second system. Appendix 3 contains the example problem's storage allocation cost formulation.

As in the first example problem, the results were computed as if no reallocation had been done since the last run. Without dynamic allocation, the total cost would be estimated at $1730 [(1[2.sup.*][20.sup.*]4)+770]. Optimally solving the second dynamic example problem, we obtain a storage cost of $1450 producing a cost saving of $280.

We note that our results now indicate that the capacity constraints for the service rates have zero slack. Thus, we have reached the maximum number of program usages that can be handled by the simple system.

COST IMPLICATIONS FOR VARYING ALPHA AND GAMMA

The examples presented in the above sections demonstrate that cost savings can be realized with the consideration of dynamic usage frequencies for storage allocation, and present the impact of savings from the effect of sensitive changes to alpha and gamma. In order to perform cost savings sensitivity analyses, simulation programming with increasing alpha rates and decreasing gamma rates was designed and implemented. The combination of an increasing alpha and decreasing gamma represents an intuitive scenario of program usage behavior. Table 5 below summarizes the results. The simulation program was executed at six different intervals with varying degrees of alpha and gamma. For each of these executions, two storage devices were again assumed (M = 2).

The results indicate that total cost savings continued to increase as the values of alpha increased, although the cost savings per program decreased slightly as the total number of programs increased. This is an intuitive result. Additionally, as the total number of programs increases, the limit for total block capacity is decreasing. Once the storage capacity for both devices reaches their limits, there would be no more savings until capacity is expanded.

CONCLUSIONS

We introduced the modeling methodology: Dynamic Storage Reallocation with Cost Minimization, and demonstrated the implications dynamic usage frequencies have on storage allocation. Viewing program storage allocation as a dynamic process will not only affect storage costs, but also affect the placement of programs under some service time constraint.

The methodology proposed in this research is a natural extension of the work by P. Chen and Ramamoorthy & Chandy, though it does have significant differences from those developed by the aforementioned authors. We have not taken into consideration mean response time constraints for individual storage requests, nor have we considered the possibility of allocating blocks of one file to different storage media. Both considerations present opportunities for future research.

The results of this research confirm that improvements in storage allocation costs can be made if files are allocated according to dynamic usage frequencies. In our simplistic example and simulation execution we studied a small sampling of programs for brevity in model formulation. Naturally, a problem of this type that models a complex, real storage allocation situation could grow to become quite large but can be handled by this approach.

APPENDIX 1

Minimize Cost

[4.sup.*][40.sup.*][X.sub.11] + [2.sup.*][40.sup.*][X.sub.12] +
[4.sup.*][20.sup.*][X.sub.21] + [2.sup.*][20.sup.*][X.sub.22] +
[4.sup.*][30.sup.*][X.sub.31] + [2.sup.*][30.sup.*][X.sub.32] +
[4.sup.*][40.sup.*][X.sub.41] + [2.sup.*][40.sup.*][X.sub.42] +
[4.sup.*][75.sup.*][X.sub.51] + [2.sup.*][75.sup.*][X.sub.52]

Subject To

[X.sub.11] + [X.sub.12] = 1
[X.sub.21] + [X.sub.22] = 1
[X.sub.31] + [X.sub.32] = 1
[X.sub.41] + [X.sub.42] = 1
[X.sub.51] + [X.sub.52] = 1
[40.sup.*][X.sub.11] + [20.sup.*][X.sub.21] + [30.sup.*][X.sub.31] +
[40.sup.*][X.sub.41] + [75.sup.*][X.sub.51] [less than or equal to] 200
[40.sup.*][X.sub.12] + [20.sup.*][X.sub.22] + [30.sup.*][X.sub.32] +
[40.sup.*][X.sub.42] + [75.sup.*][X.sub.52] [less than or equal to] 500
[17.sup.*][X.sub.11] + [9.sup.*][X.sub.21] + [5.sup.*][X.sub.31] +
[5.sup.*][X.sub.41] + [13.sup.*][X.sub.51] [less than or equal to] 30
[17.sup.*][X.sub.12] + [9.sup.*][X.sub.22] + [5.sup.*][X.sub.32] +
[5.sup.*][X.sub.42] + [13.sup.*][X.sub.52] [less than or equal to] 20

APPENDIX 2

Minimize Cost

[4.sub.*][40.sup.*][X.sub.11] + [2.sup.*][40.sup.*][X.sub.12] +
[4.sub.*][20.sup.*][X.sub.21] + [2.sup.*][20.sup.*][X.sub.22] +
[4.sup.*][30.sup.*][X.sub.31] + [2.sup.*][30.sup.*][X.sub.32] +
[4.sup.*][40.sup.*][X.sub.41] + [2.sup.*][40.sup.*][X.sub.42] +
[4.sup.*][75.sup.*][X.sub.51] + [2.sup.*][75.sup.*][X.sub.52] +
[4.sup.*][30.sup.*][X.sub.61] + [2.sup.*][30.sup.*][X.sub.62] +
[4.sup.*][30.sup.*][X.sub.71] + [2.sup.*][30.sup.*][X.sub.72] +
[4.sup.*][30.sup.*][X.sub.81] + [2.sup.*][30.sup.*][X.sub.82]

Subject To

[X.sub.11] + [X.sub.12] = 1
[X.sub.21] + [X.sub.22] = 1
[X.sub.31] + [X.sub.32] = 1
[X.sub.41] + [X.sub.42] = 1
[X.sub.51] + [X.sub.52] = 1
[X.sub.61] + [X.sub.62] = 1
[X.sub.71] + [X.sub.72] = 1
[X.sub.81] + [X.sub.82] = 1
[40.sup.*][X.sub.11] + [20.sup.*][X.sub.21] + [30.sup.*][X.sub.31] +
[40.sup.*][X.sub.41] + [75.sup.*][X.sub.51] + [30.sup.*][X.sup.61] +
[30.sup.*][X.sub.71] + [30.sup.*][X.sub.81] [less than or equal to] 200
[40.sup.*][X.sub.12] + [20.sup.*][X.sub.22] + [30.sup.*][X.sub.32] +
[40.sup.*][X.sub.42] + [75.sup.*][X.sub.52] + [30.sup.*][X.sub.62] +
[30.sup.*][X.sub.72] + [30.sup.*][X.sub.82] [less than or equal to] 500
[10.sup.*][X.sub.11] + [6.sup.*][X.sub.21] + [4.sup.*][X.sub.31] +
[3.sup.*][X.sub.41] + [5.sup.*][X.sub.51] + [4.sup.*][X.sub.61] +
[6.sup.*][X.sub.71] + [2.sup.*][X.sub.81] [less than or equal to] 30
[10.sup.*][X.sub.12] + [6.sup.*][X.sub.22] + [4.sup.*][X.sub.32] +
[3.sup.*][X.sub.42] + [5.sup.*][X.sub.52] + [4.sup.*][X.sub.62] +
[6.sup.*][X.sub.72] + [2.sup.*][X.sub.82] [less than or equal to] 20

APPENDIX 3

Minimize Cost

[4.sub.*][40.sup.*][X.sub.11] + [2.sup.*][40.sup.*][X.sub.12] +
[4.sub.*][20.sup.*][X.sub.21] + [2.sup.*][20.sup.*][X.sub.22] +
[4.sup.*][30.sup.*][X.sub.31] + [2.sup.*][30.sup.*][X.sub.32] +
[4.sup.*][40.sup.*][X.sub.41] + [2.sup.*][40.sup.*][X.sub.42] +
[4.sup.*][75.sup.*][X.sub.51] + [2.sup.*][75.sup.*][X.sub.52] +
[4.sup.*][30.sup.*][X.sub.61] + [2.sup.*][30.sup.*][X.sub.62] +
[4.sup.*][30.sup.*][X.sub.71] + [2.sup.*][30.sup.*][X.sub.72] +
[4.sup.*][30.sup.*][X.sub.81] + [2.sup.*][30.sup.*][X.sub.82] +
[4.sup.*][20.sup.*][X.sub.91] + [2.sup.*][20.sup.*][X.sub.92] +
[4.sup.*][20.sup.*][X.sub.101] + [2.sup.*][20.sup.*][X.sub.102] +
[4.sup.*][20.sup.*][X.sub.111] + [2.sup.*][20.sup.*][X.sub.112] +
[4.sup.*][20.sup.*][X.sub.121] + [2.sup.*][20.sup.*][X.sub.122] +
[4.sup.*][20.sup.*][X.sub.131] + [2.sup.*][20.sup.*][X.sub.132] +
[4.sup.*][20.sup.*][X.sub.141] + [2.sup.*][20.sup.*][X.sub.142] +
[4.sup.*][20.sup.*][X.sub.151] + [2.sup.*][20.sup.*][X.sub.152] +
[4.sup.*][20.sup.*][X.sub.161] + [2.sup.*][20.sup.*][X.sub.162] +
[4.sup.*][20.sup.*][X.sub.171] + [2.sup.*][20.sup.*][X.sub.172] +
[4.sup.*][20.sup.*][X.sub.181] + [2.sup.*][20.sup.*][X.sub.182] +
[4.sup.*][20.sup.*][X.sub.191] + [2.sup.*][20.sup.*][X.sub.192] +
[4.sup.*][20.sup.*][X.sub.201] + [2.sup.*][20.sup.*][X.sub.202]

Subject To

[X.sub.11] + [X.sub.12] = 1
[X.sub.21] + [X.sub.22] = 1
[X.sub.31] + [X.sub.32] = 1
[X.sub.41] + [X.sub.42] = 1
[X.sub.51] + [X.sub.52] = 1
[X.sub.61] + [X.sub.62] = 1
[X.sub.71] + [X.sub.72] = 1
[X.sub.81] + [X.sub.82] = 1
[X.sub.91] + [X.sub.92] = 1
[X.sub.101] + [X.sub.102] = 1
[X.sub.111] + [X.sub.112] = 1
[X.sub.121] + [X.sub.122] = 1
[X.sub.131] + [X.sub.132] = 1
[X.sub.141] + [X.sub.142] = 1
[X.sub.151] + [X.sub.152] = 1
[X.sub.161] + [X.sub.162] = 1
[X.sub.171] + [X.sub.172] = 1
[X.sub.181] + [X.sub.182] = 1
[X.sub.191] + [X.sub.192] = 1
[X.sub.201] + [X.sub.202] = 1
[40.sup.*][X.sub.11] + [20.sup.*][X.sub.21] + [30.sup.*][X.sub.31] +
[40.sup.*][X.sub.41] + [75.sup.*][X.sub.51] + [30.sup.*][X.sub.61] +
[30.sup.*][X.sub.71] + [30.sup.*][X.sub.81] + [20.sup.*][X.sub.91] +
[20.sup.*][X.sub.101] + [20.sup.*][X.sub.111] + [20.sup.*][X.sub.121] +
[20.sup.*][X.sub.131] + [20.sup.*][X.sub.141] + [20.sup.*][X.sub.151] +
[20.sup.*][X.sub.161] + [20.sup.*][X.sub.171] + [20.sup.*][X.sub.181] +
[20.sup.*][X.sub.191] + [20.sup.*][X.sub.201] [less than or equal to] 200

[40.sup.*][X.sub.12] + [20.sup.*][X.sub.22] + [30.sup.*][X.sub.32] +
[40.sup.*][X.sub.42] + [75.sup.*][X.sub.52] + [30.sup.*][X.sub.62] +
[30.sup.*][X.sub.72] + [30.sup.*][X.sub.82] + [20.sup.*][X.sub.92] +
[20.sup.*][X.sub.102] + [20.sup.*][X.sub.112] + [20.sup.*][X.sub.122] +
[20.sup.*][X.sub.132] + [20.sup.*][X.sub.142] + [20.sup.*][X.sub.152] +
[20.sup.*][X.sub.162] + [20.sup.*][X.sub.172] + [20.sup.*][X.sub.182] +
[20.sup.*][X.sub.192] + [20.sup.*][X.sub.202] [less than or equal to] 500

[9.sup.*][X.sub.11] + [5.sup.*][X.sub.21] + [4.sup.*][X.sub.31] +
[3.sup.*][X.sub.41] + [3.sup.*][X.sub.51] + [3.sup.*][X.sub.61] +
[5.sup.*][X.sub.71] + [1.sup.*][X.sub.81] + [2.sup.*][X.sub.91] +
[3.sup.*][X.sub.101] + [2.sup.*][X.sub.111] + [2.sup.*][X.sub.121] +
[1.sup.*][X.sub.141] + [1.sup.*][X.sub.151] + [1.sup.*][X.sub.161] +
[1.sup.*][X.sub.171] + [1.sup.*][X.sub.181] + [1.sup.*][X.sub.191] +
[1.sup.*][X.sub.201] [less than or equal to] 30

[9.sup.*][X.sub.12] + [5.sup.*][X.sub.22] + [4.sup.*][X.sub.32] +
[3.sup.*][X.sub.42] + [3.sup.*][X.sub.52] + [3.sup.*][X.sub.62] +
[5.sup.*][X.sub.72] + [1.sup.*][X.sub.82] + [2.eup.*][X.sub.92] +
[3.sup.*][X.sub.10]2 + [2.sup.*][X.sub.112] + [2.sup.*][X.sub.122] +
[1.sup.*][X.sub.142] + [1.sup.*][X.sub.152] + [1.sup.*][X.sub.162] +
[1.sup.*][X.sub.172] + [1.sup.*][X.sub.182] + [1.sup.*][X.sub.192] +
[1.sup.*][X.sub.202] [less than or equal to] 20

REFERENCES

Chen, P. (1973). Optimal File Allocation in Multi-level Storage Systems. AFIPS Conference Proceedings, Vol. 4, 277-282.

Chen, Y. S., P. P. Chong, & M. Y. Tong (1993). Theoretical Foundation of the 80/20 Rule. Scientometrics, 28(2), 183-203.

Ghandeharizadeh, S., D. Ierardi & R. Zimmermann (1994). Management of Space in Hierarchical Storage Systems. USC Department of Computer Science Technical Paper, September 29.

Ramamoorthy, C. & K. Chandy (1970). Optimization of Memory Hierarchies in Multiprogrammed Systems. Journal of the Association for Computing Machinery, 17(3) July, 426-445.

Simon, H. A. (1991). Models of My Life. Basic Books (Harper Collins).

Simon, H. A. (1968). On Judging the Plausibility of Theories, in Logic, Methodology and Philosophy of Sciences, Vol. III. Amsterdam: North-Holland.

Simon, H. A., & T. A. Van Wormer (1963). Some Monte Carlo Estimates of the Yule Distribution. Behavior Science, 1(8), 203-210.

Willet, S. (1994). Running Out of Room? Infoworld, 16(31), 49-52.

Rachelle F. Cope, Southeastern Louisiana University Robert F. Cope III, Southeastern Louisiana University Yvette B. Baldwin, Southeastern Louisiana University

Table 1
Block Sizes and Usage Frequencies for
the Static Storage Allocation Example

i [N.sub.i] [f.sub.i]

1 40 17
2 20 9
3 30 5
4 40 5
5 75 13

Table 2
Storage Resource Allocation Results
Optimal Objective Function Value (Cost) = $640

Variable Value

[X.sub.1] 1
[X.sub.12] 0
[X.sub.21] 0
[X.sub.22] 1
[X.sub.31] 0
[X.sub.32] 1
[X.sub.41] 0
[X.sub.42] 1
[X.sub.51] 1
[X.sub.52] 0

Table 3
Block Sizes and Usage Frequencies for the
Dynamic Storage Allocation Example (a = 0.20)

i [N.sub.i] [f.sub.i]

1 40 10
2 20 6
3 30 4
4 40 3
5 75 5
6 30 4
7 30 6
8 30 2

Table 4
Block Sizes and Usage Frequencies for the Second
Dynamic Storage Allocation Example (a = 0.40)

 i [N.sub.i] [f.sub.i]

 1 40 9
 2 20 5
 3 30 4
 4 40 3
 5 75 3
 6 30 3
 7 30 5
 8 30 1
 9 20 2
10 20 3
11 20 2
12 20 2
13 20 1
14 20 1
15 20 1
16 20 1
17 20 1
18 20 1
19 20 1
20 20 1

Table 5
Summary of Cost Saving Results from Simulation Execution
with varying degrees of a and ?

 Number Optimal
 of Cost
[alpha] [gamma] Programs (Dynamic)

 0.10 0.95 3 $100
 0.41 0.90 6 $180
 0.59 0.76 12 $360
 0.72 0.62 20 $660
 0.84 0.46 28 $1040
 0.95 0.28 37 $1480

 Cost
 Optimal Total Saving
 Cost Cost per
[alpha] (Static) Saving Program

 0.10 $200 $100 $33.33
 0.41 $300 $120 $20
 0.59 $540 $180 $15
 0.72 $1020 $360 $18
 0.84 $1500 $460 $16.43
 0.95 $2040 $640 $15.13