On Optimization of Storage Hierarchies

A simple model of the storage hierarchies is formulated with the assumptions that the effect of the storage management strategy is characterized by the hit ratio fqnction. The hit ratio function and the device technology-cost function are assumed to be representable by power functions (or piece-wise power functions). The optimization of this model is a geometric programming problem. An explicit formula for the minimum hierarchy access time is derived; the cqpacity and technology of each storage level are determined. The opfimal number of storage levels in a hierarchy is shown to be directly proportional to the logarithm of the systems capacity with the constant of propottionality dependent upon the technolagy and hit ratio characteristics. The optimal cost ratio of adjacent storage levels is constant, as are the ratios of the device access'times and storage capacities of the adjacent levels. An illustration of the effect of overhead cost and level-dependent cost, such as the cost per "box" and coq for managing memory faults is given and several generalizations are presented. . . Introduction The general trend in the developlrent of large computer systems is toward increasing the use of storage hierarchies. 'A linear storage hierarchy model consists of n levels, M,, M,; . ., M,, connected in cascade as shown in Fig. 1. The convention is that thg, higher is the level the lower is its index. Generally, thq higher is the level, the faster is its speed, the higher is iis cost per byte, and the smaller is its capacity. Information transfers are between adjacent levels and are entirely controlled by the activity in the first level M,. The rules of operation are 1. Whenever'a page is stored in-level M i , there is a copy of it in each of the lower levels, Mi+,; . ., M,. 2. Whenever a referenced page is not found in M , , a reguest for it is sent to the successive lower levels until it is found in the say Mi level. 3. Whenever M ; is full and a new page is to be brought in from Mi+,, a replacement policy, usually the Least Recently Used (LRU) policy, i? invoked to select a page to be deleted from M i (since there is already a copy'in Mi+,, there is no need to move the displaced page into Mt+, ) [ 11. The principal advantage of this storage organization is that a program's Working Set accumulates rapidly in the fastest level M,, thus, accesses are completed at nearly the speed of M , , but the total cost of the storage system approaches that of the lowest level. A second advantage is that the mechanism can be readily implemented, requiring very little operating system intervention [ 11. The most notable examples are the cache memory [ 2 ] on the IBM System 360 Model 85 and Model 195. These systems use three levels ( n = 3 ) ; a seven-level system is illustrated in a book by Lorin [ 31. Several' papers [4-81 describe some of the techniques used for cost-performance evaluation of storage hierarchies.'These papers are concerned with storage hierarchies of two orthree levels. Typically, their algorithms evaluate, for a given hierarchy configuration (given number. of storage levels, device characteristics, etc. ), its cost-performance in terms of the total system cost per memory access for different capacities and page sizes at each level, and select the configuration with the lowest cost per access. These studies and numerical results have prompted some fundamental questions: How is the system performance affected by the cost and the required capacity? What is the minimum hierarchy access time? How'should the cost be allocated to each storage level? What are the optimum Capacity and technology of each level? What is the optimum number of the levels in the hierarchy? To provide some answers to these questions, numerical computations .and simulations, important as they are, are not sufficient: They must be supplemented by analysis, and the functional relations among key system parameters are called for. Unfortunately, hierarchical storage systems are complex and difficult for mathematical analysis, and few theoretical results of general nature are available. If one starts with an all-inclusive model of the system, it is doubtful that analytic solutions can be IBM J . RES. DEVELOP, achieved. We therefore begin with bare essentials and formulate the problem in a mathematically tractable yet, hopefully practical, and'meaningful way. A simple model is formulated with the following assumptions: 1) The effect of the storage management strategy is characterized by the memory hit ratio function. 2) The device technology is specified by the device access time and the cost per unit of storage (byte or other unit). 3 ) The hit ratio function and the device technologyc&t function -are representable by power functions (or piece-wise power functions). Under these assumptions the optimization of such a system becomes a geometric programming problem. An explicit formula for the minimum hierarchy access time ,[ Eqs. (3 1 ) and (32) ] is obtained as a function of the hierarchy cost and the required capacity. The capacity and device technology are determined for each storage lev$l [Eqs. (43), (44), (47) and (48)]. Theoptimalcost ratio of adajacent levels is constant (39) and, for an optimal configuration the ratios between device access tirqes and of adajacent levels, as well as the ratios between their capacities, are also constant. [Eqs. (45) and (4611. The optimal number of storage levels in a hierarchy is shown to be direcdy proportional to the logarithm of the system's capacity, with the constant of proportionality dependent upon the technology and hit ratio charaFteristic powers [ Eqs. (57 ) 'and (60 ) ]. Model and assumptiqns The hierarchical 'storage system under consideration is shown in Fig. 1 with the operational rules described in the Introduction. 'It is a.linear hierarchy of n levels, M,, M,, . . ., M,. The two major factors that determine the performance of a storage hierarchy are the storage device characteristics of each'level and the storage management strategy. Since copies.of the information stored in the higher levels are found in all lower levels, the system storage capacity, namely the maximum amount of information that can be stored in the'systmn, is equal to the capacity of. the lowest level. In our analysis the following assumptions are made: A 1 . Each storage level, Mi, is characterized by its access time ti and capacity Ci. A2. The storage management strategy is completely characterized by the success function or hit ratio H, which is a function of the storage capacity C . A3. The device technologies are characterized by their cost function b ( t ) , which represents either purchase or rental, per storage unit (say byte) of the technology giving the access time t. The cost function is always a monotonic decreasing function .in access time; a sample curve is shown in Fig. 2 . MAY 1974 CTI Processor P Storage level Device Storage access time capacity Cost per byte