LRU Stack Processing

Stack processing, and in particular stack processing for the least recently used replacement algorithms, may present computational problems when it is applied to a sequence of page references with many different pages. This paper describes a new technique for LRU stack processing that permits efficient processing of these sequences. An analysis of the algorithm and a comparison of its running times with those of the conventional stack processing algorithms are presented. Finally we discuss a multipass implementation, which was found necessary to process trace data from a large data base system. Introduction Storage hierarchy evaluation is often accomplished by simulating the hierarchy under loads determined by “representative” address traces. Stack processing as proposed by Mattson, Gecsei, Slutz, and Traiger [ l ] allows efficient evaluation of multilevel hierarchies for a class of replacement algorithms called stack algorithms. Of these algorithms, least recently used (LRU) is the most extensively simulated. The original LRU stack processing algorithm proposed by Mattson et al. calculates for one page size a histogram of stack distances, which determines the frequency of access to each level of a multilevel linear hierarchy for any set of level capacities. The method involves the conversion of each address to a page reference and the maintaining of a list of pages called an LRU stack, wherein the pages are in order of the most recent reference. For each reference the stack distance, the position in the stack of the current referenced page, is obtained. To maintain the LRU stack and to obtain the stack distance for each reference, Mattson et al. proposed a concurrent search and update from the top down. The current page is placed on top of the stack, and each page in the stack is down-shifted by one until the current page is encountered. That position in the stack is recorded as the stack distance. If the current page is not found, i.e., if it has not occurred before, then downshifting proceeds to the bottom of the stack and a stack distance of infinity is recorded. The number of tests for a match with the current page is equal to the stack distance or, for a new page, to the number of distinct pages encountered so far. Thus, stackprocessing a trace that has a large number of distinct pages or a large average stack distance may require exJULY 1975 cessive computing time. Traiger and Slutz [ 2 ] showed that for a little additional overhead this method could also produce stack distances for page sizes that are successive multiples of the basic page size. However, for some data base reference traces and some program address traces the method was found not to be feasible for the page sizes of interest. In this paper we describe a new algorithm for LRU stack processing; this algorithm is much more efficient for the analysis of trace data for a single page size when the number of pages and the average stack distance are large, but separate page sizes require essentially separate calculations. Second, we present an analysis of the algorithm and a comparison of running times. Finally, we discuss a multipass implementation of the algorithm, which we have found necessary in order to efficiently process trace data from a large data base system. New LRU stack processing algorithm The purpose of the algorithm described here is to determine the stack distance for each reference. Rather than regarding the stack distance as the position in an LRU stack, we observe that the distance is equal to one plus the number of distinct pages that have been referenced since the current page was last referenced. If we denote the page referenced at time t by xf and the stack distance by d,, then we have dl = 1 + c(xp+l , . . ., X t l ) , where p = max i < t such that xi = xt , i.e., p is the position of the most recent past reference to page xt and C ( S ) is the number of distinct pages in S. 353 LRU STACK PROCESSING