Footmark: A New Formulation for Working Set Statistics

  • Liang YuanEmail author
  • Wesley Smith
  • Sicong Fan
  • Zixu Chen
  • Chen Ding
  • Yunquan Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11882)


The working set (WS) model pioneered by Denning and others is the foundation for analyzing memory performance and optimizing memory management. An important measure is the average working set size (WSS). In 1968, Denning derived a recursive formula to compute the average WSS. The Denning recursion was originally derived for infinitely long program executions and later adapted to use on finite length traces. All previous adaptations, however, have had to modify the Denning recursion for boundary correction.

This paper presents footmark, which redefines average WSS for finite length traces. It has three benefits. By definition, footmark satisfies a new type of symmetry. Mathematically, the paper gives four equivalent formulas for computing footmark including one that is identical to the Denning recursion. The mathematical simplicity is beneficial in both formal and practical analysis of working sets. Based on the new formulas, the paper proves a previously unknown equivalence relation between two working set definitions. Finally, the paper evaluates three WSS definitions using six test programs from the SPEC 2017 benchmark suite.


Locality Footprint Working set 



The authors wish to thank Peter Denning for the feedback and suggestions on the presentation of the paper’s contributions and William Wilson for help on the use of the loca tool. The funding was provided in part by National Key R&D Program of China (2016YFB0200803) and NFSC (61432018, 61402441, 61521092, 61502450, 61602443), by the National Science Foundation of the United States (Contract No. CCF-1717877 and CCF-1629376), by an IBM CAS Faculty Fellowship, and by Guangdong Province Key Laboratory of Popular High Performance Computers (2017B030314073).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Liang Yuan
    • 1
    Email author
  • Wesley Smith
    • 2
  • Sicong Fan
    • 3
  • Zixu Chen
    • 3
  • Chen Ding
    • 3
  • Yunquan Zhang
    • 1
  1. 1.SKL of Computer Architecture, ICT, CASBeijingChina
  2. 2.University of EdinburghEdinburghUK
  3. 3.University of RochesterRochesterUSA

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