Algebraic approach to time borrowing

This paper is about a novel application of linear algebra to the timing of digital hardware. In particular we describe a rigorous, algorithmic approach to �time borrowing�. Time borrowing is a technique whereby the use of a multiphase clock can allow for a more flexible, efficient use of time. In this approach the system is clocked periodically, but within each clock cycle processes are allowed to interact asynchronously allowing longer processes to be juxtaposed with shorter processes. We show that this problem can be solved completely using linear algebra defined over the max-plus semi-ring, and that the method so obtained conforms with an earlier, heuristic approach to the problem.

[1]  P. Butkovic.  Max-linear Systems: Theory and Algorithms , 2010 .

[2]  Josep Torrellas,et al.  ReCycle:: pipeline adaptation to tolerate process variation , 2007, ISCA '07.

[3]  Jeremy Gunawardena.  Timing Analysis of Digital Circuits and the Theory of Min-Max Functions , 2001 .

[4]  Lynn Conway,et al.  Introduction to VLSI systems , 1978 .

[5]  P. Cameron.  Combinatorics: Topics, Techniques, Algorithms , 1995 .

[6]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[7]  Jian Ruan,et al.  Latency Estimation of the Asynchronous Pipeline Using the Max-Plus Algebra , 2007, International Conference on Computational Science.

[8]  Baris Taskin,et al.  Performance optimization of single-phase level-sensitive circuits using time borrowing and non-zero clock skew , 2002, TAU '02.

[9]  T. G. Szymanski,et al.  Verifying clock schedules , 1992, 1992 IEEE/ACM International Conference on Computer-Aided Design.

[10]  Jason Helge Anderson,et al.  Latch-Based Performance Optimization for FPGAs , 2011, 2011 21st International Conference on Field Programmable Logic and Applications.

[11]  Chita R. Das,et al.  A case for dynamic frequency tuning in on-chip networks , 2009, 2009 42nd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO).

[12]  Cheng-Kok Koh,et al.  Performance analysis of latency-insensitive systems , 2006, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[13]  Peter Butkovi,et al.  Introduction to max-algebra , 2008 .

[14]  Richard M. Karp,et al.  A characterization of the minimum cycle mean in a digraph , 1978, Discret. Math..

[15]  Wolfgang Reisig.  Petri Nets: An Introduction , 1985, EATCS Monographs on Theoretical Computer Science.