Markov processes in APL

ABSTRACT

Using examples from two areas of application, ion channel models and systems reliability, we illustrate the power of APL as an executable notation for the mathematics of finite-state Markov processes. We stress, in particular, the algebra of partitioned matrices and second-generation features of DYALOG APL which allow a simple natural way of creating certain structured matrices.

REFERENCES

• 1: D.R. Cox and H.D. Miller, The theory of stochastic processes, Methuen, 1965.
• 2: S. Karlin and H.M. Taylor, A first course in stochastic processes (2nd edn.), Academic Press, 1975.
• 3: C. Moler and C. van Loan, 'Nineteen dubious ways to compute the exponential of a matrix', SIAM Reviews, vol. 20 no. 4, 1978, 801-836.
• 4: A. G. Hawkes, 'Complex numbers in APL', Vector, vol. 1, 1984, 107-120,
• 5: A.G. Hawkes and AM Sykes, *Equilibrium distributions of finite-state Markov processes', IEEE Trans. Reliability, in press.
• 6: B.B. Fawzi and A.G. Hawkes, 'Availability of Markov systems with spares and repair facilities*, Proceedings of Reliability '89, Brighton, U.K., 1989, 2B/4/1-39.
• 7: D. Colquhoun and A.G. Hawkes, 'On the stochastic properties of bursts of single ion channel openings and of clusters of bursts', Phil. Trans. R. Sot. Lund. B 300, 1982, l-59.
• 8: D. Colquhoun and A.G. Hawkes, *A note on correlations in single ion channels', Proc. R. Sot. Lond. B 230, 1987, 15-52.
• 9: D. Colquhoun, and A.G. Hawkes, 'Stochastic properties of ion channel openings and bursts in a membrane patch that contains two channels: Evidence concerning the presence of two channels when a record containing only single openings is observed* University of Wales, Swansea working paper to appear in Proc. R. sot. Lond., 1990.
• 10: D.C. Ogden, S.A. Siegelbaum and D. Colquhoun, 'Block of acetylcholine activated ion channels by an uncharged anaesthetic' Nature, 289, 1981, 596-598.

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Probabilistic algorithms (including Monte Carlo)

General Terms:
Algorithms, Languages, Reliability, Theory