Adaptive Transient Solver

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Adaptive transient solver

The adaptive transient solver (ats) solves for instant-of-time variables with t < \infty using randomization (also known as uniformization). It calculates the mean and variance of each performability variable at the time points specified in the reward model. The method used by ats is based on the same method described above for the transient solver, trs. However, ars is more efficient than trs for stiff models in which there are large (orders of magnitude) differences in the rates of actions. This method works by dividing the computation into time domains of slow and fast rates and adapting the uniformization rates to the time domains. Initially, “adaptive uniformization”[1] is used until a “switching time.” After that time, standard uniformization is used. In effect, this method attempts to reduce the number of iterations needed to compute the solution. The means and variances are given in textual format in an output file. <xr id="fig:atsSolver" /> shows the editor for the adaptive transient solver and its available options and adjustable parameters.

<figure id="fig:atsSolver">


<xr id="fig:atsSolver" nolink />: Adaptive Transient Solver ats and available options and parameters.

The options are as follows:

  • The Verbosity (n) sets a trace level of intermediate output. The default is no intermediate output. If n > 0, then an intermediate statement is printed after computation of every n columns of the power transition matrix.
  • The Fraction of (Maximum) AU Rate is a floating point value between (0, 1.0) for determining the range of rates to use in the adaptive uniformization algorithm. The default value is 0.9.

The output file contains the means and variances of the performability variables. It also contains the following information:

  • The rate of the Poisson process used to do the uniformization.
  • The number of time points.
  • For each time point, the error due to adaptive uniformization, the number of iterations, and the total error.

Pitfalls and Hints

  • Because this solver is somewhat similar to trs, the pitfalls and hints listed for that solver apply to this solver also.
  • Additionally, the transient time points of interest should be short relative to the steady-state time; otherwise, this solver will be inefficient relative to the steady-state solvers or the transient solver trs due to overheads in the adaptive uniformization computation. For example, in the failure-repair dependability model mentioned earlier (see Section 4.3.7), the time points of interest should be on the order of the failure times.
  • ATS combines the adaptive uniformization (AU) and standard uniformization (SU) algorithms. The idea is that initially AU can be used to make big jumps through activities having low rates to speed up the computation. Eventually, the AU rates converge to the SU rate. Computing AU rates as they converge toward the SU is relatively more expensive as compared to the benefits derived from using AU. By using the the parameter Fraction of (Maximum) AU Rate, you can control the range of AU rates that AU will consider. That is, if ATS takes longer to solve a model than TRS, you may want to decrease this parameter in order to lower the overhead of computing the AU rates.


  1. A. P. A. van Moorsel and W. H. Sanders. Adaptive Uniformization. ORSA Communications in Statistics: Stochastic Models, 10(3):6199–648, August 1994.