Transient Solver

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Transient solver

The transient solver (trs) 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 for the time points defined for the reward variable within the reward model. Uniformization is based on the idea of subordinating a Markov chain to a Poisson process. It is computationally efficient, preserves matrix sparsity, and solves to user-specified tolerances. Furthermore, computation of state probabilities in the uniformized Markov chain and computation of Poisson probabilities can both be done in a numerically stable manner. The means and variances are given in textual format in an output file. <xr id="fig:trsSolver" /> shows the editor for the transient solver and its available options and adjustable parameters.

<figure id="fig:trsSolver">


<xr id="fig:trsSolver" nolink />: Transient Solver trs 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 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 states with positive rewards.
  • The number of time points.
  • For each time point, the left truncation point, number of iterations and error.

Pitfalls and Hints

  • The computation time of trs is determined primarily by the number of iterations. A simple way to estimate the number of iterations is to multiply the required time instant by the rate of the Poisson process. The rate of the Poisson process is equal to the highest outgoing rate over all the states of the Markov process (the outgoing rate of a state is given by the sum of all the exponential rates of transitions out of the state). As a consequence, the time complexity of the algorithm increases linearly with t.
  • From the previous item, it follows that trs will be more time-consuming for models with high rates of the exponential distribution relative to the time points of interest. A class of models that has that kind of stiffness can be found in reliability evaluation if repairs occur relatively fast and failures occur rarely. In such models, the rate of the Poisson process is dictated by the fast repairs, but the time points of interest are often in the scale of the times between failures. For example, for a system in which component failures occur on the average once every ten days and repairs take on the order of an hour, one’s interest will typically be in the transient behavior over relatively long periods (e.g., the probability the system is up at the end of the year).
  • For large values of t, the result becomes identical to the steady-state result, and will not change further if t increases. Use the iss solver to determine if this is occurring.
  • At time t = 0, the SAN model is in the initial marking with probability 1. In Möbius it is not possible to specify another initial distribution. To change the state at t = 0, alter the initial marking of places in the SANs.