Computation of bounds for transient measures of large rewarded Markov models using regenerative randomization

Abstract

In this paper we generalize a method (called regenerative randomization) for the transient solution of continuous time Markov models. The generalized method allows to compute two transient measures (the expected transient reward rate and the expected averaged reward rate) for rewarded continuous time Markov models with a structure covering bounding models which are useful when a complete, exact model has unmanageable size. The method has the same good properties as the well-known (standard) randomization method: numerical stability, well-controlled computation error, and ability to specify the computation error in advance, and, for large enough models and long enough times, is signiÿcantly faster than the standard randomization method. The method requires the selection of a regenerative state and its performance depends on that selection. For a class of models, class C , including typical failure=repair models with exponential failure and repair time distributions and repair in every state with failed components, a natural selection for the regenerative state exists, and results are available assessing approximately the performance of the method for that natural selection in terms of "visible" model characteristics. Those results can be used to anticipate when the method can be expected to be signiÿcantly faster than standard randomization for models in that class. The potentially superior e ciency of the regenerative randomization method compared to standard randomization for models not in class C is illustrated using a large performability model of a fault-tolerant multiprocessor system. Scope and purposeRewarded continuous time Markov models are widely used for performance, dependability and performability analysis of computer and telecommunication systems. Realistic modeling of such systems usually yields Markov models whose size exceeds the available memory resources. An approach to deal with the "largeness" problem is the use of bounding Markov models. However, even those bounding models can have very * Tel.: +34-3-4016652; fax: +34-3-4017785.E-mail address: carrasco@eel.upc.es (J.A. Carrasco). so that it can be used for the computation of bounds for the expected transient reward rate and the expected averaged reward rate transient measures with general reward rate structures including arbitrary non-negative reward rates associated with the states of the model. Examples of such measures are the unavailability of a fault-tolerant system at time t and the expected interval unavailability of a fault-tolerant system at time t. The method has the same good properties as the well-known standard randomization (also called uniformization) method: numerical stability, well-controlled computation error, and ability to specify the computation error in advance, and, for large models and long mission times, can be signiÿcantly faster than that method. Furthermore, for a class of models, class C , including typical failure=repair models with exponential failure and repair time distributions and repair in every s...

A Generalized Method for the Transient Analysis of Markov Models of Fault-Tolerant Systems with Deferred Repair

“…In that case RR compares worse with SR than it did in Case 1. The reason is that when the initial probability distribution is not concentrated in the regenerative state (the state without failed components), the truncated transformed model built in RR is larger than when that initial probability distribution is concentrated in the regenerative state (Carrasco, 2003). The performance of RQD is, however, very similar to the performance of that method in Case 1.…”

“…Condition C10 can be easily circumvented in practice by adding, in case r j = 0 for all j ∈ E , a tiny transition rate ≤ 10 −10 / 2r max t max from r to some state in E , where is the allowed error, r max = max i∈ r i , and t max is the largest time at which the measure has to be computed, introducing an error ≤ 10 −10 in both ETRR t and EARR t , t ≤ t max (see Carrasco, 2003). Also, if X has a single recurrent class of states C ⊂ S, by conditions C5 and C10, C ≥ 2, since C = 1 would imply through condition C5 that r would be absorbing, in contradiction with condition C10.…”

“…and increasing values of m. Efficient and numerically stable procedures for computing S m , S m , and S m are described in Carrasco (2002b) and Carrasco (2003). Since S m = S m + 1 , an efficient and numerically stable procedure for computing S m can be obtained easily by adapting the procedure for computing S m .…”

“…Steady-state detection is useful for models which reach their steady-state before the largest time at which the measure has to be computed. Another recently proposed randomization-based method is regenerative randomization (Carrasco, 2002b(Carrasco, , 2003. That method covers rewarded CTMC models X with finite state space = S ∪ f 1 f 2 f A , A ≥ 0, satisfying some conditions.…”

Solving large interval availability models using a model transformation approach

“…The CTMC V has, in most cases, an inÿnite state space; however, it will be shown in the next section how its state space can be truncated to obtain a CTMC with ÿnite state space which has with some arbitrarily small error the same interval availability distribution as X . The transformation follows ideas similar to those used in the regenerative randomization method [13,14]. The basic idea to perform the transformation is to characterize the behavior of X from S up to state r or, if existent, the absorbing state f, and from r until next hit of r or, if existent, the absorbing state f, while keeping track of the amount of time spent in U S .…”