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# 数学代写|随机过程Stochastic Porcesses代考|MA53200 Case study: Hardware availability through CTMCs

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## 数学代写|随机过程代写Stochastic Porcesses代考|Case study: Hardware availability through CTMCs

In recent years, there has been increasing interest in reliability, availability, and maintainability (RAM) analyses of hardware (HW) systems and, in particular, safety critical systems. Sometimes such systems can be modeled using CTMCs, which, in this context, describe stochastic processes which evolve through a discrete set of states, some of which correspond to ON configurations and the rest to OFF configurations. Transition from an ON to an OFF state entails a system failure, whereas a transition from an OFF to an ON system implies a repair. Here, we shall emphasize availability, which is a key performance parameter for information technology systems. Indeed, there are many hardware configurations aimed at attaining very high system availability, for example, $99.999 \%$ of time, through transfer of workload when one, or more, system components fail, or intermediate failure states with automated recovery; see Kim et al. (2005) for details. Thus, we are concerned with hardware systems, which we assume can be modeled through a CTMC. We shall consider that states ${1,2, \ldots, l}$ correspond to operational (ON) configurations, whereas states ${l+1, \ldots, K}$ correspond to OFF configurations.

A classical approach to availability estimation of CTMC HW systems would calculate maximum likelihood estimates for the CTMC parameters and then compute the equilibrium distribution given these, and finally, estimate the long-term fraction of time that the system remains in ON configurations. A shortfall of this approach is that it does not account for parameter uncertainty, whereas the fully Bayesian framework we adopt here automatically incorporates this uncertainty. Also, both short-term and long-term forecasting can be carried out.

Initially, we shall consider steady-state prediction of the system. In this case, the availability is the sum of the equilibrium probabilities for the $\mathrm{ON}$ states, conditional on the rates and transition probabilities, $\boldsymbol{v}, \mathbf{P}$, that is
$$A\left|v, P=\sum_{i=1}^l \pi_i\right| \boldsymbol{v}, \mathbf{P} .$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Semi-Markovian processes

In this section, we analyze semi-Markovian process (SMP) models. These generalize CTMCs by assuming that the times between transitions are not necessarily exponential. Formally, a semi-Markovian process is defined as follows. Let $\left{X_t\right}_{t \in T}$ be a continuous time stochastic process which evolves within a finite state space with states $E={1,2, \ldots, K}$. When the process enters a state $i$, it remains there for a random time $T_i$, with parameters $\boldsymbol{v}_i$, which is positive with probability 1 . Let $f_i\left(. \mid \boldsymbol{v}_i\right)$ and $F_i\left(. \mid \boldsymbol{v}_i\right)$ be the density and distribution functions of $T_i$, respectively, and define $\mu_i=E\left[T_i \mid \boldsymbol{v}_i\right]$. When leaving state $i$, we assume, as earlier, that the process moves to state $j$ with probability $p_{i j}$, with $\sum_j p_{i j}=1, \forall i$, and $p_{i i}=0$. As for CTMCs, the transition probability matrix, $\mathbf{P}=\left(p_{i j}\right)$, defines an embedded DTMC.

Thus, the parameters for the SMP will be $\mathbf{P}=\left(p_{i j}\right)$ and $v=\left(v_i\right)$. Given that both the states at transition and the times between transitions are observed, inference about P follows the same procedure as before. We shall assume that we have available the posterior distribution of $v$, possibly through a sample.

## 数学代写|随机过程代写Stochastic Porcesses代考|Case study: Hardware availability through CTMCs

$$A\left|v, P=\sum_{i=1}^l \pi_i\right| \boldsymbol{v}, \mathbf{P} .$$

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