Stochastic Processes

Introduction

A number of stochastic processes are available to determine the state probabilities of an std:

AbstractStochasticProcess
  |- AbstractMarkovProcess
      |- SteadyStateProcess
      |- MarkovProcess
  |- AbstractSemiMarkovProcess
      |- SemiMarkovProcess

Solving a stochastic process may be accomplished through excecuting: solve(std, cls; tsim, dt, tol), where cls denotes the class of stochastic process you want to use.

Steady State Process

A steady state process determines the state-state probability of the state space associated with a time-homogeneous Markov process/chain, i.e., where t→∞.

https://www.maplesoft.com/support/help/maple/view.aspx?path=examples/SteadyStateMarkovChain

Markov Process

A Markov process is described by a random variable $X_t$, where $t$ denotes the calendar time. The possible values of $X_t$ are represented by the discrete state-space 𝓢 of the state transition diagram std.

A Markov process respects the Markov property, which means it respects

\[ℙ(X_t ∈ 𝓢 | 𝓕_s) = ℙ(X_t ∈ 𝓢 | X_s), ∀ s,t ∈ 𝕀: s < t,\]

where 𝓕$_s$ represents a filtration of a probability space (Ω,𝓕,ℙ) and 𝕀 a totally ordered index set. A Markov process is described by Kolmogorov equations, more specifically the Kolmogorov forward equations:

\[ δp_{ij}(s;t)/δt = ∑_k p_{ik}(s;t) ⋅ A_{kj}(t), ∀ i,j ∈ 𝓢, s,t ∈ 𝕀: s < t,\]

where $A(t)$ represents the transition matrix, syn., generator matrix. The latter may be translated into an initial value problem for finding the state probabilities, given transition rates ρ$_{ij}$(t) and initial values δ$_{i}$:

\[dp_i(t)/dt = - ∑_j ρ_{ij}(t)p_i(t) + ∑_j ρ_{ji}(t)p_j(t), ∀ i ∈ 𝓢.\]

Semi-Markov Process

A semi-Markov process is described by a random variable $X_t$, where $t$ denotes the calendar time. The possible values of $X_t$ are represented by the discrete state-space 𝓢 of the state transition diagram std.

A semi-Markov process...

Van Acker Process

• T. Van Acker, and D. Van Hertem (2018). Stochastic Process for the Availability Assessment of Single-Feeder Industrial Energy System Sections. IEEE Trans. on Rel., 67(4), 1459-1467.

An Van Acker process is described by a random variable $X_{t,φ}$, where $t$ denotes the calendar time. The possible values of $X_{t,φ}$ are represented by the semi-continuous state-space $𝓢$ of the state transition diagram std and $φ_s$ denotes its state sojourn time.

An Van Acker process is described by a single PDDE with non-local boundary condition for the normal operation state $n ∈ 𝓝$, from which all state probabilies of the other states may be derived.

\[\begin{aligned} \frac{∂p_{n}}{∂φ_{n}} + \frac{∂p_{n}}{∂t} =& - \sum_{f ∈ 𝓕} λ_{f}(t,φ_{n}) p_{n}(t,φ_{n}) \\ ~& + \sum_{f ∈ 𝓕^{\text{min}}} (𝖿_{f} * λ_{f}p_{n})(t,φ_{n}) \\ p_{n}(t,0) =& \sum_{f ∈ 𝓕^{\text{per}}} \int_{0}^{∞} (𝖿_{f} * λ_{f}p_{n})(t,φ_{n}) 𝖽φ_{f} \end{aligned}\]

where, $𝓕$, $𝓕^{\text{min}}$ and $𝓕^{\text{per}}$ denote the overall failure set and the failure sets respecively involving minimal and perfect maintenance. The parameters $λ_{f}$ and $𝖿_{f}$ respectively denote the failure rate and restoration pdf associated with a specific failure $f ∈ 𝓕$. The restoration pdf is the sum of the convolutions of transition pdf's along all paths involving a specific failure $f ∈ 𝓕$ but excluding that failure's transition pdf.

!!! The normal operation state is selected by setting that state's initial value :init to 1.0.

The problem structure permits descretization of the solution space into cohorts $a ∈ 𝓐: t = φ_{n} + t_{a}$, where $t_{a}$ is the time for which a cohort $a$ has a zero sojourn time $φ_{n}$; translating the PDDE into an non-homogeneous first order ODE.

\[\begin{aligned} \frac{𝖽p_{a,n}}{𝖽φ_{n}} =& - \sum_{f ∈ 𝓕} λ_{a,f}(φ_{n}) p_{a,n}(φ_{n}) \\ ~& + \sum_{f ∈ 𝓕^{\text{min}}} \sum_{x ∈ 𝓧} \bar{𝖿}_{f,x} λ_{a-x,f}(φ_{n}) p_{a-x,n}(φ_{n}) \\ p_{a,n}(0) =& \sum_{f ∈ 𝓕^{\text{per}}} \sum_{x ∈ 𝓧} \bar{𝖿}_{f,x} λ_{a-x,f}(φ_{f}) p_{a-x,n}(φ_{f}) 𝖽φ_{f} \end{aligned}\]

Using the solution for the normal operation state probability, all other state probabilities $p_{p}(t,\varphi_{p}),~p ∈ 𝓟$ may be determined.

\[\begin{aligned} p_{p}(t,\varphi_{p}) =& \sum_{c ∈ 𝓒_{p}: f ∈ c} (𝖿^{\text{pre}}_{c,p}*p_{f})(t-φ_{p}) R_{c,p}(φ_{p}) \\ p_{f}(t) =& \int_{0}^{\infty} λ_{f}(t,φ_{n}) p_{n}(t,\varphi_{n})𝖽φ_{n} \end{aligned}\]

where $𝓒_{p}$ is the set of simple cycles going through the state $p$ and the pdf $𝗳^{\text{pre}}_{c,p}$ convolutions of the transition pdf's of a cycle starting from the normal operation state $n$ up to state $p$ excluding the failure transition.