Equilibrium Analysis of Markov Regenerative Processes

Abstract

We present a solution to compute equilibrium probability density functions (PDFs) for the continuous component of the state in Markov regenerative processes, a class of non-Markovian processes. Equilibrium PDFs are derived as closed-form analytical expressions by applying the Key Renewal Theorem to stochastic state classes computed between regenerations. The solution, evaluated experimentally through the development of an analysis tool, provides the basis to analyze system properties from the equilibrium.

A Stochastic Time Petri Nets

STPNs are a formal model of concurrent timed systems where: transitions (depicted as vertical bars) represent activities; places (depicted as circles) represent discrete components of the logical state, with values encoded by a number of tokens (depicted as dots); directed arcs from input places to transitions and from transitions to output places represent token moves triggered by the firing of transitions. A transition is enabled when all its input places contain at least one token; its firing removes a token from each input place and adds a token to each output place. The time from the enabling to the firing of a transition is a random variable, and the choice between transitions with equal time to fire is solved by a random switch determined by transition weights. Moreover, STPNs can: (1) restrict the enabling of a transition using general constraints on token counts (called enabling functions); (2) execute additional updates of token counts after a transition firing (specified by update functions); (3) restart selected transitions after a firing (using reset sets); (4) impose priorities among immediate or deterministic transitions.

Definition 2

(Syntax). An STPN is a tuple $\langle P,T,A^-,A^+,B,U,R,EFT,LFT,$ $F,W,Z\rangle$ where: P and T are disjoint sets of places and transitions, respectively; $A^-\subseteq P\times T$ and $A^+\subseteq T\times P$ are precondition and post-condition relations, respectively; B, U, and R associate each transition $t\in T$ with an enabling function $B(t):M \rightarrow \{\textsc {true},\textsc {false}\}$, an update function $U(t):M \rightarrow M$, and a reset set $R(t)\subseteq T$, respectively, where M is the set of reachable markings $m:P\rightarrow \mathbb {N}$; EFT and LFT associate each transition $t \in T$ with an earliest firing time $EFT(t) \in \mathbb {Q}_{\geqslant 0}$ and a latest firing time $LFT(t) \in \mathbb {Q}_{\geqslant 0}\cup \{\infty \}$ such that $EFT(t) \leqslant LFT(t)$; $F$, $W$, and Z associate each transition $t \in T$ with a Cumulative Distribution Function (CDF) $F_t$ for its duration $\tau (t) \in [EFT(t), LFT(t)]$ (i.e., $F_t(x) = P\{\tau (t) \leqslant x\}$, with $F_{t}(x) = 0$ for $x<EFT(t)$, $F_{t}(x) = 1$ for $x>LFT(t)$), a weight $W(t)\in \mathbb {R}_{> 0}$, and a priority $Z(t)\in \mathbb {N}$, respectively.

A place p is said to be an input or output place for a transition t if $(p,t) \in A^-$ or $(t,p) \in A^+$, respectively. Following the usual terminology of stochastic Petri nets, a transition t is called immediate (IMM) if $EFT(t)=LFT(t)=0$ and timed otherwise; a timed transition is called exponential (EXP) if $F_{t}(x) = 1-\exp (-\lambda x)$ for some rate $\lambda \in {\mathbb R}_{>0}$, or general (GEN) if its time to fire has a non-exponential distribution; as a special case, a GEN transition t is deterministic (DET) if $EFT(t)=LFT(t)>0$. For each transition t with $EFT(t)<LFT(t)$, we assume that $F_{t}$ can be expressed as the integral function of a probability density function (PDF) $f_{t}$, i.e., $F_{t}(x) = \int _0^x f_{t}(y) \, dy$. The same notation is also adopted for an IMM or DET transition $t\in T$, which is associated with a Dirac impulse function $f_{t}(y)=\delta (y-\overline{y})$ with $\overline{y}=EFT(t)=LFT(t)$.

A marking $m \in M$ assigns a natural number of tokens to each place of an STPN. A transition t is enabled by m if m assigns at least one token to each of its input places and the enabling function B(t)(m) evaluates to true. The set of transitions enabled by m is denoted as E(m).

Definition 3

(State). The state of an STPN is a pair $\langle m, \vec {\tau }\rangle$ where $m\in M$ is a marking and vector $\vec {\tau }\in \mathbb {R}_{\geqslant 0}^{|E(m)|}$ assigns a time to fire $\vec {\tau }(t)\in \mathbb {R}_{\geqslant 0}$ to each enabled transition $t\in E(m)$.

Definition 4

(Semantics). Given an initial marking $m_0$, an execution of the STPN is a (finite or infinite) path $\omega = s_0 \overset{\gamma _1}{\longrightarrow }\ s_1 \overset{\gamma _2}{\longrightarrow }\ s_2 \overset{\gamma _3}{\longrightarrow }\ \cdots$ such that: $s_0=\langle m_0, \vec {\tau }_0 \rangle$ is the initial state, where the time to fire $\vec {\tau }_0(t)$ of each enabled transition $t \in E(m_0)$ is sampled according to the distribution $F_t$; $\gamma _{i} \in T$ is the ith fired transition; $s_{i}=\langle m_{i}, \vec {\tau }_{i} \rangle$ is the state reached after the firing of $\gamma _{i}$.

In each state $s_i$:

• The next transition $\gamma _{i+1}$ is selected from the set of enabled transitions with minimum time to fire and maximum priority according to the distribution given by weights: if $E_{\text {min}} = \arg \min _{t\in E(m_i)} \vec {\tau }_i(t)$ and $E_{\text {prio}} = \arg \max _{t\in E_{\text {min}}} Z(t)$, then $t\in E_{\text {prio}}$ is selected with probability $p_t = W(t)/\left( \sum _{u \in E_{\text {prio}}}W(u)\right)$.

• After the firing of $\gamma _{i+1}$, the new marking $m_{i+1}$ is derived by (1) removing a token from each input place of $\gamma _{i+1}$, (2) adding a token to each output place of $\gamma _{i+1}$, and (3) applying the update function $U(\gamma _{i+1})$ to the resulting marking. A transition t enabled by $m_{i+1}$ is termed persistent if it is distinct from $\gamma _{i+1}$, it is not contained in $R(\gamma _{i+1})$, and it is enabled also by $m_{i}$ and by the intermediate markings after steps (1) and (2); otherwise, t is termed newly enabled (thus, transitions in the reset set of $\gamma _{i+1}$ are newly enabled if enabled after the firing).

• For each newly enabled transition t, the time to fire $\vec {\tau }_{i+1}(t)$ is sampled according to the distribution $F_t$; for each persistent transition t, the time to fire in $s_{i+1}$ is reduced by the sojourn time in the previous marking, i.e., $\vec {\tau }_{i+1}(t) = \vec {\tau }_{i}(t)-\vec {\tau }_{i}(\gamma _{i+1})$.

When features are omitted for a transition $t \in T$, default values are assumed as follows: an always-true enabling function $B(t)(m)=\textsc {true}$; an identity update function $U(t)(m)=m$ for all $m\in M$; an empty reset set $R(t)=\varnothing$; a weight $W(t)=1$; and, a priority $Z(t)=0$.

Arc cardinalities greater than 1 can also be introduced in STPN syntax and semantics, letting the firing of a transition remove an arbitrary number of tokens from each input place or add an arbitrary number of tokens to each output place.