Computing the concurrency threshold of sound free-choice workflow nets

A Petri net $N$ is a tuple $(P,T,F)$ where $P$ is a finite set of places, $T$ is a finite set of transitions ($P\cap T=\emptyset$), and $F\subseteq(P\times T)\cup(T\times P)$ is a set of arcs. The preset of $x\in P\cup T$ is ${{}^{\bullet}x}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\left\{y\mid(y,x)\in F\right\}$ and its postset is ${x^{\bullet}}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\left\{y\mid(x,y)\in F\right\}$. We extend the definition of presets and postsets to sets of places and transitions $X\subseteq P\cup T$ by ${{}^{\bullet}X}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\bigcup_{x\in X}{{}^{\bullet}x}$ and ${X^{\bullet}}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\bigcup_{x\in X}{x^{\bullet}}$. A net is acyclic if the relation $F^{*}$ is a partial order, denoted by $\preceq$ and called the causal order. A node $x$ of an acyclic net is causally maximal if no node y satisfies $x\prec y$.

A marking of a Petri net is a function $M\colon P\to\mathbb{N}$, representing the number of tokens in each place. For a set of places $S\subseteq P$, we define $M(S)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{p\in S}M(p)$. Further, for a set of places $S\subseteq P$, we define by $M_{S}$ the marking with $M_{S}(p)=1$ for $p\in S$ and $M_{S}(p)=0$ for $p\notin S$.

A transition $t$ is enabled at a marking $M$ if for all $p\in{{}^{\bullet}t}$, we have $M(p)\geq 1$. If $t$ is enabled at $M$, it may occur, leading to a marking $M^{\prime}$ obtained by removing one token from each place of ${{}^{\bullet}t}$ and then adding one token to each place of ${t^{\bullet}}$. We denote this by $M\xrightarrow{t}M^{\prime}$. Let $\sigma=t_{1}t_{2}\ldots t_{n}$ be a sequence of transitions. For a marking $M_{0}$, $\sigma$ is an occurrence sequence if $M_{0}\xrightarrow{t_{1}}M_{1}\xrightarrow{t_{2}}\ldots\xrightarrow{t_{n}}M_{n}$ for some markings $M_{1},\ldots,M_{n}$. We say that $M_{n}$ is reachable from $M_{0}$ by $\sigma$ and denote this by $M_{0}\xrightarrow{\sigma}M_{n}$. The set of all markings reachable from $M$ in $N$ by some occurrence sequence $\sigma$ is denoted by $\mathcal{R}^{N}\left(M\right)$. A system is a pair $(N,M)$ of a Petri net $N$ and a marking $M$. A system $(N,M)$ is live if for every $M^{\prime}\in\mathcal{R}^{N}\left(M\right)$ and every transition $t$ some marking $M^{\prime\prime}\in\mathcal{R}^{N}\left(M^{\prime}\right)$ enables $t$. The system is 1-safe if $M^{\prime}(p)\leq 1$ for every $M^{\prime}\in\mathcal{R}^{N}\left(M\right)$ and every place $p\in P$.
Convention: Throughout this paper we assume that systems are 1-safe, i.e., we identify “system” and “1-safe system”.

A net $N=(P,T,F)$ is a marked graph if $|{{}^{\bullet}p}|\leq 1$ and $|{p^{\bullet}}|\leq 1$ for every place $p\in P$, and a free-choice net if for any two places $p_{1},p_{2}\in P$ either ${p_{1}^{\bullet}}\cap{p_{2}^{\bullet}}=\emptyset$ or ${p_{1}^{\bullet}}={p_{2}^{\bullet}}$.

An $(A,B)$-labeled Petri net is a tuple $N=(P,T,F,\lambda,\mu)$, where $\lambda\colon P\rightarrow A$ and $\mu\colon T\rightarrow B$ are labeling functions over alphabets $A,B$. The nonsequential processes of a 1-safe system $(N,M)$ are acyclic, $(P,T)$-labeled marked graphs. Say that a set $P^{\prime\prime}$ of places of a $(P,T)$-labeled acyclic net enables $t\in T$ if all the places of $P^{\prime\prime}$ are causally maximal, carry pairwise distinct labels, and $\lambda(P^{\prime\prime})={{}^{\bullet}t}$.

As usual, we say that two processes are isomorphic if they are the same up to renaming of the places and transitions (notice that we rename only the names of the places and transitions, not their labels).

Fig. 2 shows two processes of the workflow net in Fig. 1(a). (The figure does not show the names of places and transitions, only their labels.) The net containing the white and grey nodes only is already a process, and the grey places are causally maximal places that enable $t_{6}$. Therefore, according to the definition we can extend the process with the green nodes to produce another process. On the right we extend the same process in a different way, with the transition $t_{4}$.

The following is well known. Let $(P^{\prime},T^{\prime},F^{\prime},\lambda,\mu)$ be a process of $(N,M)$:

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  For every linearization $\sigma=t_{1}^{\prime}\ldots t_{n}^{\prime}$ of $T^{\prime}$ respecting the causal order $\preceq$, the sequence $\mu(\sigma)=\mu(t_{1}^{\prime})\ldots\mu(t_{n}^{\prime})$ is a firing sequence of $(N,M)$. Further, all these firing sequences lead to the same marking. We call it the final marking of $\Pi$, and say that $\Pi$ leads from $M$ to its final marking.
  For example, in Fig. 2 the sequences of the right process labeled by $t_{1}t_{2}t_{3}t_{4}$ and $t_{1}t_{3}t_{2}t_{4}$ are firing sequences leading to the marking $M=\{p_{2},p_{5},p_{7}\}$.

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  For every firing sequence $t_{1}\cdots t_{n}$ of $(N,M)$ there is a process $(P^{\prime},T^{\prime},F^{\prime},\lambda,\mu)$ such that $T^{\prime}=\{t_{1}^{\prime},\ldots,t_{n}^{\prime}\}$, $\mu(t_{i}^{\prime})=t_{i}$ for every $1\leq i\leq n$, and $\mu(t_{i}^{\prime})\preceq\mu(t_{j}^{\prime})$ implies $i\leq j$.

We slightly generalize the definition of workflow net as presented in e.g. [1] by allowing multiple initial and final places. A workflow net is a Petri net with two distinguished sets $I$ and $O$ of input places and output places such that (a) ${{}^{\bullet}I}=\emptyset={O^{\bullet}}$ and (b) for all $x\in P\cup T$, there exists a path from some $i\in I$ to some $o\in O$ passing through $x$. The markings $M_{I}$ and $M_{O}$ are called initial and final markings of $N$. A workflow net $N$ is sound if

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  $\forall M\in\mathcal{R}^{N}\left(M_{I}\right):M_{O}\in\mathcal{R}^{N}\left(M\right)$,

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  $\forall M\in\mathcal{R}^{N}\left(M_{I}\right):(M(O)\geq|O|)\Rightarrow(M=M_{O})$, and

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  $\forall t\in T:\exists M\in\mathcal{R}^{N}\left(M_{I}\right):t$ is enabled at $M$.

It is well-known that every sound free-choice workflow net is a 1-safe system with the initial marking $M_{I}$ [2, 7]. Given a workflow net according to this definition one can construct another one with one single input place $i$ and output place $o$ and two transitions $t_{i},t_{o}$ with ${{}^{\bullet}t_{i}}=\{i\},{t_{i}^{\bullet}}=I$ and ${{}^{\bullet}t_{o}}=O,{t_{o}^{\bullet}}=\{o\}$. For all purposes of this paper these two workflow nets are equivalent.

Given a workflow net $N$, we say that a process $\Pi$ of $(N,M_{I})$ is a run if it leads to $M_{O}$. For example, the net in Fig. 1(b) is a run of the net in Fig. 1(a).

We consider Petri nets in which, intuitively, when a token arrives in a place $p$ it has to execute a task taking $\tau(p)$ time units before the token can be used to fire any transition. Formally, we consider tuples $N=(P,T,F,\tau)$ where $(P,T,F)$ is a net and $\tau\colon P\rightarrow\mathbb{N}$.

Intuitively, $f(p^{\prime})$ describes the starting time of the task executed at $p^{\prime}$. Condition (1) states that if $p_{1}^{\prime}\preceq p_{2}^{\prime}$, then the task associated to $p_{2}^{\prime}$ can only start after the task for $p_{1}^{\prime}$ has ended; condition (2) states that all tasks are done by time $t$, and condition (3) that at any moment in time at most $k$ tasks are being executed. As an example, the process in Fig. 1(b) can be executed with two resources in time $6$ with the schedule $i,p_{1},p_{2}\mapsto 0$; $p_{3},p_{4}\mapsto 1$; $p_{7},p_{6}\mapsto 3$, and $p_{8},p_{9}\mapsto 4$.

Given a process $\Pi=(P^{\prime},T^{\prime},F^{\prime},\lambda,\mu)$ of $(N,M)$ we define the schedule $f_{\min}$ as follows: if $p^{\prime}\in\min(\Pi)$ then $f_{\min}(p^{\prime})=0$, otherwise define $f_{\min}(p^{\prime})=\max\{f_{\min}(p^{\prime\prime})+\tau(\lambda(p^{\prime\prime}))\mid p^{\prime\prime}\preceq p^{\prime}\}$. Further, we define the minimal execution time $t_{\min}(\Pi)=\max\{f(p^{\prime})+\tau(\lambda(p^{\prime\prime}))\mid p^{\prime}\in\max(\Pi)\}$. In the process in Fig. 1(b), the schedule $f_{\min}$ is the function that assigns $i,p_{1},p_{2},p_{7}~\mapsto~0$, $p_{3},p_{4}~\mapsto~1$, $p_{6},p_{8}~\mapsto~3$, $p_{9}~\mapsto~4$, and $o~\mapsto~6$, and so $t_{\min}(\Pi)=6$. We have: