Computing the Expected Execution Time of Probabilistic Workflow Nets
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Abstract
FreeChoice Workflow Petri nets, also known as Workflow Graphs, are a popular model in Business Process Modeling.
In this paper we introduce Timed Probabilistic Workflow Nets (TPWNs), and give them a Markov Decision Process (MDP) semantics. Since the time needed to execute two parallel tasks is the maximum of the times, and not their sum, the expected time cannot be directly computed using the theory of MDPs with rewards. In our first contribution, we overcome this obstacle with the help of “earliestfirst” schedulers, and give a single exponentialtime algorithm for computing the expected time.
In our second contribution, we show that computing the expected time is \({\textsc {\#P}{}}\)hard, and so polynomial algorithms are very unlikely to exist. Further, \({\textsc {\#P}{}}\)hardness holds even for workflows with a very simple structure in which all transitions times are 1 or 0, and all probabilities are 1 or 0.5.
Our third and final contribution is an experimental investigation of the runtime of our algorithm on a set of industrial benchmarks. Despite the negative theoretical results, the results are very encouraging. In particular, the expected time of every workflow in a popular benchmark suite with 642 workflow nets can be computed in milliseconds. Data or code related to this paper is available at: [24].
1 Introduction
Workflow Petri Nets are a popular model for the representation and analysis of business processes [1, 3, 7]. They are used as backend for different notations like BPMN (Business Process Modeling Notation), EPC (Eventdriven Process Chain), and UML Activity Diagrams.
There is recent interest in extending these notations with quantitative information, like probabilities, costs, and time. The final goal is the development of tool support for computing performance metrics, like the average cost or the average runtime of a business process.
In a former paper we introduced Probabilistic Workflow Nets (PWN), a foundation for the extension of Petri nets with probabilities and rewards [11]. We presented a polynomial time algorithm for the computation of the expected cost of freechoice workflow nets, a subclass of PWN of particular interest for the workflow process community (see e.g. [1, 10, 13, 14]). For example, 1386 of the 1958 nets in the most popular benchmark suite in the literature are freechoice Workflow Nets [12].
In this paper we introduce Timed PWNs (TPWNs), an extension of PWNs with time. Following [11], we define a semantics in terms of Markov Decision Processes (MDPs), where, loosely speaking, the nondeterminism of the MDP models absence of information about the order in which concurrent transitions are executed. For every scheduler, the semantics assigns to the TPWN an expected time to termination. Using results of [11], we prove that this expected time is actually independent of the scheduler, and so that the notion “expected time of a TPWN” is well defined.
We then proceed to study the problem of computing the expected time of a sound TPWN (loosely speaking, of a TPWN that terminates successfully with probability 1). The expected cost and the expected time have a different interplay with concurrency. The cost of executing two tasks in parallel is the sum of the costs (cost models e.g. salaries of power consumption), while the execution time of two parallel tasks is the maximum of their individual execution times. For this reason, standard rewardbased algorithms for MDPs, which assume additivity of the reward along a path, cannot be applied.
Our solution to this problem uses the fact that the expected time of a TPWN is independent of the scheduler. We define an “earliestfirst” scheduler which, loosely speaking, resolves the nondeterminism of the MDP by picking transitions with earliest possible firing time. Since at first sight the scheduler needs infinite memory, its corresponding Markov chain is infinitestate, and so of no help. However, we show how to construct another finitestate Markov chain with additive rewards, whose expected reward is equal to the expected time of the infinitestate chain. This finitestate Markov chain can be exponentially larger than the TPWN, and so our algorithm has exponential complexity. We prove that computing the expected time is \({\textsc {\#P}{}}\)hard, even for freechoice TPWNs in which all transitions times are either 1 or 0, and all probabilities are 1 or \(\nicefrac {1}{2}\). So, in particular, the existence of a polynomial algorithm implies Open image in new window .
In the rest of the paper we show that, despite these negative results, our algorithm behaves well in practice. For all 642 sound freechoice nets of the benchmark suite of [12], computing the expected time never takes longer than a few milliseconds. Looking for a more complicated set of examples, we study a TPWN computed from a set of logs by process mining. We observe that the computation of the expected time is sensitive to the distribution of the execution time of a task. Still, our experiments show that even for complicated distributions leading to TPWNs with hundreds of transitions and times spanning two orders of magnitude the expected time can be computed in minutes.
All missing proofs can be found in the Appendix of the full version [19].
2 Preliminaries
We introduce some preliminary definitions. The full version [19] gives more details.
Workflow Nets. A workflow net is a tuple \(\mathbf {N}=(P,T,F,i,o)\) where P and T are disjoint finite sets of places and transitions; \(F \subseteq (P\times T) \cup (T \times P)\) is a set of arcs; \(i, o \in P\) are distinguished initial and final places such that i has no incoming arcs, o has no outgoing arcs, and the graph \((P \cup T, F \cup \{ (o, i) \})\) is strongly connected. For \(x \in P \cup T\), we write \({{^\bullet x}}\) for the set \(\{ y \mid (y,x) \in F\}\) and \({x^\bullet }\) for \(\{ y \mid (x, y) \in F\}\). We call \({{^\bullet x}}\) (resp. \({x^\bullet }\)) the preset (resp. postset) of x. We extend this notion to sets \(X \subseteq P \cup T\) by \({{^\bullet X}\,{\mathop {=}\limits ^{\scriptscriptstyle \text {def}}}\,\cup _{x \in X} {^\bullet x}}\) resp. \({X^\bullet }\,{\mathop {=}\limits ^{\scriptscriptstyle \text {def}}}\,\cup _{x \in X} {x^\bullet }\). The notions of marking, enabled transitions, transition firing, firing sequence, and reachable marking are defined as usual. The initial marking (resp. final marking) of a workflow net, denoted by Open image in new window (resp. Open image in new window ), has one token on place i (resp. o), and no tokens elsewhere. A firing sequence \(\sigma \) is a run if Open image in new window , i.e. if it leads to the final marking. \( Run_\mathbf {N} \) denotes the set of all runs of \(\mathbf {N}\).
Soundness and 1safeness. Well designed workflows should be free of deadlocks and livelocks. This idea is captured by the notion of soundness [1, 2]: A workflow net is sound if the final marking is reachable from any reachable marking.^{1} Further, in this paper we restrict ourselves to 1safe workflows: A marking M of a workflow net \(\mathcal {W}\) is 1safe if \(M(p) \le 1 \) for every place p, and \(\mathcal {W}\) itself is 1safe if every reachable marking is 1safe. We identify 1safe markings M with the set \(\{ p \in P \mid M(p) = 1 \}\).
Independence, concurrency, conflict [22]. Two transitions \(t_1\), \(t_2\) of a workflow net are independent if \({{^\bullet t_1} \cap {^\bullet t_2} = \emptyset }\), and dependent otherwise. Given a 1safe marking M, two transitions are concurrent at M if M enables both of them, and they are independent, and in conflict at M if M enables both of them, and they are dependent. Finally, we recall the definition of Mazurkiewicz equivalence. Let \(\mathbf {N}=(P,T,F,i,o)\) be a 1safe workflow net. The relation \(\equiv _1 \subseteq T^* \times T^*\) is defined as follows: \(\sigma \equiv _1 \tau \) if there are independent transitions \(t_1, t_2\) and sequences \(\sigma ',\sigma '' \in T^*\) such that \(\sigma = \sigma ' \, t_1 \, t_2 \sigma ''\) and \(\tau = \sigma ' \, t_2 \, t_1 \sigma ''\). Two sequences \(\sigma , \tau \in T^*\) are Mazurkiewicz equivalent if \(\sigma \equiv \tau \), where \(\equiv \) is the reflexive and transitive closure of \(\equiv _1\). Observe that \(\sigma \in T^*\) is a firing sequence iff every sequence \(\tau \equiv \sigma \) is a firing sequence.
Confusionfreeness, freechoice workflows. Let t be a transition of a workflow net, and let M be a 1safe marking that enables t. The conflict set of t at M, denoted C(t, M), is the set of transitions in conflict with t at M. A set U of transitions is a conflict set of M if there is a transition t such that \(U = C(t,M)\). The conflict sets of M are given by \(\mathcal {C}(M)\,{\mathop {=}\limits ^{\scriptscriptstyle \text {def}}}\,\cup _{t \in T} C(t, M)\). A 1safe workflow net is confusionfree if for every reachable marking M and every transition t enabled at M, every transition u concurrent with t at M satisfies \({C(u, M) = C(u, M \setminus {^\bullet t}) = C(u, (M \setminus {^\bullet t}) \cup {t^\bullet })}\). The following result follows easily from the definitions (see also [11]):
Lemma 1
[11]. Let \(\mathbf {N}\) be a 1safe workflow net. If \(\mathbf {N}\) is confusionfree then for every reachable marking M the conflict sets \(\mathcal {C}(M)\) are a partition of the set of transitions enabled at M.
A workflow net is freechoice if for every two places \(p_1,p_2\), if \({p_1^\bullet } \cap {p_2^\bullet } \ne \emptyset \), then \({p_1^\bullet } = {p_2^\bullet }\). Any freechoice net is confusionfree, and the conflict set of a transition t enabled at a marking M is given by \({C(t, M) = {\left( {^\bullet t}\right) ^\bullet }}\) (see e.g. [11]).
3 Timed Probabilistic Workflow Nets
In [11] we introduced a probabilistic semantics for confusionfree workflow nets. Intuitively, at every reachable marking a choice between two concurrent transitions is resolved nondeterministically by a scheduler, while a choice between two transitions in conflict is resolved probabilistically; the probability of choosing each transition is proportional to its weight. For example, in the net in Fig. 1a, at the marking \(\{p_1,p_3\}\), the scheduler can choose between the conflict sets \(\{t_2,t_3\}\) and \(\{t_4\}\), and if \(\{t_2,t_3\}\) is chosen, then \(t_2\) is chosen with probability \(\nicefrac {1}{5}\) and \(t_3\) with probability \(\nicefrac {4}{5}\). We extend Probabilistic Workflow Nets by assigning to each transition t a natural number \(\tau (t)\) modeling the time it takes for the transition to fire, once it has been selected.^{2}
Definition 1
(Timed Probabilistic Workflow Nets). A Timed Probabilistic Workflow Net (TPWN) is a tuple \(\mathcal {W}= (\mathbf {N},w, \tau )\) where \(\mathbf {N}=(P,T,F,i,o)\) is a 1safe confusionfree workflow net, \(w :T \rightarrow \mathbb {Q}_{> 0}\) is a weight function, and \(\tau :T \rightarrow \mathbb {N}\) is a time function that assigns to every transition a duration.
Example 1
The net in Fig. 1a is a TPWN. Weights are shown in red next to transitions, and times are written in blue into the transitions. For the sequence \(\sigma _1 = t_1 t_3 t_4 t_5\), we have \( tm (\sigma _1) = 9\), and for \(\sigma _2 = t_1 t_2 t_3 t_4 t_5\), we have \( tm (\sigma _2) = 10\). Observe that the time taken by the sequences is not equal to the sum of the durations of the transitions.
Markov Decision Process semantics. A Markov Decision Process (MDP) is a tuple \(\mathcal {M} = (Q, q_0, { Steps})\) where Q is a finite set of states, \(q_0\in Q\) is the initial state, and \({ Steps} :Q \rightarrow 2^{dist(Q)}\) is the probability transition function. Paths of an MDP, schedulers, and the probability measure of paths compatible with a scheduler are defined as usual (see the Appendix of the full version [19]).

if M enables no transitions, then \({ Steps}(M)\) contains exactly one distribution, which assigns probability 1 to M, and 0 to all other states.

if M enables at least one transition, then \({ Steps}(M)\) contains a distribution \(\lambda \) for each conflict set C of M. The distribution is defined by: \(\lambda (M, t) = P_{M,C}(t)\) for every \(t \in C\), and \(\lambda (s)=0\) for every other state s.
For every \((M, t) \in \mathcal{E}\), \({ Steps}(M,t)\) contains one single distribution that assigns probability 1 to the marking \(M'\) such that Open image in new window , and probability 0 to every other state.
Example 2
Figure 1b shows a graphical representation of the MDP of the TPWN in Fig. 1a. Black nodes represent states, white nodes probability distributions. A black node q has a white successor for each probability distribution in \({ Steps}(q)\). A white node \(\lambda \) has a black successor for each node q such that \(\lambda (q) > 0\); the arrow leading to this black successor is labeled with \(\lambda (q)\), unless \(\lambda (q)=1\), in which case there is no label. States (M, t) are abbreviated to t.
Schedulers. Given a TPWN \(\mathcal {W}\), a scheduler of \({ MDP}_W\) is a function \(\gamma : T^* \rightarrow 2^T\) assigning to each firing sequence Open image in new window with \(\mathcal {C}(M) \ne \emptyset \) a conflict set \(\gamma (\sigma ) \in \mathcal {C}(M)\). A firing sequence Open image in new window is compatible with a scheduler \(\gamma \) if for all partitions \(\sigma = \sigma _1 t \sigma _2\) for some transition t, we have \(t \in \gamma (\sigma _1)\).
Example 3
In the TPWN of Fig. 1a, after firing \(t_1\) two conflict sets become concurrently enabled: \(\{t_2, t_3\}\) and \(\{t_4\}\). A scheduler picks one of the two. If the scheduler picks \(\{t_2, t_3\}\) then \(t_2\) may occur, and in this case, since firing \(t_2\) does not change the marking, the scheduler chooses again one of \(\{t_2, t_3\}\) and \(\{t_4\}\). So there are infinitely many possible schedulers, differing only in how many times they pick \(\{t_2, t_3\}\) before picking \(t_4\).
Definition 2
In [11] we proved a result for Probabilistic Workflow Nets (PWNs) with rewards, showing that the expected reward of a PWN is independent of the scheduler (intuitively, this is the case because in a confusionfree Petri net the scheduler only determines the logical order in which transitions occur, but not which transitions occur). Despite the fact that, contrary to rewards, the execution time of a firing sequence is not the sum of the execution times of its transitions, the proof carries over to the expected time with only minor modifications.
Theorem 1
Let \(\mathcal {W}\) be a TPWN.
 (1)
There exists a value \(ET_\mathcal {W}\) such that for every scheduler S of \(\mathcal {W}\), the expected time \(ET^S_\mathcal {W}\) of \(\mathcal {W}\) under S is equal to \(ET_\mathcal {W}\).
 (2)
\(ET_\mathcal {W}\) is finite iff \(\mathcal {W}\) is sound.
By this theorem, the expected time \(ET_\mathcal {W}\) can be computed by choosing a suitable scheduler S, and computing \(ET^S_\mathcal {W}\).
4 Computation of the Expected Time
We show how to compute the expected time of a TPWN. We fix an appropriate scheduler, show that it induces a finitestate Markov chain, define an appropriate reward function for the chain, and prove that the expected time is equal to the expected reward.
4.1 EarliestFirst Scheduler
Consider a firing sequence Open image in new window . We define the starting time of a conflict set \(C \in \mathcal {C}(M)\) as the earliest time at which the transitions of C become enabled. This occurs after all tokens of \({{^\bullet C}}\) arrive^{3}, and so the starting time of C is the maximum of \(\mu (\sigma )_p\) for \(p \in {^\bullet C}\) (recall that \(\mu (\sigma )_p\) is the latest time at which a token arrives at p while firing \(\sigma \)).
Example 4
Figure 2a shows the Markov chain induced by the “earliestfirst” scheduler defined above in the MDP of Fig. 1b. Initially we have a token at Open image in new window with arrival time 0. After firing \(t_1\), which takes time 1, we obtain tokens in \(p_1\) and \(p_3\) with arrival time 1. In particular, the conflict sets \(\{t_2,t_3\}\) and \(\{t_4\}\) become enabled at time 1. The scheduler can choose any of them, because they have the same starting time. Assume it chooses \(\{t_2,t_3\}\). The Markov chain now branches into two transitions, corresponding to firing \(t_2\) and \(t_3\) with probabilities \(\nicefrac {1}{5}\) and \(\nicefrac {4}{5}\), respectively. Consider the branch in which \(t_2\) fires. Since \(t_2\) starts at time 1 and takes 4 time units, it removes the token from \(p_1\) at time 1, and adds a new token to \(p_1\) with arrival time 5; the token at \(p_3\) is not affected, and it keeps its arrival time of 1. So we have \(\mu (t_1t_2) = \left\{ \genfrac{}{}{0.0pt}{}{p_1}{5},\genfrac{}{}{0.0pt}{}{p_3}{1} \right\} \) (meaning \(\mu (t_1t_2)_{p_1}=5\), \(\mu (t_1t_2)_{p_3}=1\), and \(\mu (t_1t_2)_{p}= \bot \) otherwise). Now the conflict sets \(\{t_2,t_3\}\) and \(\{t_4\}\) are enabled again, but with a difference: while \(\{t_4\}\) has been enabled since time 1, the set \(\{t_2,t_3\}\) is now enabled since time \(\mu (t_1 t_2)_{p_1} = 5\). The scheduler must now choose \(\{t_4\}\), leading to the marking that puts tokens on \(p_1\) and \(p_4\) with arrival times \(\mu (t_1 t_2 t_4)_{p_1} = 5\) and \(\mu (t_1 t_2 t_4)_{p_4} = 6\). In the next steps the scheduler always chooses \(\{t_2,t_3\}\) until \(t_5\) becomes enabled. The final marking Open image in new window can be reached after time 9, through \(t_1 t_3 t_4 t_5\) with probability \(\nicefrac {4}{5}\), or with times \(10 + 4k\) for \(k \in \mathbb {N}\), through \(t_1 t_2 t_4 t_2^k t_3 t_5\) with probability \(\left( \nicefrac {1}{5}\right) ^{k+1} \cdot \nicefrac {4}{5}\) (the times at which the final marking can be reached are written in blue inside the final states).
Theorem 2 below shows that the earliestfirst scheduler only needs finite memory, which is not clear from the definition. The construction is similar to those of [6, 15, 16]. However, our proof crucially depends on TPWNs being confusionfree.
Theorem 2
Example 5
Figure 2b shows the finitestate Markov chain induced by the “earliestfirst” scheduler computed using the abstraction \(\nu \). Consider the firing sequence \(t_1t_3\). We have \(\mu (t_1t_3) = \left\{ \genfrac{}{}{0.0pt}{}{p_2}{3},\genfrac{}{}{0.0pt}{}{p_3}{1} \right\} \), i.e. the tokens in \(p_2\) and \(p_3\) arrive at times 3 and 1, respectively. Now we compute \(\nu (t_1t_3)\), which corresponds to the local arrival times of the tokens, i.e. the time elapsed since the last transition starts to fire until the token arrives. Transition \(t_3\) starts to fire at time 1, and so the local arrival times of the tokens in \(p_2\) and \(p_3\) are 2 and 0, respectively, i.e. we have \(\nu (t_1t_3) = \left\{ \genfrac{}{}{0.0pt}{}{p_2}{2},\genfrac{}{}{0.0pt}{}{p_3}{0} \right\} \). Using these local times we compute the local starting time of the conflict sets enabled at \(\{p_2, p_3\}\). The scheduler always chooses the conflict set with earliest local starting time. In Fig. 2b the earliest local starting time of the state reached by firing \(\sigma \), which is denoted \(r(\nu (\sigma ))\), is written in blue inside the state. The theorem above shows that this scheduler always chooses the same conflict sets as the one which uses the function \(\mu \), and that the time of a sequence can be obtained by adding the local starting times. This allows us to consider the earliest local starting time of a state as a reward associated to the state; then, the time taken by a sequence is equal to the sum of the rewards along the corresponding path of the chain. For example, we have \( tm (t_1 t_2 t_4 t_3 t_5) = 0 + 1 + 0 + 4 + 2 + 3 = 10\).
Finally, let us see how \(\nu (\sigma t)\) is computed from \(\nu (\sigma )\) for \(\sigma = t_1 t_2 t_4\) and \(t=t_2\). We have \(\nu (\sigma ) = \left\{ \genfrac{}{}{0.0pt}{}{p_1}{4},\genfrac{}{}{0.0pt}{}{p_4}{5} \right\} \), i.e. the local arrival times for the tokens in \(p_1\) and \(p_4\) are 4 and 5, respectively. Now \(\{t_2,t_3\}\) is scheduled next, with local starting time \(r(\nu (\sigma )) = \nu (\sigma )_{p_1} = 4\). If \(t_2\) fires, then, since \(\tau (t_2)=4\), we first add 4 to the time of \(p_1\), obtaining \(\left\{ \genfrac{}{}{0.0pt}{}{p_1}{8},\genfrac{}{}{0.0pt}{}{p_4}{5} \right\} \). Second, we subtract 4 from all times, to obtain the time elapsed since \(t_2\) started to fire (for local times the origin of time changes every time a transition fires), yielding the final result \(\nu (\sigma t_2)=\left\{ \genfrac{}{}{0.0pt}{}{p_1}{4},\genfrac{}{}{0.0pt}{}{p_4}{1} \right\} \).
4.2 Computation in the Probabilistic Case
Given a TPWN and its corresponding MDP, in the previous section we have defined a finitestate earliestfirst scheduler and a reward function of its induced Markov chain. The reward function has the following property: the execution time of a firing sequence compatible with the scheduler is equal to the sum of the rewards of the states visited along it. From the theory of Markov chains with rewards, it follows that the expected accumulated reward until reaching a certain state, provided that this state is reached with probability 1, can be computed by solving a linear equation system. We use this result to compute the expected time \(ET_W\).
Lemma 2
Let \(\mathcal {W}\) be a sound TPWN. Then the system of linear equations (4) has a unique solution \(\varvec{X}\), and \(ET_\mathcal {W}= \varvec{X}_{\nu (\epsilon )}\).
Theorem 3
Let \(\mathcal {W}\) be a TPWN. Then \(ET_\mathcal {W}\) is either \(\infty \) or a rational number and can be computed in single exponential time.
Proof
We assume that the input has size n and all times and weights are given in binary notation. Testing whether \(\mathcal {W}\) is sound can be done by exploration of the state space of reachable markings in time \(\mathcal {O}(2^n)\). If \(\mathcal {W}\) is unsound, we have \(ET_\mathcal {W}= \infty \).
Now assume that \(\mathcal {W}\) is sound. By Lemma 2, \(ET_\mathcal {W}\) is the solution to the linear equation system (4), which is finite and has rational coefficients, so it is a rational number. The number of variables \(\left \varvec{X} \right \) of (4) is bounded by the size of \({[H]}_\bot ^P\), and as \(H = \max _{t \in T} \tau (t)\) we have \(\left \varvec{X} \right \le (1 + H)^{\left P \right } \le \left( 1 + 2^n\right) ^n \le 2^{n^2 + n}\). The linear equation system can be solved in time \(\mathcal {O}\left( n^2 \cdot \left \varvec{X} \right ^3\right) \) and therefore in time \(\mathcal {O}(2^{p(n)})\) for some polynomial p.
5 Lower Bounds for the Expected Time
We analyze the complexity of computing the expected time of a TPWN. Botezano et al. show in [5] that deciding if the expected time exceeds a given bound is Open image in new window hard. However, their reduction produces TPWNs with weights and times of arbitrary size. An open question is if the expected time can be computed in polynomial time when the times (and weights) must be taken from a finite set. We prove that this is not the case unless Open image in new window , even if all times are 0 or 1, all weights are 1, the workflow net is sound, acyclic and freechoice, and the size of each conflict set is at most 2 (resulting only in probabilities 1 or \(\nicefrac {1}{2}\)). Further, we show that even computing an \(\epsilon \)approximation is equally hard. These two results above are a consequence of the main theorem of this section: computing the expected time is #Phard [23]. For example, counting the number of satisfying assignments for a boolean formula (#SAT) is a #Pcomplete problem. Therefore a polynomialtime algorithm for a #Phard problem would imply Open image in new window .
The problem used for the reduction is defined on PERT networks [9], in the specialized form of twostate stochastic PERT networks [17], described below.
Definition 3
A twostate stochastic PERT network is a tuple \(\mathbf {PN}= (G, s, t, \varvec{p})\), where \(G = (V,E)\) is a directed acyclic graph with vertices V, representing events, and edges E, representing tasks, with a single source vertex s and sink vertex t, and where the vector \(\varvec{p} \in \mathbb {Q}^E\) assigns to each edge \(e \in E\) a rational probability \(p_e \in [0,1]\). We assume that all \(p_e\) are written in binary.
Example 6
Figure 3a shows a small PERT network (without \(\varvec{p}\)), where the project duration depends on the paths \(\varPi = \left\{ e_1 e_3 e_6, e_1 e_4 e_7, e_2 e_5 e_7\right\} \).
First reduction: 0/1 times, arbitrary weights. We reduce the problem above to computing the expected time of an acyclic TPWN with 0/1 times but arbitrary weights. Given a twostate stochastic PERT network \(\mathbf {PN}\), we construct a timed probabilistic workflow net \(\mathcal {W}_\mathbf {PN}\) as follows:Given: A twostate stochastic PERT network \(\mathbf {PN}\).
Compute: The expected project duration \({{\,\mathrm{\mathbb {E}}\,}}(PD(\mathbf {PN}))\).

For each edge \(e = (u,v) \in E\), add the “gadget net” shown in Fig. 3b. Assign \(w(t_{e,0}) = 1p_e\), \(w(t_{e,1}) = p_e\), \(\tau (t_{e,0}) = 0\), and \(\tau (t_{e,1}) = 1\).

For each vertex \(v \in V\), add a transition \(t_v\) with arcs from each [e, v] such that \(e = (u,v) \in E\) for some u and arcs to each [v, e] such that \(e = (v,w) \in E\) for some w. Assign \(w(t_v) = 1\) and \(\tau (t_v) = 0\).

Add the place i with an arc to \(t_s\) and the place o with an arc from \(t_t\).
The result of applying this construction to the PERT network from Fig. 3a is shown in Fig. 3d. It is easy to see that this workflow net is sound, as from any reachable marking, we can fire enabled transitions corresponding to the edges and vertices of the PERT network in the topological order of the graph, eventually firing \(t_t\) and reaching Open image in new window . The net is also acyclic and freechoice.
Lemma 3
Let \(\mathbf {PN}\) be a twostate stochastic PERT network and let \(\mathcal {W}_\mathbf {PN}\) be its corresponding TPWN by the construction above. Then \(ET_{\mathcal {W}_\mathbf {PN}} = {{\,\mathrm{\mathbb {E}}\,}}(PD(\mathbf {PN}))\).
Second reduction: 0/1 times, 0/1 weights. The network constructed this way already uses times 0 and 1, however the weights still use arbitrary rational numbers. We now replace the gadget nets from Fig. 3b by equivalent nets where all transitions have weight 1. The idea is to use the binary encoding of the probabilities \(p_e\), deciding if the time is 0 or 1 by a sequence of coin flips. We assume that \(p_e = \sum _{i=0}^k 2^{i} p_i\) for some \(k \in \mathbb {N}\) and \(p_i \in \left\{ 0, 1\right\} \) for \(0 \le i \le k\). The replacement is shown in Fig. 3c for \(p_e = \nicefrac {5}{8} = {(0.101)}_2\).
Approximating the expected time is #Phard. We show that computing an \(\epsilon \)approximation for \(ET_\mathcal {W}\) is #Phard [17, 20].
Theorem 4
Given: A sound, acyclic and freechoice TPWN \(\mathcal {W}\) where all transitions t satisfy \(w(t) = 1\), \(\tau (t) \in \left\{ 0, 1 \right\} \) and \({\left {({^\bullet t})^\bullet } \right \le 2}\), and an \(\epsilon > 0\).
Compute: A rational r such that \(r  \epsilon< ET_W < r + \epsilon \).
6 Experimental Evaluation
We have implemented our algorithm to compute the expected time of a TPWN as a package of the tool \({\texttt {ProM}{}}\)^{4}. It is available via the package manager of the latest nightly build under the package name WorkflowNetAnalyzer.
We evaluated the algorithm on two different benchmarks. All experiments in this section were run on the same machine equipped with an Intel Core i76700K CPU and 32 GB of RAM. We measure the actual runtime of the algorithm, split into construction of the Markov chain and solving the linear equation system, and exclude the time overhead due to starting \({\texttt {ProM}{}}\) and loading the plugin.
6.1 IBM Benchmark
We evaluated the tool on a set of 1386 workflow nets extracted from a collection of five libraries of industrial business processes modeled in the IBM WebSphere Business Modeler [12]. All of the 1386 nets in the benchmark libraries are freechoice and therefore confusionfree. We selected the sound and 1safe nets among them, which are 642 nets. Out of these, 409 are marked graphs, i.e. the size of any conflict set is 1. Out of the remaining 233 nets, 193 are acyclic and 40 cyclic.
As these nets do not come with probabilistic or time information, we annotated transitions with integer weights and times chosen uniformly from different intervals: (1) \(w(t) = \tau (t) = 1\), (2) \(w(t),\tau (t) \in [1,10^3]\) and (3) \(w(t),\tau (t) \in [1,10^6]\). For each interval, we annotated the transitions of each net with random weights and times, and computed the expected time of all 642 nets.
Analysis times and size of the state space \(\left \varvec{X} \right \) for the 4 nets with the highest analysis times, given for each of the three intervals \([1],[10^3],[10^6]\) of possible times. Here, Open image in new window denotes the number of reachable markings of the net.
Net  Net info & size  Analysis time (ms)  \(\left \varvec{X} \right \)  

cyclic  \(\left P \right \)  \(\left T \right \)  [1]  \([10^3]\)  \([10^6]\)  [1]  \([10^3]\)  \([10^6]\)  
m1.s30_s703  no  264  286  6117  40.3  44.6  43.8  304  347  347 
m1.s30_s596  yes  214  230  623  21.6  24.4  23.6  208  232  234 
b3.s371_s1986  no  235  101  \(2\cdot 10^{17}\)  16.8  16.4  16.5  101  102  102 
b2.s275_s2417  no  103  68  237626  14.2  17.8  15.9  355  460  431 
6.2 Process Mining Case Study
As a second benchmark, we evaluated the algorithm on a model of a loan application process. We used the data from the BPI Challenge 2017 [8], an event log containing 31509 cases provided by a financial institute, and took as a model of the process the final net from the report of the winner of the academic category [21], a simple model with high fitness and precision w.r.t. the event log.
Expected time, analysis time and state space size for the net in Fig. 4 for various distributions, where Open image in new window denotes reaching the memory limit.
Distribution  \(\left T \right \)  \(ET_W\)  \(\left \varvec{X} \right \)  Analysis time  

Total  Construction  Solving  
Deterministic  19  24 d  1 h  33  40 ms  18 ms  22 ms 
Histogram/12 h  141  24 d  18 h  4054  244 ms  232 ms  12 ms 
Histogram/6 h  261  24 d  21 h  15522  2.1 s  1.8 s  0.3 s 
Histogram/4 h  375  24 d  22 h  34063  10 s  6 s  4 s 
Histogram/2 h  666  24 d  23 h  122785  346 s  52 s  294 s 
Histogram/1 h  1117  —  422614  —  12.7 min 
For a first analysis, we simply set the execution time of each transition deterministically to its mean waiting time. However, note that the two transitions “O_Create Offer” and “W_Complete application” are executed in parallel, and therefore the distribution of their execution times influences the total expected time. Therefore we also annotated these two transitions with a histogram of possible execution times from each case. Then we split them up into multiple transitions by grouping the times into buckets of a given interval size, where each bucket creates a transition with an execution time equal to the beginning of the interval, and a weight equal to the number of cases with a waiting time contained in the interval. The times for these transitions range from 6 ms to 31 days. As bucket sizes we chose 12, 6, 4, 2 and 1 hour(s). The net always has 14 places and 15 reachable markings, but a varying number of transitions depending on the chosen bucket size. For the net with the mean as the deterministic time and for the nets with histograms for each bucket size, we then analyzed the expected execution time using our algorithm.
The results are given in Table 2. They show that using the complete distribution of times instead of only the mean can lead to much more precise results. When the linear equation system becomes very large, the solver time dominates the construction time of the system. This may be because we chose to use an exact solver for sparse linear equation systems. In the future, this could possibly be improved by using an approximative iterative solver.
7 Conclusion
We have shown that computing the expected time to termination of a probabilistic workflow net in which transition firings have deterministic durations is Open image in new window hard. This is the case even if the net is freechoice, and both probabilities and times can be written down with a constant number of bits. So, surprisingly, computing the expected time is much harder than computing the expected cost, for which there is a polynomial algorithm [11].
We have also presented an exponential algorithm for computing the expected time based on earliestfirst schedulers. Its performance depends crucially on the maximal size of conflict sets that can be concurrently enabled. In the most popular suite of industrial benchmarks this number turns out to be small. So, very satisfactorily, the expected time of any of these benchmarks, some of which have hundreds of transitions, can still be computed in milliseconds.
Footnotes
 1.
In [2], which examines many different notions of soundness, this is called easy soundness.
 2.
The semantics of the model can be defined in the same way for both discrete and continuous time, but, since our results only concern discrete time, we only consider this case.
 3.
This is proved in Lemma 7 in the Appendix of the full version [19].
 4.
Notes
Acknowledgements
We thank Hagen Völzer for input on the implementation and choice of benchmarks.
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