Compositional Safe Approximation of Response Time Distribution of Complex Workflows

Abstract

We propose a compositional technique for efficient evaluation of the cumulative distribution function of the response time of complex workflows, consisting of activities with generally distributed stochastic durations composed through sequence, choice/merge, split/join, and repetition blocks, with unbalanced split and join constructs that break the structure of well-formed nesting. Workflows are specified using a formalism defined in terms of stochastic Petri nets, that permits decomposition of the model into a hierarchy of sub-workflows with positively correlated response times, which guarantees a stochastically ordered approximation of the end-to-end response time when intermediate results are approximated by stochastically ordered distributions and when dependencies are simplified by replicating activities appearing in multiple sub-workflows. This opens the way to an efficient hierarchical solution that manages complex models by recursive application of Markov regenerative analysis and numerical composition of monovariate distributions.

6 Appendix

We recall syntax and semantics of STPNs (Sect. 6.1) and numerical analysis of well-structured workflows (Sect. 6.2), and we report theorem proofs (Sect. 6.3) and a graphical representation of analysis heuristics (Sect. 6.4).

1.1 6.1 Formal Syntax and Semantics of STPNs

Syntax. An STPN is a tuple $\langle P,T,A^-,A^+,EFT,LFT,F,W,Z\rangle$ where: P and T are disjoint sets of places and transitions, respectively; $A^-\subseteq P\times T$ and $A^+\subseteq T\times P$ are pre-condition and post-condition relations, respectively; 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 $x<EFT(t)$ and $F_{t}(x) = 1$ for $x>LFT(t)$), a weight $W(t)\in \mathbb {R}_{> 0}$, and a priority $Z(t)\in \mathbb {N}$, respectively.

As usual in stochastic Petri nets, a transition t is called immediate (IMM) if $EFT(t)=LFT(t)=0$ and timed otherwise; a timed transition t is called exponential (EXP) if $F_{t}(x) = 1-\exp (-\lambda x)$ for some rate $\lambda \in {\mathbb R}_{>0}$, or general (GEN) otherwise. For each GEN transition t, we assume that $F_{t}$ can be expressed as the integral function of a probability density function (PDF) $f_{t}$, i.e., $F_{t}(x) = \int _0^x f_{t}(y) \, dy$. Similarly, an IMM transition $t\in T$ is associated with a generalized PDF represented by the Dirac impulse function $f_{t}(y)=\delta (y-\overline{y})$ with $\overline{y}=EFT(t)=LFT(t)$. A place $p \in P$ is called an input or output place for a transition $t \in T$ if $(p,t) \in A^-$ or $(t,p) \in A^+$, respectively.

Semantics. The state of an STPN is a pair $s=\langle m, \tau \rangle$, where $m: P \rightarrow \mathbb {N}$ is a marking assigning a number of tokens to each place and $\tau : T \rightarrow \mathbb {R}_{\geqslant 0}$ associates each transition with a time-to-fire. A transition is enabled by a marking if each of its input places contains at least one token. The next transition t to fire in a state s is selected from the set E of enabled transitions having time-to-fire equal to zero and maximum priority with probability $W(t) / \sum _{t_i \in E}W(t_i)$. When t fires, s is replaced with $s' = \langle m', \tau ' \rangle$, where: $m'$ is derived from m by removing a token from each input place of t, yielding an intermediate marking $m_{\mathrm {tmp}}$, and adding a token to each output place of t; $\tau '$ is derived from $\tau$ by: i) reducing the time-to-fire of each persistent transition (i.e., enabled by m, $m_\mathrm {tmp}$ and $m'$) by the time elapsed in s; ii) sampling the time-to-fire of each newly-enabled transition $t_n$ (i.e., enabled by $m'$ but not by $m_\mathrm {tmp}$) according to $F_{t_n}$; and, iii) removing the time-to-fire of each disabled transition (i.e., enabled by m but not by $m'$).

1.2 6.2 Numerical Analysis of Well-Structured Workflows

We derive the numerical form of the response time CDF of a block by combining bottom-up the numerical forms of the response time CDFs of the blocks that it contains, provided that the block has a well-formed structure, i.e., it is not a DAG block and it does not contain DAG blocks, and that it does not contain REPEAT blocks. Specifically, given n blocks $b_1, \ldots , b_n$ with response time CDF $\varPhi _1(t), \ldots , \varPhi _n(t)$ and PDF $\phi _1(t), \ldots , \phi _n(t)$, respectively:

• the response time CDF $\varPhi _{\mathrm {seq}}(t)$ of a SEQ block made of $b_1, \ldots , b_n$ is derived by performing subsequent convolutions of $\phi _1(t),\ldots ,\phi _n(t)$ $\forall \,t\in [0,t_{\mathrm {max}}]$:

  $$\begin{aligned} \begin{array}{l} \varPhi _{\mathrm {seq}}(t)=\varPhi ^{1,n}(t) \\ \\ \varPhi ^{1,i}(t)=\displaystyle \int _0^t \int _0^\tau \phi ^{1,i-1}(x) \, \phi _i(\tau -x) \, dx \, d\tau ~\forall \,i\in \{2,\ldots ,n\} \\ \\ \phi ^{1,i-1}(t)=\displaystyle \frac{d}{dt}\varPhi ^{1,i-1}(t)~~\forall \,i\in \{2,\ldots ,n-1\},~~\phi ^{1,1}(t)=\phi _1(t) \end{array} \end{aligned}$$

  (2)

• the response time CDF $\varPhi _{\mathrm {and}}(t)$ of an AND block made of $b_1, \ldots , b_n$ is the CDF of the maximum among the response times of $b_1,\ldots ,b_n$, which is derived as the product of $\varPhi _1(t),\ldots ,\varPhi _n(t)$ $\forall \,t\in [0,t_{\mathrm {max}}]$ due to the fact that the response times of $b_1,\ldots ,b_n$ are independent random variables:

  $$\begin{aligned} \varPhi _{\mathrm {and}}(t)=\displaystyle \varPhi _1(t) \cdot \ldots \cdot \varPhi _n(t) \end{aligned}$$

  (3)

• the response time CDF $\varPhi _{\mathrm {xor}}(t)$ of an XOR block made of $b_1, \ldots , b_n$ is derived as the weighted sum of $\varPhi _1(t), \ldots , \varPhi _n(t)$ $\forall \,t\in [0,t_{\mathrm {max}}]$:

  $$\begin{aligned} \varPhi _{\mathrm {xor}}(t)=\displaystyle p_1 \, \varPhi _1(t) + \ldots + p_n \, \varPhi _n(t) \end{aligned}$$

  (4)

1.3 6.3 Theorem Proofs

Proof of Lemma 2. By construction, $\hat{F}(x) \leqslant F(x)$ $\forall \,x\in D \,\cap \, [d,\infty )$, and $\hat{F}(x)=0$ $\forall \,x<d$. Starting from the point with abscissa a, the PDF of F(x) may be either increasing or decreasing: If the PDF is increasing, then it will reach a maximum point that comprises an inflection point for the CDF F(x), where the concavity changes from upward to downward (see Fig. 5b). By construction, the tangent line to the inflection point intersects the x-axis in a point whose abscissa d is larger than a, otherwise the CDF should have downward concavity before the inflection point, which contradicts the hypothesis. Otherwise (i.e., if the PDF is decreasing), F(x) has downward concavity (see Fig. 5a), d is selected equal to a, and stochastic order is verified for $\forall \,x\in D \,\cap \, [a,\infty )$. Therefore, $\hat{X} \geqslant _{st} X$.    $\square$

Fig. 5.

Stochastic upper bound CDF: concavity of approximated CDF.

Proof of Lemma 3. The duration of the sub-workflow associated with any node m (SEQ, AND, XOR, REPEAT, DAG) is a monotone nondecreasing function of the durations of the sub-workflows associated with its children; respectively, the sum (SEQ), max (AND), random mixture (XOR), series (REPEAT), max over all paths from the initial to the final node (DAG). By definition of stochastic order, if a child n is replaced with $n'$ s.t. $T(n) \leqslant _{st} T(n')$, then $T(m) \leqslant _{st} T(m')$ for the new node $m'$. By recursion, $T(n_0) \leqslant _{st} T(n'_0)$ for the new root $n'_0$.    $\square$

Proof of Lemma 4. Since DAG edges denote AND-join dependencies, the response time of a vertex v is $T(v) = D(v) + \max (T(k_1), \dots , T(k_n))$ where D(v) is the duration of the block associated with v and $T(k_1), \dots , T(k_n)$ are the response times of its predecessors. By visiting the vertices of G in topological order, we can evaluate the response time $T(v_F)$ of the DAG as an expression combining nonnegative block durations $D(v) \;\forall v\in V$ through monotone nondecreasing operators (i.e., summation and maximum). The intermediate values of this expression obtained during the visit are the response times $T(\cdot )$ of the nodes of G, which, by construction, are positively correlated. In the evaluation of $T(v_F')$ in $G'$, the random variable T(k) of each node $k \in K$ is replaced with the independent replica $T(k') \sim T(k)$. Then, by Lemma 1, we obtain $T(v_F') \geqslant _{st} T(v_F)$.    $\square$

1.4 6.4 Analysis Actions

Figure 6 illustrates the application of the sequence of actions 3 and 4 on the structure tree presented in Fig. 1b. In particular, in Fig. 6a, a red and a green box identify two sub-structures on which actions 3 and 4 are applied, respectively:

• Action 3: is applied as follows: some action (depending on the considered analysis heuristic) is used to evaluate the response time of the sub-structure in the red box, which is then replaced with an activity block $T_\text {new}$ associated with the evaluated response time CDF (see Fig. 6b).

• Action 4 evaluates the sub-structure in the green box independently of the rest of the DAG: the blocks that are shared with the rest of the DAG (i.e., block R) are replicated (i.e., block $R_\text {bis}$ is added), also adding two fictitious zero-duration nodes $v_I$ and $v_F$ (see Fig. 6c); then, this sub-structure is evaluated though some action (depending on the considered analysis heuristic); finally, the sub-structure is replaced with an activity block $QRT_\text {new}$ associated with the evaluated response time CDF (see Fig. 6d).

Fig. 6.

A graphical representation for actions 3 and 4. (a) The structure tree presented in Fig. 1b, where the red and the green boxes indicate the sub-structures subject to actions 3 and 4, respectively. (b) The structure tree after the application of action 3, which replaces the sub-structure in the red box with an activity block. (c) The structure tree after the replication of block R during action 4. (d) The structure tree after the execution of action 4, which replaces the sub-structure in the green box with an activity block. (Color figure online)