CoBTx-Net: A model for business collaboration reliability verification

Abstract

Collaborative business process can become unreliable when business partners collaborate in a peer- based fashion without central control. Therefore, reliability checking becomes an important issue that needs to be dealt with for any generic solution in managing business collaboration. In this paper, we propose a novel Choreographical Business Transaction Net (CoBTx-Net) to model collaborative business process and to manage the collaboration by individual participants. Furthermore three reliability properties named Time-embedded dead marking freeness, Inter-organizational dead marking freeness, and Collaborative soundness are defined based on CoBTx-Net to verify (1) the violation of time constraint, (2) collaborative logic conflicts, and (3) the improper termination from individual organizations.

Appendix

Theorem 1

A CoBTx-Net is time-embedded dead marking free iff

$$ \exists t_h\in T^{CoN},\ P_j^{r_{t_h}}(time)=\sum\limits_{i=1}^m \frac{r_{t_h}}{d_{t_h}} c_i\geq 1 $$

where $M^{\ast t}_k(time)\!=\!M_{k-1}\!\cdot\! L_{BP}^{\ast t},\ M^{\ast t}_k(time_j)\!=\!P_j(time)\!=\sum_{h=1}^n p_{j}^{r_{t_h}}(time).$

Proof

If a CoBTx-Net is time-embedded dead marking free, then within specified time there exists transition t _h to be executed. Therefore, the time constraint is not violated so that the transition can be fired. Obviously, there exists $r_{t_h}=1$ and q pre-places in M _k − 1 as c _i  ≥ 1 to accumulate enough tokens to activate the transition t _h firing(d ≤ q ≤ m). Hence $P_j^{r_{t_h}}(time)\geq 1$. If there exists $P_j^{r_{t_h}}(time)=\sum_{i=1}^m \frac{r_{t_h}}{d}c_i\geq 1$, then $r_{t_h}$=1 and q (d ≤ q ≤ m) pre-places as c _i are non-zero. It means that at specific time, there exists a transition t _h to be fired by tokens in pre-places c _i without violation on time constraint to finally deposit tokens in post-place p _j . Hence, the net is time-embedded dead marking free. □

Theorem 3

The solution for $\left[\bigtriangleup M^{AO}, \bigtriangleup M^{MO}\right]=$ $A^T\! \sum_{j=1}^{n}\left[\!u_j^{AO}\!, u_j^{MO}\!\right]$ exists and is qualified iff $B_{\!f}\big[\!\!\bigtriangleup\!\! M^{AO}\!,\bigtriangleup M^{MO}\big]=0$ and the fire of a transition in the solution to transfer an AO-Token must transfer a specified MO-Token together, where: $B_{\!f}=\big[ I_u:-A_{11}^T\big(A_{12}^T\big)^{-1}\big]$ , and r is the rank of incident matrix A and partitions A as:

$$ A=\left[\begin{array}{ll} A_{11}^{r \times m-r} & A_{12}^{r \times r}\\[7pt] A_{21}^{n-r \times m-r} & A_{22}^{n-r \times r}\\ \end{array} \right] $$

Proof

Firstly, we transform the equation into two equivalent linea equations as:

$$ \left\{\begin{array}{l} \bigtriangleup M^{AO}=A^T \sum_{j=1}^{n}u_j^{AO} \\[8pt] \bigtriangleup M^{MO}=A^T \sum_{j=1}^{n} u_j^{MO} \end{array}\right. $$

According to theorem in Murata (1989), if $B_f\big[\bigtriangleup M^{AO},\bigtriangleup M^{MO}\big]=0$, then there exists solutions for the combined equation. Since AO-Token will move with MO-Token, the qualified solutions for the equation will satisfy the condition that, if the transition $u_j^{AO}(x)\neq\emptyset$ (transferring an AO-Token), then it must also transfer a MO-Token together as $u_j^{MO}(x)\neq\emptyset$. □