Message response time analysis for automotive cyber-physicalsystems with uncertain delay: An M/PH/1 queue approach.

The analysis problem of message response time is a key and challenging problem for in-vehicle networks that has gained much research attention in recent years. The arbitrated networked control system (ANCS), which is a special cyber–physicalsystem (CPS), was recently designed for scheduling or arbitrating networks in a control system. A dominant feature of this CPS is the arbitration of messages on a shared communication medium. The control applications in an ANCS are sensitive to the end-to-end delay of message responses from sensors to actuators. In this study, a multi-hierarchical flexible TDMA/fixed priority scheduling policy is first adopted to configure the bus in an ANCS, which is based on the event trigger protocol. Then, an M/PH/1 queue model with random-sized batch arrivals is used to model the control application in the ANCS, and to obtain the stationary probability for each control application. On the basis of the model, a constrained optimization problem is formulated to minimize the nonlinear cost function of the ANCS, and the probability density function of the uncertain response delay for each message is generated. The use of an optimal minislot algorithm that is based on the DFP variable-scale method and a parameter estimation algorithm of the pdfs of the uncertain response delays is proposed to obtain optimal values of the bus minislot and matrix parameters of PH distribution, respectively. Furthermore, the probability transformed GM (1,1) (PTGM (1,1)) with two parameters, which is a novel grey model (1,1) (GM (1,1)), is proposed for predicting the response delay for each message, in which a two-parameters optimization algorithm of PTGM (1,1) that is based on the DFP-method is provided to minimize the mean absolute percentage error and obtain the optimal values of the two parameters for each message. Extensive numerical results are provided to illustrate the usefulness of the proposed algorithms and show the accuracy of the predicted results.