A Coupled Laplace–Bessel Approach for Transient and Steady-State Analysis of Multi-Stage Wireless Sensor Networks

In this paper, we consider a multi-stage network with transition between intermediate states and relate it to packet movement in wireless sensor networks. Both transient-state and steady-state analyses are carried out. A continuous-time Markov chain model is employed for analysis. The model best resembles a network that can be used to study transient behavior in a packet-switched network, for example, routers in the internet backbone. The state probabilities are analytically derived and numerically computed. The transient-state analysis section explores the system probability in various states. In addition, we explore the average number of packets at a particular time and at a particular state. The transient-state and steady-state solutions help to identify retransmission after a failed attempt, allowing us to study the influence of wireless channel quality. Transition entropy is employed to quantify the uncertainty and dynamic variability in state transitions of the system. Finally, numerical illustrations are provided in order to visualize the network behavior, especially the intensity of data traffic over a time period t. The numerical section captures the transient packet dynamics and evaluates the mean and variance of the packet population. Sensitivity analysis is also performed, showing that higher phase-wise arrival rates increase packet count variability, while experiments with synthetic event logs demonstrate the model’s applicability for parameter estimation.