Ismail A Mageed - PhD, AIMMA, IEEE, IAENG, School of Computer Science, AI, and Electronics, Faculty of Engineering and Digital Technologies, University of Bradford, United Kingdom.
Abdul Raheem Nazir - Department of Computing, Sheffield Hallam University, United Kingdom.

This research investigates the unexplored domain of negative k parameters within dynamic systems, focusing on their influence across stability, traffic intensity, and chaotic phases. Using an iterative computational framework, we examine the sigma function (σ) to characterize system responses under varying conditions of k and ρ. Visualizations and insights demonstrate transitions between phases for a first time-ever exploration for negative values of the number of sets of phases, namely k, with novel findings extending foundational studies. This work establishes a baseline for further explorations of negative parameter spaces in complex systems. It is to be noted that, the current work provides new contributions to Ismail’s contemporary pointwise stationary fluid approximation theory, through offering a comprehensive computational framework for exploring dynamic system behaviours across varying parameters.

Pointwise stationary fluid approximation, The non-stationary E_k/M/۱ queue, k sets of phases