A new disk-based technique for solving the largeness problem of stochastic modeling formalisms

Stochastic modeling formalisms such as stochastic Petri nets, generalized stochastic Petri nets, and stochastic reward nets can be used to model and evaluate the dynamic behavior of realistic computer systems. Once we translate the stochastic system model to the underlying corresponding Markov Chain (MC), the developed MC grows wildly to several hundred thousands states. This problem is known as the largeness problem. To tolerate the largeness problem of Markov models, several iterative and direct methods have been proposed in the literature. Although the iterative methods provide a feasible solution for most realistic systems, a major problem appears when these methods fail to reach a solution. Unfortunately, the direct method represents an undesirable numerical technique for tolerating large matrices due to the fill-in problem. In order to solve such problem, in this paper, we develop a Disk-Based Segmentation (DBS) technique based on modifying the Gauss Elimination (GE) technique. The proposed technique has the capability of solving the consequences of the fill-in problem without making assumptions about the underlying structure of the Markov processes of the developed model. The DBS technique splits the matrix into a number of vertical segments and uses the hard disk to store these segments. Using the DBS technique, we can greatly reduce the memory required as compared to that of the GE technique. To minimize the increase in the solution time due to the disk accessing processes, the DBS utilizes a clever management technique for such processes. The effectiveness of the DBS technique has been demonstrated by applying it to a realistic model for the Kanban manufacturing system.