A hybrid approach to minimize state space explosion problem for the solution of two stage tandem queues

Two stage open queuing networks are used for modelling the subsystem-behaviour in computers and communication networks, mass storage devices, memory servers, and queuing analysis of wireless mobile cellular networks. The queuing analysis of wireless systems are essential in order to quantify the impacts of different factors on quality of service (QoS) performance measures so that wireless protocols can be designed and/or tuned in an optimal manner. In that sense two stage open queuing systems are particularly important to model handoff phenomena, especially for the integration of two different systems such as cellular and wireless local area networks (WLANs). Analytical solutions for two dimensional Markov processes suffer from the state space explosion problem. The numerical difficulties caused by large state spaces, makes it difficult to handle multiple servers at the second stage of a tandem queuing system together with server failures and repairs. This study presents a new approach to analytical modelling of open networks offering improvements in alleviating this problem. The proposed solution is a hybrid version, which combines well known spectral expansion, and hierarchical Markov reward rate approaches. Using this approach, two-stage open networks with multiple servers, break-downs, and repairs at the second stage and feedback can be modelled as three dimensional Markov processes and solved for performability measures. Comparative results show that the new algorithm used for solution, provides high degree of accuracy, and it is computationally more efficient than the existing approaches. The proposed model is capable of solving other three dimensional Markov processes.