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Queueing networks, which consist of several service stations, are frequently used in modeling to represent particular structures of real systems with large number of resources, such as computer and communication networks. In queueing networks at least two service stations are connected to each other. These service stations, or nodes in a queueing network, represent resources in a real system. Generally, jobs can be transferred between any two nodes of a queueing network; particularly, a job can be directly returned to the node it has just left.
A queueing network is open when jobs can enter the network from and leave the network to the outside. A queueing network is closed when jobs can neither enter nor leave the network. The number of jobs in a closed network is constant. Performance measure of a queueing network could be obtained while computing the steady-state probabilities (Hassin and Haviv, 2003; Lokshina, 2016; Lokshina and Bartolacci, 2012; Lokshina and Bartolacci, 2007; Lokshina, Zhong and Lanting, 2020; Radev, Lokshina and Denchev, 2007; Radev, Lokshina and Radeva, 2007).
Developing computational procedures to obtain the steady-state probabilities for all possible states in a queueing network is a central issue in the queueing theory. The means for many other important network performance measures can be easily obtained after having these algorithms developed (Dugas, Chapados, Ducharme, Saint-Mleux and Vincent; 2011; Hassin and Haviv, 2003; Lokshina, 2016; Lokshina and Bartolacci, 2012; Lokshina and Bartolacci, 2007).
Markov processes provide flexible, powerful, and efficient means for evaluation and analysis of dynamic systems. Performance and reliability measures of a queueing network can be derived and evaluated with the steady-state analysis of Discrete-Time Markov Chains (DTMC) and Continuous-Time Markov Chains (CTMC). Generally, each queueing system can be considered as a specific case of Markov process, and then mathematically evaluated in terms of the process (Hassin and Haviv, 2003; Liu, Ma and Li, 2012; Lokshina, 2016; Lokshina and Bartolacci, 2012; Zhang, 2009). Direct and iterative methods can be used to obtain numerical solutions in the steady-state analysis of Markov chains (Bhalai, 2002; Liu, Ma and Zhang, 2015; Lokshina, 2016; Lokshina and Bartolacci, 2012; Lokshina and Bartolacci, 2007; Radev, Lokshina and Denchev, 2007; Radev, Lokshina and Radeva, 2007).
Direct methods, as they are applied, can modify the parameter matrix. They can also require fixed amount of computational time, independently to parameter values. At the same time, direct methods are subject to accumulation of round-off errors and create significant difficulties with sparse storage (Hassin and Haviv, 2003; Mehdi, 2003; Lokshina, 2016; Radev, Lokshina and Denchev, 2007).