Analysis of a semi-open queueing network with Markovian arrival process

A semi-open queueing network having a finite number of nodes is considered. The nodes are modeled by single-server queueing systems with a finite buffer and an exponential service time distribution. Customers arrive to the network according to a Markovian arrival process. The number of customers, which can be processed in the network simultaneously, is restricted by a threshold. If the number of customers in the network is less than this threshold, when a new customer arrives, the customer is processed in the network. Choice of the first and the subsequent nodes for service is performed randomly according to a fixed stochastic vector and a transition probability matrix. If the number of customers in the network at the customer arrival epoch is equal to the threshold, the customer is queued into an input buffer with an infinite capacity. Customers in the input buffer are impatient. The stationary behavior of network states is analyzed. The Laplace-Stieltjes transform of the distribution of the customer's waiting time in the input buffer is obtained. Expressions for computing performance measures of the network are derived. Numerical results are presented. The model is suitable, e.g., for analysis and optimization of wireless telecommunication networks and manufacturing systems with a finite number of machines and workers.