A heterogeneous two-server network system with balking and a Bernoulli vacation schedule

In this paper, we study a two-server Markovian network system with balking and a Bernoulli schedule under a single vacation policy, where servers have different service rates. After every service, only one server may take a vacation or continue to stay in the system. The vacation time follows an exponential distribution. An arriving customer finding both servers free will choose the faster server. If the customer finds only one server is free, this customer chooses this free server. If the customer finds both servers are not free, then this customer may join the system or balk. For this system, we obtain the steady state condition, the stationary distribution of the number of customers in the system, and the mean system size by using a matrix-geometric method. Some special cases are deduced, which match with earlier exiting results. Extensive numerical illustrations are provided. Motivation for this system model also comes from some computer communication networks with different types of traffic such as real-time traffic and non-real-time traffic, where messages can be processed by two channels (servers) with different transmission rates. The behavior of abandoning messages can be equated with the balking of customers in this system model.