Approximate Analysis of Tandem Blocking Queueing Networks with Correlated Arrivals and Services

Many real systems such as production lines are modeled as queueing networks with finite buffers. In such a queueing model, blocking occurs if a customer attempts to enter the next queue whose buffer is full and not available. In this case, the customer is forced to wait until the next queue can be entered. General queueing networks with blocking do not have a product-form solution. Therefore, it is very difficult to obtain the exact solution because of the explosion of state space. Accordingly, many approximation methods have been proposed so far. Most of them are based on the node decomposition method, where the queueing network is decomposed into several subsystems, and each subsystem is analyzed separately. Also, most of approximation methods proposed in the past consider renewal arrivals and services in each node. However, they can be correlated in real systems. It is known that the correlations in arrivals and services have a great impact on the performance of queueing networks. In this thesis, we develop an approximate method for tandem queueing networks with finite buffers and blocking, taking account of correlations in arrivals and services. We assume that the arrival process is a two-state Markov Arrival Process (MAP), which can represent correlation in interarrival times. In addition, we apply MAP to the service process in each node. In our method, a tandem queueing network with n nodes is decomposed into n−2 subsystems, each of which consists of three nodes in tandem. We develop an efficient algorithmic procedure for analyzing each subsystem and propose an iterative procedure for computing the steady-state probabilities of the number of customers in each node. Through numerical experiments, we examine the accuracy of the approximation and confirm that our method well approximates the performance of tandem queueing networks with correlated arrivals and services.