On the Exact Inter-departure, Inter-start, and Cycle Time Distribution of Closed Queueing Networks Subject to Blocking

Abstract

This paper presents a method to determine the exact inter-departure, inter-start and cycle time distribution of closed queueing networks that can be modeled as Continuous-Time Markov Chains with finite state space. The method is based on extending the state space to determine the transitions that lead to a departure or to an arrival of a part on a station. Once these transitions are identified and represented in an indicator matrix, a first passage time analysis is utilized to determine the exact distributions of the inter-departure, inter-start, and cycle time. In order to demonstrate the methodology, we consider closed-loop production lines with phase-type service time distributions and finite buffers. We discuss the methodology to automatically generate the state space and to obtain the transition rate matrices for the considered distributions. We use the proposed method to analyze the effects of the system parameters on the inter-departure, inter-start time, and cycle time distributions numerically for various cases. The proposed methodology allows the exact analysis of the inter-departure, inter-start, and cycle time distributions of a wide range of production systems with phase-type servers that can be modeled as Continuous-Time Markov Chains in a unified way.