Derivation of the N-step interdeparture time distribution in GI/G/1 queueing systems |
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| Institution: | 1. State Key Laboratory of Materials-Orient Chemical Engineering, College of Materials Science and Engineering, Nanjing Tech University, Nanjing, 211816, China;2. Jiangsu National Synergetic Innovation Center for Advanced Materials (SICAM), Nanjing Tech University, Nanjing, 211816, China;3. Jiangsu Collaborative Innovation Center for Advanced Inorganic Function Composites, Nanjing Tech University, Nanjing, 211816, China;1. College of Management and Economics, Tianjin University, No.92, Weijin Road, Nankai District, Tianjin 300072, China;2. Systems Management and Strategy Department, Business School, University of Greenwich, SE10 9LS, UK;3. Department of Industrial Engineering, Tsinghua University, Haidian District, Beijing, China |
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| Abstract: | The departure process of a queueing system has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this paper, we derive a closed form expression for the distribution of interdeparture time in a GI/G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general KM/G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model. |
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