INTRODUCTION
Level crossing methods for obtaining probability distributions in stochastic models such as queues and inventories were originated by Brill (1975, 1976, 1979) and elucidated further in Brill and Posner (1974, 1975, 1977, 1981), and Cohen (1976, 1977). These methods began as an essential part of system point theory and are also known as system point analysis, sample path analysis, or level crossing technique, approach, theory, or analysis in the literature (Brill, 1975). Level crossing methods are very useful rate conservation techniques for stochastic models (Miyazawa, 1994).
MODEL AND STATIONARY DISTRIBUTION
Consider a stochastic process {W(t), t ≥ 0} where both the parameter set and state space are continuous. The random variable W(t) at time point t may denote the content of a dam with general efflux, the stock on hand in an < s,S > or < r,nQ > inventory system with stock decay, or the virtual wait or workload in a queue. Assume that upward jumps of {W(t)} occur at...
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Azoury, K. and Brill, P.H. (1986). “An Application of the System-Point Method to Inventory Models under Continuous Review.” Jl. Applied Probability, 23, 778–789.
Brill, P.H. (1975). “System Point Theory in Exponential Queues,” Ph.D. Dissertation, University of Toronto.
Brill, P.H. (1976). “Embedded Level Crossing Processes in Dams and Queues.” WP #76-022, Department of Industrial Engineering, University of Toronto.
Brill, P.H. (1979). “An Embedded Level Crossing Technique for Dams and Queues.” Jl. Applied Probability, 16, 174–186.
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Brill, P.H. and Posner, M.J.M. (1981). “The System Point Method in Exponential Queues: A Level Crossing Approach.” Mathematics Operations Research, 6, 31–49.
Cohen, J.W. (1976). On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag. New York.
Cohen, J.W. (1977). “On Up and Down Crossings.” Jl. Applied Probability, 14, 405–410.
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© 2001 Kluwer Academic Publishers
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Brill, P.H. (2001). Level crossing methods . In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_529
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DOI: https://doi.org/10.1007/1-4020-0611-X_529
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