# Level crossing methods

**DOI:**https://doi.org/10.1007/1-4020-0611-X_529

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## 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...

## References

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