# Level crossing methods

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

- 3 Citations
- 28 Downloads

## 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

- [1]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.Google Scholar
- [2]Brill, P.H. (1975). “System Point Theory in Exponential Queues,” Ph.D. Dissertation, University of Toronto. Google Scholar
- [3]Brill, P.H. (1976). “Embedded Level Crossing Processes in Dams and Queues.” WP #76-022, Department of Industrial Engineering, University of Toronto. Google Scholar
- [4]Brill, P.H. (1979). “An Embedded Level Crossing Technique for Dams and Queues.” Jl. Applied Probability, 16, 174–186.Google Scholar
- [5]Brill, P.H. (1991). “Estimation of Stationary Distributions in Storage Processes Using Level Crossing Theory.” Proc. Statist. Computing Section, Amer. Statist. Assn., 172–177. Google Scholar
- [6]Brill, P.H. and Posner, M.J.M. (1974). “On the Equilibrium Waiting Time Distribution for a Class of Exponential Queues.” WP #74-012, Department of Industrial Engineering, University of Toronto. Google Scholar
- [7]Brill, P.H. and Posner, M.J.M. (1975). “Level Crossings in Point Processes Applied to Queues.” WP #75-009, Department of Industrial Engineering, University of Toronto. Google Scholar
- [8]Brill, P.H. and Posner, M.J.M. (1977). “Level Crossings in Point Processes Applied to Queues: Single Server Case.” Operations Research, 25, 662–673.Google Scholar
- [9]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.Google Scholar
- [10]Cohen, J.W. (1976). On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag. New York.Google Scholar
- [11]Cohen, J.W. (1977). “On Up and Down Crossings.” Jl. Applied Probability, 14, 405–410.Google Scholar
- [12]Miyazawa, M. (1994). “Rate Conservation Laws: A Survey.” Queueing Systems: Theory & Applics., 18, 1–58.Google Scholar
- [13]Ross, S. (1985). Introduction to Probability Models, 4th edition, Academic Press, Inc. Google Scholar