On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

Let X = {X t : t ≥ 0} be a stationary piecewise continuous R d -valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ R d be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the in...

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Bibliographic Details
Published in:Journal of applied probability 2012-06, Vol.49 (2), p.351-363
Main Authors: Borovkov, K., Last, G.
Format: Article
Language:English
Subjects:
60G55
60J75
Curves
Density
Integrals
Level crossing
Markov processes
Mathematical analysis
Mathematical models
Mathematical vectors
Multivariate analysis
Network servers
Networks
Palm probability
piecewise-deterministic process
Queues
Queuing theory
Railroad grade crossings
Rice
Rice's formula
stationarity
stochastic network
Stochastic processes
Studies
Symbols
Trajectories
Vector fields
Vectors (mathematics)
Workloads
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Summary:Let X = {X t : t ≥ 0} be a stationary piecewise continuous R d -valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ R d be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X 0. Our result is illustrated by examples relating to queueing networks and stress release network models.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1339878791