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Sunset over Brownistan. (English) Zbl 0757.60073

Let \(B_ t\) be a standard Brownian motion starting at zero and \(X_{t,a}=B_ t-at\) be a Brownian motion with the downward drift of rate \(a>0\). The author investigates \(Z_ a=\max_{t\geq 0} X_{t,a}\) and \(D_ a=\sup\{t: B_ t-at=Z_ a\}\) as the stochastic processes indexed by \(a\). Using simple geometric observations he discovers the relation between \(Z_ a\) and the concave majorant of \(B_ t\) and establishes the characterization of \(Z_ a\) and \(D_ a\) in terms of Poisson random mmeasure. This gives the opportunity to obtain some distributional results for \(Z_ a\) and \(D_ a\) which are used in queueing theory.

MSC:

60J65 Brownian motion
60G57 Random measures
60K25 Queueing theory (aspects of probability theory)
60G17 Sample path properties
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References:

[1] Aldous, D., Some interesting processes arising as heavy traffic limits in an M/M/∞ storage process, Stochastic Process. Appl., 22, 291-313 (1986) · Zbl 0607.60086
[2] Coffman, E. G.; Kadota, T. T.; Shepp, L. A., A stochastic model of fragmentation in dynamic storage allocation, SIAM J. Comput., 14, 416-425 (1985) · Zbl 0605.68021
[3] Groeneboom, P., The concave majorant of Brownian motion, Ann. Probab., 11, 1016-1027 (1983) · Zbl 0523.60079
[4] Newell, G. F., The M/M/∞ service system with ranked servers in heavy traffic, (Lecture Notes in Econ. and Math. Syst. No. 231 (1984), Springer: Springer New York) · Zbl 0543.90040
[5] Pitman, J. W., Remarks on the convex minorant of Brownian motion, (Seminar on Stochastic Processes, 1982 (1983), Birkhäuser: Birkhäuser Boston, MA), 219-227
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