Çinlar, Erhan Sunset over Brownistan. (English) Zbl 0757.60073 Stochastic Processes Appl. 40, No. 1, 45-53 (1992). 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. Reviewer: N.M.Zinchenko (Kiev) Cited in 8 Documents MSC: 60J65 Brownian motion 60G57 Random measures 60K25 Queueing theory (aspects of probability theory) 60G17 Sample path properties Keywords:Brownian motion; Brownian motion with the down-ward drift; concave majorant; Poisson random measure; queueing theory PDFBibTeX XMLCite \textit{E. Çinlar}, Stochastic Processes Appl. 40, No. 1, 45--53 (1992; Zbl 0757.60073) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.