Abstract
The paper reviews recent results of D. Perry, W. Stadje and S. Zacks, on functionals of stopping times and the associated compound Poisson process with lower and upper linear boundaries. In particular, formulae of these functionals are explicitly developed for the total expected discounted cost of discarded service in an M/G/1 queue with restricted accessibility; for the expected total discounted waiting cost in an M/G/1 restricted queue; for the shortage, holding and clearing costs in an inventory system with continuous input; for the risk in sequential estimation and for the transform of the busy period when the upper boundary is random.
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References
K. Borovkov, and Z. Burq, “Kendall’s identity for the first crossing time revisited,” Electronic Communications in Probability vol. 6 pp. 91–94, 2001.
M. Ghosh, N. Mukhopadhyay, and P. K. Sen, Sequential Estimation, Wiley: New York, 1997.
E. P. C. Kao, An Introduction to Stochastic Processes, Duxbury: New York, 1977.
D. Perry, W. Stadje, and S. Zacks, “Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility,” Operations Research Letters vol. 27 pp. 163–174, 2000.
D. Perry, W. Stadje, and S. Zacks, “The first rendezvous time of a Brownian motion and compound Poisson-type processes,” Journal of Applied Probability vol. 41 pp. 1059–1070, 2004.
D. Perry, W. Stadje, and S. Zacks, “Sporadic and continuous clearing policies for a production/inventory system under M/G demand process,” Mathematics of Operations Research vol. 30 pp. 354–368, 2005a.
D. Perry, W. Stadje, and S. Zacks, “A two-sided first-exist problem for a compound Poisson process with a random upper boundary,” Methodology and Computing in Applied Probability vol. 7 pp. 51–62, 2005b.
D. Perry, W. Stadje, and S. Zacks, “Hysteretic capacity switching for M/G/1 queues," Stochastic Models 2007 (in print).
W. Stadje, and S. Zacks, “Upper first-exit times of compound Poisson processes revisited,” Probability in the Engineering and Informational Sciences vol. 17 pp. 459–465, 2003.
W. Stadje, and S. Zacks, “Telegraph processes with random velocities,” Journal of Applied Probability vol. 41 pp. 665–678, 2004.
N. Starr, and M. Woodroofe, “Further remarks on sequential estimation: the exponential case,” Annals of Mathematical Statistics vol. 43 pp. 1147–1154, 1972.
M. Woodroofe, “Second-order approximations for sequential point and interval estimation,” Annali di Statistica vol. 5 pp. 984–995, 1977.
M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis, SIAM: Philadelphia, 1982.
S. Zacks, “Exact determination of the run length distribution of a one-sided CUSUM procedure applied on an ordinary Poisson process,” Sequential Analysis vol. 23 pp. 159–178, 2004a.
S. Zacks, “Generalized integrated telegraph processes and the distribution of related stopping times,” Journal of Applied Probability vol. 41 pp.497–507, 2004b.
S. Zacks, “Some recent results on the distributions of stopping times of compound Poisson processes with linear boundaries,” Journal of Statistics Inference and Planning vol. 130 pp. 95–109, 2005.
S. Zacks, and N. Mukhopadhyay, “Exact risks of sequential point estimators of the exponential parameter,” Sequential Analysis vol. 25 pp. 203–226, 2006.
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Zacks, S. Review of Some Functionals of Compound Poisson Processes and Related Stopping Times. Methodol Comput Appl Probab 9, 343–356 (2007). https://doi.org/10.1007/s11009-006-9015-1
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DOI: https://doi.org/10.1007/s11009-006-9015-1
Keywords
- Compound Poisson process
- M/G/1 queue with restricted accessibility
- Expected cost functionals
- Stopping times
- Production/Inventory systems
- Sequential estimation
- Random boundaries