Review of Some Functionals of Compound Poisson Processes and Related Stopping Times

  • S. ZacksEmail author


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.


Compound Poisson process M/G/1 queue with restricted accessibility Expected cost functionals Stopping times Production/Inventory systems Sequential estimation Random boundaries 

AMS 2000 Subject Classification

60G17 60G40 60K25 60J75 90B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. K. Borovkov, and Z. Burq, “Kendall’s identity for the first crossing time revisited,” Electronic Communications in Probability vol. 6 pp. 91–94, 2001.MathSciNetGoogle Scholar
  2. M. Ghosh, N. Mukhopadhyay, and P. K. Sen, Sequential Estimation, Wiley: New York, 1997.zbMATHGoogle Scholar
  3. E. P. C. Kao, An Introduction to Stochastic Processes, Duxbury: New York, 1977.Google Scholar
  4. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  8. D. Perry, W. Stadje, and S. Zacks, “Hysteretic capacity switching for M/G/1 queues," Stochastic Models 2007 (in print).Google Scholar
  9. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  10. W. Stadje, and S. Zacks, “Telegraph processes with random velocities,” Journal of Applied Probability vol. 41 pp. 665–678, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  11. N. Starr, and M. Woodroofe, “Further remarks on sequential estimation: the exponential case,” Annals of Mathematical Statistics vol. 43 pp. 1147–1154, 1972.MathSciNetzbMATHGoogle Scholar
  12. M. Woodroofe, “Second-order approximations for sequential point and interval estimation,” Annali di Statistica vol. 5 pp. 984–995, 1977.zbMATHMathSciNetCrossRefGoogle Scholar
  13. M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis, SIAM: Philadelphia, 1982.Google Scholar
  14. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  15. S. Zacks, “Generalized integrated telegraph processes and the distribution of related stopping times,” Journal of Applied Probability vol. 41 pp.497–507, 2004b.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  17. S. Zacks, and N. Mukhopadhyay, “Exact risks of sequential point estimators of the exponential parameter,” Sequential Analysis vol. 25 pp. 203–226, 2006.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA

Personalised recommendations