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Level Crossings in Point Processes Applied to Queues: Single-Server Case

Author

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  • P. H. Brill

    (University of Toronto, Toronto, Ontario)

  • M. J. M. Posner

    (University of Toronto, Toronto, Ontario)

Abstract

This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

Suggested Citation

  • P. H. Brill & M. J. M. Posner, 1977. "Level Crossings in Point Processes Applied to Queues: Single-Server Case," Operations Research, INFORMS, vol. 25(4), pages 662-674, August.
    • Handle: RePEc:inm:oropre:v:25:y:1977:i:4:p:662-674
    DOI: 10.1287/opre.25.4.662
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    Cited by:

    1. Ivo Adan & Brett Hathaway & Vidyadhar G. Kulkarni, 2019. "On first-come, first-served queues with two classes of impatient customers," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 113-142, February.
    2. P. H. Brill & C. M. Harris, 1992. "Waiting times for M/G/1 queues with service‐time or delay‐dependent server vacations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(6), pages 775-787, October.
    3. D. Perry & M. J. M. Posner, 1990. "Control of input and demand rates in inventory systems of perishable commodities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(1), pages 85-97, February.
    4. Pengfei Guo & Paul Zipkin, 2008. "The effects of information on a queue with balking and phase‐type service times," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(5), pages 406-411, August.
    5. Azoury, Katy S. & Miyaoka, Julia, 2020. "Optimal and simple approximate solutions to a production-inventory system with stochastic and deterministic demand," European Journal of Operational Research, Elsevier, vol. 286(1), pages 178-189.
    6. Roni, Mohammad S. & Eksioglu, Sandra D. & Jin, Mingzhou & Mamun, Saleh, 2016. "A hybrid inventory policy with split delivery under regular and surge demand," International Journal of Production Economics, Elsevier, vol. 172(C), pages 126-136.
    7. Urtzi Ayesta & Peter Jacko & Vladimir Novak, 2017. "Scheduling of multi-class multi-server queueing systems with abandonments," Journal of Scheduling, Springer, vol. 20(2), pages 129-145, April.
    8. Yutaka Sakuma & Onno Boxma & Tuan Phung-Duc, 2021. "An M/PH/1 queue with workload-dependent processing speed and vacations," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 373-405, August.
    9. Ken’ichi Kawanishi & Tetsuya Takine, 2016. "MAP/M/c and M/PH/c queues with constant impatience times," Queueing Systems: Theory and Applications, Springer, vol. 82(3), pages 381-420, April.
    10. Brill, Percy H. & Yu, Kaiqi, 2011. "Analysis of risk models using a level crossing technique," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 298-309.
    11. Ben A. Chaouch, 2007. "Inventory control and periodic price discounting campaigns," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(1), pages 94-108, February.
    12. Charles S. Tapiero & Morton J. Posner, 1988. "Warranty reserving," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(4), pages 473-479, August.
    13. Gérard Hébuterne & Catherine Rosenberg, 1999. "Arrival and departure state distributions in the general bulk‐service queue," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(1), pages 107-118, February.
    14. Yoshiaki Inoue & Onno Boxma & David Perry & Shelley Zacks, 2018. "Analysis of $$\hbox {M}^{\mathrm {x}}/\hbox {G}/1$$ M x / G / 1 queues with impatient customers," Queueing Systems: Theory and Applications, Springer, vol. 89(3), pages 303-350, August.
    15. Vyacheslav M. Abramov, 2023. "Crossings States and Sets of States in Random Walks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    16. J. George Shanthikumar, 1989. "Level crossing analysis of priority queues and a conservation identity for vacation models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 36(6), pages 797-806, December.
    17. Wolfgang Stadje, 1998. "Level-Crossing Properties of the Risk Process," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 576-584, August.
    18. Mohebbi, Esmail & Posner, Morton J. M., 1999. "A lost-sales continuous-review inventory system with emergency ordering," International Journal of Production Economics, Elsevier, vol. 58(1), pages 93-112, January.
    19. Mohebbi, Esmail & Hao, Daipeng, 2008. "An inventory model with non-resuming randomly interruptible lead time," International Journal of Production Economics, Elsevier, vol. 114(2), pages 755-768, August.
    20. Mohebbi, Esmail & Hao, Daipeng, 2006. "When supplier's availability affects the replenishment lead time--An extension of the supply-interruption problem," European Journal of Operational Research, Elsevier, vol. 175(2), pages 992-1008, December.
    21. Esmail Mohebbi & Morton J.M. Posner, 1998. "A continuous‐review inventory system with lost sales and variable lead time," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(3), pages 259-278, April.
    22. Boxma, Onno & Frostig, Esther & Perry, David & Yosef, Rami, 2017. "A state dependent reinsurance model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 170-181.

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