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The Stationary Distribution of a Stochastic Clearing Process

Author

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  • Ward Whitt

    (Bell Laboratories, Holmdel, New Jersey)

Abstract

This research grew out of an investigation of utilization in capacity expansion. The utilization at any time is the demand divided by the capacity. When there is uncertainty about the evolution of demand, it is appropriate to model the demand as a stochastic process, and thus the utilization also becomes a stochastic process. It was found that a utilization stochastic process associated with exponentially growing stochastic demand is closely related to the stochastic clearing processes introduced and investigated by Stidham. Interest in the impact of uncertainty on utilization led to this study of the impact of uncertainty on the stationary distribution of a stochastic clearing process. Stidham showed for a large class of clearing processes that the stationary distribution is never the uniform distribution, which is characteristic of deterministic models with continuous linear input. Here it is shown for a larger class of clearing processes that the stationary distribution is always stochastically less than or equal to the uniform distribution in the sense of second-order stochastic dominance (characterized by the expected value of all nondecreasing concave functions). For various special cases, stronger stochastic order relations are established. For a related capacity expansion model, it is shown that greater uncertainty lowers the expected utilization.

Suggested Citation

  • Ward Whitt, 1981. "The Stationary Distribution of a Stochastic Clearing Process," Operations Research, INFORMS, vol. 29(2), pages 294-308, April.
    • Handle: RePEc:inm:oropre:v:29:y:1981:i:2:p:294-308
    DOI: 10.1287/opre.29.2.294
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    Cited by:

    1. Rahul R. Marathe & Sarah M. Ryan, 2009. "Capacity expansion under a service‐level constraint for uncertain demand with lead times," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(3), pages 250-263, April.
    2. Tunay I. Tunca & Weiming Zhu, 2018. "Buyer Intermediation in Supplier Finance," Management Science, INFORMS, vol. 64(12), pages 5631-5650, December.
    3. Perera, Sandun & Gupta, Varun & Buckley, Winston, 2020. "Management of online server congestion using optimal demand throttling," European Journal of Operational Research, Elsevier, vol. 285(1), pages 324-342.
    4. Kumar Muthuraman & Sridhar Seshadri & Qi Wu, 2015. "Inventory Management with Stochastic Lead Times," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 302-327, February.
    5. AlShelahi, Abdullah & Wang, Jingxing & You, Mingdi & Byon, Eunshin & Saigal, Romesh, 2020. "Data-driven prediction for volatile processes based on real option theories," International Journal of Production Economics, Elsevier, vol. 226(C).
    6. Jacobovic, Royi & Kella, Offer, 2020. "Minimizing a stochastic convex function subject to stochastic constraints and some applications," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 7004-7018.
    7. Royi Jacobovic & Offer Kella, 2019. "Asymptotic independence of regenerative processes with a special dependence structure," Queueing Systems: Theory and Applications, Springer, vol. 93(1), pages 139-152, October.
    8. Sarah M. Ryan, 2004. "Capacity Expansion for Random Exponential Demand Growth with Lead Times," Management Science, INFORMS, vol. 50(6), pages 740-748, June.
    9. Gupta, Varun & Perera, Sandun, 2021. "Managing surges in online demand using bandwidth throttling: An optimal strategy amid the COVID-19 pandemic," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 151(C).

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