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Stability and Chaos in Input Pricing for a Service Facility with Adaptive Customer Response to Congestion

Author

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  • Christopher M. Rump

    (Department of Industrial Engineering, State University of New York at Buffalo, Buffalo, New York 14260-2050)

  • Shaler Stidham

    (Department of Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599-3180)

Abstract

We consider the stability of the equilibrium arrival rate and equilibrium admission price at a service facility, using a generalization of an input-pricing model introduced by Dewan and Mendelson and further examined by Stidham. At the equilibrium, the marginal value of service equals the admission price, that is, the sum of the admission fee and the expected delay cost. Stability means (roughly) that the system returns to the equilibrium after a perturbation, assuming the customers base their join/balk decisions on previous prices. We extend the discrete-time, dynamic-system pricing model of Stidham to allow adaptive expectations in which customers predict the future price based on a convex combination of the current price and the previous prediction. We show that this can lead to chaotic behavior when the equilibrium is unstable. That is, the price and arrival rate can follow aperiodic orbits, which appear to be completely random. Our results suggest an alternative explanation for observed variations in the mean arrival rate to a queueing system, which are often modeled by means of a random exogenous (e.g., Markovian) environment process.

Suggested Citation

  • Christopher M. Rump & Shaler Stidham, 1998. "Stability and Chaos in Input Pricing for a Service Facility with Adaptive Customer Response to Congestion," Management Science, INFORMS, vol. 44(2), pages 246-261, February.
    • Handle: RePEc:inm:ormnsc:v:44:y:1998:i:2:p:246-261
    DOI: 10.1287/mnsc.44.2.246
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    References listed on IDEAS

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    1. Sanjeev Dewan & Haim Mendelson, 1990. "User Delay Costs and Internal Pricing for a Service Facility," Management Science, INFORMS, vol. 36(12), pages 1502-1517, December.
    2. Ward Whitt, 1993. "Large Fluctuations in a Deterministic Multiclass Network of Queues," Management Science, INFORMS, vol. 39(8), pages 1020-1028, August.
    3. Chiarella, Carl, 1988. "The cobweb model: Its instability and the onset of chaos," Economic Modelling, Elsevier, vol. 5(4), pages 377-384, October.
    4. Shaler Stidham, 1992. "Pricing and Capacity Decisions for a Service Facility: Stability and Multiple Local Optima," Management Science, INFORMS, vol. 38(8), pages 1121-1139, August.
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    Citations

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    Cited by:

    1. Addabbo, Tommaso & Kocarev, Ljupco, 2009. "Periodic dynamics in queuing networks," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2178-2192.
    2. Christian Haxholdt & Erik R. Larsen & Ann van Ackere, 2003. "Mode Locking and Chaos in a Deterministic Queueing Model with Feedback," Management Science, INFORMS, vol. 49(6), pages 816-830, June.
    3. Fang Wu & Junhai Ma, 2023. "Research Trend, Logical Structure and Outlook on Complex Economic Game," Mathematics, MDPI, vol. 11(5), pages 1-16, February.
    4. Yuan, Xuchuan & Brian Hwarng, H., 2023. "Examining the dynamics of reactive capacity allocation through a chaos lens," European Journal of Operational Research, Elsevier, vol. 308(2), pages 912-928.
    5. Qian Wang & Rajan Batta & Christopher M. Rump, 2004. "Facility location models for immobile servers with stochastic demand," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(1), pages 137-152, February.
    6. Gérard P. Cachon & Patrick T. Harker, 2002. "Competition and Outsourcing with Scale Economies," Management Science, INFORMS, vol. 48(10), pages 1314-1333, October.
    7. Anthony M. Kwasnica & Euthemia Stavrulaki, 2008. "Competitive location and capacity decisions for firms serving time‐sensitive customers," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(7), pages 704-721, October.
    8. Wu, Y. & Zhang, D.Z., 2007. "Demand fluctuation and chaotic behaviour by interaction between customers and suppliers," International Journal of Production Economics, Elsevier, vol. 107(1), pages 250-259, May.
    9. Bitran, Gabriel R. & Rocha e Oliveira, Paulo & Schilkrut, Ariel, 2008. "Managing customer relationships through price and service quality," IESE Research Papers D/750, IESE Business School.
    10. Wang, Shoou-Jiun & Batta, Rajan & Rump, Christopher M., 2005. "Stability of a crime level equilibrium," Socio-Economic Planning Sciences, Elsevier, vol. 39(3), pages 229-244, September.
    11. Pavlin, J. Michael, 2017. "Dual bounds of a service level assignment problem with applications to efficient pricing," European Journal of Operational Research, Elsevier, vol. 262(1), pages 239-250.
    12. Qian Liu & Garrett van Ryzin, 2011. "Strategic Capacity Rationing when Customers Learn," Manufacturing & Service Operations Management, INFORMS, vol. 13(1), pages 89-107, September.
    13. Edward Anderson & Frank Kelly & Richard Steinberg, 2006. "A Contract and Balancing Mechanism for Sharing Capacity in a Communication Network," Management Science, INFORMS, vol. 52(1), pages 39-53, January.

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