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An Overloaded Multiclass FIFO Queue with Abandonments

Author

Listed:
  • Otis B. Jennings

    (Graduate School of Business, Columbia University, New York, New York 10027)

  • Josh E. Reed

    (Leonard N. Stern School of Business, New York University, New York, New York 10012)

Abstract

In this paper we consider a single-server queue fed by K independent renewal arrival streams, each representing a different job class. Jobs are processed in a FIFO fashion, regardless of class. The total amount of work arriving to the system exceeds the server's capacity. That is, the nominal traffic intensity of the system is assumed to be greater than one. Jobs arriving to the system grow impatient and abandon the queue after a random amount of time if service has not yet begun. Interarrival, service, and abandonment times are assumed to be generally distributed and class specific.We approximate this system using both fluid and diffusion limits. To this end, we consider a sequence of systems indexed by n in which the arrival and service rates are proportional to n ; the abandonment distribution remains fixed across the sequence. In our first main result, we show that in the limit as n tends to infinity, the virtual waiting time process converges to a limiting deterministic process. This limit may be characterized as the solution to a first-order ordinary differential equation (ODE). Specific examples are then presented for which the ODE may be explicitly solved. In our second main result, we refine the deterministic fluid approximation by showing that the fluid-centered and diffusion-scaled virtual waiting time process weakly converges to an Ornstein-Uhlenbeck process whose drift and infinitesimal variance both vary over time. This process may also be solved for explicitly, thus yielding approximations to the transient as well as steady-state behavior of the virtual waiting time process.

Suggested Citation

  • Otis B. Jennings & Josh E. Reed, 2012. "An Overloaded Multiclass FIFO Queue with Abandonments," Operations Research, INFORMS, vol. 60(5), pages 1282-1295, October.
  • Handle: RePEc:inm:oropre:v:60:y:2012:i:5:p:1282-1295
    DOI: 10.1287/opre.1120.1095
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    References listed on IDEAS

    as
    1. Xuanming Su & Stefanos Zenios, 2004. "Patient Choice in Kidney Allocation: The Role of the Queueing Discipline," Manufacturing & Service Operations Management, INFORMS, vol. 6(4), pages 280-301, June.
    2. Xuanming Su & Stefanos A. Zenios, 2006. "Recipient Choice Can Address the Efficiency-Equity Trade-off in Kidney Transplantation: A Mechanism Design Model," Management Science, INFORMS, vol. 52(11), pages 1647-1660, November.
    3. Ward Whitt, 2006. "Fluid Models for Multiserver Queues with Abandonments," Operations Research, INFORMS, vol. 54(1), pages 37-54, February.
    4. Ward Whitt, 2004. "Efficiency-Driven Heavy-Traffic Approximations for Many-Server Queues with Abandonments," Management Science, INFORMS, vol. 50(10), pages 1449-1461, October.
    5. Francis de Véricourt & Otis B. Jennings, 2011. "Nurse Staffing in Medical Units: A Queueing Perspective," Operations Research, INFORMS, vol. 59(6), pages 1320-1331, December.
    6. Rishi Talreja & Ward Whitt, 2008. "Fluid Models for Overloaded Multiclass Many-Server Queueing Systems with First-Come, First-Served Routing," Management Science, INFORMS, vol. 54(8), pages 1513-1527, August.
    7. Mor Armony & Nahum Shimkin & Ward Whitt, 2009. "The Impact of Delay Announcements in Many-Server Queues with Abandonment," Operations Research, INFORMS, vol. 57(1), pages 66-81, February.
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    Cited by:

    1. Otis B. Jennings & Jamol Pender, 2016. "Comparisons of ticket and standard queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 145-202, October.
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    3. Barış Ata & Anton Skaro & Sridhar Tayur, 2017. "OrganJet: Overcoming Geographical Disparities in Access to Deceased Donor Kidneys in the United States," Management Science, INFORMS, vol. 63(9), pages 2776-2794, September.

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