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Recent developments in the queueing problem

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Abstract

A group of agents must be served in a facility. The facility can serve only one agent at a time and agents incur waiting costs. The queueing problems is concerned with finding the order to serve agents and the monetary transfers. It can be solved by taking various approaches: the cooperative game theoretic approach, the normative approach, the strategic approach, the bargaining approach, and the combination of these approaches. In this paper, we provide a survey on the recent developments in the queueing problem.

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Notes

  1. See Chun (2016) for a survey of the literature on the queueing problem.

  2. For any set A,  |A| denotes the cardinality of A.

  3. \(\mathbb {R}_{+}\) denotes the non-negative orthant of the real line.

  4. Also, see Curiel et al. (1989) for a sequencing problem with an initial queue and Chun (2011) for a sequencing problem with bilateral transfers.

  5. The family of VCG rules is due to Vickrey (1961), Clarke (1971), and Groves (1973).

  6. For example, the bankruptcy problem discussed in Thomson (2003).

  7. Kar et al. (2009), by taking a general set of queueing games that includes all convex combinations of the optimistic queueing game and the pessimistic queueing game, obtained the coincidence between the Shapley value and the prenucleolus.

  8. Moulin (2007) makes the same observation for the scheduling problem.

  9. Since we assume an existence of a tie-breaking rule, we always have a unique queue satisfying queue-efficiency. As a consequence, we do not need either Pareto-indifference or anonymity in the statement of our theorems.

  10. For this, a position in a queue is considered as an indivisible good.

  11. Group no-envy extends the notion of no-envy to groups. See Svensson (1983) for details.

  12. These are allocations that can be supported as Walrasian equilibrium with an equal implicit income.

  13. Object-efficiency requires that there is no feasible allocation which makes every agent better off and at least one agent strictly better off.

  14. The literature on strategy-proofness is too large to give a comprehensive list of references. A recent review of this literature, along with a list of references, can be found in Barberà (2011) and Thomson (2013).

  15. Since t depends on the choice of the queue, we should denote the transfers by \(t(\sigma (\theta ))\) instead of \(t(\theta )\). Note that our (single-valued) rule chooses a unique queue which in turn determines the unique transfers. Therefore, we abuse the notation and write \(t(\theta )\).

  16. For various bargaining protocols implementing the Shapley value, see Gul (1989), Hart and Mas-Colell (1996), Ju (2013), Ju and Wettstein (2009), and Pérez-Castrillo and Wettstein (2001).

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Authors and Affiliations

  1. Department of Economics, Seoul National University, Seoul, South Korea

    Youngsub Chun

  2. Economic Research Unit, Indian Statistical Institute, Kolkata, India

    Manipushpak Mitra

  3. Economics Division, School of Business, University of Leicester, Leicester, LE1 7RH, UK

    Suresh Mutuswami

Authors
  1. Youngsub Chun
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  2. Manipushpak Mitra
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  3. Suresh Mutuswami
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Correspondence to Youngsub Chun.

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This invited paper is discussed in the comments available at https://doi.org/10.1007/s11750-019-00500-w, https://doi.org/10.1007/s11750-019-00501-9, https://doi.org/10.1007/s11750-019-00502-8.

We are grateful to Gustavo Bergantiños for his comments. Chun’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016S1A3A2924944).

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Chun, Y., Mitra, M. & Mutuswami, S. Recent developments in the queueing problem. TOP 27, 1–23 (2019). https://doi.org/10.1007/s11750-019-00499-0

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