It is sometimes argued that bribery is inefficient because bureaucrats may cause delays for attracting more bribes. This hypothesis is examined in the context of a queue where customers having different values of time are ranked by their bribe payments to the queue's server. The Nash equilibrium strategies of the customers are derived. It is shown that the server is unlikely to slow down the allocation process when bribery is allowed. The model does not have stringent informational requirements, and the equilibrium outcome minimizes the average value of time costs of the queue. It also suggests a useful auctioning procedure.

An Equilibrium Queuing Model of Bribery

Preface. 1. Introduction. 2. Observable Queues. 3. Unobservable Queues. 4. Priorities. 5. Reneging and Jockeying. 6. Schedules and Retrials. 7. Competition Among Servers. 8. Service Rate Decisions. Index.

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To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems

https://www-sop.inria.fr/members/Eitan.Altman/PAPERS/srvUpdate.pdf

A survey on networking games in telecommunications

Consider a system that is modeled as an M/M/1 queueing system with multiple user classes. Each class is characterized by its delay cost per unit of time, its expected service time and its demand function. This paper derives a pricing mechanism which is optimal and incentive-compatible in the sense that the arrival rates and execution priorities jointly maximize the expected net value of the system while being determined, on a decentralized basis, by individual users. A closed-form expression for the resulting price structure is presented and studied.

https://www.jstor.org/stable/pdfplus/171046.pdf

Optimal incentive-compatible priority pricing for the M/M/1 queue

Scale economies are commonplace in operations, yet because of analytical challenges, relatively little is known about how firms should compete in their presence. This paper presents a model of competition between two firms that face scale economies; (i.e., each firm's cost per unit of demand is decreasing in demand). A general framework is used, which incorporates competition between two service providers with price- and time-sensitive demand (a queuing game), and competition between two retailers with fixed-ordering costs and pricesensitive consumers (an Economic Order Quantity game). Reasonably general conditions are provided under which there exists at most one equilibrium, with both firms participating in the market. We demonstrate, in the context of the queuing game, that the lower cost firm in equilibrium may have a higher market share and a higher price, an enviable situation. We also allow each firm to outsource their production process to a supplier. Even if the supplier's technology is no better than the firms' technology and the supplier is required to establish dedicated capacity (so the supplier's scale can be no greater than either firm's scale), we show that the firms strictly prefer to outsource. We conclude that scale economies provide a strong motivation for outsourcing that has not previously been identified in the literature.

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Competition and Outsourcing with Scale Economies

SOME DISCUSSION has arisen recently as to whether the imposition of an "entrance fee" on arriving customers who wish to be serviced by a station and hence join a waiting line is a rational measure. Not much of this discussion has appeared in print; indeed this author is aware of only three short communications, representing an exchange of arguments between Leeman [1, 2] and Saaty [3]. The ideas advanced there were of qualitative character and no attempt was made to quantify the arguments. The problem under consideration is obviously analogous to one that arises in connection with the control of vehicular traffic congestion on a road network. It has been argued2 by traffic economists that the individual car driver on making an optimal routing choice for himself-does not optimize the system at large. The purpose of this communication is to demonstrate that, indeed, analogous conclusions can be drawn for queueing models if two basic conditions are satisfied:

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P. Naor

Papers

On the regulation of queue size by levying tolls