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THE OPTIMAL ADMISSION THRESHOLD IN OBSERVABLE QUEUES WITH STATE DEPENDENT PRICING

Published online by Cambridge University Press:  13 December 2013

Christian Borgs ,
Jennifer T. Chayes ,
Sherwin Doroudi ,
Mor Harchol-Balter  and
Kuang Xu
Christian Borgs
Affiliation:
Microsoft Research, Cambridge, MA. E-mail: borgs@microsoft.com, jchayes@microsoft.com
Jennifer T. Chayes
Affiliation:
Microsoft Research, Cambridge, MA. E-mail: borgs@microsoft.com, jchayes@microsoft.com
Sherwin Doroudi
Affiliation:
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA E-mail: sdoroudi@andrew.cmu.edu
Mor Harchol-Balter
Affiliation:
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA E-mail: harchol@cs.cmu.edu
Kuang Xu
Affiliation:
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA E-mail: kuangxu@mit.edu
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Abstract

We consider the social welfare model of Naor [20] and revenue-maximization model of Chen and Frank [7], where a single class of delay-sensitive customers seek service from a server with an observable queue, under state dependent pricing. It is known that in this setting both revenue and social welfare can be maximized by a threshold policy, whereby customers are barred from entry once the queue length reaches a certain threshold. However, no explicit expression for this threshold has been found. This paper presents the first derivation of the optimal threshold in closed form, and a surprisingly simple formula for the (maximum) revenue under this optimal threshold. Utilizing properties of the Lambert W function, we also provide explicit scaling results of the optimal threshold as the customer valuation grows. Finally, we present a generalization of our results, allowing for settings with multiple servers.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 1 , January 2014 , pp. 101 - 119
Copyright
Copyright © Cambridge University Press 2013 

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