Optimal customer behavior in observable and unobservable discrete-time queues

Veena Goswami; Gopinath Panda

doi:10.3934/jimo.2019112

Article Contents

2021, Volume 17, Issue 1: 299-316. Doi: 10.3934/jimo.2019112

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Optimal customer behavior in observable and unobservable discrete-time queues

Veena Goswami^1, and
Gopinath Panda^2,

1.
School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar-751024, India
2.
Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Rd, Singapore 487372

Received: November 30, 2018

Revised: March 31, 2019

Early access: September 2019

Published: January 2021

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Abstract

This paper studies the effect of information suppression on Naor's model as well as on Edelson and Hildebrand's model under geometric distribution. We set the suitable non-cooperative games and search for their Nash equilibria under the observable and unobservable system. In each case, we analyze the effects of information level on the customers' equilibrium and socially optimal balking strategies as well as on the profit maximization of the system manager. The socially optimal behavior and the inefficiency of the equilibrium strategies are quantified via the price of anarchy measure. We discuss a comparison study of the profit maximization and social welfare under an imposed admission fee. Also, the impact of information on the selfish and social optimal joining rates is examined. Numerical results are presented to exemplify the impact of system parameters on the optimal behavior of customers under different information levels.

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Mathematics Subject Classification: Primary: 60K25, 68M20, 90B22; Secondary: 91A10, 91A80.

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Figure 1. Various time epochs in late-arrival system with delayed access (LAS-DA)

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Figure 2. State transition diagram of the Geo/Geo/1/$ n_e $ queueing model

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Figure 3. State transition diagram of the Geo/Geo/1 queue with joining probability $ f $

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Figure 4. Dependence of threshold strategies on $ R/C $ for $ \lambda = 0.2, \mu = 0.5 $

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Figure 5. Dependence of threshold strategies on $ \mu $ for $ \lambda = 0.2, R = 30, C = 1 $

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Figure 6. Dependence of threshold strategies on $ \lambda $ for $ R = 30, \mu = 0.5, C = 1 $

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Figure 7. PoA vs $ \lambda $ in the observable queue with parameters $ R = 30, C = 1, \mu = 0.5 $

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Figure 8. PoA vs $ \mu $ in the observable queue with parameters $ \lambda = 0.5, R = 30, C = 1 $

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Figure 9. $ R/C $ vs mixed strategies for the unobservable case with $ \lambda = 0.2, \mu = 0.5 $

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Figure 10. $ \mu $ vs mixed strategies for the unobservable case with $ \lambda = 0.5, R = 30, C = 1 $

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Figure 11. $ \lambda $ vs mixed strategies for the unobservable case with $ R = 10, \mu = 0.6, C = 5 $

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Figure 12. $ \lambda $ vs socially equilibrium benefit for the observable case with $ R = 30, C = 1, \mu = 0.5 $

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Figure 13. Social welfare under a profit maximizing fee for the observable case with $ R = 30, C = 1, \mu = 0.5 $

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Figure 14. Selfish optimal joining rate comparison

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Figure 15. Socially optimal joining rate comparison

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Table 1. Equilibrium joining strategy

Case	$ f_e $	$ \lambda_e $	$ W_e $
$ \lambda\le \mu-\frac{C\bar{\mu}}{R-C} $	$ 1 $	$ \lambda $	$ \frac{\bar{\lambda}}{\mu-\lambda} $
$ 0\le \mu-\frac{C\bar{\mu}}{R-C} < \lambda $	$ \frac{1}{\lambda}(\mu-\frac{C\bar{\mu}}{R-C}) $	$ \mu-\frac{C\bar{\mu}}{R-C} $	$ \frac{R}{C} $
$ \mu-\frac{C\bar{\mu}}{R-C} <0 $	0	0	$ \frac{1}{\mu} $

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Table 2. Socially optimal joining strategy

Case	$ f_s $	$ \lambda_s $	$ W_s $
$ \lambda\le \mu-\sqrt{\frac{C\mu\bar{\mu}}{R-C}} $	$ 1 $	$ \lambda $	$ \frac{\bar{\lambda}}{\mu-\lambda} $
$ \lambda > \mu-\sqrt{\frac{C\mu\bar{\mu}}{R-C}} $	$ \frac{\mu-\sqrt{\frac{C\mu\bar{\mu}}{R-C}}}{\lambda} $	$ \mu-\sqrt{\frac{C\mu\bar{\mu}}{R-C}} $	$ 1+\bar{\mu}\sqrt{\frac{R-C}{C\mu\bar{\mu}}} $

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