Customers' joining behavior in an unobservable GI/Geo/m queue

Veena Goswami; Gopinath Panda

doi:10.3934/naco.2021059

Article Contents

2023, Volume 13, Issue 1: 139-153. Doi: 10.3934/naco.2021059

This issue Previous Article A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces Next Article Robust optimum design of tuned mass dampers for high-rise buildings subject to wind-induced vibration

Customers' joining behavior in an unobservable GI/Geo/m queue

Veena Goswami^1, and
Gopinath Panda^2,

1.
School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar-751024, India
2.
Department of Electrical and Computer Engineering, University of Central Florida, USA

Received: June 2021

Revised: October 2021

Early access: November 2021

Published: March 2023

Abstract Full Text(HTML) Figure(10) / Table(2) Related Papers Cited by

Abstract

This paper studies the equilibrium balking strategies of impatient customers in a discrete-time multi-server renewal input queue with identical servers. Arriving customers are unaware of the number of customers in the queue before making a decision whether to join or balk the queue. We model the decision-making process as a non-cooperative symmetric game and derive the Nash equilibrium mixed strategy and optimal social strategies. The stationary system-length distributions at different observation epochs under the equilibrium structure are obtained using the roots method. Finally, some numerical examples are presented to show the effect of the information level together with system parameters on the equilibrium and social behavior of impatient customers.

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

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Figure 1. A schematic representation of an ATM switch

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Figure 2. Various time epochs in early-arrival system (EAS)

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Figure 3. λ vs mixed strategies with m = 2, µ = 0.2, R = 6, C = 1

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Figure 4. R vs mixed strategies with m = 2, λ = 0.4, µ = 0.2, C = 1

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Figure 5. C vs mixed strategies with m = 2, λ = 0.3, µ = 0.2, R = 40

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Figure 6. λ vs benefit with m = 2, µ = 0.2, R = 6, C = 1

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Figure 7. C vs benefit with m = 2, λ = 0.3, µ = 0.2, R = 40

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Figure 8. µ vs expected waiting time with λ = 0.3

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Figure 9. R vs PoA with m = 2, λ = 0.5, µ = 0.4, C = 1

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Figure 10. λ vs PoA with m = 2, µ = 0.4, R = 6, C = 1

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Table 1. Survey on queueing models related to game-theoretic analysis

Reference	Model	Buffer size	Findings
[29]	M/M/1	Finite	Individual and social
			optimal behavior
[6]	M/M/1	Infinite	Individual and social
			optimal behavior
[8]	M/M/1	Finite	Price of Anarchy
[11]	Geo/Geo/1	Finite & infinite	Individual & social optimal
			behavior and PoA
[34]	GI/M/s	finite & infinite	Self & social optimization
[19]	M/M/s	Infinite	Individual & social optimization
			behavior of customers under a
			non-linear holding cost
[23]	M/M/s	Infinite	Individual & social optimization
			behavior of customers under a
			linear holding cost
[1]	M/G/s	Infinite	Individual & social optimization
			with holding cost
[14]	GI/M/c	Infinite	Equilibrium balking strategy
			with reneging
Present study	GI/Geo/m	Infinite	Individual & social optimal
			behavior and PoA

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Table 2. Notations and model parameters

Operational parameters
$ 1/\lambda $	mean arrival times
$ 1/\mu $	mean service time of each server
$ m $	number of independent homogeneous servers
$ d \in [0,1] $	probability of joining in unobservable case
Economic parameters
$ R $	customers gets a reward after completion of service
$ C $	waiting cost per time unit in the system
Performance measures
$ W_s $	average sojourn time in the system
$ L_s $	mean system-length
$ \Delta_e(d) $	net benefit of the tagged customer
$ \Delta_s(d) $	social benefit per time unit
$ PoA $	price of anarchy
$ d_e $	equilibrium joining probability
$ d^* $	socially optimal joining probability

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