# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021059
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

 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

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.

Citation: Veena Goswami, Gopinath Panda. Customers' joining behavior in an unobservable GI/Geo/m queue. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021059
##### References:

show all references

##### References:
A schematic representation of an ATM switch
Various time epochs in early-arrival system (EAS)
λ vs mixed strategies with m = 2, µ = 0.2, R = 6, C = 1
R vs mixed strategies with m = 2, λ = 0.4, µ = 0.2, C = 1
C vs mixed strategies with m = 2, λ = 0.3, µ = 0.2, R = 40
λ vs benefit with m = 2, µ = 0.2, R = 6, C = 1
C vs benefit with m = 2, λ = 0.3, µ = 0.2, R = 40
µ vs expected waiting time with λ = 0.3
R vs PoA with m = 2, λ = 0.5, µ = 0.4, C = 1
λ vs PoA with m = 2, µ = 0.4, R = 6, C = 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
 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
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
 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
 [1] Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial & Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065 [2] Veena Goswami, Gopinath Panda. Synchronized abandonment in discrete-time renewal input queues with vacations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021163 [3] Gopinath Panda, Veena Goswami. Effect of information on the strategic behavior of customers in a discrete-time bulk service queue. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1369-1388. doi: 10.3934/jimo.2019007 [4] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [5] Tzu-Hsin Liu, Jau-Chuan Ke. On the multi-server machine interference with modified Bernoulli vacation. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1191-1208. doi: 10.3934/jimo.2014.10.1191 [6] Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073 [7] Pradeep Dubey, Rahul Garg, Bernard De Meyer. Competing for customers in a social network. Journal of Dynamics & Games, 2014, 1 (3) : 377-409. doi: 10.3934/jdg.2014.1.377 [8] Wenlian Lu, Fatihcan M. Atay, Jürgen Jost. Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays. Networks & Heterogeneous Media, 2011, 6 (2) : 329-349. doi: 10.3934/nhm.2011.6.329 [9] Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851 [10] Ali Delavarkhalafi. On optimal stochastic jumps in multi server queue with impatient customers via stochastic control. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021030 [11] Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111 [12] Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489 [13] Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253 [14] Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435 [15] Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 [16] Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics & Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009 [17] Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial & Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843 [18] Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2243-2264. doi: 10.3934/jimo.2020067 [19] Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3309-3332. doi: 10.3934/jimo.2020120 [20] Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066

Impact Factor:

## Tools

Article outline

Figures and Tables