# 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 and Optimization, doi: 10.3934/naco.2021059
##### References:
 [1] C. E. Bell and S. Stidham Jr, Individual versus social optimization in the allocation of customers to alternative servers, Management Science, 29 (1983), 831-839. [2] W. Chan and D. Maa, The GI/Geom/N queue in discrete time, INFOR: Information Systems and Operational Research, 16 (1978), 232-252.  doi: 10.1080/03155986.1978.11731705. [3] M. L. Chaudhry, U. C. Gupta and V. Goswami, Relations among the distributions at different epochs for discrete-time GI/Geom/m and continuous-time GI/M/m queues, International Journal of Information and Management Sciences, 12 (2001), 71-82. [4] J. P. Cosmas, G. H. Petit, R. Lehnert, C. Blondia, K. Kontovassilis, O. Casals and T. Theimer, A review of voice, data and video traffic models for atm, European Transactions on Telecommunications, 5 (1994), 139-154. [5] M. De Prycker, Asynchronous Transfer Mode solution for broadband ISDN, Prentice Hall International (UK) Ltd., 1995. [6] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415. [7] S. Gao and J. Wang, Equilibrium balking strategies in the observable Geo/Geo/1 queue with delayed multiple vacations, RAIRO-Operations Research, 50 (2016), 119-129.  doi: 10.1051/ro/2015019. [8] G. Gilboa-Freedman, R. Hassin and Y. Kerner, The price of anarchy in the Markovian single server queue, IEEE Transactions on Automatic Control, 59 (2013), 455-459.  doi: 10.1109/TAC.2013.2270872. [9] V. Goswami, Analysis of discrete-time multi-server queue with balking, International Journal of Management Science and Engineering Management, 9 (2014), 21-32.  doi: 10.1155/2014/358529. [10] V. Goswami and G. Panda, Optimal information policy in discrete-time queues with strategic customers, Journal of Industrial & Management Optimization, 15 (2019), 689-703.  doi: 10.3934/jimo.2018065. [11] V. Goswami and G. Panda, Optimal customer behavior in observable and unobservable discrete-time queues, Journal of Industrial & Management Optimization, 17 (2021), 299-316.  doi: 10.3934/jimo.2019112. [12] A. Gravey and G. Hébuterne, Simultaneity in discrete-time single server queues with bernoulli inputs, Performance Evaluation, 14 (1992), 123-131.  doi: 10.1016/0166-5316(92)90014-8. [13] D. Guha, A. D. Banik, V. Goswami and S. Ghosh, Equilibrium balking strategy in an unobservable GI/M/c queue with customers impatience, in Distributed Computing and Internet Technology, Springer, (2014), 188–199. [14] D. Guha, V. Goswami and A. Banik, Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging, Applied Mathematical Modelling, 40 (2016), 4199-4219.  doi: 10.1016/j.apm.2015.11.018. [15] R. Hassin, Rational Queueing, CRC press, 2016.  doi: 10.1201/b20014. [16] R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. [17] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, vol. 1, Academic Press, 1983. [18] M. Jeffrey, Asynchronous transfer mode: the ultimate broadband solution, Electronics & Communication Engineering Journal, 6 (1994), 143-151. [19] N. C. Knudsen, Individual and social optimization in a multiserver queue with a general cost-benefit structure, Econometrica: Journal of the Econometric Society, 40 (1972), 515-528.  doi: 10.2307/1913182. [20] P. J. Kuehn, Reminder on queueing theory for atm networks, Telecommunication Systems, 5 (1996), 1-24. [21] J.-Y. Le Boudec, The asynchronous transfer mode: a tutorial, Computer Networks and ISDN Systems, 24 (1992), 279-309. [22] D. H. Lee, A note on the optimal pricing strategy in the discrete-time Geo/Geo/1 queuing system with sojourn time-dependent reward, Operations Research Perspectives, 4 (2017), 113-117.  doi: 10.1016/j.orp.2017.08.001. [23] S. A. Lippman and S. Stidham Jr, Individual versus social optimization in exponential congestion systems, Operations Research, 25 (1977), 233-247.  doi: 10.1287/opre.25.2.233. [24] R. Lotfi, N. Mardani and G. W. Weber, Robust bilevel programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. [25] R. Lotfi, B. Kargar, S. H. Hoseini, S. Nazari, S. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research. [26] Y. Ma, W.-q. Liu and J.-h. Li, Equilibrium balking behavior in the Geo/Geo/1 queueing system with multiple vacations, Applied Mathematical Modelling, 37 (2013), 3861-3878.  doi: 10.1016/j.apm.2012.08.017. [27] Y. Ma and Z. Liu, Pricing analysis in Geo/Geo/1 queueing system, Mathematical Problems in Engineering, 2015, Article ID 181653. doi: 10.1155/2015/181653. [28] G. Martin and L. Pankoff, Optimal customer decisions in a G/M/c queue, Mathematical and Computer Modelling, 10 (1988), 251-256.  doi: 10.1016/0895-7177(88)90003-9. [29] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. [30] G. Panda and V. Goswami, Effect of information on the strategic behavior of customers in a discrete-time bulk service queue, Journal of Industrial & Management Optimization, 16 (2020), 1369-1388.  doi: 10.3934/jimo.2019007. [31] Y. A. Ra'ed and H. T. Mouftah, Survey of ATM switch architectures, Computer Networks and ISDN Systems, 27 (1995), 1567-1613. [32] Y. Tang, P. Guo and Y. Wang, Equilibrium queueing strategies of two types of customers in a two-server queue, Operations Research Letters, 46 (2018), 99-102.  doi: 10.1016/j.orl.2017.11.009. [33] T. Yang, J. Wang and F. Zhang, Equilibrium balking strategies in the Geo/Geo/1 queues with server breakdowns and repairs, Quality Technology & Quantitative Management, 11 (2014), 231-243. [34] U. Yechiali, Customers' optimal joining rules for the GI/M/s queue, Management Science, 18 (1972), 434-443.  doi: 10.1287/mnsc.18.7.434. [35] M. Yu and A. S. Alfa, Strategic queueing behavior for individual and social optimization in managing discrete time working vacation queue with bernoulli interruption schedule, Computers & Operations Research, 73 (2016), 43-55.  doi: 10.1016/j.cor.2016.03.011.

show all references

##### References:
 [1] C. E. Bell and S. Stidham Jr, Individual versus social optimization in the allocation of customers to alternative servers, Management Science, 29 (1983), 831-839. [2] W. Chan and D. Maa, The GI/Geom/N queue in discrete time, INFOR: Information Systems and Operational Research, 16 (1978), 232-252.  doi: 10.1080/03155986.1978.11731705. [3] M. L. Chaudhry, U. C. Gupta and V. Goswami, Relations among the distributions at different epochs for discrete-time GI/Geom/m and continuous-time GI/M/m queues, International Journal of Information and Management Sciences, 12 (2001), 71-82. [4] J. P. Cosmas, G. H. Petit, R. Lehnert, C. Blondia, K. Kontovassilis, O. Casals and T. Theimer, A review of voice, data and video traffic models for atm, European Transactions on Telecommunications, 5 (1994), 139-154. [5] M. De Prycker, Asynchronous Transfer Mode solution for broadband ISDN, Prentice Hall International (UK) Ltd., 1995. [6] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415. [7] S. Gao and J. Wang, Equilibrium balking strategies in the observable Geo/Geo/1 queue with delayed multiple vacations, RAIRO-Operations Research, 50 (2016), 119-129.  doi: 10.1051/ro/2015019. [8] G. Gilboa-Freedman, R. Hassin and Y. Kerner, The price of anarchy in the Markovian single server queue, IEEE Transactions on Automatic Control, 59 (2013), 455-459.  doi: 10.1109/TAC.2013.2270872. [9] V. Goswami, Analysis of discrete-time multi-server queue with balking, International Journal of Management Science and Engineering Management, 9 (2014), 21-32.  doi: 10.1155/2014/358529. [10] V. Goswami and G. Panda, Optimal information policy in discrete-time queues with strategic customers, Journal of Industrial & Management Optimization, 15 (2019), 689-703.  doi: 10.3934/jimo.2018065. [11] V. Goswami and G. Panda, Optimal customer behavior in observable and unobservable discrete-time queues, Journal of Industrial & Management Optimization, 17 (2021), 299-316.  doi: 10.3934/jimo.2019112. [12] A. Gravey and G. Hébuterne, Simultaneity in discrete-time single server queues with bernoulli inputs, Performance Evaluation, 14 (1992), 123-131.  doi: 10.1016/0166-5316(92)90014-8. [13] D. Guha, A. D. Banik, V. Goswami and S. Ghosh, Equilibrium balking strategy in an unobservable GI/M/c queue with customers impatience, in Distributed Computing and Internet Technology, Springer, (2014), 188–199. [14] D. Guha, V. Goswami and A. Banik, Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging, Applied Mathematical Modelling, 40 (2016), 4199-4219.  doi: 10.1016/j.apm.2015.11.018. [15] R. Hassin, Rational Queueing, CRC press, 2016.  doi: 10.1201/b20014. [16] R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. [17] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, vol. 1, Academic Press, 1983. [18] M. Jeffrey, Asynchronous transfer mode: the ultimate broadband solution, Electronics & Communication Engineering Journal, 6 (1994), 143-151. [19] N. C. Knudsen, Individual and social optimization in a multiserver queue with a general cost-benefit structure, Econometrica: Journal of the Econometric Society, 40 (1972), 515-528.  doi: 10.2307/1913182. [20] P. J. Kuehn, Reminder on queueing theory for atm networks, Telecommunication Systems, 5 (1996), 1-24. [21] J.-Y. Le Boudec, The asynchronous transfer mode: a tutorial, Computer Networks and ISDN Systems, 24 (1992), 279-309. [22] D. H. Lee, A note on the optimal pricing strategy in the discrete-time Geo/Geo/1 queuing system with sojourn time-dependent reward, Operations Research Perspectives, 4 (2017), 113-117.  doi: 10.1016/j.orp.2017.08.001. [23] S. A. Lippman and S. Stidham Jr, Individual versus social optimization in exponential congestion systems, Operations Research, 25 (1977), 233-247.  doi: 10.1287/opre.25.2.233. [24] R. Lotfi, N. Mardani and G. W. Weber, Robust bilevel programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. [25] R. Lotfi, B. Kargar, S. H. Hoseini, S. Nazari, S. Safavi and G. W. Weber, Resilience and sustainable supply chain network design by considering renewable energy, International Journal of Energy Research. [26] Y. Ma, W.-q. Liu and J.-h. Li, Equilibrium balking behavior in the Geo/Geo/1 queueing system with multiple vacations, Applied Mathematical Modelling, 37 (2013), 3861-3878.  doi: 10.1016/j.apm.2012.08.017. [27] Y. Ma and Z. Liu, Pricing analysis in Geo/Geo/1 queueing system, Mathematical Problems in Engineering, 2015, Article ID 181653. doi: 10.1155/2015/181653. [28] G. Martin and L. Pankoff, Optimal customer decisions in a G/M/c queue, Mathematical and Computer Modelling, 10 (1988), 251-256.  doi: 10.1016/0895-7177(88)90003-9. [29] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. [30] G. Panda and V. Goswami, Effect of information on the strategic behavior of customers in a discrete-time bulk service queue, Journal of Industrial & Management Optimization, 16 (2020), 1369-1388.  doi: 10.3934/jimo.2019007. [31] Y. A. Ra'ed and H. T. Mouftah, Survey of ATM switch architectures, Computer Networks and ISDN Systems, 27 (1995), 1567-1613. [32] Y. Tang, P. Guo and Y. Wang, Equilibrium queueing strategies of two types of customers in a two-server queue, Operations Research Letters, 46 (2018), 99-102.  doi: 10.1016/j.orl.2017.11.009. [33] T. Yang, J. Wang and F. Zhang, Equilibrium balking strategies in the Geo/Geo/1 queues with server breakdowns and repairs, Quality Technology & Quantitative Management, 11 (2014), 231-243. [34] U. Yechiali, Customers' optimal joining rules for the GI/M/s queue, Management Science, 18 (1972), 434-443.  doi: 10.1287/mnsc.18.7.434. [35] M. Yu and A. S. Alfa, Strategic queueing behavior for individual and social optimization in managing discrete time working vacation queue with bernoulli interruption schedule, Computers & Operations Research, 73 (2016), 43-55.  doi: 10.1016/j.cor.2016.03.011.
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 and 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 and 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 and 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 and 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 and 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 and Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073 [7] Lixuan Zhang, Xuefei Yang. On pole assignment of high-order discrete-time linear systems with multiple state and input delays. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022022 [8] Pradeep Dubey, Rahul Garg, Bernard De Meyer. Competing for customers in a social network. Journal of Dynamics and Games, 2014, 1 (3) : 377-409. doi: 10.3934/jdg.2014.1.377 [9] 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 and Heterogeneous Media, 2011, 6 (2) : 329-349. doi: 10.3934/nhm.2011.6.329 [10] 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 and Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851 [11] Ali Delavarkhalafi. On optimal stochastic jumps in multi server queue with impatient customers via stochastic control. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021030 [12] 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 and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111 [13] Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489 [14] Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253 [15] 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 and Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435 [16] Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 [17] Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics and Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009 [18] Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial and Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843 [19] Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2243-2264. doi: 10.3934/jimo.2020067 [20] Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066

Impact Factor:

## Tools

Article outline

Figures and Tables