doi: 10.3934/naco.2021059
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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:
[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.   Google Scholar

[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.  Google Scholar

[3]

M. L. ChaudhryU. 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.   Google Scholar

[4]

J. P. CosmasG. H. PetitR. LehnertC. BlondiaK. KontovassilisO. Casals and T. Theimer, A review of voice, data and video traffic models for atm, European Transactions on Telecommunications, 5 (1994), 139-154.   Google Scholar

[5]

M. De Prycker, Asynchronous Transfer Mode solution for broadband ISDN, Prentice Hall International (UK) Ltd., 1995. Google Scholar

[6]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415.  Google Scholar

[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.  Google Scholar

[8]

G. Gilboa-FreedmanR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

D. GuhaV. 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.  Google Scholar

[15] R. Hassin, Rational Queueing, CRC press, 2016.  doi: 10.1201/b20014.  Google Scholar
[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.  Google Scholar

[17] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, vol. 1, Academic Press, 1983.   Google Scholar
[18]

M. Jeffrey, Asynchronous transfer mode: the ultimate broadband solution, Electronics & Communication Engineering Journal, 6 (1994), 143-151.   Google Scholar

[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.  Google Scholar

[20]

P. J. Kuehn, Reminder on queueing theory for atm networks, Telecommunication Systems, 5 (1996), 1-24.   Google Scholar

[21]

J.-Y. Le Boudec, The asynchronous transfer mode: a tutorial, Computer Networks and ISDN Systems, 24 (1992), 279-309.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. LotfiN. Mardani and G. W. Weber, Robust bilevel programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534.   Google Scholar

[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. Google Scholar

[26]

Y. MaW.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.   Google Scholar

[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.  Google Scholar

[31]

Y. A. Ra'ed and H. T. Mouftah, Survey of ATM switch architectures, Computer Networks and ISDN Systems, 27 (1995), 1567-1613.   Google Scholar

[32]

Y. TangP. 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.  Google Scholar

[33]

T. YangJ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

M. L. ChaudhryU. 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.   Google Scholar

[4]

J. P. CosmasG. H. PetitR. LehnertC. BlondiaK. KontovassilisO. Casals and T. Theimer, A review of voice, data and video traffic models for atm, European Transactions on Telecommunications, 5 (1994), 139-154.   Google Scholar

[5]

M. De Prycker, Asynchronous Transfer Mode solution for broadband ISDN, Prentice Hall International (UK) Ltd., 1995. Google Scholar

[6]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415.  Google Scholar

[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.  Google Scholar

[8]

G. Gilboa-FreedmanR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

D. GuhaV. 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.  Google Scholar

[15] R. Hassin, Rational Queueing, CRC press, 2016.  doi: 10.1201/b20014.  Google Scholar
[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.  Google Scholar

[17] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, vol. 1, Academic Press, 1983.   Google Scholar
[18]

M. Jeffrey, Asynchronous transfer mode: the ultimate broadband solution, Electronics & Communication Engineering Journal, 6 (1994), 143-151.   Google Scholar

[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.  Google Scholar

[20]

P. J. Kuehn, Reminder on queueing theory for atm networks, Telecommunication Systems, 5 (1996), 1-24.   Google Scholar

[21]

J.-Y. Le Boudec, The asynchronous transfer mode: a tutorial, Computer Networks and ISDN Systems, 24 (1992), 279-309.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. LotfiN. Mardani and G. W. Weber, Robust bilevel programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534.   Google Scholar

[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. Google Scholar

[26]

Y. MaW.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.   Google Scholar

[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.  Google Scholar

[31]

Y. A. Ra'ed and H. T. Mouftah, Survey of ATM switch architectures, Computer Networks and ISDN Systems, 27 (1995), 1567-1613.   Google Scholar

[32]

Y. TangP. 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.  Google Scholar

[33]

T. YangJ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A schematic representation of an ATM switch
Figure 2.  Various time epochs in early-arrival system (EAS)
Figure 3.  λ vs mixed strategies with m = 2, µ = 0.2, R = 6, C = 1
Figure 4.  R vs mixed strategies with m = 2, λ = 0.4, µ = 0.2, C = 1
Figure 5.  C vs mixed strategies with m = 2, λ = 0.3, µ = 0.2, R = 40
Figure 6.  λ vs benefit with m = 2, µ = 0.2, R = 6, C = 1
Figure 7.  C vs benefit with m = 2, λ = 0.3, µ = 0.2, R = 40
Figure 8.  µ vs expected waiting time with λ = 0.3
Figure 9.  R vs PoA with m = 2, λ = 0.5, µ = 0.4, C = 1
Figure 10.  λ vs PoA with m = 2, µ = 0.4, R = 6, C = 1
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
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
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
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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