American Institute of Mathematical Sciences

Advanced
January  2021, 17(1): 299-316. doi: 10.3934/jimo.2019112

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  December 2018 Revised  April 2019 Published  September 2019

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.

Keywords: Discrete-time queue, equilibrium behavior, strategic customers, profit maximization, social optimization, price of anarchy.
Mathematics Subject Classification: Primary: 60K25, 68M20, 90B22; Secondary: 91A10, 91A80.
Citation: Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112
References:
[1]

Y. Bixuan, H. Zhenting, W. Jinbiao and L. Zaiming, Analysis of the equilibrium strategies in the Geo/Geo/1 queue with multiple working vacations, Quality Technology & Quantitative Management, 15 (2018), 663-685.  doi: 10.1080/16843703.2017.1335488.  Google Scholar

[2]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715.  doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[3]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 56 (2007), 213-228.  doi: 10.1007/s11134-007-9036-7.  Google Scholar

[4]

H. Chen and M. Frank, Monopoly pricing when customers queue, IIE Transactions, 36 (2004), 569-581.  doi: 10.1080/07408170490438690.  Google Scholar

[5]

Y. Dimitrakopoulos and A. N. Burnetas, Customer equilibrium and optimal strategies in an M/M/1 queue with dynamic service control, European Journal of Operational Research, 252 (2016), 477-486.  doi: 10.1016/j.ejor.2015.12.029.  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. T. 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-Freedman, R. Hassin and Y. Kerner, The price of anarchy in the Markovian single server queue, IEEE Transactions on Automatic Control, 59 (2014), 455-459.  doi: 10.1109/TAC.2013.2270872.  Google Scholar

[9]

V. Goswami and G. Panda, Mixed equilibrium and social joining strategies in Markovian queues with Bernoulli-schedule-controlled vacation and vacation interruption, Quality Technology & Quantitative Management, (2018), 531–559. doi: 10.1080/16843703.2018.1480266.  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.   Google Scholar

[11]

P. F. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 59 (2011), 986-997.  doi: 10.1287/opre.1100.0907.  Google Scholar

[12]

R. Hassin, Consumer information in markets with random product quality: The case of queues and balking, Econometrica, 54 (1986), 1185-1195.  doi: 10.2307/1912327.  Google Scholar

[13] R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b20014.  Google Scholar
[14]

R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, International Series in Operations Research & Management Science, 59. Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[15] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, Vol. 1. Operations Research and Industrial Engineering, Academic Press, Inc., New York, 1983.   Google Scholar
[16]

R. Ibrahim, Sharing delay information in service systems: A literature survey, Queueing Systems, 89 (2018), 49-79.  doi: 10.1007/s11134-018-9577-y.  Google Scholar

[17]

L. Li, J. T. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757.  doi: 10.1016/j.cie.2013.09.023.  Google Scholar

[18]

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.  Google Scholar

[19]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar

[20]

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, 708–715. doi: 10.3934/jimo.2019007.  Google Scholar

[21]

G. Panda, V. Goswami and A. D. Banik, Equilibrium and socially optimal balking strategies in Markovian queues with vacations and sequential abandonment, Asia-Pacific Journal of Operational Research, 33 (2016), 1650036, 34 pp. doi: 10.1142/S0217595916500366.  Google Scholar

[22]

G. Panda, V. Goswami and A. D. Banik, Equilibrium behaviour and social optimization in Markovian queues with impatient customers and variant of working vacations, RAIRO-Operations Research, 51 (2017), 685-707.  doi: 10.1051/ro/2016056.  Google Scholar

[23]

R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141.  doi: 10.1016/j.ejor.2012.12.016.  Google Scholar

[24]

W. Sun, S. Y. Li and E. Cheng-Guo, Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and $N$-policy, Applied Mathematical Modelling, 40 (2016), 284-301.  doi: 10.1016/j.apm.2015.04.045.  Google Scholar

[25]

T. T. Yang, J. T. 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.  doi: 10.1080/16843703.2014.11673341.  Google Scholar

[26]

M. 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]

Y. Bixuan, H. Zhenting, W. Jinbiao and L. Zaiming, Analysis of the equilibrium strategies in the Geo/Geo/1 queue with multiple working vacations, Quality Technology & Quantitative Management, 15 (2018), 663-685.  doi: 10.1080/16843703.2017.1335488.  Google Scholar

[2]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715.  doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[3]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 56 (2007), 213-228.  doi: 10.1007/s11134-007-9036-7.  Google Scholar

[4]

H. Chen and M. Frank, Monopoly pricing when customers queue, IIE Transactions, 36 (2004), 569-581.  doi: 10.1080/07408170490438690.  Google Scholar

[5]

Y. Dimitrakopoulos and A. N. Burnetas, Customer equilibrium and optimal strategies in an M/M/1 queue with dynamic service control, European Journal of Operational Research, 252 (2016), 477-486.  doi: 10.1016/j.ejor.2015.12.029.  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. T. 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-Freedman, R. Hassin and Y. Kerner, The price of anarchy in the Markovian single server queue, IEEE Transactions on Automatic Control, 59 (2014), 455-459.  doi: 10.1109/TAC.2013.2270872.  Google Scholar

[9]

V. Goswami and G. Panda, Mixed equilibrium and social joining strategies in Markovian queues with Bernoulli-schedule-controlled vacation and vacation interruption, Quality Technology & Quantitative Management, (2018), 531–559. doi: 10.1080/16843703.2018.1480266.  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.   Google Scholar

[11]

P. F. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 59 (2011), 986-997.  doi: 10.1287/opre.1100.0907.  Google Scholar

[12]

R. Hassin, Consumer information in markets with random product quality: The case of queues and balking, Econometrica, 54 (1986), 1185-1195.  doi: 10.2307/1912327.  Google Scholar

[13] R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b20014.  Google Scholar
[14]

R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, International Series in Operations Research & Management Science, 59. Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[15] J. J. Hunter, Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory, Vol. 1. Operations Research and Industrial Engineering, Academic Press, Inc., New York, 1983.   Google Scholar
[16]

R. Ibrahim, Sharing delay information in service systems: A literature survey, Queueing Systems, 89 (2018), 49-79.  doi: 10.1007/s11134-018-9577-y.  Google Scholar

[17]

L. Li, J. T. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757.  doi: 10.1016/j.cie.2013.09.023.  Google Scholar

[18]

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.  Google Scholar

[19]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar

[20]

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, 708–715. doi: 10.3934/jimo.2019007.  Google Scholar

[21]

G. Panda, V. Goswami and A. D. Banik, Equilibrium and socially optimal balking strategies in Markovian queues with vacations and sequential abandonment, Asia-Pacific Journal of Operational Research, 33 (2016), 1650036, 34 pp. doi: 10.1142/S0217595916500366.  Google Scholar

[22]

G. Panda, V. Goswami and A. D. Banik, Equilibrium behaviour and social optimization in Markovian queues with impatient customers and variant of working vacations, RAIRO-Operations Research, 51 (2017), 685-707.  doi: 10.1051/ro/2016056.  Google Scholar

[23]

R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141.  doi: 10.1016/j.ejor.2012.12.016.  Google Scholar

[24]

W. Sun, S. Y. Li and E. Cheng-Guo, Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and $N$-policy, Applied Mathematical Modelling, 40 (2016), 284-301.  doi: 10.1016/j.apm.2015.04.045.  Google Scholar

[25]

T. T. Yang, J. T. 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.  doi: 10.1080/16843703.2014.11673341.  Google Scholar

[26]

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 14.  Selfish optimal joining rate comparison
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 15.  Socially optimal joining rate comparison
Figure Options

Download full-size image

Download as PowerPoint slide

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} $
Table Options

Download as excel

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} $
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}}} $
Table Options

Download as excel

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}}} $
[1]

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
[2]

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
[3]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121
[4]

Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial & Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467
[5]

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
[6]

Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times. Journal of Industrial & Management Optimization, 2014, 10 (1) : 131-149. doi: 10.3934/jimo.2014.10.131
[7]

Michiel De Muynck, Herwig Bruneel, Sabine Wittevrongel. Analysis of a discrete-time queue with general service demands and phase-type service capacities. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1901-1926. doi: 10.3934/jimo.2017024
[8]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008
[9]

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
[10]

Biao Xu, Xiuli Xu, Zhong Yao. Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1599-1615. doi: 10.3934/jimo.2018113
[11]

Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial & Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779
[12]

Lasse Kliemann, Elmira Shirazi Sheykhdarabadi, Anand Srivastav. Price of anarchy for graph coloring games with concave payoff. Journal of Dynamics & Games, 2017, 4 (1) : 41-58. doi: 10.3934/jdg.2017003
[13]

Fan Sha, Deren Han, Weijun Zhong. Bounds on price of anarchy on linear cost functions. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1165-1173. doi: 10.3934/jimo.2015.11.1165
[14]

Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
[15]

Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049
[16]

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
[17]

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
[18]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264
[19]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[20]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457

2019 Impact Factor: 1.366

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences