我們為不耐煩顧客之重試等候系統的平穩機率提供一個新的上界。如
果模型滿足某些條件，則會給出更好的上界。以此上界，我們可以用有限
矩陣計算平穩機率，並用數值實驗驗證論文中提出的定理。此外，我們提
出了該定理的進一步推廣形式，任何滿足條件的模型都可以應用這個定理。

We present a new upper bound of the stationary probability of retrial queueing systems with impatient customers. If the model satisfies some conditions, it gives a better upper bound. Furthermore, we can calculate the stationary probability with a finite matrix. Numerical experiments to verify the theorems are presented in the thesis. In addition, we propose a further generalization form of the theorem. Any model satisfying the condition could apply this theorem.

[1] V.V. Anisimov and J.R. Artalejo. Approximation of multiserver retrial queues by means of generalized truncated models. Top, 10(1):51–66, 2002.
[2] J.R. Artalejo. A classified bibliography of research on retrial queues: progress in 1990–1999. Top, 7(2):187–211, 1999.
[3] J.R. Artalejo and M. Pozo. Numerical calculation of the stationary distribution of the main multiserver retrial queue. Annals of Operations Research, 116(1):41–56, 2002.
[4] H. Baumann and W. Sandmann. Numerical solution of level dependent quasi-birth-anddeath processes. Procedia Computer Science, 1(1):1561–1569, 2010.
[5] A. Gómez-Corral. A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Annals of Operations Research, 141(1):163–191, 2006.
[6] B.K. Kumar, R.N. Krishnan, R. Sankar, and R. Rukmani. Analysis of dynamic service system between regular and retrial queues with impatient customers. Journal of Industrial & Management Optimization, 18(1):267, 2022.
[7] G. Latouche, V. Ramaswami, and Society for Industrial and Applied Mathematics. Introduction to matrix analytic methods in stochastic modeling. Society for Industrial and Applied Mathematics, 1999.
[8] J. Liu and J.T. Wang. Strategic joining rules in a single server markovian queue with bernoulli vacation. Operational Research, 17(2):413–434, 2017.
[9] H.P. Luh and P.C. Song. Matrix analytic solutions for m/m/s retrial queues with impatient customers. In International Conference on Queueing Theory and Network Applications, pages 16–33. Springer, 2019.
[10] M.F. Neuts. Matrix-geometric solutions in stochastic models. Johns Hopkins series in the mathematical sciences. Johns Hopkins University Press, Baltimore, MD, July 1981.
[11] E. Onur, H. Deliç, C. Ersoy, and M. Çaǧlayan. Measurement-based replanning of cell capacities in gsm networks. Computer Networks, 39(6):749–767, 2002.
[12] V. Ramaswami and P.G. Taylor. Some properties of the rate perators in level dependent quasi-birth-and-death processes with countable number of phases. Stochastic Models, 12(1):143–164, 1996.
[13] A. Remke, B.R. Haverkort, and L. Cloth. Uniformization with representatives: comprehensive transient analysis of infinite-state qbds. In Proceeding from the 2006 workshop on Tools for solving structured Markov chains, pages 7–es, 2006.
[14] J.F. Shortle, J.M. Thompson, D. Gross, and C.M. Harris. Fundamentals of queueing theory, volume 399. John Wiley & Sons, 2018.
[15] P.D. Tuan, M. Hiroyuki, K. Shoji, and T. Yutaka. A simple algorithm for the rate matrices of level-dependent qbd processes. In Proceedings of the 5th international conference on queueing theory and network applications, pages 46–52, 2010.
[16] K.Z. Wang, N. Li, and Z.B. Jiang. Queueing system with impatient customers: A review. In Proceedings of 2010 IEEE international conference on service operations and logistics, and informatics, pages 82–87. IEEE, 2010.
[17] W.S. Yang and S.C. Taek. M/M/s queue with impatient customers and retrials. Applied Mathematical Modelling, 33(6):2596–2606, 2009