文章研究了具有不耐烦顾客的MAP/PH/1排队系统,其中顾客的到达过程是马尔可夫到达过程,顾客的服务时间服从位相型分布,顾客的不耐烦时间服从指数分布.针对这个排队系统,文章构建了一个水平相依的拟生灭过程.首先,文章利用平均漂移技术给出了排队系统的稳定性条件.其次,文章借助于马氏过程的RG-分解方法,提供了拟生灭过程的平稳概率向量,并得到排队系统稳态队长的概率分布和平均稳态队长.再次,为了分析任意一个到达顾客在系统中的逗留时间,文章建立了一个具有吸收状态的马氏过程,给出这个逗留时间的概率分布和平均逗留时间.最后,文章使用数值算例分析了一些关键参数对系统性能指标的影响.
Abstract
This paper considers an MAP/PH/1 queueing system with impatient customers. Customers arrive at the queueing system according to a Markovian arrival process, each of the customers requiring a service that is of phase type, and the impatient time of customers is assumed to be exponentially distributed. For analyzing the queueing system, a level dependent quasi-birth-and-death process is constructed. Firstly, the stability condition of the system is obtained by using the mean drift technique. Then the stationary probability vector of the quasi-birth-and-death process is given by using the RG-factorization of Markov process. Based on the stationary probability vector, the probability distribution of the stationary queue length and the average stationary queue length are obtained. Moreover, a Markov process with an absorption state is built for analyzing the sojourn time of any arriving customer in the system, and the probability distribution of the sojourn time and the average sojourn time are given. Finally, the effect of some crucial parameters on the performance measures of the system is analyzed by means of numerical examples.
关键词
排队系统 /
不耐烦顾客 /
马尔可夫到达过程 /
位相型分布 /
RG-分解
{{custom_keyword}} /
Key words
Queueing system /
impatient customer /
Markovian arrival process /
phase type distribution /
RG-factorization
{{custom_keyword}} /
中图分类号：
60K25
90B22
{{custom_clc.code}}
({{custom_clc.text}})
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Stanford R E. Reneging phenomena in single channel queues. Mathematics of Operations Research, 1979, 4(2):162-178.
[2] Stanford R E. On queues with impatience. Advances in Applied Probability, 1990, 22(3):768-769.
[3] Wang K, Li N, Jiang Z. Queueing system with impatient customers:A review. Proceedings of 2010 IEEE International Conference on Service Operations and Logistics, and Informatics, 2010, 82-87.
[4] Lee E Y, Lim K E. The analysis of the M/M/1 queue with impatient customers. Communications for Statistical Applications and Methods, 2000, 7(2):489-497.
[5] Choi B D, Kim B, Chung J. M/M/1 queue with impatient customers of higher priority. Queueing Systems, 2001, 38(1):49-66.
[6] Perel N, Yechiali U. Queues with slow servers and impatient customers. European Journal of Operational Research, 2010, 201(1):247-258.
[7] Yu X, Wu D, Wu L, et al. An M/M/1 queue system with single working vacation and impatient customers. The 19th International Conference on Industrial Engineering and Engineering Management, 2013, 767-776.
[8] Manoharan P, Jeeva T. Impatient customers in an M/M/1 working vacation queue with a waiting server and setup time. Journal of Computer and Mathematical Sciences, 2019, 10(5):1189-1196.
[9] Ammar S I, Jiang T, Ye Q. Transient analysis of impatient customers in an M/M/1 disasters queue in random environment. Engineering Computations, 2020, 37(6):1945-1965.
[10] Zhang M, Gao S. The disasters queue with working breakdowns and impatient customers. RAIRO-Operations Research, 2020, 54(3):815-825.
[11] De Kok A G, Tijms H C. A queueing system with impatient customers. Journal of Applied Probability, 1985, 22(3):688-696.
[12] Rao S S. The effect of postponable interruptions on an M/G/1 system with impatient customers. Australian & New Zealand Journal of Statistics, 1967, 9(1):16-34.
[13] Bae J, Kim S, Lee E Y. The virtual waiting time of the M/G/1 queue with impatient customers. Queueing Systems, 2001, 38(4):485-494.
[14] Boxma O, Perry D, Stadje W, et al. The busy period of an M/G/1 queue with customer impatience. Journal of Applied Probability, 2010, 47(1):130-145.
[15] Boxma O, Prabhu B. Analysis of an M/G/1 queue with customer impatience and adaptive arrival process. International Conference on Network Games, Control and Optimization, 2011, 1-4.
[16] Stihi N, Djellab N. Approximation of the steady state system state distribution of the M/G/1 retrial queue with impatient customers. Yugoslav Journal of Operations Research, 2012, 22(2):285-296.
[17] Bae J, Kim S. The stationary workload of the G/M/1 queue with impatient customers. Queueing Systems, 2010, 64(3):253-265.
[18] Finch P D. Deterministic customer impatience in the queueing system GI/M/1. Biometrika, 1960, 47(1-2):45-52.
[19] Teghem J. Use of discrete transforms for the study of a GI/M/S queue with impatient customer phenomena. Zeitschrift Für Operations Research, 1979, 23(3):95-106.
[20] Kim J, Kim J. M/PH/1 queue with deterministic impatience time. Communications of the Korean Mathematical Society, 2013, 28(2):383-396.
[21] Sakuma Y, Takine T. Multi-class M/PH/1 queues with deterministic impatience times. Stochastic Models, 2017, 33(1):1-29.
[22] Rabia S I. An improved truncation technique to analyze a Geo/PH/1 retrial queue with impatient customers. Computers & Operations Research, 2014, 46:69-77.
[23] Van Houdt B, Lenin R B, Blondia C. Delay distribution of (im) patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times. Queueing Systems, 2003, 45(1):59-73.
[24] Van Velthoven J, Van Houdt B, Blondia C. Response time distribution in a D-MAP/PH/1 queue with general customer impatience. Stochastic Models, 2005, 21(2-3):745-765.
[25] Klimenok V I, Orlovsky D S, Dudin A N. A BMAP/PH/N system with impatient repeated calls. Asia-Pacific Journal of Operational Research, 2007, 24(3):293-312.
[26] Van Houdt B. Analysis of the adaptive MMAP[K]/PH[K]/1 queue:A multi-type queue with adaptive arrivals and general impatience. European Journal of Operational Research, 2012, 220(3):695-704.
[27] D'Arienzo M P, Dudin A N, Dudin S A, et al. Analysis of a retrial queue with group service of impatient customers. Journal of Ambient Intelligence and Humanized Computing, 2020, 11(6):2591-2599.
[28] Dudin A N, Dudin S A, Dudina O S, et al. Analysis of queueing model with processor sharing discipline and customers impatience. Operations Research Perspectives, 2018, 5:245-255.
[29] Combé M B. Impatient customers in the MAP/G/1 queue. Report-Department of Operations Research, Statistics, and System Theory, 1994, (13):1-11.
[30] Li Q L. Constructive Computation in Stochastic Models with Applications:The RGFactorizations. Berlin:Springer, 2010.
[31] Neuts M F. Matrix-Geometric Solutions in Stochastic Models:An Algorithmic Approach. Baltimore City, USA:Johns Hopkins University Press, 1981.
[32] Ramaswami V, Taylor P G. Some properties of the rate operators in level dependent quasibirth-and-death processes with a countable number of phases. Stochastic Models, 1996, 12(1):143-164.
[33] Li Q L, Cao J. Two types of RG-factorizations of quasi-birth-and-death processes and their applications to stochastic integral functionals. Stochastic Models, 2004, 20(3):299-340.
[34] Li Q L, Liu L. An algorithmic approach for sensitivity analysis of perturbed quasi-birth-and-death processes. Queueing Systems, 2004, 48(3):365-397.
[35] Li Q L, Zhao Y Q. The RG-factorizations in block-structured Markov renewal processes. Observation, Theory and Modeling of Atmospheric Variability. Singapore:World Scientific Publishing, 2004.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
国家自然科学基金重点项目(71932002), 国家自然科学基金面上项目(71671158)资助课题
{{custom_fund}}
PDF(503 KB)
