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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (1): 25-35.doi: 10.3969/j.issn.1005-3085.2016.01.003

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对服务率可变的T-SPH$/M/1/N$排队基于广义特征值方法的分析

张宏波,  杨宪立,  封平华   

  1. 河南教育学院数学系，郑州 450046
  • 收稿日期:2014-11-17 接受日期:2015-05-07 出版日期:2016-02-15 发布日期:2016-04-15
  • 基金资助:
    国家自然科学基金 (61174160)；中南大学博士后基金 (125011)；河南省高等学校青年骨干教师资助项目 (2014GGJS-136)；河南教育学院应用数学重点学科.

A Generalized Eigenvalue Approach to Analyzing T-SPH$/M/1/N$ Queue with State-dependent Service Rate

ZHANG Hong-bo,  YANG Xian-li,  FENG Ping-hua   

  1. Department of Mathematics, Henan Institute of Education, Zhengzhou 450046
  • Received:2014-11-17 Accepted:2015-05-07 Online:2016-02-15 Published:2016-04-15
  • Supported by:
    The National Natural Science Foundation of China (61174160); the Postdoctoral Science Foundation of Central South University (125011); the Foundation for University Key Teacher of Henan Province (2014GGJS-136); the Key Discipline of Applied Mathematics in Henan Institute of Education.

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摘要： 本文讨论服务率依赖于当前系统中顾客数的有限T-SPH$/M/1/N$排队，其中T-SPH表示连续时间可数状态吸收生灭过程吸收时间的分布．对该排队模型，可以用水平无限位相有限的拟生灭(QBD)过程进行建模．通过用广义特征值方法对该QBD过程进行分析，得到了T-SPH$/M/1/N$排队的平稳到达队长分布．另外，为了说明我们方法的有效性，还用几个数值例子对模型进行了分析，以刻画参数变化对系统性能的影响．

关键词: T-SPH$/M/1/N$排队, QBD过程, 广义特征值问题, 平稳队长

Abstract:

In this paper, we analyze a finite T-SPH$/M/1/N$ queue model with state-dependent service rate, where T-SPH denotes the continuous time phase type distribution defined on a birth and death process with countable number of states. The queue system investigated can be described by a quasi-birth-and-death (QBD) process with infinite levels and finite number of phases. By analyzing the QBD process with the method of generalized eigenvalues, we derive the analytic expression of the stationary queue length distribution of the queue model. Meanwhile, to explain the validity of our method, we also present several numerical examples to illustrate the effect of the varying parameters on the system performance.

Key words: T-SPH$/M/1/N$ queue, QBD process, generalized eigenvalues, stationary queue length

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