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2010 Chinese Control and Decision Conference

作者： Dequan Yue~1,Ling Wang~1,Tingting Xu~1,Haiying Li~1 1.Department of Statistics,College of Science,Yanshan University,Qinhuangdao 066004,China

＜正＞In this paper,we study a repairable queueing system with two different servers,where Server 1 is perfectly reliable and Server 2 is subject to breakdown.The service times of two servers are assumed to follow phase type（PH） distribution and exponential distribution,respectively.By establishing the quasi-birth-and-death（QBD） process of the system states,we first derive the equilibrium condition of the system,and then obtain the matrix-geometric solution for the steady-state probability vectors of the system.Finally,numerical results are presented.

关键词： Queueing system Breakdowns matrix-geometric solution PH distribution Mean system size

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THE M/PH/1 QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

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Journal of Systems Science and Systems Engineering 2010年第4期19卷 496-503页

作者： Yutaka BABA Faculty of Education and Human Sciences Yokohama National University

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.

关键词： Working vacation vacation interruption phase type distribution quasi birth and death process matrix-geometric solution stochastic decomposition

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A New Theoretical Framework of Pyramid Markov Processes for Blockchain Selfish Mining

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Journal of Systems Science and Systems Engineering 2021年第6期30卷 667-711页

作者： Quanlin Li Yanxia Chang Xiaole Wu Guoqing Zhang School of Economics and Management Beijing University of TechnologyBeijing 100124China School of Management Fudan UniversityShanghai 200433China Department of Mechanical Automotive&Materials EngineeringUniversity of WindsorWindsorOntarioCanada

In this paper,we provide a new theoretical framework of pyramid Markov processes to solve some open and fundamental problems of blockchain selfish mining under a rigorous mathematical setting.We first describe a more general model of blockchain selfish mining with both a two-block leading competitive criterion and a new economic incentive mechanism.Then we establish a pyramid Markov process and show that it is irreducible and positive recurrent,and its stationary probability vector is matrix-geometric with an explicitly representable rate matrix.Also,we use the stationary probability vector to study the influence of orphan blocks on the waste of computing resource.Next,we set up a pyramid Markov reward process to investigate the long-run average mining profits of the honest and dishonest mining pools,respectively.As a by-product,we build one-dimensional Markov reward processes and provide some new interesting interpretation on the Markov chain and the revenue analysis reported in the seminal work by Eyal and Sirer(2014).Note that the pyramid Markov(reward)processes can open up a new avenue in the study of blockchain selfish mining.Thus we hope that the methodology and results developed in this paper shed light on the blockchain selfish mining such that a series of promising research can be developed potentially.

关键词： Blockchain Proof of Work selfish mining main chain pyramid Markov process pyramid Markov reward process phase-type distribution matrix-geometric solution

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THE M/M/1 QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTIONS

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Journal of Systems Science and Systems Engineering 2007年第1期16卷 121-127页

作者： Jihong LI Naishuo TIAN College of Economics and Management Yanshan University College of Sciences Yanshan University Qinhuangdao Hebei 066004 China

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy：, the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.

关键词： Working vacation vacation interruption matrix-geometric solution stochastic decomposition

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离散时间可修排队MAP/geometric(geometric/PH)/1

离散时间可修排队MAP/Geometric(Geometric/PH)/1

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中国运筹学会第六届学术交流会

作者：禹海波周家良聂赞坎西安交通大学理学院

本文研究离散时间可修排队系统MAP/geometric(geometric/PH)/1,其中顾客到达过程为离散马尔可夫到达过程(MAP)[6],服务台对顾客的服务时间和服务台寿命均服从几何分布,修理时间为离散PH变量。首先我们考虑广义服务时间,证明它是离散PH变... 详细信息

本文研究离散时间可修排队系统MAP/geometric(geometric/PH)/1,其中顾客到达过程为离散马尔可夫到达过程(MAP)[6],服务台对顾客的服务时间和服务台寿命均服从几何分布,修理时间为离散PH变量。首先我们考虑广义服务时间,证明它是离散PH变量,然后运用矩阵几何解理论给出了系统的稳态队长和等待时间分布,同时给出了系统争态可用度和稳态故障频度等可靠性指标。离散时间排队和服务台可修的排队模型广泛应用于计算机和通讯网络,逮昭义等文[1]中把计算机网络(环形LAN)中的竞争-冲突等实际问题用离散时间排队模型来处理,并给出计算机仿真结果,在离散时间排队模型的应用方面做了一些非常有意义的工作。在离散时间排队中考虑服务台可能失效和修理的离散时间可修排队系统的研究具有较强的实际意义,然而至今很少见有文献讨论过,如[4],基于以上的考虑,我们给出了本文的对照连续时间经典排队模型讲座了离散时间排队模型,侯玉梅在文[4]中讲座了离散时间可修正排队模型,但是他们均假定顾客到达过程为更新程,即到达间隔i.i.d,而实际情况不一定是这样的。本文假定顾客到达过程为离散马尔可夫到达过程(MAP),它是一类易于计算的马尔可夫更亲过程,其中顾客到达间隔非i.i.d,这在理论上具有广泛性,国为MAP包含PH更新过程为其特殊情形,同时本文讲座的模型更具有实际意义。另一方面,西文使用Neuts发展的矩阵几何解理论,给出的结果均以矩阵形式表示,其中矩阵R虽不能得到显式表示,但可以使用有效的数值迭代方法,计算程序和误差估计见[7]。

关键词： Discrete-time queueing system Repairable Phase type distribution Marko- vian arrival process matrix-geometric solution

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