【作者】 余君；

【导师】 岳德权；

【作者基本信息】 燕山大学 ， 计算数学， 2009， 硕士

【摘要】 多服务员特别是两个不同服务员的排队系统是排队论中的一类典型模型。与此同时,服务员的故障或休假以及顾客的止步或中途退出等现象对系统的性能指标和经济效益有着重要影响。因此,研究综合上述机制的两个不同服务员的排队系统具有重要的理论意义和应用价值。论文考虑了两个不同服务员的具有止步或中途退出等现象的可修排队模型和休假排队模型。首先,研究了等待空间有限且带有止步和中途退出的两个不同服务员的M/M/2/N可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用马尔可夫过程的方法建立了稳态概率满足的方程组,通过分块矩阵的解法求出了稳态概率向量简便而又明显的迭代计算公式,进而得到了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。其次,研究了等待空间有限且止步依赖于故障过程的两个不同服务员的M/M/2可修排队系统,其中一个服务员不可靠,并且忙时与闲时具有不同的故障率。利用拟生灭过程的方法推导出了系统稳态平衡存在的条件和稳态概率向量的矩阵几何解,进而给出了系统的一些性能指标和不可靠服务员的可靠性指标,并对结果进行了数值分析。最后,研究了两个不同服务员的M/M/2异步多重休假的排队系统。利用拟生灭过程的方法,得到了率阵和边界概率向量的明显表达式,还推导出系统队长和服务员状态的联合分布。此外,给出了两个服务员均忙时,系统中等待队长和到达顾客等待时间的解析表达式,并证明了其条件随机分解特性。更多还原

【Abstract】 The queuing systems of multi-server especially two heterogeneous servers are typical models of the queuing theory. Meantime, the breakdowns or vacations of the servers, and the phenomena of the customers’balking or reneging have a important impact on the performance measures and the economic profit of the system. Thus, the study on the queuing systems of two heterogeneous servers with integrated above mechanism has important theoretical significance and application value.In this paper, we consider the queuing model of two heterogeneous servers with breakdowns and the phenomena of balking or reneging, as well as the model with vacation.Firstly, we investigate a finite waiting room M/M/2/N repairable queuing system with balking, reneging and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. By Markov process method, we develop the steady-state probability equations, and obtain the simple and obvious iterative formulas of the steady-state probability vectors by using a method of blocking matrix. Then we obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given.Secondly, we study an infinite waiting room M/M/2 repairable queuing system that balking depends on breakdown process and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. Using the quasi-birth-and-death process method, we derive the existing condition of steady-state equilibrium, and the matrix-geometric solutions of the steady-state probability vectors. We then obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given. Finally, we study the M/M/2 queuing system with asynchronous multiple vacation and heterogeneous servers. By using the quasi-birth-and-death process method, we obtain the explicit expression of the rate matrix and the boundary probability vectors, and derive the joint distribution of queue length and the state of servers. Furthermore, we give the analytical expression of the waiting queue length and waiting time of the arrival customers when all of the servers are busy, and also prove the conditional stochastic decomposition properties of them.更多还原