一种带有灾难和伯努利机制的M/M/1队列

A M/M/1 Queue with Disasters and Bernoulli Mechanism

针对带有灾难和伯努利机制的模型在实际生活中的应用问题,提出了一种伯努利机制下具有灾难、延迟维修、反馈和休假的单工作台队列。当工作台运行时,灾难才会影响系统,此时,系统需要被维修,在场的所有顾客从系统中永远离开;工作人员对顾客完成一次服务后,可以选择休假或者继续服务;而接受这次服务的顾客,离开系统或者回到队首等待下次服务。利用马氏链方法,对稳态下系统进行分析,得到平衡方程;对平衡方程求解,导出稳态下队列中顾客人数的PGF,工作人员分别处于休假期、忙期、延迟期、维修期和空闲期的概率;根据强马尔可夫性求出稳态下逗留时间的分布;最后利用数值实验解释一些参数对系统中平均顾客人数的影响,验证了模型与方法的正确性。

Aiming at the application of the model with disasters and Bernoulli mechanism in real life,a single workbench queue with disasters,delayed maintenance,feedback and vacation under Bernoulli mechanism is proposed.The disaster affects the system only when the workbench is running,and the system needs to be repaired and all the customers on the site leave the system forever.After completing a service,the staff can choose to take a vacation or continue to serve;the customer who accepts this service will leave the system or go back to the head of the queue to wait for the next service.By using the Markov Chain method,the steady-state system is analyzed and the equilibrium equation is obtained.The PGF of the number of customers in the system under steady-state are derived by solving the balance equations.The probability of the waiter being on vacation,busy period,delay period,maintenance period and idle period are obtained respectively.The distribution of stay time in steady-state is obtained according to the strong Markov property.Finally,numerical simulation is used to explain the influence of some parameters on the average number of customers in the system,and the correctness of the model and method is verified.