带不耐烦顾客Bernoulli反馈和慢服务期的排队

Bernoulli feedback queue with slow service period and impatient customers

考虑一个带有不耐烦顾客的具有两相位(快速期和慢速期)、Bernoulli反馈的M/M/1排队模型.系统处于两相位的时间,以及服务时间均服从负指数分布.当系统处于慢速服务期时,顾客变得不耐烦,并激活一个服从负指数分布的计时器.如果在计时器到期之前,系统没能从慢速期转到快速期,则该顾客将放弃排队,永不再来.而完成服务的顾客以概率σ(0<σ≤1)离开系统,以概率1-σ反馈到队尾寻求再次服务.由系统的平衡方程组,建立关于队长概率母函数的微分方程,运用解析的方法得到系统的队长概率母函数;此外,由平衡方程得到系统的平均队长.

Consider an M/M/1 queue with impatient customers and Bernoulli feedback in a two-phase（fast and slow） random enviroment.The system resides in a two-phase exponentially distributed random time,and the service time in two phases is subject to exponential distribution.When the system is in the slow phase,customers become impatient and activate a timer subject to exponential distribution.If the system′s environment does not change from slow to fast until the expiration of the timer,the customer will abandon the queue and never return again.Just after completion of his service,a customer may leave the system with probabilty σ（0σ≤1）,or feedback with probability 1-σ.From the system′s equilibrium equations,we build a differential equation about the probabilty generating function of queue-length,and derive the result by using analytic solutions.In addition,we also derive the mean queue size from another equilibrium equation.