We consider single-server queues with exponentially distributed service times in which the arribal process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type l customer who meets upon his arrival k customers present in the SMP/M/1/PS queue is identical with the waiting time distribution of a type l customer who meets upon his arrival k+1 customers present in the SMP/M/1/RS queue. The Laplace-Stieltjes transforms of the sojourn time distribution for an arbitrary customer in the SMP/M/1/PS queue and the waiting time distribution for an arbitrary customer in the SMP/M/1/RS queue are derived. We also consider a special case of the SMP in which the interarrival time distriburion is determined only by the type of the next arrival.
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机译：我们考虑服务时间呈指数分布的单服务器队列，其中进入过程由半马尔可夫过程（SMP）控制。研究了两个服务准则，即处理器共享（PS）和随机服务（RS）。我们注意到，在SMP / M / 1 / PS队列中到达的k类客户的到来时间停留时间分布与在到达k + 1时满足的l类客户的等待时间分布相同。 SMP / M / 1 / RS队列中的客户。推导了SMP / M / 1 / PS队列中任意客户的停留时间分布的Laplace-Stieltjes变换和SMP / M / 1 / RS队列中任意客户的等待时间分布。我们还考虑了SMP的一种特殊情况，其中到达时间间隔仅由下次到达的类型决定。
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