冗余模型的稳定性
On the stability of redundancy models
摘要：
研究了冗余$d$多服务器系统的稳定性条件。每个服务器都有自己的队列，并实现流行的调度规则，如先来先服务（FCFS），处理器共享（PS）和随机服务顺序（ROS）。新作业按照泊松过程到达，每个作业的副本被发送到随机均匀选择的$d$服务器。假设工件的服务时间服从指数分布。作业的一个副本完成服务后，作业立即离开。假设所有$d$拷贝都是i.i.d.，我们表明，对于PS和ROS（对于FCFS，这是已知的），发送冗余副本不会减小稳定区域。假设$d$拷贝是相同的，我们表明，（i）ROS不减少稳定区域，（ii）FCFS减少稳定区域，其特征在于通过相关的饱和系统，和（iii）PS严重减少稳定区域，这与系统中所有的拷贝必须提供\emph{fully}。的证据是基于仔细刻画的基本随机过程的标度极限。通过模拟，我们获得有趣的见解系统的性能为非指数服务时间分布和异构服务器的速度。
英文摘要：
We investigate the stability condition of redundancy-$d$ multi-server systems. Each server has its own queue and implements popular scheduling disciplines such as First-Come-First-Serve (FCFS), Processor Sharing (PS), and Random Order of Service (ROS). New jobs arrive according to a Poisson process and copies of each job are sent to $d$ servers chosen uniformly at random. The service times of jobs are assumed to be exponentially distributed. A job departs as soon as one of its copies finishes service. Under the assumption that all $d$ copies are i.i.d., we show that for PS and ROS (for FCFS it is already known) sending redundant copies does not reduce the stability region. Under the assumption that the $d$ copies are identical, we show that (i) ROS does not reduce the stability region, (ii) FCFS reduces the stability region, which can be characterized through an associated saturated system, and (iii) PS severely reduces the stability region, which coincides with the system where all copies have to be \emph{fully} served. The proofs are based on careful characterizations of scaling limits of the underlying stochastic process. Through simulations we obtain interesting insights on the system's performance for non-exponential service time distributions and heterogeneous server speeds.
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