New computing and communications paradigms will result in traffic loads in information server systems that fluctuate over much broader ranges of time scales than current systems. In addition, these fluctuation time scales may only be indirectly known or even be unknown. However, we should still be able to accurately design and manage such systems. This paper addresses this issue: we consider an M/M/1 queueing system operating in a random environment (denoted M/M/1(R)) that alternates between HIGH and LOW phases, where the load in the HIGH phase is higher than in the LOW phase. Previous work on the performance characteristics of M/M/1(R) systems established fundamental properties of the shape of performance curves. In this paper, we extend monotonicity results to include convexity and concavity properties, provide a partial answer to an open problem on stochastic ordering, develop new computational techniques, and include boundary cases and various degenerate M/M/1(R) systems. The basis of our results are novel representations for the mean number in system and the probability of the system being empty. We then apply these results to analyze practical aspects of system operation and design; in particular, we derive the optimal service rate to minimize mean system cost and provide a bias analysis of the use of customer-level sampling to estimate time-stationary quantities.

新的计算和通信范式将导致信息服务器系统中的流量负载在比当前系统更广泛的时间范围内波动。此外，这些波动时间尺度可能只是间接知道甚至是未知的。但是，我们仍然应该能够准确地设计和管理这样的系统。本文解决了这个问题：我们考虑一个在随机环境中运行的M / M /1 排队系统（表示为M / M /1( R )），它在 HIGH 和 LOW 阶段之间交替，其中 HIGH 阶段的负载高于在低电平阶段。以前关于M / M /1( R的性能特征的工作) 系统建立了性能曲线形状的基本属性。在本文中，我们将单调性结果扩展到包括凸性和凹性属性，为随机排序的开放问题提供部分答案，开发新的计算技术，并包括边界情况和各种退化M / M /1( R ) 系统。我们的结果的基础是系统中的平均数和系统为空的概率的新表示。然后，我们将这些结果应用于分析系统操作和设计的实际方面；特别是，我们推导出最佳服务率以最小化平均系统成本，并提供使用客户级抽样来估计时间固定数量的偏差分析。