Almost all queueing analysis assumes i.i.d. arrivals and service. In reality, arrival and service rates fluctuate over time. In particular, it is common for real systems to intermittently experience overload, where the arrival rate temporarily exceeds the service rate, which an i.i.d. model cannot capture. We consider the MAMS system, where the arrival and service rates each vary according to an arbitrary finite-state Markov chain, allowing intermittent overload to be modeled. We derive the first explicit characterization of mean queue length in the MAMS system, with explicit bounds for all arrival and service chains at all loads. Our bounds are tight in heavy traffic. We prove even stronger bounds for the important special case of two-level arrivals with intermittent overload. Our key contribution is an extension to the drift method, based on the novel concepts of relative arrivals and relative completions. These quantities allow us to tractably capture the transient correlational effect of the arrival and service processes on the mean queue length.

翻译：几乎所有的排队分析都假设到达与服务过程是独立同分布的。然而现实中，到达率与服务率会随时间波动。特别地，实际系统常会间歇性地经历过载状态，即到达率暂时超过服务率，这是独立同分布模型无法描述的。我们研究MAMS系统，其中到达率与服务率各自按照任意有限状态马尔可夫链变化，从而能够对间歇性过载进行建模。我们首次推导出MAMS系统中平均队列长度的显式表征，并针对所有负载条件下的任意到达链与服务链给出了显式边界。我们的边界在重负载条件下是紧致的。针对具有间歇性过载的双水平到达这一重要特例，我们证明了更强的边界。本研究的核心贡献是基于相对到达与相对完成这两个新概念，对漂移方法进行了扩展。这些量使我们能够以可处理的方式捕捉到达过程与服务过程对平均队列长度产生的瞬态相关性效应。