论文标题
批处理到达的非平稳队列
Non-Stationary Queues with Batch Arrivals
论文作者
论文摘要
由涉及设定适当人员配备级别的多服务器排队系统的应用程序的激励，我们对排队长度过程$ \ {q（t）进行了详尽的研究； t \ geq 0 \} $，出发过程$ \ {d（t）; t \ geq 0 \} $和工作负载过程$ \ {w（t）; T \ geq 0 \} $与M $ _ {t}^{b_ {t}} $/g $ _ {t} $/$ \ infty $ queuing Systems关联。通过两个基本假设（非平稳）泊松到达和无限多个服务器，我们否则维护一个高度一般的模型，其中服务持续时间和批量分布可能取决于时间，此外，批处理内的服务持续时间可能是任意依赖的。尽管如此，我们发现泊松和无限的服务器假设足以表明，对于每个$ t> 0 $，$ q（t）$的定律是相互独立的泊松随机变量的加权总和。我们进一步调用这种类型的分解，以得出各种关节拉普拉斯 - 斯泰尔杰斯转换与队列长度和出发过程相关的。接下来，我们研究工作负载过程的时间依赖性行为，并通过确定队列长度和工作负载过程（正确缩放时）几乎融合到两个不同的射击噪声过程，从而提高了前面显示的弱收敛结果。
Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t \geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system. With two fundamental assumptions of (non-stationary) Poisson arrivals and infinitely many servers, we otherwise maintain a highly general model, in which the service duration and batch size distributions may depend on time and, moreover, where the service durations within a batch may be arbitrarily dependent. Nevertheless, we find that the Poisson and infinite server assumptions are enough to show that for each $t > 0$, the law of $Q(t)$ is that of a weighted sum of mutually independent Poisson random variables. We further invoke this type of decomposition to derive various joint Laplace-Stieltjes transforms associated with the queue-length and departure processes. Next, we study the time-dependent behavior of the workload process, and we conclude by establishing almost sure convergence of the queue-length and workload processes (when properly scaled) to two different shot-noise processes, elevating the weak convergence results shown previously.
