We consider the Erlang A model, or \(M/M/m+M\) queue, with Poisson arrivals, exponential service times, and m parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth–death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green’s functions and contour integrals related to hypergeometric functions. Our results are specialized to the \(M/M/\infty \) queue, the M / M / m queue, and the M / M / m / m loss model. We also obtain some corresponding results for diffusion approximations to these models.

中文翻译：

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Erlang A模型的瞬态分析。

我们考虑带有Poisson到达，指数服务时间和m个并行服务器的Erlang A模型或\（M / M / m + M \）队列，以及等待客户在指数时间后放弃队列的属性。在这种情况下，队列长度过程是出生-死亡过程，为此，我们获得了与时间相关的分布和首次通过时间的拉普拉斯变换的显式表达式。通常认为这两个瞬时特性是棘手的。拉普拉斯变换的求解涉及使用格林函数和与超几何函数有关的轮廓积分。我们的结果专门用于\（M / M / \ infty \）队列，即M  /  M  /  m队列，以及M  /  M  /  m  /  m损失模型。我们还获得了这些模型的扩散近似的一些相应结果。