Abstract A Markovian network of two queues, with finite size batch Poisson arrivals and departures, is solved approximately, but to arbitrary accuracy, for its equilibrium state probabilities. Below a pair of thresholds on the queue lengths, a modification of the Spectral Expansion Method is used to construct a semi-product-form at all lengths of one queue in a finite lattice strip defined by the threshold of the other queue. No additional special arrival streams are required, for example at empty queues, from which it is already known that a product-form can be constructed. Hence the first exact closed form solution for the equilibrium probabilities in an unmodified Markovian queueing network with batches is obtained, the only constraint being finiteness of the batches. The method is illustrated numerically, first in a tandem network and then in a two-node network with feedback. Simulation results confirm convincingly the precision of the method, and partial batch forwarding and discarding are thereby compared quantitatively. Response time distributions are derived using the generating function method, which complements well the semi-product form of the equilibrium state probabilities. Again, agreement with simulation is excellent. The regenerative simulation method was used, so that no warm-up period was needed, and the statistical estimates for the 95% confidence bands are explained for the different cases of state occupancy and response time prediction at equilibrium.

中文翻译：

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具有有限批次的一对队列的半产品形式：均衡状态概率和响应时间密度

摘要 两个队列的马尔可夫网络，具有有限大小批量泊松到达和离开，对其平衡状态概率进行了近似求解，但具有任意精度。在队列长度的一对阈值之下，频谱扩展方法的修改用于在由另一个队列的阈值定义的有限格子带中构造一个队列的所有长度的半积形式。不需要额外的特殊到达流，例如在空队列中，已知可以从中构造产品形式。因此，获得了带有批次的未修改马尔可夫排队网络中均衡概率的第一个精确封闭形式解，唯一的约束是批次的有限性。该方法用数字说明，首先在串联网络中，然后在具有反馈的双节点网络中。仿真结果令人信服地证实了该方法的精度，从而定量比较了部分批量转发和丢弃。响应时间分布是使用生成函数方法导出的，它很好地补充了平衡状态概率的半积形式。同样，与模拟的一致性非常好。使用再生模拟方法，因此不需要预热期，并且针对平衡状态下的状态占用和响应时间预测的不同情况解释了 95% 置信带的统计估计。响应时间分布是使用生成函数方法导出的，它很好地补充了平衡状态概率的半积形式。同样，与模拟的一致性非常好。使用再生模拟方法，因此不需要预热期，并且针对平衡状态下的状态占用和响应时间预测的不同情况解释了 95% 置信带的统计估计。响应时间分布是使用生成函数方法导出的，它很好地补充了平衡状态概率的半积形式。同样，与模拟的一致性非常好。使用再生模拟方法，因此不需要预热期，并且针对平衡状态下的状态占用和响应时间预测的不同情况解释了 95% 置信带的统计估计。