在有顾客弃权的马尔可夫排队系统中，基于固定加入顾客数计算瞬态概率
Computing transient probabilities in Markovian queues with balking conditional on a fixed number of joined customers
中文总结 AI 辅助
该研究针对有限时间内恰好K名顾客加入的带弃权马尔可夫排队系统，推导概率生成函数相关方程与线性微分方程，开发数值方法计算相关矩，解释了条件下的反直觉行为。
中文摘要 AI 辅助
我们分析在有限时间区间[0, T]内恰好有K名顾客加入系统的情况下，带弃权（不执行）顾客的马尔可夫排队系统的瞬态行为。研究的关键量是弃权顾客的累计数量，我们针对该量考虑了概率生成函数（PGF）并推导了它满足的方程。该方法规避了直接计算弃权顾客累计数量、加入顾客总数量以及系统内顾客数量的联合分布时，处理高维马尔可夫系统带来的计算负担。为高效计算(t, T]内的状态转移，我们研究了给定T-u时刻系统状态时T时刻的条件状态概率，证明其满足关于u的线性微分方程。对于分段常数到达率，我们开发了一种数值方法，用于在K名顾客加入的条件下，评估弃权顾客总数量的矩，以及加入顾客累计数量和系统内顾客数量的矩。我们还给出了数值示例，这些示例凸显了该条件下出现的反直觉行为，并解释了其潜在机制。
英文摘要
We analyze the transient behavior of a Markovian queue with balking, conditional on exactly $K$ customers joining the system in a finite time interval $[0, T]$. A key quantity of interest is the cumulative number of balking customers, for which we consider the probability generating function (PGF) and derive an equation it satisfies. This approach circumvents the computational burden arising from dealing with high-dimensional Markovian system, which arises when attempting to directly compute the joint distribution of the cumulative number of balking customers together with the total number of joining customers and the number of customers in the system. To compute state transitions in $(t, T]$ efficiently, we examine the conditional state probability at time $T$ given the system state at time $T - u$, and show that it satisfies a linear differential equation in $u$. For piecewise-constant arrival rates, we develop a numerical procedure for evaluating the moment of the total number of balking customers, jointly with the cumulative number of joining customers and the number of customers in the system, under the condition that $K$ customers joined. We also present numerical examples that highlight counterintuitive behaviors arising from this conditioning, along with explanations of the underlying mechanisms.
