We consider a general k -dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k -dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

中文翻译：

---

具有半马尔可夫多元折扣输入的无限服务器队列分析

我们考虑一个通用的 k 维折扣无限服务器排队过程（或者，一个已发生但未报告的索赔过程），其中多元输入（索赔）由 k 维有限状态马尔可夫链给出，并且到达遵循更新过程。在给定时间 0 时输入的初始状态的情况下，将矩生成函数的多维积分方程推导出到时间 t 时的输入状态，然后提供该过程的第一和第二（矩阵）矩的渐近结果。特别是，当到达间隔或服务时间呈指数分布时，获得前两个时刻的瞬态表达式。此外，具有确定性到达时间的过程的矩生成函数被认为提供了更明确的表达。最后，