This paper investigates the following general problem relating to ordered random processes: given $n$ independent but not necessarily identical random processes, how frequently, on average, does any given process become one of the $p{rm th}$ ($p=1,2,ldots, n-1$) largest processes? This is a fundamental problem arising in the design and analysis of contemporary multidimensional wireless communication systems (e.g., multiantenna, multiuser) employing opportunistic selection. We formulate this problem as one involving the level crossing rate (LCR) of a carefully defined ordered random process across the zero threshold, which we solve by developing a new mathematical framework based on the theory of permanents. For the case where the processes correspond to time-varying Rayleigh fading channels, we present exact closed-form formulas for the LCR, simplified tight upper bounds, as well as asymptotic results for $n$ and $p$ approaching infinity but with fixed ratio. These results reveal interesting fundamental limits for the LCR, and are shown to given meaningful insight even for small values of $n$ and $p$ . We further use our mathematical framework to characterize the required per-branch and overall switching rate of a generalized selection combining diversity receiver, allowing for different average powers for each branch. With the aid of majorization theory, we demonstrate that the overall switching rate is maximized when the power delay profile is uniform.
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机译：本文研究了与有序随机过程有关的以下一般问题：给定$ n $独立但不一定相同的随机过程，平均而言，任何给定过程成为$ p {rm th} $（$ p = 1,2，ldots，n-1 $）最大的过程？这是在采用机会选择的现代多维无线通信系统（例如，多天线，多用户）的设计和分析中出现的基本问题。我们将此问题表述为涉及跨零阈值精心定义的有序随机过程的平交率（LCR）的问题，我们通过基于永久性理论开发新的数学框架来解决该问题。对于过程对应于随时间变化的瑞利衰落信道的情况，我们给出了LCR的精确封闭式公式，简化的紧上限以及$ n $和$ p $接近无穷大但具有固定比率的渐近结果。这些结果揭示了LCR的有趣基本限制，并且即使对于$ n $和$ p $的较小值，也显示出有意义的见解。我们进一步使用我们的数学框架来描述组合分集接收器的通用选择的每分支所需的总切换速率，并为每个分支提供不同的平均功率。借助主化理论，我们证明了当功率延迟曲线均匀时，总开关速率最大。
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