论文标题
$ m^{[x]}/m/1 $处理器共享队列（ii）
Sojourn time in a $M^{[X]}/M/1$ Processor Sharing Queue with batch arrivals (II)
论文作者
论文摘要
对于$ m^{[x]}/m/1 $处理器共享队列与批处理，研究了批处理的sojourn Time $ω$。 我们首先表明$ω$的分布通常可以从无限线性差异系统中获得。当进一步假设批处理大小具有给定参数$ q \ [0,1 [$ [$）的几何分布时，通过相关的双变量生成函数$（x，x，x，u，v）\ mapsto e（x，x，u，v）$进一步分析了这种差异系统。具体而言，用$ s \ mapsto e^*（s，u，v）$ $ e（\ cdot，u，v）$和定义$$φ（s，u，v）= p（s，s，u）\，（1- v）\，（1- v）\，（1- v） \ vert <1，$$对于某些已知的多项式$ p（s，u）$以及其中$$ f^*（s，s，u，v）= \ frac {e^*（s，s，u，u，v）-e^*（s，s，s，q，q，q，v）} {u-q} \ frac {\ partialφ} {\ partial u} - \ left [\ frac {\ q} {p（s，s，s，u）} \右] v（1- v）\，\ frac {\ frac {\partialφ} $ \ ell（s，u，v）$涉及$ e^*（s，q，v）$和第一阶派生$ \ partial e^*（s，s，q，v）/\ partial v $在边界点$ u = q $。通过其特征曲线和所需的分析性属性以$φ$解决此PDE最终确定单方面的拉普拉斯变换$ e^*$。 通过对此转换$ e^*$的拉普拉斯反转，然后以积分形式给出了批量的sojourn时间$ω$的分布函数。最终得出了Sojourn Time $ω$的分布的尾巴行为。
For the $M^{[X]}/M/1$ processor Sharing queue with batch arrivals, the sojourn time $Ω$ of a batch is investigated. We first show that the distribution of $Ω$ can be generally obtained from an infinite linear differential system. When further assuming that the batch size has a geometric distribution with given parameter $q \in [0,1[$, this differential system is further analyzed by means of an associated bivariate generating function $(x,u,v) \mapsto E(x,u,v)$. Specifically, denoting by $s \mapsto E^*(s,u,v)$ the one-sided Laplace transform of $E(\cdot,u,v)$ and defining $$ Φ(s,u,v) = P(s,u) \, (1-v) \, F^*(s,u,uv), \quad 0 < \vert u \vert < 1, \, \vert v \vert < 1, $$ for some known polynomial $P(s,u)$ and where $$ F^*(s,u,v) = \frac{E^*(s,u,v)-E^*(s,q,v)}{u-q}, $$ we show that the function $Φ$ verifies an inhomogeneous linear partial differential equation (PDE) $$ \frac{\partial Φ}{\partial u} - \left [ \frac{u - q}{P(s,u)} \right ] v(1-v) \, \frac{\partial Φ}{\partial v} + \ell(s,u,v) = 0 $$ for given $s$, where the last term $\ell(s,u,v)$ involves both $E^*(s,q,v)$ and the first order derivative $\partial E^*(s,q,v)/\partial v$ at the boundary point $u = q$. Solving this PDE for $Φ$ via its characteristic curves and with the required analyticity properties eventually determines the one-sided Laplace transform $E^*$. By means of a Laplace inversion of this transform $E^*$, the distribution function of the sojourn time $Ω$ of a batch is then given in an integral form. The tail behavior of the distribution of sojourn time $Ω$ is finally derived.
