F. Guillemin | Searcharxiv

arXiv2020

Sojourn time in a $M^{[X]}/M/1$ Processor Sharing Queue with batch arrivals (II)

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.

math.PR↗

arXiv2019

An inversion formula with hypergeometric polynomials and application to singular integral operators

Given parameters $x \notin \mathbb{R}^- \cup \{1\}$ and $ν$, $\mathrm{Re}(ν) < 0$, and the space $\mathscr{H}_0$ of entire functions in $\mathbb{C}$ vanishing at $0$, we consider the family of operators $\mathfrak{L} = c_0 \cdot δ\circ \mathfrak{M}$ with constant $c_0 = ν(1-ν)x/(1-x)$, $δ= z \, \mathrm{d}/\mathrm{d}z$ and integral operator $\mathfrak{M}$ defined by $$ \mathfrak{M}f(z) = \int_0^1 e^{- \frac{z}{x}t^{-ν}(1-(1-x)t)} \, f \left ( \frac{z}{x} \, t^{-ν}(1-t) \right ) \, \frac{\mathrm{d}t}{t}, \qquad z \in \mathbb{C}, $$ for all $f \in \mathscr{H}_0$. Inverting $\mathfrak{L}$ or $\mathfrak{M}$ proves equivalent to solve a singular Volterra equation of the first kind. The inversion of operator $\mathfrak{L}$ on $\mathscr{H}_0$ leads us to derive a new class of linear inversion formulas $T = A(x,ν) \cdot S \Leftrightarrow S = B(x,ν) \cdot T$ between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in \mathbb{N}^*}$, where the infinite lower-triangular matrix $A(x,ν)$ and its inverse $B(x,ν)$ involve Hypergeometric polynomials $F(\cdot)$, namely $$ \left\{ \begin{array}{ll} A_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,-nν;-n;x), B_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,kν;k;x) \end{array} \right. $$ for $1 \leqslant k \leqslant n$. Functional relations between the ordinary (resp. exponential) generating functions of the related sequences $S$ and $T$ are also given. These relations finally enable us to derive the integral representation $$ \mathfrak{L}^{-1}f(z) = \frac{1-x}{2iπx} \, e^{z} \int_{(0+)}^1 \frac{e^{-xtz}}{t(1-t)} \, f \left ( xz \, (-t)^ν(1-t)^{1-ν} \right ) \, \mathrm{d}t, \quad z \in \mathbb{C}, $$ for the inverse $\mathfrak{L}^{-1}$ of operator $\mathfrak{L}$ on $\mathscr{H}_0$, where the integration contour encircles the point 0.

math.CA↗