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

Abstract

For the $M^{[X]}/M/1$ processor Sharing queue with batch arrivals, the sojourn time $\Omega$ of a batch is investigated. We first show that the distribution of $\Omega$ 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 $s\mapsto E^{\ast }(s,u,v)$ the one-sided Laplace transform of $E(\cdot ,u,v)$ and defining $$\Phi (s,u,v)=P(s,u)(1-v)F^{\ast }(s,u,uv),0<|u|<1,|v|<1,$$ for some known polynomial $P(s,u)$ and where $$F^{\ast }(s,u,v)=\frac{E^{\ast }(s,u,v)-E^{\ast }(s,q,v)}{u-q},$$ we show that the function $\Phi$ verifies an inhomogeneous linear partial differential equation (PDE) $$\frac{\partial \Phi }{\partial u}-\left[\frac{u-q}{P(s,u)}\right]v(1-v)\frac{\partial \Phi }{\partial v}+\ell (s,u,v)=0$$ for given $s$, where the last term $\ell (s,u,v)$ involves both $E^{\ast }(s,q,v)$ and the first order derivative $\partial E^{\ast }(s,q,v)/\partial v$ at the boundary point $u=q$. Solving this PDE for $\Phi$ via its characteristic curves and with the required analyticity properties eventually determines the one-sided Laplace transform $E^{\ast }$. By means of a Laplace inversion of this transform $E^{\ast }$, the distribution function of the sojourn time $\Omega$ of a batch is then given in an integral form. The tail behavior of the distribution of sojourn time $\Omega$ is finally derived.

22 pages

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