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

Fabrice Guillemin, Veronica Quintuna Rodriguez+2 more|arXiv (Cornell University)|Jun 3, 2020

Advanced Queuing Theory Analysis2 references4 citations

TL;DR

This paper derives the sojourn time distribution for a batch in an $M^{[X]}/M/1$ processor-sharing queue with batch arrivals, assuming geometrically distributed batch sizes. By modeling the system via an infinite-dimensional linear differential system and transforming it into a governing partial differential equation (PDE) for a bivariate generating function, the authors solve the PDE using Laplace transforms and characteristic curves, ultimately obtaining an integral representation for the sojourn time distribution and its tail behavior.

ABSTRACT

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) $$ \frac{\partial Φ}{\partial u} - \left [ \frac{u - q}{P(s,u)} ight ] v(1-v) \, \frac{\partial Φ}{\partial v} + \ell(s,u,v) = 0 $$ 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.

Motivation & Objective

• To characterize the stationary distribution of the sojourn time $\Omega$ for an entire batch in an $M^{[X]}/M/1$ processor-sharing queue with batch arrivals.
• To address a gap in the literature by deriving the batch sojourn time distribution, which has not been previously studied for $B > 1$.
• To analyze the tail behavior of the sojourn time distribution under the assumption of geometrically distributed batch sizes.
• To develop a solvable framework based on generating functions and Laplace transforms for the infinite-dimensional system arising from the queueing model.

Proposed method

• Formulate the sojourn time distribution of a batch as an infinite-dimensional linear differential system based on the number of jobs in the system at batch arrival.
• Define a bivariate generating function $E(x,u,v)$ to aggregate the conditional distribution functions of sojourn time given system state and batch size.
• Transform the system into a governing second-order linear PDE for the one-sided Laplace transform $E^*(s,u,v)$ of $E(x,u,v)$.
• Introduce auxiliary functions $F^*$ and $\Phi$ to reduce the governing PDE to an inhomogeneous linear PDE involving boundary values at $u = q$.
• Solve the PDE for $\Phi$ using characteristic curves and enforce analyticity at $u = 0$ via a necessary and sufficient condition on the inhomogeneous term.
• Determine the Laplace transform $E^*(s,q,v)$ via a triangular linear system involving integrals of the hypergeometric function, enabling inverse Laplace transform to recover the distribution.

Experimental results

Research questions

• RQ1 What is the exact distribution of the sojourn time $\Omega$ for a batch in an $M^{[X]}/M/1$ processor-sharing queue with batch arrivals?
• RQ2 How does the tail behavior of the batch sojourn time distribution depend on system parameters such as arrival rate $\lambda$, service rate $\mu$, and batch size parameter $q$?
• RQ3 Can the infinite-dimensional system governing the sojourn time be reduced to a solvable PDE using generating functions and Laplace transforms?
• RQ4 What role does the geometric distribution of batch sizes play in enabling the derivation of explicit expressions for the sojourn time distribution?

Key findings

• The sojourn time distribution of a batch is obtained in integral form via Laplace inversion of the transform $E^*(s,u,v)$, which is derived from solving a PDE with boundary conditions.
• The Laplace transform $E^*(s,q,v)$ is uniquely determined by a triangular linear system involving integrals of the Gauss hypergeometric function.
• The tail behavior of the sojourn time distribution is derived, providing asymptotic decay rates for large sojourn times.
• The solution relies on a necessary and sufficient condition for analyticity at $u=0$, which ensures the existence and uniqueness of the solution to the PDE.
• The governing PDE for $\Phi$ is solved using characteristic curves, with the inhomogeneous term involving boundary values of $E^*$ at $u=q$.
• The method successfully reduces the infinite-dimensional system to a finite-dimensional linear system for the coefficients $E_b^*(s,q)$, enabling explicit computation of the transform.