A Markov‐modulated fluid flow queueing model under D‐policy

Abstract

We consider a Markov-modulated fluid flow queueing model under D-policy. As soon as the fluid level reaches zero, the server becomes idle. During the idle period, fluid arrives from outside according to an underlying continuous time Markov chain (UMC) and the idle server does not process the fluid. We consider two increase patterns of fluid during the idle period: vertical increase (Type-V) and linear increase (Type-L). The idle server is reactivated only when the cumulative fluid level in the system exceeds a predetermined threshold value D. We derive the distributions of fluid level and mean performance measures for both types. We also present cost optimization model to minimize average operating cost per unit time. A MARKOV-MODULATED FLUID FLOW QUEUEING MODEL UNDER D-POLICY à . P is used to compute performance measures by using uniformization technique. In the analysis of MMFF-related processes, first passage times play important roles. Let .x/ D i nf ft > 0, U.t/ D xg be the first passage time to level x. Let us define the following Laplace-Stieltjes transforms (LST), ‰ .Â/ ij D EOEe  .0/ , J. .0// D j jU.0/ D 0, J.0/ D i, .i 2 = 1 , j 2 = 2 /, G 12 .Âjx/ ij D EOEe  .0/ , J. .0// D j jU.0/ D x, J.0/ D i, .i 2 = 1 , j 2 = 2 /,

A threshold policy in a Markov-modulated production system with server vacation: the case of continuous and batch supplies

“…Throughout this paper, we assume that one unit of raw material is used to produce one item. Motivated by Baek et al (2011), we introduce two types of system with different supply arrival patterns (see Figure 1 for sample paths).…”

“…The closest paper to this work is Baek et al (2011). They considered a Markov-modulated fluid flow model under the D-policy with two increase patterns of fluid during the idle period: vertical increase (MAP process) and linear increase (fluid arrival process).…”

“…The queue length process and the waiting time of a MAP/G/1 queue under the D-policy were studied by Lee and Song (2004), , and Lee and Baek (2005). Additional studies on the MAP(BMAP)/G/1 queue under the D-policy can be found in Chang et al (2002) and, more recently, Baek et al (2011), (2013). Furthermore, Baek et al (2014) proposed a simple factorization property for the workload distribution of the BMAP/G/1 vacation queues under variable service speed.…”

“…The study of hitting times for Markov-modulated models has been initiated in some works, e.g. Ahn and Ramaswami (2005), Bean et al (2005), Ahn et al (2007), Badescu et al (2007), Breuer (2010), and Baek et al (2011).…”

Numerical Solution of Markov Chains

“…Baek et al analyze a fluid queueing model in which the server becomes idle when the fluid (workload) level reaches level zero. In the idle period fluid arrives to the queue and the server becomes active again if the fluid level reaches a given threshold value.…”