Matrix equations in Markov modulated Brownian motion: theoretical properties and numerical solution

A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic Models, 1995) proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, Ahn and Ramaswami (Stochastic Models, 2017) derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE

Analyses of the Markov modulated fluid flow with one-sided phtype jumps using coupled queued and the completed graphs

In this paper, we analyze Markov modulated fluid flow rocesses with one-sided ph-type jumps using the completed graph and also through the limits of coupled queueing processes to be constructed. For the models, we derive various results on time-dependent distributions and distributions of first passage times, and present the Riccati equations that transform matrices of the first return times to 0 satisfy. The Riccati equations enable us to compute the transform matrices using Newton’s method which is known very fast and stable. Finally, we present some duality results between the model with ph-type downward jumps and the model with ph-type upward jumps. This paper contains extended results of Ahn (2009) and probabilistic interpretations given by the completed graphs.Accepted versio

A workload factorization for BMAP/G/1 vacation queues under variable service speed

This paper proposes a simple factorization property for the workload distribution of the BMAP/G/1/ vacation queues under variable service speed. The server provides service at different service speeds depending on the phases of the underlying Markov chain. Using the factorization principle, the workload distribution at any arbitrary time point can be easily derived only by obtaining the distribution during the idle period. We prove the factorization property and the moments formula. Lastly, we provide some applications of our factorization principle

A MAP-modulated fluid flow model with multiple vacations

We consider a MAP-modulated fluid flow queueing model with multiple vacations. As soon as the fluid level reaches zero, the server leaves for repeated vacations of random length V until the server finds any fluid in the system. During the vacation period, fluid arrives from outside according to the MAP (Markovian Arrival Process) and the fluid level increases vertically at the arrival instance. We first derive the vector Laplace–Stieltjes transform (LST) of the fluid level at an arbitrary point of time in steady-state and show that the vector LST is decomposed into two parts, one of which the vector LST of the fluid level at an arbitrary point of time during the idle period. Then we present a recursive moments formula and numerical examples.Accepted versio

A factorization property for BMAP/G/1 vacation queues under variable service speed

This paper proposes a simple factorization principle that can be used efficiently and effectively to derive the vector generating function of the queue length at an arbitrary time of the BMAP/G/1/ queueing systems under variable service speed. We first prove the factorization property. Then we provide moments formula. Lastly we present some applications of the factorization principle.Accepted versio