## Two-sided Reflection of Markov-modulated Brownian Motion (2010)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Ivanovs10two-sidedreflection,

author = {J. Ivanovs and O. Kella and M. Mandjes},

title = {Two-sided Reflection of Markov-modulated Brownian Motion},

year = {2010}

}

### OpenURL

### Abstract

In this paper we consider Markov-modulated Brownian motion on which a two-sided reflection is imposed (which can be interpreted as a finite-buffer queue with Markov-modulated Brownian input). Our approach heavily uses spectral properties of the matrix polynomial associated with the generator of the free (that is, non-reflected) process. These we use to compute for the doubly-reflected process the Laplace transform of the stationary distribution, as well as the average loss rates at both barriers. This work extends previous partial results; our framework allows the spectrum of the generator to be non-semi-simple, and it also covers the delicate case where the asymptotic drift of the free process is zero.

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Citation Context ...for an arbitrary Jordan chain v0, . . . , vn−1 of A(z) corresponding to some eigenvalue λ. Letting b T (z) = ∑K i=1 fi(z)uT i we have k∑ j=0 1 j! bT(j) (λ) vk−j = 0 T k = 0, . . . , n − 1, 6see e.g. =-=[10]-=-. It only remains to observe that the columns of V fi(Γ) are given by according to (12). k∑ j=0 1 (j) f i (λ)vk−j k = 0, . . . , n − 1, j! 4 The Matrix Exponent The matrix exponent F (α) defined in (2... |

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Citation Context ...an input) was considered; it was shown how to derive the stationary distribution under the (restrictive) assumption that the Brownian component be not subject to modulation. Relaxing this assumption, =-=[4]-=- applied a martingale technique to express the Laplace transform of the stationary distribution in terms of the average loss rates at the two boundaries. They managed to compute these average loss rat... |

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Citation Context ...I ± and M ∈ M 1{±κ≥0}, which satisfy the matrix equations 1 2 ∆2 σ P M 2 ∓ ∆a P M + Q P = O ± . (8) The above result has appeared in different degrees of generality in various previous works, such as =-=[3, 5, 14, 18]-=-. As an aside, we mention here that the uniqueness part is not required for the present work; in fact, this uniqueness follows from the analysis presented in Section 4. 3 Matrix Polynomials In this se... |

2 |
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Citation Context ...ution of (W, J), in terms of its Laplace transform, through Eqn. (4). This result completes the analysis of finite-buffer mmbm-driven queues contained in [4, Section 9] and extends the previous works =-=[12, 13, 18]-=-. 2.2 First Passage Process The study of the doubly-reflected process is closely related with the analysis of the exit times for the process X(t) from an interval, see for example [3, Prop. XIV.3.7]. ... |

1 |
Kella and M. Mandjes (2010): First passage of a Markov additive process and generalized Jordan chains
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Citation Context ... on a detailed understanding of the structure of the spectrum of the matrix exponent. More specifically, we use the methodology of generalized Jordan chains, which we introduced for the first time in =-=[6]-=- as a tool to analyze the first passage times of spectrally one-sided Markov additive processes (where spectrally one-sided indicates that the process has either only positive jumps or only negative j... |

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Citation Context ...orbing state corresponding to J(∞), we note that J(τ ± (x)), x ≥ 0 lives on E±∪{∂}, because X(t) can not hit a new maximum (resp. minimum) when J(t) is in a state belonging to E↓ (resp. E↑), see also =-=[16]-=-. Let Λ ± be the N± ×N± dimensional transition rate matrices of J(τ ± (x)) restricted to E±. In addition define the matrices of initial distributions Π ± as the N × N± matrices with the (i, j) element... |