Approximations for Time-Dependent Distributions in Markovian Fluid Models

Abstract

In this paper we study the distribution of the level at time θ of Markovian fluid queues and Markovian continuous time random walks, the maximum (and minimum) level over [0, θ], and their joint distributions. We approximate θ by a random variable T with Erlang distribution and we use an alternative way, with respect to the usual Laplace transform approach, to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration.

On the exponential of semi-infinite quasi-Toeplitz matrices

“…The problem of computing the matrix exponential of Toeplitz and quasi-Toeplitz matrices is encountered in diverse applications, like the Erlangian approximation of Markovian fluid queues [10], [5], or the discretization of integro-differential equations with a shift-invariant kernel which describe the pricing of single-asset options modeled by jump-diffusion processes [16], [17], [19], where matrices are finite but have huge dimensions since their size has an asymptotic meaning. Another simple example comes from the numerical solution of the heat equation ∂u ∂t = γ ∂ 2 u ∂x 2 where the second derivative in space is discretized with the three-point finite difference formula, so that the equation is reduced to an ordinary differential equation of the kind v ′ = Av, being v the vector function of the values of u(x, t) at the discretization nodes.…”

Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

“…Example 2. This problem is inspired from [12]. The level normally evolves in Phases 1 or 2; occasionally it is in Phase 3 for short periods of time.…”

Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues

“…The test matrix T (U ) is taken from two real world problems concerning the Erlangian approximation of a Markovian fluid queue [10]. The block-size n of T (U ) is usually very large since a bigger n leads to a better Erlangian approximation, while the size m of the blocks is equal to 2 for both problems.…”

“…, n − 1, are m × m matrices. Our interest stems from the analysis in Dendievel and Latouche [10] of the Erlangization method for Markovian fluid models, but the story goes further back in time.…”