Stochastic Fluid Flows with Upward Jumps and Phase Transitions: Analysis Through Matrix-Analytic Methods

Abstract

We consider a stochastic fluid model {(X(t), J(t)) : t ≥ 0} with level variable
X(t) ≥ 0, phase variable J(t) and some fixed ‘jump’ levels q_N > . . . > q₁ > 0. The
process is driven by a continuous-time Markov chain {J(t) : t ≥ 0} with state space S,
generator T, and real-valued fluid rates c_i ∈ R for all i ∈ S. The evolution of the level
variable X(t) is such that dX(t)/dt = c_J(t) whenever X(t) > 0, and as soon as the process
hits X(t) = 0, the phase variable J(t) transitions to some phase in S, while the level
variable X(t) may jump to some level q_n > 0, remain at the boundary q₀ = 0 or reflect
from it, and a phase transition may also occur. The process was previously analysed using
various algebraic methods in a special case with N = 1 and without special behaviour at the
boundary q₀. Here, for the first time, we analyse this process using matrix-analytic methods,
which is a powerful methodology in the field of applied probability, suitable for convenient
numerical analysis. We present methodology for the computation of the stationary and transient
distribution of the key performance measures of the model under general assumptions
and illustrate the theory with numerical examples.