On Moments of Discrete Phase-Type Distributions

Abstract

Abstract.Recently, an efficient and stable method to compute moments of first passage times from a subset of states classified as safe to the other states in ergodic discrete-time Markov chains (DTMCs) has been proposed. This paper shows that the same method can be used to compute moments of discrete phase-type (DPH) distributions, analyzes its complexity on various acyclic DPH (ADPH) distributions, and presents results on a set of DPH distributions arising in a test suite of DTMCs.

Formal Techniques for Computer Systems and Business Processes

“…In this section, we present results that illustrate the computation of moments of DPH distributions. We have implemented the proposed method in C using double-precision IEEE floating-point arithmetic [3], performed and timed all experiments under Cygwin 1.5.15-1 on a Pentium IV 3.4 GHz processor and a 1 GB main memory running Windows XP. In each problem, the degree of coupling, p S ∞ , and the average degree of coupling, p S 1 /n S (see (1)), are reported so as to indicate the inherent difficulty associated with computing the moments accurately [4].…”

On model‐based detectors for linear time‐invariant stochastic systems under sensor attacks

“…Let denote the number of iterations needed to reach the absorbing state from . The random variable follows a discrete phase‐type distribution with and as defined in (27), see [44]. By construction, is a finite state approximation of the continuous Markov chain formed by the CUSUM sequence , driven by , .…”

Characterization of Model-Based Detectors for CPS Sensor Faults/Attacks

“…Let K ∈ N denote the number of iterations needed to reach the absorbing state E N from E 0 . The random variable K follows a discrete phase-type distribution with E[ K] = µ 1 and µ 1 as defined in (27), see [42]. By construction, M is a finite state approximation of the continuous Markov chain formed by the CUSUM sequence…”

“…where We linearize the nonlinear model introduced in [18] about the origin x(t) = 0 4×1 and then discretize it with sampling time h = 0.05. The resulting discrete-time linear system is given by ( 3)-( 8) with matrices as given in (42). The original model in [18] does not consider sensor/actuator noise, we have included noise to increase the complexity of our simulation experiments.…”