Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

Abstract

Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, timedependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

On the Bounds for a Two-Dimensional Birth-Death Process with Catastrophes

“…Here we apply the approach from [18]. One can rewrite the original system for the unperturbed process in the form: dp dt =B(t)p(t) +f(t) +B(t)p(t) +f(t).…”

“…Such models are widely used in queueing theory and biology, particularly, for simulations in hight-performance computing. In some recent papers, the authors deal with more or less special birth-death processes with additional transitions from and to origin [9][10][11][12][13][18][19][20]. In [22], a general class of Markovian queueing models with possible catastrophes is analyzed and some bounds on the rate of convergence are obtained.…”

Ergodicity and truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

“…But transitions from each state i > 0 happen with intensities µ i (t), λ i (t) and β i (t) and can be either to state (i − 1) or (i + 1) or 0, respectively. Such subclass of processes finds its application in the study of queueing systems with catastrophes and bulk arrivals (see, for example, [7,1,3,2,8,5,9,11,4,10]). For more details, concerning possible applications, one can refer to [11] and references therein.…”

“…The motivation for this research was given by papers [11]- [14], where authors studied transient behaviour of various inhomogeneous birth and death processes being a subclass of X(t). Specifically their results concerned ergodicity and perturbation bounds, and bounds of truncation 1 of the processes (for example, probability of being in a particular state at time t, or expected value of the process at time t, which started from any given state).…”

“…A good alternative is to have bounds for performance characteristics of interest, which can be computed fast and are tight enough to make results meaningful. With respect to this observation, the direction of research, indicated by papers [11]- [14], looks promising.…”