Some universal limits for nonhomogeneous birth and death processes

Abstract

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X (t), t ≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP X N . Finally we present some examples where these bounds are used in order to approximate the double mean.

On truncations for weakly ergodic inhomogeneous birth and death processes

“…The method is based on the following two components: the logarithmic norm of a linear operator and a special similarity transformation of the matrix of intensities of the Markov chain considered, see the corresponding definitions, bounds, references and other details in the works of Van Doorn et al (2010), Granovsky and Zeifman (2004), Zeifman (1985;1995b;1995a) or Zeifman et al (2006).…”

“…The best previous bounds contain an additional factor of t on the right-hand sides of (34) and (35), (see Zeifman et al, 2006).…”

“…On the other hand, if we deal with an inhomogeneous Markovian model, then we must approximately calculate, in addition to that, the limiting probability characteristics of the process. The problem of existence and construction of limiting characteristics for inhomogeneous (in time) birth and death processes is important for queueing applications (see, e.g., Di Crescenzo and Nobile, 1995;Di Crescenzo et al, 2003;Granovsky and Zeifman, 2004;Mandelbaum and Massey, 1995;Massey and Whitt, 1994;Massey and Pender, 2013;Olwal et al, 2012;Tan et al, 2013;Zeifman et al, 2006). This is the second main problem.…”

“…This is the second main problem. A general approach and related bounds for the study on the convergence rate was considered by Zeifman (1995a), who also first mentioned computation of the limiting characteristics for the process via truncations (Zeifman, 1988), and later considered it in detail (Zeifman et al, 2006). The first results for more general Markovian queueing models have been obtained recently by Zeifman et al (2014).…”

“…At first, we recall the definition of the logarithmic norm and the related bound (for details, see Van Doorn et al, 2010;Granovsky and Zeifman, 2004;Zeifman et al, 2006) Let B (t) , t ≥ 0 be a one-parameter family of bounded linear operators on a Banach space B and let I denote the identity operator. For a given t ≥ 0, the number…”