Two-Sided Truncations Of Inhomogeneous Birth-Death Processes

We consider a class of inhomogeneous birth-death queueing models and obtain uniform approximation bounds of two-sided truncations. Some examples are considered. Our approach to truncations of the state space can be used in modeling information flows related to high-performance computing. INTRODUCTION It is well known that explicit expressions for the probability characteristics of stochastic birth-death queueing models can be found only in a few special cases. Therefore, the study of the rate of convergence as time t → ∞ to the steady state of a process is one of two main problems for obtaining the limiting behavior of the process. If the model is Markovian and stationary in time, then, as a rule, the stationary limiting characteristics provide sufficient or almost sufficient information about the model. On the other hand, if one deals with inhomogeneous Markovian model then, in addition, the limiting probability characteristics of the process must be approximately calculated. The problem of existence and construction of limiting characteristics for time-inhomogeneous birth and death processes is important for queueing and some other applications, see for instance, [1], [3], [5], [8], [15], [16]. General approach and related bounds for the rate of convergence was considered in [13]. Calculation of the limiting characteristics for the process via truncations was firstly mentioned in [14] and was considered in details in [15], uniform in time bounds have been obtained in [17]. As a rule, the authors dealt with the so-called northwest truncations (see also [9]), namely they studied the truncated processes with the same first states 0, 1, . . . , N In the present paper we consider a more general approach and deal with truncated processes on state space N1, N1 + 1, . . . , N2 for some natural N1, N2 > N1. Let X = X(t), t ≥ 0 be a birth and death process (BDP) with birth and death rates λn(t), μn(t) respectively. Let pij(s, t) = Pr {X(t) = j |X(s) = i} for i, j ≥ 0, 0 ≤ s ≤ t be the transition probability functions of the process X = X(t) and pi(t) = Pr {X(t) = i} be the state probabilities. Throughout the paper we assume that P (X (t+ h) = j|X (t) = i) = = qij (t)h+ αij (t, h) if j ̸= i, 1− ∑ k ̸=i qik (t)h+ αi (t, h) if j = i, (1) where all αi(t, h) are o(h) uniformly in i, i. e. supi |αi(t, h)| = o(h). Here all qi,i+1 (t) = λi(t), i ≥ 0, qi,i−1 (t) = μi(t) i ≥ 1, and all other qij(t) ≡ 0. The probabilistic dynamics of the process is represented by the forward Kolmogorov system of differential equations:  dp0 dt = −λ0(t)p0 + μ1(t)p1, dpk dt = λk−1(t)pk−1 − (λk(t) + μk(t)) pk+ +μk+1(t)pk+1, k ≥ 1. (2) By p(t) = (p0(t), p1(t), . . . ) , t ≥ 0, we denote the column vector of state probabilities and by A(t) = (aij(t)) , t ≥ 0 the matrix related to (2). One can see that A (t) = Q⊤ (t), where Q(t) is the intensity (or infinitesimal) matrix for X(t). We assume that all birth and death intensity functions λi(t) and μi(t) are linear combinations of a finite number of functions which are locally integrable on [0,∞). Moreover, we suppose that λn(t) ≤ Λn ≤ L < ∞, μn(t) ≤ ∆n ≤ L < ∞, (3) Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) for almost all t ≥ 0. Throughout the paper by ∥ · ∥ we denote the l1-norm, i. e. ∥x∥ = ∑ |xi|, and ∥B∥ = supj ∑ i |bij | for B = (bij)i,j=0. Let Ω be a set all stochastic vectors, i. e. l1 vectors with nonnegative coordinates and unit norm. Then we have ∥A(t)∥ ≤ 2 sup(λk(t) + μk(t)) ≤ 4L, for almost all t ≥ 0. Hence the operator function A(t) from l1 into itself is bounded for almost all t ≥ 0 and locally integrable on [0;∞). Therefore we can consider the system (2) as a differential equation dp dt = A (t)p, p = p(t), t ≥ 0, (4) in the space l1 with bounded operator function A(t). It is well known (see, for instance, [2]) that the Cauchy problem for differential equation (1) has unique solutions for arbitrary initial condition, and moreover p(s) ∈ Ω implies p(t) ∈ Ω for t ≥ s ≥ 0. Therefore, we can apply the general approach to employ the logarithmic norm of a matrix for the study of the problem of stability of Kolmogorov system of differential equations associated with nonhomogeneous Markov chains. 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 [4], [5], [13], [15], [17]. Definition. A Markov chain X(t) is called weakly ergodic, if ∥p∗(t) − p∗∗(t)∥ → 0 as t → ∞ for any initial conditions p∗(0),p∗∗(0). Here p∗(t) and p∗∗(t) are the corresponding solutions of (4). Put Ek(t) = E {X(t) |X(0) = k } ( then the corresponding initial condition of system (4) is the k − th unit vector ek). Definition. Let X(t) be a Markov chain. Then φ(t) is called the limiting mean of X(t) if lim t→∞ (φ(t)− Ek(t)) = 0