On Truncations For A Class Of Finite Markovian Queuing Models

We consider a class of finite Markovian queueing models and obtain uniform approximation bounds of truncations. INTRODUCTION It is well known that explicit expressions for the probability characteristics of stochastic models can be found only in a few special cases, moreover, if we deal with an inhomogeneous Markovian model, then we must approximately calculate the limiting probability characteristics of the process. The problem of calculation of the limiting characteristics for inhomogeneous birth-death process via truncations was firstly mentioned in (Zeifman 1991) and was considered in details in (Zeifman et al. 2006). In (Zeifman et al. 2014b) we have proved uniform (in time) error bounds of truncation this class of Markov chains. First uniform bounds of truncations for the class of Markovian time-inhomogeneous queueing models with batch arrivals and group services (SZK models) introduced and studied in our recent papers (Satin et al. 2013, Zeifman et al. 2014a), were obtained in (Zeifman et al. 2014c). In this note we deal with approximations of finite SZK model via the same models with smaller state space and obtain the correspondent bounds of error of truncation bounds. Consider a time-inhomogeneous continuous-time Markovian queueing model on the state space E = {0, 1, . . . , r} with possible batch arrivals and group services. Let X(t), t ≥ 0 be the queue-length process for the queue, pij(s, t) = P {X(t) = j |X(s) = i}, i, j ≥ 0, 0 ≤ s ≤ t, be transition probabilities for X = X(t), and pi(t) = P {X(t) = i} be its state probabilities. Throughout the paper we assume that P (X (t+ h) = j|X (t) = i) = = { qij (t)h+ αij (t, h) , if j 6= i, 1− ∑ k 6=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). We also assume qi,i+k (t) = λk(t), qi,i−k (t) = μk(t) for any k > 0. In other words, we suppose that the arrival rates λk(t) and the service rates μk(t) do not depend on the queue length. In addition, we assume that λk+1(t) ≤ λk(t) and μk+1(t) ≤ μk(t) for any k and almost all t ≥ 0. Hence, X(t) is a so-called SZK model, which was studied in (Satin et al. 2013, Zeifman et al. 2014a, 2014c). We suppose that all intensity functions are locally integrable on [0,∞), and λk(t) ≤ λk, μk(t) ≤ μk, (2) for any k and almost all t ≥ 0, and put