Читать книгу Queueing Theory 2 - Nikolaos Limnios - Страница 16

Our objective here is to establish stochastic boundedness of the process Q when the traffic rate ρ < 1. Under some additional assumptions providing the regenerative structure of the process Q, this property has its stability as a consequence.

THEOREM 1.2.– Let conditions 1.4and 1.5 and condition 1.1 (1.2) for the continuoustime (for the discrete-time) case be fulfilled. If ρ < 1, then Q is a stochastically bounded process.

PROOF.– Because of condition 1.4 there are two possible cases: Qn = Q(Tn) is either stochastically bounded or Assume that the second case takes place and ρ < 1. Because of condition 1.5 for there is nϵ such that for n > nϵ

that contradicts our assumption that ■

In the next sections, we discuss some examples to verify our results and to compare them with previous works. First, we consider two queueing models with service interruptions. These models occur in numerous applications and there is extensive literature concerning queueing system with interruptions. Let us mention some papers in which detail description of the literature in this sphere is given (Krishnamoorthy et al. 2012; Fiems and Bruneel 2013; Pechinkin et al. 2009; Morozov et al. 2011). However, to the best of our knowledge there are no papers that study the stability problem for multichannel systems with heterogeneous servers for the non-Markovian case: with general input flow and general distribution of blocked and available periods.