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Let Q(t) be the number of customers in the system S including the customers on the servers at time t so that
CONDITION 1.4.– There are two possible cases:
1 i) Q(t) is a stochastically bounded process, i.e.
2 ii)
CONDITION 1.5.– If
1.4. Instability result for the case ρ ≥ 1
THEOREM 1.1.– Let conditions 1.3 and 1.1 (1.2) for the continuous-time (for the discrete-time) case be fulfilled. If ρ ≥ 1, then
[1.9]
PROOF.– Consider the embedded process Qn = Q(Tn) and denote
Define the auxiliary sequence
Because of the equality
and in distribution
For ρ ≥ 1, the sequence
1.5. Stochastic boundedness for the case ρ < 1
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
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.
1.6. Queueing system with unreliable servers and preemptive resume service discipline
We consider a continuous-time queueing system with regenerative input flow X and m heterogeneous servers that may be not available for operation from time to time. We also propose that the velocity of the service may be dependent on the state of the server. Assume that for the ith server a stochastic process ni(t) with state space
It is possible that an unavailable period starts while a customer is receiving service. Then service of the customer is immediately interrupted. There are various disciplines for continuation of the service after restoration (Gaver 1962). Here, we consider the preemptive resume service discipline assuming that interrupted service continues when the server returns from a blocked period and the service velocity is the next state of the process ni(t).
CONDITION 1.6.–The stochastic process
It follows from condition 1.6 and Smith’s (1955) theorem that there exist the limits
and
