Now we may calculate the traffic rate ρ and under some assumptions we get the necessary and sufficient stability condition for the system based on theorems 1.1 and 1.2. As an example, we consider the famous case (Morozov et al. 2011) when
i.e. a server may be in an available or unavailable state. Let
denote the length of the nth blocked and the nth available period for the ith server, respectively,
The sequence
consists of iid random vectors (for all
) and these sequences do not depend on the input flow X and service times. Let
be the length of the nth cycle for the server i. A cycle consists of a blocked period followed by an available period. We assume that
We put ni(t) = 0 if the ith server is in an unavailable state at time t and ni(t) = 1, otherwise
If a blocked period
has an exponential phase, i.e.
where
are independent random variables and
has an exponential distribution with a parameter αi, then we may define the sequence
of regeneration points for the regenerative process
as above. Therefore, condition 1.6 holds. Under condition 1.7, the auxiliary process Y is strongly regenerative and we can construct the common points of regeneration
for X and Y and apply theorems 1.1 and 1.2 for this model. Since
If bi = b, then we get the same stability condition as obtained in Morozov et al. (2011) for a queueing system GI|G|m with a common distribution function of service times for all servers.
COROLLARY 1.1.– For a queueing system with
if ρ > I.
Under condition 1.4, the process is stochastically bounded if ρ < 1.
PROOF.– Let, as before,
be the number of customers actually served on the ith server up to time t. It is evident that stochastic inequality
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