and
CONDITION 1.7.– Service times have the first exponential phase, i.e.
where
and independent random variables and
As regeneration points for Y we take subsequence
of the sequence
such that at time
interrupted services for processes
are in the exponential phase. Because of conditions 1.6 and 1.7,
Y is a strongly regenerative flow and we may define the common sequence
of regeneration points for both processes
X and
Y with the help of formula
[1.4]. We need only to take
instead of
Because of [1.10] we can easily obtain from the renewal theory the formula for the rate of the auxiliary process
[1.11]
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
be moments of breakdowns and
moments of restorations for the
ith server. Here
[1.12]
Then
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
we have from [1.11]
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
for t > 0 takes place and hence
Since
To prove the second statement, we first assume that conditions 1.6 and 1.7 hold. Then condition 1.1 for the process Y takes place. We also may organize the performance of the systems S and S0 in such a way that inequality [1.8] is realized when
Thus, conditions 1.1, 1.4 and 1.5 are satisfied and because of theorem 1.2 the process
Q is stochastically bounded.
If conditions 1.6 and 1.7 (or one of them) are not valid, we construct a system Sδ satisfying conditions 1.6 and 1.7 and majorising our system S, so that in distribution
[1.13]
