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      and

      CONDITION 1.7.– Service times have the first exponential phase, i.e.

      where

and independent random variables and

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

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

      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

      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.

      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

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 satisfying conditions 1.6 and 1.7 and majorising our system S, so that in distribution