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      Here, (t) is the number of customers in the system at instant t. Let us introduce independent sequences

of iid random variables with exponential distribution with a rate δ. Assume that repair time
in the system has the form
and service time
by the ith server has the form

      Then satisfies conditions 1.6 and 1.7. Since

and
we may choose δ so that ρδ < 1.

      Let us note that condition 1.4 may be provided in various ways. For instance, assume that blocked (or available) period has an exponential phase and

that is a subsequence of the sequence
such that
and all servers are in the exponential phase of their blocked (or available) periods. Now condition 1.4 follows directly from theorem 1 in Afanasyeva and Tkachenko (2014). We also note that in this case Q is a stable process if ρ < 1. If only assumption [1.14] takes place with the help of the majorising system , we obtain the stochastic boundedness Q when ρ < 1. ■

and moments of restorations
for the ith server satisfy [1.12]. The input flow X is an aperiodic discrete-time regenerative flow with rate λX.

      We consider the preemptive repeat different service discipline that means that the service is repeated from the start after restoration of the server and the new service time is independent of the original service time (Gaver 1962).

      To define the process Yi for the ith server in the auxiliary system S0, we introduce the collection

of independent sequences
consisting of iid random variables with distribution function Bi. Of course, we assume that
Let
be the counting process associated with the sequence
be the number of cycles for the ith server during [0,t], i.e.
Then the process Yi is defined by the relation

      [1.15]

We denote by Hi(t) the renewal function for

      LEMMA 1.2.– There exists the limit

      The proof easily follows from the evident inequalities

      where

the strong law of large numbers and convergence

      From lemma 1.2, we have

      [1.16]

      We introduce the counting processes

      CONDITION 1.8.– The counting processes

are aperiodic.

      Then Y is a regenerative aperiodic flow with points of regeneration

is a point of regeneration of Y if all the servers get out of the order simultaneously at this moment. Taking into account condition 1.8, we conclude from lemma 1.1 that
Now we construct the sequence
of common points of regeneration for processes X and Y with the help of [1.3]. Because of lemma 1.1
and the traffic rate ρ of the system is defined by [1.7].

      1 i)

      2 ii) Q(t) is a stochastically bounded process if ρ < 1.

      PROOF.– The first statement follows from theorem 1.1 since conditions 1.2 and 1.3 are realized.

      Let ρ < 1. For the system S, we introduce the embedded process

where Qn is the number of customers in the system on time Tn and ζi(n) = 1 if there is a customer on the ith server and ζi (n) = 0 otherwise. In a view

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