Queueing Theory 2. Nikolaos Limnios

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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]

      Here, Qδ(t) is the number of customers in the system Sδ 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 Sδ has the form and service time by the ith server has the form

      Then Sδ satisfies conditions 1.6 and 1.7. Since

and we may choose δ so that ρδ < 1.

The proof of [1.13] is based on the “so-called” probability space method (Belorusov 2012).

      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

[1.14]

Then Q is a regenerative process with points of regeneration

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 Sδ, we obtain the stochastic boundedness Q when ρ < 1. ■

1.7. Discrete-time queueing system with interruptions and preemptive repeat different service discipline

Here, we consider the system with interruptions described in the previous section for the discrete-time case. The moments of breakdowns

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]

and

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

In other words,

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,

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