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Queueing Theory 2. Nikolaos Limnios
Читать онлайн.Название Queueing Theory 2
Год выпуска 0
isbn 9781119755227
Автор произведения Nikolaos Limnios
Жанр Математика
Издательство John Wiley & Sons Limited
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 formThen 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
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 forLEMMA 1.2.– There exists the limit
The proof easily follows from the evident inequalities
where
the strong law of large numbers and convergenceFrom 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, 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].COROLLARY 1.2.– Let condition 1.8 be fulfilled. Then
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