Queueing Theory 2. Nikolaos Limnios

Queueing Theory 2

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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, 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

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