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1

GI/GI/1 queues with infinite means of service time and interarrival time

Szczotka W.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 2

279--293

EN

The main results deal with the GI/GI/1 queues with Infinite means of the service times and interarrival times. Theorem 3.1 gives an asymptotic, in a heavy traffic situation, of the sequence of waiting times of the consecutive customers. Theorem 4.1 gives an asymptotic of stationary waiting times in a heavy traffic situation. In a special case, the asymptotic stationary waiting times have an exponential distribution (Corollary 4.1).

2

Chain dependent continuous time random walk

Szczotka W., Żebrowski P.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

239--261

EN

An asymptotic behavior of a continuous time random walk is investigated in the case when the sequence of pairs of jump vectors and times between jumps is chain dependent.

3

Two types of Markov property

Kasprzyk A., Szczotka W.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 1

75--90

EN

The paper gives some insight into the relations between two types of Markov processes – in the strict sense and in the wide sense – as well as into two aspects of periodicity. It concerns Markov processes with finite state space, the elements of which are complex numbers. Firstly it is shown that under some assumptions this space can be transformed in such a way that the resulting Markov process is also Markov in the wide sense. Next, sufficient conditions are given under which periodic homogeneous Markov chain is a periodically correlated process.

4

Tightness of stationary waiting times in heavy traffic for G1/G1/1 queues with thick tails

Czystołowski M., Szczotka W.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 1

109--123

EN

Recently, a Heavy Traffic Invariance Principle was proposed by Szczotka and Woyczyński to characterize the heavy traffie limiting distribution of normalized stationary waiting times of G/G/X queues in terms of an appropriate convergence to a Levy process. It has two important assumptions. The first of them deals with a convergence to a Levy process of appropriate processes which is well investigated in the literature. The second one states that the sequence of appropriate normalized stationary waiting times is tight. In the present paper we characterize the tightness condition for the case of GI/GI/1 queues in terms of the first condition.

5

Heavy-tailed dependent queues in heavy traffic

Szczotka W., Woyczyński W. A.

Probability and Mathematical Statistics

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2004

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Vol. 24, Fasc. 1

67--96

EN

The paper studies G/G/1 queues with heavy-tailed probability distributions of the service times and/or the interarrival times. It relies on the fact that the heavy traffic limiting distribution of the normalized stationary waiting times for such queues is equal to the distribution of the supremum M = sup0 ≤ t < ∞ (X(t)−βt), where X is a Lévy process. This distribution turns out to be exponential if the tail of the distribution of interarrival times is heavier than that of the service times, and it has a more complicated non-exponential shape in the opposite case; if the service times have heavy-tailed distribution in the domain of attraction of a one-sided α-stable distribution, then the limit distribution is Mittag-Leffler’s. In the case of a symmetric α-stable process X, the Laplace transform of the distribution of the supremum M is also given. Taking into account the known relationship between the heavy-traffic-regime distribution of queue length and its waiting time, asymptotic results for the former are also provided. Statistical dependence between the sequence of service times and the sequence of interarrival times, as well as between random variables within each of these two sequences, is allowed. Several examples are provided.

6

Distributions of suprema of Lévy processes via the Heavy Traffic Invariance Principle

Szczotka W., Woyczyński W. A.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

251--272

EN

We study the relationship between the distribution of the supremum functional MX = sup0 ≤ t < ∞ (X(t) − βt) for a process X with stationary, but not necessarily independent increments, and the limiting distribution of an appropriately normalized stationary waiting time for G/G/l queues in heavy traffic. As a by-product we obtain explicit expressions for the distribution of MX in several special cases of Lévy processes.