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1

Diffusion Limits for Shortest Remaining Processing Time Queues with Multiple Customer Types

Gieroba Robert, Kruk Łukasz

Annals of Computer Science and Information Systems

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2023

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Vol. 35

119--130

EN

We consider a single-server queueing system with multiple customer types in which users are scheduled according to the Shortest Remaining Processing Time (SRPT) discipline, with First In First Out (FIFO) as the tie-breaker. We use probabilistic methods to find, under typical heavy traffic assumptions, a suitable approximation of the workload and queue length processes after a long time has passed and show that these processes are divided among the customer classes according to specific proportions, depending on their arrival rates and distributions of initial service times. Our results are confirmed by simulations.

2

Formulas for average transition times between states of the Markov birth-death process

Zhernovyi Yuriy, Kopytko Bohdan

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 4

99--110

EN

In this paper, we consider Markov birth-death processes with constant intensities of transitions between neighboring states that have an ergodic property. Using the exponential distributions properties, we obtain formulas for the mean time of transition from the state i to the state j and transitions back, from the state j to the state i. We found expressions for the mean time spent outside the given state i, the mean time spent in the group of states (0,...,i-1) to the left from state i, and the mean time spent in the group of states (i+1,i+2,...) to the right. We derive the formulas for some special cases of the Markov birth-death processes, namely, for the Erlang loss system, the queueing systems with finite and with infinite waiting room and the reliability model for a recoverable system.

3

Modelowanie analitycznie wielousługowych, niepełnodostępowych systemów kolejkowych

Hanczewski S., Kmiecik D., Weissenberg J.

Przegląd Telekomunikacyjny + Wiadomości Telekomunikacyjne

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2016

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nr 8-9

874--878, CD

PL

W artykule przedstawiony został nowy, przybliżony model wielousługowego, niepełnodostęnego systemu kolejkowego. Może być on wykorzystywany do modelowania systemów sieciowych o ograniczonym dostępie do zasobów. Ze względu na fakt, że model ten jest modelem przybliżonym, rezultaty uzyskane za jego pomocą zostały porównane z wynikami eksperymentów symulacyjnych. Porównanie to pozwoliło na potwierdzić słuszność przyjętych założeń modelu.

EN

The paper presents a new approximate analytical model of a multi-service, non-full-availability queueing system. The model can be use to model the access nodes of the network. Taking into consideration that the proposed model is approximate, the results of the analytical calculations obtained on the basis of the proposed in the paper model are compared with the simulation data. The simulation study has confirmed that the proposed model is characterised by high accuracy.

4

Comparison of firefly and cockroach algorithms in selected discrete and combinatorial problems

Kwiecień J., Filipowicz B.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2014

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Vol. 62, nr 4

797--804

EN

In recent years, newer algorithms inspired by nature have been created and used to solve various problems. Therefore, in the paper we present the application of firefly and cockroach algorithms to optimize two queueing systems and permutation flow shop problems with the objective of minimizing the makespan. The article briefly describes these algorithms to solve selected problems and their results. Because these algorithms were originally developed for continuous optimization problems, we introduce a new formula to transform the position of ith individual to solve the discrete problems.

5

Firefly algorithm in optimization of queueing systems

Kwiecień J., Filipowicz B.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2012

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Vol. 60, nr 2

363-368

EN

Queueing theory provides methods for analysis of complex service systems in computer systems, communications, transportation networks and manufacturing. It incorporates Markovian systems with exponential service times and a Poisson arrival process. Two queueing systems with losses are also briefly characterized. The article describes firefly algorithm, which is successfully used for optimization of these queueing systems. The results of experiments performed for selected queueing systems have been also presented.

6

M/M/n/m queueing systems with non-identical servers

Ziółkowski M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

163--168

EN

M/M/n/m queueing systems with identical servers are well known in queueing theory and its applications. The analysis of these systems is very simple thanks to the fact that the number of customers ɳ(t) in the system at arbitrary time instant t forms a Markov chain. The main purpose of this paper is to analyse the M/M/n/m system under assumption that its servers are different, i.e. they have different parameters of service time.

7

Stationary characteristics of M θ /M/1 queue with switching of service modes

Kopytko B., Zhernovyi K.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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Vol. 10, nr 2

117-128

EN

For an M θ /M/1 queue with a threshold switching of service modes at the start of the service of the next customer an algorithm for determining the stationary distribution of the number of customers and stationary characteristics (average queue length, average waiting time, variance of queue length) is proposed. In the case the minimum number of incoming customers in the group is comparable to threshold value h, the stationary characteristics are found in an explicit form. The results are verified by simulation models constructed with the assistance of GPSS World.

8

Queueing systems and networks. Models and applications

Filipowicz B., Kwiecień J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2008

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Vol. 56, nr 4

379-390

EN

This article describes queueing systems and queueing networks which are successfully used for performance analysis of different systems such as computer, communications, transportation networks and manufacturing. It incorporates classical Markovian systems with exponential service times and a Poisson arrival process, and queueing systems with individual service. Oscillating queueing systems and queueing systems with Cox and Weibull service time distribution as examples of non-Markovian systems are studied. Jackson’s, Kelly’s and BCMP networks are also briefly characterized. The model of Fork-Join systems applied to parallel processing analysis and the FES approximation making possible of Fork-Join analysis is also presented. Various types of blocking representing the systems with limited resources are briefly described. In addition, examples of queueing theory applications are given. The application of closed BCMP networks in the health care area and performance evaluation of the information system is presented. In recent years the application of queueing systems and queueing networks to modelling of human performance arouses researchers’ interest. Hence, in this paper an architecture called the Queueing Network-Model Human Processor is presented.

9

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.

10

On global maxima in multiphase queues

Minkevicius S., Steisunas S.

Control and Cybernetics

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2005

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Vol. 34, no 2

575-588

EN

The target of this research in the queueing theory is to prove the law of the iterated logarithm (LIL) under the conditions of heavy traffic in multiphase queueing systems. In this paper, the LIL for global maxima is proved in the phases of a queueing system studied for an important probability characteristic of the system (total waiting time of a customer and waiting time of a customer).

11

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.

12

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.

13

Zastosowanie otwartych sieci wykładniczych BCMP do modelowania struktur organizacyjnych w służbie zdrowia

Kwiecień J.

Elektrotechnika i Elektronika

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2001

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T. 20, z. 2

83--87

PL

W pracy przedstawiono zastosowanie sieci kolejkowych BCMP do modelowania struktur organizacyjnych w służbie zdrowia na przykładzie wybranej przychodni lekarskiej. Opisano sposób jej funkcjonowania oraz przedstawiono model matematyczny odpowiadającej jej sieci kolejkowej. Wyliczono wielkości charakteryzujące pracę sieci: średnie liczby klientów w poszczególnych stacjach, czasy przebywania pacjentów w stacjach i w całym systemie. Zaproponowano zmiany w strukturze przychodni prowadzące do usprawnienia jej funkcjonowania.

EN

This paper presents the application of the queueing networks BCMP in modelling of health seryice organizing structure for example selected out-patient clinic. Analogous mathematical model of queueing network and the way of this clinic work have been presented. Characteristic work quantity o f network as the average number of patients in each service station, the average waiting time perpatient in each service station and the total time that a patient spends in the network are calculated. Suggested changes in structure of out-patient clinic making improvements to this clinic efficiency have been presented.

14

Zastosowanie rozkładu Coxa do analizy systemów kolejkowych

Filipowicz B.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2001

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T. 5, z. 1/2

193-201

PL

W pracy omówiono własności rozkładu Coxa L-tego rzędu. Zaprezentowano model systemu kolejkowego M/CoxL/1/FIFO/∞. Podano przykłady zastosowań tego modelu do aproksymacji innych systemów kolejkowych występujących w praktyce, dla których rozwiązania analityczne są trudne do uzyskania.

EN

Properties of L-th order Cox's distribution are described in the paper. The MCoxL/1/FLFO/∞ queueing system model is also presented. Some examples of application model to other queueing systems approximation appearing in practice, when analytical solutions are difficult to obtain.

15

Optymalizacja systemów kolejkowych z użyciem metod gradientowych

Filipowicz B., Rotter P.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2001

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T. 5, z. 1/2

203-211

PL

Optymalizacja systemu kolejkowego polega na znalezieniu maksimum funkcji zysku lub minimum funkcji kosztów w zależności od podstawowych parametrów opisujących system. Parametry te - a przynajmniej ich część - przyjmują wartości dyskretne. Przykładowo liczba kanałów obsługi m może przyjmować wartości ze zbioru liczb naturalnych, natomiast dla pozostałych m funkcja zysku (strat) systemu nie jest określona. W niniejszej pracy zaproponowano podejście alternatywne w stosunku do użycia procedur optymalizacji dyskretnej, polegające na uogólnieniu funkcji zysku bądź strat na dowolne rzeczywiste wartości parametrów. Pozwala to na użycie efektywnych algorytmów poszukiwania ekstremów funkcji ciągłych, dostępnych w postaci gotowych pakietów (np. Optimization Toolbox w języku Matlab). W pracy zaproponowano uogólnienie spełniające warunek różniczkowalności, dzięki czemu możliwe jest zastosowanie metod gradientowych.

EN

Optimization of queueing systems consists in finding maximum of gain function or minimum of cost function in dependence of system parameters, which are integers. Exemplary, m - number of service channels can take values from set of natural numbers, but for other m gain or loss function is not defined. Alternative approach to the usage of discrete optimization procedures based on generalization of gain or loss function onto the real parameter values is suggested. It allows for a usage of effective algorithms of searching extremum of continuous functions available as ready files (e.g. Optimization Toolbox in Matlab). Generalization of gain function fulfilling differentiability condition is introduced, so the application of gradient method is possible.

16

The Summarized Volume Distribution in GI/M/1/ ∞Queueing System

Tikhonenko O.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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1999

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Vol. 6

116--121

17

Symulacyjne badanie niezawodności pewnej klasy systemów kolejkowych

Filipowicz B.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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1999

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T. 3, z. 1

93-103

PL

W gospodarce rynkowej niezawodność szeroko rozumianych środków technicznych nabiera podstawowego znaczenia. Wynika to z faktu, iż obecnie coraz częściej mamy do czynienia z dużymi wzajemnie powiązanymi i współzależnymi systemami. Awaria nawet pojedynczego elementu takiego systemu może pociągnąć za sobą poważne straty. Dlatego też pierwszoplanową rolę odgrywa obecnie umiejętność analizy złożonych systemów pod względem niezawodności. Rozwiązanie tego zagadnienia może być przeprowadzone w oparciu o teorię kolejek. W zastosowaniach praktycznych jedyną efektywną metodą analizy złożonych systemów kolejkowych jest metoda symulacyjna. Niezależnie od wybranej metody ostatecznym celem zastosowania systemów kolejkowych w teorii niezawodności jest uzyskanie interesujących użytkownika parametrów systemu takich jak, np. oczekiwany czas bezawaryjnej pracy, czy też wykrycie najbardziej wrażliwej na uszkodzenie ścieżki. Taka analiza pozwala na ocenę pracy całego systemu, oraz na zastosowanie poprawek i ulepszeń, co w efekcie prowadzi do jego optymalizacji. W pracy zaprezentowany jest model symulacyjny bazy naprawczej, wyposażonej we własną bazę serwisową, opracowany na gruncie teorii kolejek.

EN

In economy of each country the reliability of technical means is essential. It results from the fact that great interrelated and interde-pendent systems more and morę often appear in our life. A damage even of a single element of the system can cause great losses. So that, ability of analysis of complex system in respect of reliability is the most important problem which can be solved using queueing theory. In practice a simulation method is the only effective method of analysis of complex queueing systems. Independently of the chosen method, the final purpose of the application of queueing systems in reliability theory is to obtain the required parameters of the system like expected error-free running time or to find a path, the most sen-sitive to damage. Such an analysis allows for the evaluation of the whole system, then its correction and improvement as well. Finally, it leads to the optimisation of working systems. The paper presents the simulation model of repair base which has been elaborated using ąueueing systems.