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On applications of computer algebra systems in queueing theory calculations

Ziółkowski Marcin

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2024

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Vol. 72, nr 4

art. no. e150199

EN

In the present paper, the most important aspects of computer algebra systems applications in complicated calculations for classical queueing theory models and their novel modifications are discussed. We mainly present huge computational possibilities of Mathematica environment and effective methods of obtaining symbolic results connected with the most important performance characteristics of queueing systems. First of all, we investigate effective solutions to computational problems appearing in queueing theory such as: finding final probabilities for Markov chains with a huge number of states, calculating derivatives of complicated rational functions of one or many variables with the use of classical and generalized L’Hospital’s rules, obtaining exact formulae of Stieltjes convolutions, calculating chosen integral transforms used often in the above-mentioned theory and possible applications of generalized density function of random variables and vectors in these computations. Some exemplary calculations for practical models belonging both to classical models and their generalizations are attached as well.

2

Single–server queueing system with limited queue, random volume customers and unlimited sectorized memory buffer

Ziółkowski Marcin, Tikhonenko Oleg

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 6

art. no. e143647

EN

In the present paper, we analyze the model of a single–server queueing system with limited number of waiting positions, random volume customers and unlimited sectorized memory buffer. In such a system, the arriving customer is additionally characterized by a non– negative random volume vector whose indications usually represent the portions of unchanged information of a different type that are located in sectors of unlimited memory space dedicated for them during customer presence in the system. When the server ends the service of a customer, information immediately leaves the buffer, releasing resources of the proper sectors. We assume that in the investigated model, the service time of a customer is dependent on his volume vector characteristics. For such defined model, we obtain a general formula for steady–state joint distribution function of the total volume vector in terms of Laplace-Stieltjes transforms. We also present practical results for some special cases of the model together with formulae for steady–state initial moments of the analyzed random vector, in cases where the memory buffer is composed of at most two sectors. Some numerical computations illustrating obtained theoretical results are attached as well.

3

Single-server queueing system with external and internal customers

Tikhonenko O., Ziółkowski M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 4

539--551

EN

In the paper, we investigate a single-server queueing system with unlimited memory space and non-homogeneous customers (calls) of the two following types: 1) external customers that are served by the system under consideration, 2) internal customers that arrive and interrupt the service process only when an external customer is being served. The external customers appear according to a stationary Poisson process. Customers of each of the above-mentioned types are characterized by some random volume. The customer service time depends arbitrarily on its volume. Two schemes of customer service organization are analyzed. The non-stationary and stationary distributions of the total volume of customers present in the system are determined in terms of Laplace and Laplace-Stieltjes transforms. The stationary first and second moments of total customers volume are also calculated. The obtained results are used to approximate loss characteristics in analogous systems with limited buffer space. Numerical examples illustrating theoretical results are attached as well.

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Wyznaczenie rozkładu objętości sumarycznej w jednoliniowym systemie kolejkowym z priorytetem bezwzględnym

Tikhonenko O.

Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie

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2014

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Nr 2(8)

57--75

PL

W artykule przeanalizowano jednoliniowe systemy kolejkowe z priorytetem bezwzględnym przy założeniu, że zgłoszenie każdej klasy priorytetowej posiada objetość losową, od której zależy czas obsługi tego zgłoszenia. Założono również, że charakterystyki wspólnego rozkładu objetości zgłoszenia i jego czasu obsługi mogą być różne dla różnych klas priorytetowych. Dla rozpatrywanego systemu z nieograniczoną pojemnością pamięci otrzymano charakterystyki wspólnego rozkładu objetości sumarycznej zgłoszeń różnych klas priorytetowych.

EN

One-server preemptive priority queueing systems are analyzed under assumption that a demand of each priority has some random capacity and service time of the demand depends on its capacity. It is assumed that the joint characteristics of service time and demand capacity can be different for different priorities. For the system under consideration (with unbounded total capacity), characteristics of joint distribution of each priority total demand capacity are obtained.

5

On the geometric compounding model with applications

Hu C. Y., Lin G. D.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

135--147

EN

Under the geometric compounding model, we first investigate the relationship between the compound geometric distribution and the underlying distribution, including the preservation of the infinite divisibility property. An interesting upper bound for the tail probability of the compound geometric distribution is provided by using only the mean of the underlying distribution. Secondly, we apply the obtained results to understand better the L-class of life distributions. In particular, we strengthen a surprising result of Bhattacharjee and Sengupta [5] and show that there are life distributions FєL with the following properties: (i) the support of F consists of countably infinite points, (ii) (ii) the coefficient of variation of F is equal to one, and (iii) (iii) F is not in the HNBUE class (the harmonic new better than used in expectation class).Finally, we apply geometric compounds to characterize the semi-Mittag-Leffler distribution and extend a known result about the exponential distribution.