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Order relations in the set of probability distribution functions and their applications in queueing theory

100%

Rolski T.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction......................................................................................................................................... 5 1. n-Monotonic functions on (— ∞, ∞)........................................................................................... 6 2. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts............................................................................................................ 12 2.2. Relations $≤_{1.n}, ≤_{2.n}$............................................................................................. 13 2.3. Extremal probability distribution functions........................................................................ 17 2.4. Relations $≤_{2.0}, ≤_{2.0}$............................................................................................. 18 2.5. Isotonic operators................................................................................................................. 22 2.6. Remarks about quasi-ordering relations in the set of random variables.................. 26 3. Order relationship between queueing systems................................................................. 26 3.1. Preliminary concepts, $GI^{(x)}/G^{(y)}/1$ queues.......................................................... 26 3.2. $GI^{(x)}/G/1$ queues........................................................................................................... 27 3.3. Order relationship between $GI^{(x)}/M^{(y)}/1$ and $M^{(x)}/G^{(y)}/1$ queues...... 30 4. Bounds for $GI^{(x)}/G^{(y)}/1$ queues................................................................................ 32 4.1. Introduction............................................................................................................................. 32 4.2. Bounds for $GI^{(x)}/G^{(y)}/1$ queues ........................................................................... 33 4.3. Bounds for $GI^{(x)}/M^{(y)}/1, M^{(x)}/G^{(y)}/1$ queues............................................... 36 4.4. Application of the relations $≤_{1.n} ≤_{2.n}$ in queues............................................ 37 Appendix...................................................................................................................................................... 38 References.................................................................................................................................................. 46

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A note on the history of the Poisson process

96%

Rolski T.

Antiquitates Mathematicae

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2019

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tom 13

PL

Artykuł jest poświęcony historii współczesnej koncepcji procesu Poissona (czasami nazywanej losową miarą Poissona). Zaczynamy od początków, które są zawarte w pracach F. Lundberga i W. Fellera aż do R´enyi (1967) i wreszcie, konstrukcji procesów Poissona przez procesy dwumianowe (lub Bernoulli’ego), które można znaleźć w pracach Moyala (1962), Meckego (1967) i Kingmana (1967). Również u Ryll-Nardzewskiego (1954) ta konstrukcja została wykorzystana do dowodu twierdzenia o jednorodnych procesach Poissona.

EN

This note reviews how the contemporary concept of the Poisson process (sometimes called the Poisson random measure or Poisson point process) evolved through out the time up to recent times. That is from F. Lundberg and W. Feller up to A. Renyi (1967) paper and finally, a construction of Poisson processes by binomial (or Bernoulli) processes, which can be found in J.E. Moyal (1962), J. Mecke (1967) and J.F. Kingman (1967) papers. Also in C. Ryll-Nardzewski (1953) this construction was used in the proof of Theorem for homogeneous Poisson processes.

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Approximation of the probability of insolvability of a portfolio

61%

Pusz R., Rolski T.

Mathematica Applicanda

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2005

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tom 33

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nr 47/06

EN

The authors review and criticize different approaches for calculations of ruin probabilities in one or multiperiod life insurance portfolios. They claim that precise calculations are time-consuming and in some cases (especially in the multiperiod ones) are practically inaccessible even if theoretical formulae are available. The reviewer considers that, because parameters of any such model must be estimated somehow, then precision in calculations cannot be achieved. For example, the authors assume a constant interest rate, which is not precisely the best way of doing things for the multiperiod model, and the precision must be lost anyway. Particularly, the authors show that the use of the central limit theorem is not appropriate; errors are large. The precise recursive method is difficult to perform for a large portfolio. The authors propose the importance of the Monte Carlo sampling method instead of the crude one. Their estimations are based on some probabilistic inequalities such as a version of Chernoff's (saddle point), also analyzed by S. Asmussen [Stochastic simulation with a view toward stochastic processes, Univ. Aarhus, Aarhus, 1999; Zbl 0981.65013], and are related to the approximation from D. Blackwell and J. L. Hodges, Jr. [Ann. Math. Statist. 30 (1959), 1113–1120; MR0112197].

Ograniczanie wyników

1 Antiquitates Mathematicae

1 Mathematica Applicanda

3 Rolski T.

1 Pusz R.

1 2019

1 2005

1 1976

Order relations in the set of probability distribution functions and their applications in queueing theory

A note on the history of the Poisson process

Approximation of the probability of insolvability of a portfolio