Wyniki wyszukiwania - BazTech

1

On the semi-Mittag-Leffler distributions

Lin Gwo Dong, Hu Chin-Yuan

Probability and Mathematical Statistics

|

2022

|

Vol. 42, Fasc. 1

81--9

EN

The semi-Mittag-Leffler (SML) distribution arises as the marginal of a stationary Markovian process, and is a generalization of the well-known Mittag-Leffler (ML) or positive Linnik distribution. Unlike the ML distribution, which has been well established, few properties of the SML distribution are discussed in the literature. In this paper, we derive some more characterizations of the SML and related distributions. By using stochastic inequalities, we further extend some characterizations, including Pitman and Yor’s (2003) result about the hyperbolic sine distribution.

2

Queueing systems with random volume customers and a sectorized unlimited memory buffer

Tikhonenko Oleg, Ziółkowski Marcin, Kempa Wojciech M.

International Journal of Applied Mathematics and Computer Science

|

2021

|

Vol. 31, no. 3

471--486

EN

In the present paper, we concentrate on basic concepts connected with the theory of queueing systems with random volume customers and a sectorized unlimited memory buffer. In such systems, the arriving customers are additionally characterized by a non-negative random volume vector. The vector’s indications can be understood as the sizes of portions of information of a different type that are located in the sectors of memory space of the system during customers’ sojourn in it. This information does not change while a customer is present in the system. After service termination, information immediately leaves the buffer, releasing its resources. In analyzed models, the service time of a customer is assumed to be dependent on his volume vector characteristics, which has influence on the total volume vector distribution. We investigate three types of such queueing systems: the Erlang queueing system, the single-server queueing system with unlimited queue and the egalitarian processor sharing system. For these models, we obtain a joint distribution function of the total volume vector in terms of Laplace (or Laplace-Stieltjes) transforms and formulae for steady-state initial mixed moments of the analyzed random vector, in the case when the memory buffer is composed of two sectors. We also calculate these characteristics for some practical case in which the service time of a customer is proportional to the customer’s length (understood as the sum of the volume vector’s indications). Moreover, we present some numerical computations illustrating theoretical results.

3

Performance evaluation of unreliable system with infinite number of servers

Tikhonenko O., Ziółkowski M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2020

|

Vol. 68, nr 2

289--297

EN

In the paper, we investigate queueing system M/G/∞ with non-homogeneous customers. By non-homogeneity we mean that each customer is characterized by some arbitrarily distributed random volume. The arriving customers appear according to a stationary Poisson process. Service time of a customer is proportional to his its volume. The system is unreliable, which means that all its servers can break simultaneously and then the repair period goes on for random time having an arbitrary distribution. During this period, customers present in the system and arriving to it are not served. Their service continues immediately after repair period termination. Time intervals of the system in good repair mode have an exponential distribution. For such system, we determine steady-state sojourn time and total volume of customers present in it distributions. We also estimate the loss probability for the similar system with limited total volume. An analysis of some special cases and some numerical examples are attached as well.

4

M/G→ /n/0 Erlang queueing system with heterogeneous servers and non-homogeneous customers

Ziółkowski M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2018

|

Vol. 66, nr 1

59--66

EN

In the present paper, we investigate a multi-server queueing system with heterogeneous servers, unlimited memory space, and non-homogeneous customers. The arriving customers appear according to a stationary Poisson process. Service time distribution functions may be different for every server. Customers are additionally characterized by some random volume. On every server, the service time of the customer depends on their volume. The number of customers distribution function is obtained in the classical model of the system. In the model with non-homogeneous customers, the stationary total volume distribution function is determined in the term of Laplace–Stieltjes transform. The stationary first and second moments of a total customers volume are calculated. An analysis of some special cases of the model and some numerical examples are also included.

5

Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

Kim C., Dudin A., Dudin S., Dudina O.

International Journal of Applied Mathematics and Computer Science

|

2014

|

Vol. 24, no. 3

485--501

EN

A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace–Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented.

6

A functional equation that leads to semistability

Bouzar N.

Probability and Mathematical Statistics

|

2010

|

Vol. 30, Fasc. 1

87--102

EN

Some functional equations related to the notion of semistability of probability distributions on Z+ and R+ are studied. The solution sets of these equations are fully described.