Showing papers on "Probability-generating function published in 2019"
TL;DR: In this paper, it was shown that if none of the complex zeros of the polynomials { P n ( z ) } is contained in a neighborhood of 1 ∈ C and σ n > n e for some e > 0, then X n ⁎ = ( X n − μ n ) σn − 1 is asymptotically normal as n → ∞ : that is, it tends in distribution to a random variable Z ∼ N ( 0, 1 ).
15 citations
TL;DR: In this investigation, a novel sort of retrial queueing system with working breakdown services is introduced and the probability generating functions (PGF) of the system size is found using the concepts of the supplementary variable technique (SVT).
Abstract: In this investigation, a novel sort of retrial queueing system with working breakdown services is introduced. Two distinct kinds of customers are considered, which are priority and ordinary customers. The normal busy server may become inadequate due to catastrophes at any time which cause the major server to fail. At a failure moment, the major server is sent to be fixed and the server functions at a lower speed (called the working breakdown period) during the repair period. The probability generating functions (PGF) of the system size is found using the concepts of the supplementary variable technique (SVT). The impact of parameters in system performance measures and cost optimization are examined numerically.
11 citations
TL;DR: This work describes the application of the pgf technique for modeling uni- and bi-variate distributions of polymer properties with parallelization of the model code and shows that accurate results can be achieved in very short running times, which makes the technique suitable for models to be employed in optimization and online control tasks.
9 citations
01 Jun 2019
TL;DR: In this paper, a two-parameter expectation thinning operator based on a linear fractional probability generating function was introduced and used to define a first-order integer-valued autoregressive INAR(1) process.
Abstract: We introduce a two-parameter expectation thinning operator based on a linear fractional probability generating function. The operator is then used to define a first-order integer-valued autoregressive INAR (1) process. Distributional properties of the INAR (1) process are described. We revisit the Bernoulli-geometric INAR (1) process of Bourguignon and Weis (Test 26(4):847–868, 2017. https://doi.org/10.1007/s11749-017-0536-4
) and we introduce a new stationary INAR (1) process with a compound negative binomial distribution. Lastly, we show how a proper randomization of our operator leads to a generalized notion of monotonicity for distributions on $$\mathbf{Z}_+$$
.
7 citations
TL;DR: An M[X]/G/1 queue with breakdowns, repair, Bernoulli vacation, two delays and geometric loss is considered and simplified expressions for Probability Generating Functions of the joint distribution of server state and system size are obtained.
Abstract: This paper considers an M[X]/G/1 queue with breakdowns, repair, Bernoulli vacation, two delays and geometric loss. In this paper, a special attention is given to the limiting distribution of system states. We obtain simplified expressions for the Probability Generating Functions (PGFs) of the joint distribution of server state and system size. Some performance measures were derived from the analysis of the steady state probabilities. PGF of a departure point system size distribution is developed. Particular cases of the studied system were investigated. The effect of system parameters on the main performance measures are illustrated and discussed.
5 citations
TL;DR: In this article, the authors consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice and determine the probability generating functions, the transition probabilities and the relevant moments.
Abstract: We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a 2-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behavior making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight-line. Under suitable symmetry assumptions we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.
4 citations
TL;DR: Exact analysis of a multi-server Markovian queueing system with cross selling in steady-state with optimal threshold level is presented, showing the effect of the cross selling intensity and of the threshold level on the systems performance measures.
Abstract: Exact analysis of a multi-server Markovian queueing system with cross selling in steady-state is presented. Cross selling attempt is initiated at the end of a customer’s service every time the number of customers in the system is below a threshold. Both probability generating functions (PGFs) and matrix geometric methods are employed. The relation between the methods is revealed by explicitly calculating the entries of the matrix geometric rate-matrix R. Those entries are expressed in terms of the roots of a determinant of a matrix related to the set of linear equations involving the PGFs. This is a further step towards understanding of the analytical relationship between the two methods. Numerical results are presented, showing the effect of the cross selling intensity and of the threshold level on the systems performance measures. Finally, for a given set of parameters, the optimal threshold level is determined.
3 citations
TL;DR: A discrete-time multiserver queueing system with an infinite storage capacity and deterministic service times equal to 1 slot is studied and a method to determine the pgf of the steady-state system content at various observation instants is presented.
2 citations
01 Jan 2019
TL;DR: In this article, the authors discuss the problems related to the cyclical nature of the economy in short, medium and long term and provide a review of the most well-known cycles in economic theory, in particular, the cycles of Kitchin, Juglar, Kuznets, Kondratieff.
Abstract: The article discusses the problems related to the cyclical nature of the economy in short, medium and long term. It provides the review of the most well-known cycles in economic theory, in particular, the cycles of Kitchin, Juglar, Kuznets, Kondratieff. The analysis of Kuznets cycles takes a special place in the work. The author suggests that the duration of cyclical oscillations in the economy may be described using the elements of a numerical sequence known in mathematics as the Fibonacci numbers.
2 citations
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TL;DR: In this paper, the first passage times of Markov chains are estimated using collective marks. But the first pass times of the chains are not known to be known to all the authors.
Abstract: Probability generating functions for first passage times of Markov chains are found using the method of collective marks. A system of equations is found which can be used to obtain moments of the first passage times.
2 citations
TL;DR: In this article, the first passage times of Markov chains are estimated using collective marks. But the first pass times of the chains are not directly associated with the moments of the second passage times.
Abstract: Probability generating functions for first passage times of Markov chains are found using the method of collective marks. A system of equations is found which can be used to obtain moments of the first passage times. Second passage probabilities are discussed.
12 Mar 2019
TL;DR: It is shown that homogeneous systems (i.e. systems with identical servers) are optimal for various load-related objective functions, and that Catalan's numbers, and their generalizations, arise naturally in this context.
Abstract: A tandem stochastic system is a network of n sites (queues) in series, where particles (customers, jobs, packets, etc.) move unidirectionally from one site to the next until they leave the system. When each site is an M/M/1 queue, i.e., where the buffer size of each site is unlimited and only single particles move between sites, the system is known as Tandem Jackson Network (TJN) [3]. The TJN is famous for its product-form solution of the multi-dimensional distribution function of the sites' queue-sizes (occupancies). Another well-known tandem stochastic system is the Asymmetric Simple Exclusion Process (ASEP) [1], where each site can hold at most a single particle, a constraint that causes blockings on particles' forward movements. The ASEP is a paradigmatic model in non-equilibrium statistical mechanics. In contrast, the newly introduced (Reuveni, Eliazar and Yechiali) Asymmetric Inclusion Process (ASIP) is a tandem series of sites, each with unbounded buffer capacity and with unlimited-size batch service [5]. That is, when service is completed at site k, all particles present there move simultaneously to site k + 1 and form a cluster together with the cluster of particles (if any) already residing in site k + 1. The ASIP is a showcase of complexity [6]. We analyze the innovative ASIP and show that its multi-dimensional Probability Generating Function (PGF) does not admit a product-form solution. Nevertheless, we present a method to calculate this PGF [5]. Surprisingly though, the load (total number of particles up to site k) does admit a product-form solution. It is consequently shown that homogeneous systems (i.e. systems with identical servers) are optimal for various load-related objective functions [5]. Considering the total number of particles in a site-interval of length m (m = 1, 2,..., n) that starts at site k, the corresponding probability generating functions create a discrete two-dimensional boundary-value problem which is solved explicitly [8]. Catalan's numbers, and their generalizations, arise naturally in this context [4]. It is further proved [8] that the probability of site k being positively occupied is proportional to 1/√k, while the variance in the occupancy of site k is proportional to √k. Finally, we derive limit laws (when the number of sites becomes large) for various system's variables [7]. In particular, we show that the 'load', as well as the 'draining time', each obeys a Gaussian distribution (with corresponding coefficients), while the 'inter-exit time' follows a Rayleigh distribution. An extension of the basic ASIP model with a fairly general arrival scheme, where gate opening intervals follow a Markov renewal process, is studied in [2]. The steady-state distribution of the total number of customers in the first k queues is determined.