Showing papers on "Probability-generating function published in 2018"

On Recurrence and Transience of Fractional RandomWalks in Lattices

Abstract: The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p 0 to reach ‘by chance’ a site p for the first time? Further important are recurrence and transience features of a random walk: A random walker starting at p 0 will he ever reach site p (ever return to p 0)? How often a site is visited? Here we investigate Markovian random walks generated by fractional (Laplacian) generator matrices L\( \frac{\alpha }{2} \) 2 (0 < \( \alpha \) ≤2) where L stands for ‘simple’ Laplacian matrices. This walk we refer to as ‘Fractional Random Walk’ (FRW). In contrast to classical Polya type walks where only local steps to next neighbor sites are possible, the FRW allows nonlocal long-range moves where a remarkably rich dynamics and new features arise. We analyze recurrence and transience features of the FRW on infinite d-dimensional simple cubic lattices. We deduce by means of lattice Green’s function (probability generating functions) the mean residence times (MRT) of the walker at preselected sites. For the infinite 1D lattice (infinite ring) we obtain for the transient regime (0 < \( \alpha \) < 1) closed form expressions for these characteristics. The lattice Green’s function on infinite lattices existing in the transient regime fulfills Riesz potential asymptotics being a landmark of anomalous diffusion, i.e. random motion (Levy flights) where the step lengths are drawn from a Levy \( \alpha \)-stable distribution.

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

Abstract: We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).

Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

Abstract: This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e. , an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability q i , i = 1, 2, …, d . Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

Unreliable bulk queueing model with optional services, Bernoulli vacation schedule and balking

Abstract: This paper deals with MX/G/1 queueing system in which arriving units join a single waiting line. Server provides the first essential service and one of the optional services among m available optional services, to all arriving units. After completion of both phases of services of each unit the server may take optional vacation with probability p. It is assumed that during any phase of service, server may stop working due to random failure and is sent for repair. Further it is assumed that arriving units may balk from the system when server is busy, vacation and under repair with probability -b = 1 – b. Using the probability generating functions we derive the queue size distribution at different time points as well as waiting time distribution. Finally numerical illustration is provided to analyse the sensitivity of different parameters on various performance measures.

Joint distributions of numbers of runs of specified lengths on directed trees

Abstract: In this paper, we consider exact joint distributions of numbers of success runs of specified lengths on a Markov directed tree under $$\ell $$ -overlapping enumeration schemes. Methods for deriving the probability generating functions of the joint distribution are presented. We extend the exact distribution theory of runs based on sequences to runs based on directed trees. Some numerical results for the run statistics are given in order to illustrate the computational aspects and the feasibility of our theoretical results. The reliability systems and lifetime distributions are considered. Our general results are applied to reliability systems and lifetime distributions. Finally, we discuss parametric estimation problems based on the lifetimes of the systems.

Analysis of an Infinite-Server Queue $$MAP_ k|G_ k|\infty $$ in Random Environment with k Mar kov Arrival Streams and Random Volume of Customers

Abstract: In this paper the model of an infinite-server \(MAP_k|G_k|\infty \) queue in random environment with catastrophes is considered. The transient and limiting probability generating functions (PGF) of joint distributions of number of busy servers and numbers of served customers, the joint distributions of total resource in the model and total served resource are found. All results are obtained by using collective marks method (CMM) and renewal processes.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

Abstract: We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

Infinite-server M|G|$\infty$ queueing models with catastrophes

Abstract: The infinite-server queueing models with homogeneous and non-homogeneous arrivals of customers and catastrophes are considered. The probability generating functions of joint distributions of numbers of busy servers and served customers, as well as the Laplace-Stieltjes Transforms of distribution of busy period and distribution of busy cycle for the models are found.