Showing papers on "Probability-generating function published in 2021"
TL;DR: It is shown that H and $\vec{b}(z) can be explicitly expressed in terms of the triple A0, A1, and A2, and the stability condition is readily extracted.
Abstract: Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$; and (ii) the stability condition is readily extracted.
15 citations
TL;DR: In this paper, many discrete stochastic systems are analyzed using the probability generating function (pgf) technique, which often leads to expressions in terms of the complex roots of a certain equation.
Abstract: Many (discrete) stochastic systems are analyzed using the probability generating function (pgf) technique, which often leads to expressions in terms of the (complex) roots of a certain equation. In...
2 citations
TL;DR: The busy-period distribution for discrete-time queues assuming a Bernoulli arrival process is studied, with arbitrary service-time and batch-arrival distributions, and explicit analytic formulas are derived using the Lagrange Implicit Function Theorem applied to probability generating functions.
1 citations
DOI•
16 Aug 2021
TL;DR: Steady state queue length of various states of the system in terms of probability generating functions is found and Steady state results of a few interesting special cases have been derived.
Abstract: We study a single server queue with Poisson arrivals in batches of variable size. The server provides one by one general service to customers with a set-up time of random length before starting the first service at the start of the system as well as after every idle period of the system. The set-up time has been assumed to be general. Further, the server is subject to random breakdowns. The repair time has been assumed to be deterministic with a further delay time before starting repairs. The delay time in starting repairs has been assumed to be general. We find steady state queue length of various states of the system in terms of probability generating functions. Steady state results of a few interesting special cases have been derived.
1 citations
01 Jan 2021
TL;DR: In this paper, the authors considered continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates and gave explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process.
Abstract: We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. 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 subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364].
TL;DR: In this article, the authors considered random occupancy models and the related problems based on the methods of generating functions and provided the effective computational tools for the evaluation of the probability functions by making use of the Bell polynomials.
Abstract: In this article, we consider random occupancy models and the related problems based on the methods of generating functions. The waiting time distributions associated with sequential random occupancy models are investigated through the probability generating functions. We provide the effective computational tools for the evaluation of the probability functions by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of occupancy models. Finally, we treat several examples in order to demonstrate how our theoretical results are employed for the investigation of the random occupancy models along with numerical results.
01 Jan 2021
TL;DR: In this paper, the authors considered a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability, and they considered the delay characteristics.
Abstract: In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.