Probability-generating function

About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.

On an M^x/G/1 queue with a random set up time, random breakdowns and delayed deterministic repairs

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.

Computing Probability Generating Functions of Coin Flipping and Its Generalizations

Abstract: A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different letters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probability generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.

Asymptotics of the probability of values of random Boolean expressions

Abstract: The random Boolean expressions are considered that are obtained by the random and independent substitution with the probabilities p and 1 − p of the constantly one function and constantly zero function for variables of repetition-free formulas over a given basis. The probability is studied that the expressions are equal to one. It is shown that, for each finite basis and p ∊ (0, 1), this probability tends to some finite limit P 1(p) as the length of an expression grows. Explicit representation of the probability function P 1(p) is found for all finite bases, the analytic properties of this function are studied, and its behavior is investigated in dependence on the properties of the basis.