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.

A discrete-time queue with finite-support service capacities

Abstract: In discrete-time queueing theory, the service process is traditionally modeled using the notion of service time, the time it takes the server to completely process one customer. In our research, we take a different approach and model the service process by means of two more basic quantities: service demands and service capacities. The service demands are independent and identically distributed (i.i.d.) random variables that describe the number of work units that each customer requires from the system, whereas the service capacities are i.i.d. random variables that describe the amount of work units that the server can process per timeslot. If a customer requires more work units than the server can provide in a slot, the service continues in the next slot. Conversely, if the service capacity in a slot is higher than the customer in service still requires, the remaining capacity is used for the next customer in line, and more than one customer might be served in that slot. This type of model has been studied in previous work, but with either the restriction that the service capacities follow a geometric distribution or that they are deterministically equal to a given constant. In our research, we analyze the system with the restriction that the service capacity distribution must have finite support. The numbers of customers arriving per slot and the service demands of the customers can be general i.i.d. random variables. The analysis is performed using probability generating functions (pgfs), and as a result we obtain expressions for the pgfs of the delay of a random customer, the amount of unfinished work and the number of customers in the system in a random slot. These pgfs are then used to derive expressions for the moments of these quantities, and to approximate the probability mass function of these quantities with much higher precision than simulations could provide.

The Properties Related to the Moment Generating Function of the Fuzzy Variable

Abstract: In the paper, some properties related to the moment generating function of a fuzzy variable are discussed based on uncertainty theory. And we obtain the result that the convergence of moment generating functions to an moment generating function implies convergence of credibility distribution functions. Thats, the moment generating function characterizes a credibility distribution.

Alternative expressions for probability-generating functions concerning an inherited character after a panmixia

Abstract: In any case, the generating function must be, of course, uniquely determined in a definite manner o However, our results have concerned not directly the generating functions themselves but somewhat indirectly the related functions from which the generating functions can be obtained as the constant terms of respective Laurent expansions with respect to parameters involved* Under such circumstances, it will be possible to find alternative sources of generating functions* In the present paper we shall illustrate the circumstances by deriving some alternative expressions for probability—generating functions*

On a supplementary time variable approach to battle analysis

Abstract: A supplementary time variable approach is used to analyze a battle where attackers search for defended logistical targets and the future actions of an attacker depend on its activities in the whole time interval since arrival at the operating area. As compared with the method discussed in this paper a corresponding Markov chain analysis of the problem requires a considerably greater number of mathematical relations for state transition probabilities with less possibilities in obtaining numerical solutions effectively and with less para-metrical insight. A supplementary time variable is introduced to describe the arrival instants of attackers at the operating area, and is then used to express the probability generating functions of the losses at an arbitrarily selected time instant in terms of a general description of the capabilities of an individual attacker in the time interval since arrival. These capabilities are then related to a selected set of lower level parameters describing a specific battle situation by using a small set of integral relations of the multiple convolution type with solutions expressed as Laplace transforms.