On a modification of the classical risk process

In this paper, we present the classical risk process with two-step premium function. This means that the gross risk premium rate changes if the insurer’s surplus reaches a certain threshold level. The formula for the infinite-time ruin probability is obtained. The asymptotic behaviour of the ruin probability in the case where the claim size distribution has a light tail is considered as well.     

On the loss process in a batch arrival queue

In this paper, we investigate the loss process in a finite-buffer queue with batch arrivals and total rejection discipline. In such a model, if the buffer has insufficient capacity to accept all the customers included in an arriving batch, the whole batch is blocked and lost. This scheme is especially useful in performance evaluation of buffering processes in IP (internet protocol) networks. The main result of this paper is a closed-form formula for the joint distribution of the length of the first lost series of batches and the time of the first loss. Moreover, the limiting distribution (as the buffer size grows to infinity) is shown.     

Application of the Superposition of Renewal Processes to the Study of Batch Arrival Queues

The G /G/1-type batch arrival system is considered. We deal with non-steady-state characteristics of the system like the first busy period and the first idle time, the number of customers served on the first busy period. The study is based on a generalization of Korolyuk's method which he developed for semi-Markov random walks.