The article explores the asymptotic behavior of the expected number of drawings in the Coupon Collector’s Problem with group-drawing under the uniform distribution. In this variant, each draw consists of a package of s distinct coupons selected uniformly at random from a set of n coupons. We focus on three regimes of the package size s : (i) constant s , (ii) s proportional to n , and (iii) s “very close” to n . For each case, we provide precise asymptotic expressions for the expected collection time.

This article aims to derive the Laplace-Stieltjes transform matrix for the total increment of a one-level process during the first passage of another level process to level zero in so-called the two-dimensional Markov modulated Brownian motion. The process comprises an irreducible continuous-time Markov process with a finite state space, alongside two level processes modulated by the Markov process. These paired level processes can be viewed as a two-dimensional Brownian motion, with Brownian parameters varying based on the Markov process. Due to the infeasibility of explicit computation, we formulate a nonsymmetric algebraic Riccati equation with a minimal nonnegative solution that represents the transform matrix through a matrix exponential function. To our knowledge, this achievement is innovative within the context of the two- dimensional Markov modulated Brownian motion.

. In this note, by an elementary use of Girsanov’s transform, we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Peköz and R. Righter in 2024, while the Brownian motion case is the continuous analogue as discussed in the same paper. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

We consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } , which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } . Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples.

. We consider Markov fluid models with two infinite buffers, whose fluid rates ensure that the fluid level of buffer 1 is never larger than the one of buffer 2. For the model, we derive the system of PDEs describing the transient behavior and expressions for the stationary analysis in the Laplace domain. Examples for the application of the model in the context of network access control are presented. Numerical results are compared with detailed simulation results to show that the numerical approach computes the required measures with sufficient accuracy.

. In this article, we exploit recent results for the Laplace transform of the sojourn time Ω of an entire batch in the M [ X ] / M / 1 processor sharing queue in order to derive the asymptotic behavior of the distribution function of Ω, namely the behavior of ℙ(Ω > x) when x tends to infinity. Specifically, we show that, up to a multiplying factor, ℙ(Ω > x) has the same order of magnitude as ℙ(ω > x) for large x, where ω is the sojourn time of an arbitrary job in the system. From a practical point of view, this means that if a system has to be dimensioned to guarantee the processing time for jobs, then the system can also guarantee processing times for entire batches by introducing a marginal amount of processing capacity.

. A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from the data.

The moments based matrix representation of Markovian and rational arrival processes (MAP/RAPs) with full rank marginal (FRM) is provided in Telek and Horváth[ 14 ]. MAP/RAPs with reduced rank marginal (RRM) differ in essential properties from the ones with FRM [ 13 ]. The main difficulty of the moments based matrix representation of MAP/RAPs with RRM comes from the fact that the moments needed to characterize a MAP/RAP with RRM depends on the internal structure of the MAP/RAP. In this work, we propose a general procedure for moments based matrix representation that is applicable to MAP/RAPs with both FRM and RRM, independent of their internal structures. We also show that the procedure terminates in a finite number of steps which is proportional to the order of the MAP/RAP.

In He et al.[8], continuous phase-type (PH) distributions are constructed to approximate finite discrete probability distributions and discrete PH-distributions. The approximations are based on Erlangization with a fixed number of phases. In this article, we first introduce continuous PH approximations with Erlang distributions of different orders. Then we develop an algorithm to find the continuous PH approximation with the minimum variance, among all such PH approximations with the same total number of phases. Thus, the proposed continuous PH approximations lead to a smaller gap between the variances of the Erlangization-based approximations and the original discrete random variables, which is achieved without adding more phases. The new approximations are useful to mitigate the burden in computation caused by the large number of phases needed in Erlangization approximation. Stochastic dominance is shown between the original (discrete) distributions and the approximations, which leads to bounds on the quantities of original distributions and/or stochastic models (e.g., reliability models). The approximation method is applied to analyze reliability models and a COVID-19 isolation program.