Stochastic Models

Haviv, Moshe; Kerner, Yoav

doi: 10.1080/15326349.2022.2066130pmid: N/A

Abstract We derive the mean waiting times in an M/G/1 priority queue when the decision who receives the current completed service (production) is determined at the end of the service period. We consider two variations of this scheme. The first is when the server works only when customers are present, while the second is when the server works on a nonstop basis but scraps its work if production is completed when there are no customers in line. We show that for the former variant (whose overall mean is as in the standard head-of-the-line (HOL) priority model), the gain from this scheme in comparison with the HOL case is monotone increasing with the priority level (being positive for the higher classes and negative for the lower classes).

Bergquist, Jacob; Sigman, Karl

doi: 10.1080/15326349.2022.2074458pmid: N/A

Abstract We analyze two nonwork-conserving variations of the M/G/1 preemptive last-in first-out (LIFO) queue with emphasis on deriving explicit expressions for the limiting (stationary) distributions of service times found in service by an arrival, workload and a variety of related quantities of interest. Workload is also used as a tool to derive the proportion of time that the system is busy, and stability conditions. In the first model, known as preemptive-repeat different (PRD), preempted customers are returned to the front of the queue with a new independent and identically distributed service time. In the second, known as preemptive-repeat identical (PRI), they are returned to the front of the queue with their original service time. Our analysis is based on queueing theory methods such as the Rate Conservation Law, PASTA, regenerative process theory and Little’s Law ( ). For the second model we even derive the joint distribution of age and excess of the service time found in service by an arrival, and find they are quite different from what is found in standard work-conserving models. We also give heavy-traffic limits and tail asymptotics for stationary workload for both models, as well as deriving an implicit representation for the distribution of sojourn time by introducing an alternative effective service time distribution.

Sfragara, Matteo

doi: 10.1080/15326349.2022.2074459pmid: N/A

Abstract We consider random-access networks with nodes representing transmitter-receiver pairs whose signals interfere with each other depending on their vicinity. Data packets arrive at the nodes over time and form queues. The nodes can be either active or inactive: a node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. In order to model the effects of user mobility in wireless networks, we analyze dynamic interference graphs where the edges are allowed to appear and disappear over time. We focus on bipartite graphs and study the transition time between the two states where one part of the network is active and the other part is inactive, in the limit as the queue lengths become large. Depending on the speed of the dynamics, we are able to obtain a rough classification of the effects of the dynamics on the transition time.

Gutiérrez-Pavón, Jonathan; Pacheco, Carlos G.

doi: 10.1080/15326349.2022.2074460pmid: N/A

Abstract We are proposing to study a diffusion in random environment confined in bounded interval. As for many standard diffusions, we build in natural way a stochastic process with bounded domain, but with addition of considering a random environment. We carry out this construction using the Brox diffusion and applying well-known diffusion theory in a quenched fashion, which is a natural way to deal with random environment. The outcome of this procedure is an object that we may call the killed Brox diffusion. Since the generator of this process is initially an ill-posed expression we develop a Sturm–Liouville theory for one-dimensional second-order differential operators with white-noise coefficients. Our first main result is to give a close form of the Green operator associated to the generator, i.e., the inverse of the generator. We do so by setting the Lagrange identity in this context. Then, we give explicit expressions in quenched form of the probability density function of the process; such an object is given theoretically in terms of the spectral decomposition using the eigenvalues and eigenfuntions of the infinitesimal generator of the diffusion. Moreover, we characterize the eigenvalues and eigenfunctions using some integro-differential equations.

Fahim, Arash; Zhu, Lingjiong

doi: 10.1080/15326349.2022.2080709pmid: N/A

Abstract The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for a dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividend problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.

Telek, Miklós

doi: 10.1080/15326349.2022.2081206pmid: N/A

Abstract The paper investigates the relation of random clock based and numerical inverse Laplace transformation based transient analysis of Continuous time Markov chains (CTMCs) and Markov fluid models (MFMs) and proves that these methods are identical. This identity leads to new intuitive understanding about the analysis approaches.