Probability Commons^™

(R2077) Enhancing Queue Management: Dynamic Server Allocation And Optional Services In Stochastic Modeling, G. Ayyappan, S. Sankeetha

Applications and Applied Mathematics: An International Journal (AAM)

Consider a queueing system with a single server, where customer arrivals follow a Markovian arrival process and service times follow a phase-type distribution. The main server has the capability to recruit an additional server when the number of customers in the system exceeds a certain threshold, denoted as L. Both servers provide normal service to customers, and optional service is provided upon request. The main server takes multiple vacations, with the durations following an exponential distribution with rate parameter η, until there is at least one customer in the system. This system can be represented as a Markov chain process, …

(R2072) A Method To Sample From Transition Densities Of Diffusion Processes With Unknown Densities, Yasin Kikabi, Juma Kasozi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a reliable method that can be used to sample paths of a diffusion process described by a system of stochastic differential equations. These are types of processes whose transition density is not known in closed form. This reliable method involves solving a partial differential equation of the Fokker-Planck type by rejection sampling techniques applied to the associated densities. We then use the novel method to sample from the Ornstein-Uhlenbeck (OU) process whose transition density is known and has unbounded drift. The results obtained compare excellently with the analytical results, thus rendering the method reliable and …

(R2133) Analysis Of A Bulk Queue With Balking, Adaptive Overloading Service, Multiple Vacation, Inspection And Rework, S. Karpagam, R. Lokesh

Applications and Applied Mathematics: An International Journal (AAM)

We consider a queueing system that utilizes a single server to manage product flow through bulk and overload service modes, depending on queue length. Balking occurs when the queue length reaches ‘N’. After completion of service, products undergo quality inspection, with defective products sent for rework based on specific probabilities. If the product is non-defective, the server continues processing until the queue length exceeds a certain threshold; beyond this point, the server is either routed back to bulk service or directed to an overload service. Also, the server goes into a vacation mode when the queue size is below a …

Irreversible K-Threshold Dynamics On Corona And Base-B Corona Product Graphs, Eric J. Moon, Soumya Bhoumik, Paul Flesher

SACAD: Scholarly Activities

This poster studies the irreversible k-threshold process on corona-type graph products, where a vertex becomes colored once at least k of its neighbors are colored and then remains colored permanently. We focus on corona, double corona, and base-b corona product graphs built from cycles and complete graphs, with particular attention to how graph structure affects complete activation from a minimum seed set.

A generalized reduction lemma is used to relate threshold dynamics on layered corona graphs to smaller residual graphs, yielding explicit formulas for the irreversible k-threshold conversion number on both corona and double corona families. The …

A Mechanistic Model Of The Wash Shuffle And Monte Carlo Exploration Of Its Impact On Card Shuffling In Texas Hold’Em, Michael A. Alexeev, Peter B. Chi

UNLV Gaming Research & Review Journal

In casino games using a standard deck of cards, a wash shuffle is sometimes performed prior to the rest of the card shuffling procedure. Unlike other methods of shuffling, the wash shuffle has not yet been well studied. To this end, we first develop a mechanistic model of the wash shuffle based on our observation of how cards tend to move when a wash shuffle is being performed. Then, we use this model to simulate the card shuffling procedure used in casino poker rooms, and explore the resulting impact on where the cards are dealt in the context of Texas …

Cash Or Crash: Return To Player Percentages And Expected Value Of Crash Games, Robert H. Scott Iii, Mikhail M. Sher, Jonathan Daigle

UNLV Gaming Research & Review Journal

Crash games are a new type of casino game offered on some online gambling sites. The first crash game was created by the online cryptocurrency casino Bustabit. Other cryptocurrency casinos started offering their own versions of crash games. Now mainstream online casinos have developed their own crash games—notably Rocket by DraftKings. Crash games are easy to learn and play. They offer the opportunity to increase your bet by many hundreds of multiples—though, as we show when deriving expected values, these outcomes are rare. In this paper, we study the history of crash games and calculate theoretical values of crash game …

Let's Play Uno! Training Mathematical Thinking With An Inclusive Card Game, Thierry Meyrath, Clara-Ioana Mincu, Antonella Perucca

Journal of Humanistic Mathematics

We explore probability exercises based on the game UNO. Excluding the Wild cards, the deck contains precisely 100 cards, which makes it convenient to express probabilities as percentages. We discuss the mathematical relation between cards that can be played after other cards. We also propose an algebraic exercise that relates several quantities, such as the number of cards left in the losing player’s hand. Finally, we argue that UNO is a very inclusive game: it is logistically easy to play, allows for players of different skill levels, and can be adapted to meet special needs. Beyond being playful exercises, our …

Inhomogeneous Branching Random Walks: Incorporating Genealogy And Density Effects, Lauren Ajax, Beatrice Durham, Pratima Hebbar, Cade Johnston, Jiayi Zhang

Spora: A Journal of Biomathematics

We introduce a novel framework using inhomogeneous branching random walks (BRWs) to model biological processes, specifically by introducing genealogy-dependence in branching rates and displacement distributions to model bacterial colony growth. Current stochastic models often either assume independent and identical behavior of individual agents or incorporate only spatiotemporal inhomogeneity, ignoring the effect of genealogy-based inhomogeneity on the long-time behavior of these processes. Such asymptotics are of independent mathematical interest and are crucial in understanding the emergence of patterns. We propose several inhomogeneous BRW models in 2D space where displacement distributions and branching rates vary with time, space, and genealogy. A combined …

(R2152) New Bivariate Type-2 Gumbel Distribution Based On The Farlie-Gumbel-Morgenstern Copula: Properties And Its Application In Survival Analysis, Muneeb Javed, Said Farooq Shah, Muhammad Osama, Muhammad Atif, Muhammad Farooq

Applications and Applied Mathematics: An International Journal (AAM)

We introduce a new bivariate probability distribution, termed the Bivariate FGM Type-2 Gumbel Distribution, constructed by combining the Farlie–Gumbel–Morgenstern (FGM) copula with the Type-2 Gumbel marginal distributions. This proposed distribution provides a flexible framework for modeling bivariate data and offers a viable alternative to several existing bivariate distributions, especially in scenarios where capturing dependence between variables is crucial. The theoretical properties of the distribution are thoroughly explored. We derive the marginal and conditional distributions, conditional expectations, moment generating function, and product moments. Procedures for random number generation from the distribution are discussed. Reliability-based characteristics, such as the survival function and …

(R2131) Analysis Of Map/Ph/1 Retrial Inventory Queueing System With Constant Retrial Rate, Bernoulli Vacation, Breakdown, Delayed Repair, Balking, (S, S) Policy And Emergency Replenishment, G. Ayyappan, M. Thilakavathy

Applications and Applied Mathematics: An International Journal (AAM)

A single server retrial queueing-inventory model is investigated in this study. In Bernoulli vacation, after providing service to the customer, the server may opt to avail vacation or start the service to subsequent customer. During the busy period, breakdown and balking may occur. The inventory is replenished to an (s, S) policy and the replenishing time is assumed to adopt an exponential distribution. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. We integrate the emergency replenishment into the system to ensure customer satisfaction. For our …

(R2120) Flexible Group Service Map/Ph/1 Queueing Model With Working Vacation And Optional Service, G. Ayyappan, S. Kalaiarasi

Applications and Applied Mathematics: An International Journal (AAM)

There are many uses of queues, where services are provided in groups; these types of queues are widely studied in the literature. In this paper we examine a particular queueing model wherein the services are provided in groups and the group size may be less than or equal to the size initially fixed. The arrival follows a Markovian arrival process. The service time of each individual customer follows phase type distribution. The maximum of each customer’s individual service time within a group is defined as the group’s service time. At the service completion moment if there are fewer customers than …

(R2115) Analysis Of Single Server Queueing System With Differentiated Vacations And Differentiated Breakdowns, V. Karthick, V. Suvitha

Applications and Applied Mathematics: An International Journal (AAM)

This research work considers a single server queueing model with differentiated vacations. In addition there is a possibility of two types of failures when the server is in a busy period; namely hard failure and soft failure. In the time of soft failure server may work with a slow service rate. We analyzed as a Quasi-Birth-and-Death (QBD) process, using the matrix geometric method, the steady state probability vector of the number of customers in the queue and the stability conditions are produced. Busy period analysis of the proposed model in given. The effects of various parameters on the system performance …

(R2132) A Multi Server Markovian Working Vacation Queue With Randomly Varying Environment, A. Sundaramoorthy, R. Kalyanaraman

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we consider a multi server Markovian queueing system with working vacation. During busy period, the arrival and service completion are generated by K distinct randomly varying environments. At a service completion epoch, if no customer in the system, the servers take vacation, the vacation policy is multiple vacation policy and the vacation period follows negative exponential distribution. In addition, during vacation period the servers serve customers if they arrive. Based on the vacation termination point we define two Models. For the two models, the steady state probability vector of number of customers in the queue, the stability …

The Odds Don’T Lie: Mathematical Reasoning And Societal Ignorance In Don’T Look Up, Nysa Vedwan, Shane Carey

LASER Journal

In Adam McKay’s 2021 satirical sci-fi movie Don’t Look Up, two astronomers discover a comet heading directly toward Earth. Despite overwhelming evidence and near-certainty of global extinction, their warnings are ignored and ridiculed. This paper discusses the mathematical and scientific foundations of the movie’s social and political reception, and specifically focuses on orbital prediction and probabilistic modeling as they relate to public understanding of risk. This paper shows how data is often undermined by political and social dynamics, by connecting the fictional events of the movie with real-world crises like the COVID-19 pandemic and the climate emergency. In Don’t Look …

Syntax-Enhanced Boundary-Aware Named Entity Recognition Model, Chuanming Yu, Bin Deng, Zhengang Zhang

Journal of Scientific Information Research

[Purpose/significance] This study addresses the issue of inadequate perception of entity boundaries in traditional character-level modeling-based named entity recognition models by integrating syntax information containing entity boundary features into the task using a multi-head graph attention network with dense connections. This integration enhances the effectiveness of named entity recognition.

[Method/process] This study proposes a Syntax-enhanced Boundary-aware Named Entity Recognition Model (SynBNER), which utilizes BERT for text semantic representation and integrates syntax information using a dense-connected graph attention network. This integration incorporates implicit entity boundary information from syntax information into word representations, thereby enhancing the model's entity boundary perception capability.

[Result/conclusion] …

Irreversible K-Threshold Number Ck(G) And Saturation Probability P[G] For Corona Product And Double Corona Product Graphs, Eric J. Moon, Soumya Bhoumik, Paul Flesher

SACAD: Scholarly Activities

We discuss the Irreversible k-conversion process for graphs, where a vertex becomes saturated and remains saturated indefinitely if at least k of its neighbors are saturated. We investigate sets S₀, which when initially saturated, lead to complete graph saturation. We are interested in the minimum |S₀| = C_k(G), called the k-threshold number. We consider the construction of the Corona Product Graphs (of C_n and K_p). Additionally, we extend our analysis by defining and exploring Double Corona Product Graphs (of C_n and K_p). Then we incorporate …

(R2095) On Discrete Hypoexponential Distribution And Integer-Valued Autoregressive Process, K. Krishnakumari, Dais George

Applications and Applied Mathematics: An International Journal (AAM)

Discrete distributions provide a substantive contribution in modeling real world frame work. Though a lot of discrete distributions are available in literature, they are inappropriate to model many practical situations. The conventional discrete distributions like geometric, Negative Binomial and Poisson have limited applications in modeling count and lifetime data. This paper introduces a new two parameter discrete distribution namely, discrete hypoexponential distribution by discretizing the well known hypoexponential distribution. Various distributional and structural properties of the proposed distribution are studied. We introduce a first order auto regressive process with the newly proposed distribution as marginal and study the properties. Depending …

(R2098) Dynamic Analysis Of Stochastic Leslie-Gower Biological Predator-Prey Model With Prey Cannibalism, Sada Nand Prasad, Itendra Kumar Universiry Of Delhi, India, Pawan Kumar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study the dynamical analysis of a stochastic Leslie–Gower biological predator– prey model. Earlier, the Leslie–Gower model was studied in the context of biological systems, including cases involving cannibalism. In our model, we investigate the dynamic properties of a stochastic Leslie–Gower predator–prey ecological system using the stability of invariant measures on invariant sets, where the invariant measures are shown to be ergodic. We also conduct a threshold analysis to study the stochastic persistence and extinction of species. Stochastic bifurcation is also examined. The theoretical results are supported by numerical simulations and examples. Intra-species competition is considered and …

(R2116) The Novel Garima Distribution Properties And Nuclear Data Application, Murat Aygün, Ayşe Metin Karakaş

Applications and Applied Mathematics: An International Journal (AAM)

This research introduces a new two-parameter Marshall-Olkin Garima distribution model. The novel model has many sub-models that are useful in modeling real-life data, such as the extended Garima distribution, exponentiated Garima distribution, exponential distribution, Lindley distribution, Kumaraswamy Garima distribution, and normal distribution. The proposed model demonstrates a high level of suitability in modeling both reliability and survival data. It is flexible in accommodating various failures. The quantile function, density shapes, hazard rate functions, and order statistics are a few of the statistical features that have been explored. Maximum likelihood estimation methods were employed to estimate the parameters. Using five data …

(R2117) Cost Optimization Of Queueing System With Differentiated Vacations And Reneging Of Customers, Poonam Gupta, Rajni Gupta

Applications and Applied Mathematics: An International Journal (AAM)

This manuscript deals with an infinite-capacity queueing system under multiple differentiated working vacations and customers’ impatience. The first vacation is assumed to be a working vacation where the server, instead of being idle, serves the customers at a lower rate. In contrast, the second one is considered a non-working vacation of a different duration. The customers may leave the system at any time due to long delays in service during vacations but, via some convincing mechanisms, they are retained in the system. The operating characteristics of the system are obtained in a steady state. The results obtained are illustrated numerically …

Big Two And N-Card Poker Probabilities, Brian Wu, Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of n≥5. In particular, we derive closed expressions for the probabilities of flush, straight and full house and show that the probability of a flush is less than a straight …

(R2073) Analysis Of Mmap/Ph(1), Ph(2)/1 Preemptive Priority Queueing Model With Single Vacation, Repair And Impatient Customers, S. Meena, G. Ayyappan

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we analyse a single server preemptive priority queue with phase-type vacation and repair, feedback, working breakdown, close-down and impatient customers. Customers arrive according to the Marked Markovian Arrival Process and their service time according to Phase-type distribution. If the High Priority customers need feedback, they lose their priority and join the Low Priority queue. At any instant, if the server is broken down, the server provide service with slow mode for that current customer and then the server will go into a repair process. When there are no customers present in both the queues, the server close-down …

(R2081) Analysis Of A Flexible Group Service Map/Ph/1 Queueing Model With Soft Failure And Reneging, S. Kalaiarasi, G. Ayyappan

Applications and Applied Mathematics: An International Journal (AAM)

Queueing models where services are offered in groups (or blocks or batches) have shown to be very helpful in real-world applications and these queues have been well studied in the literature. In this paper we see one such group service queueing model with soft failure and reneging; here, by soft failure, we mean an emergency arrival. The arrival process is a Markovian arrival, whereas the emergency arrival follows an exponential distribution. Customers are served in groups ranging in size from 1 to a fixed constant, let’s say N. A batch’s service time is determined by the phase-type distribution that corresponds …

Stock Market Volatility In The United Kingdom: Simulating Post-Covid-19 Recovery, Bala A. Dahiru, Mohammed Shuaibu, Najibullah Hassanov

CBN Journal of Applied Statistics (JAS)

This paper investigates the time it would take for the FTSE-100 index to reach its post-COVID-19 peak. The paper utilises an exponential generalised autoregressive conditional heteroscedasticity (EGARCH) model that accounts for leverage effect and asymmetries. The preferred models amongst competing variants was the Autoregressive Moving Average (ARMA)-EGARCH(2,1) specification and was used to predict daily FTSE-100 data from 5th January 2000 to 21st June 2024. The empirical exercise showed that the COVID-19-induced financial crisis negatively affected the United Kingdom’s stock market performance. The results show that the FTSE100 index could reach its post-pandemic peak around 27th August, 2024 (two months after …

Stability Of Predator-Prey Model For Worm Attack In Wireless Sensor Networks, Rajeev Kishore, Padam Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we propose a predator-prey mathematical model for analyzing the dynamical behaviors of the system. This system is an epidemic model, and it is capable of ascertaining the worm's spreading at the initial stage and improving the security of wireless sensor networks. We investigate different fixed points and examine the stability of the projected model.

Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

Rose-Hulman Undergraduate Mathematics Journal

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

Sensitivity Analysis Of Prior Distributions In Regression Model Estimation, Ayoade I Adewole, Oluwatoyin K. Bodunwa

Al-Bahir

Bayesian inferences depend solely on specification and accuracy of likelihoods and prior distributions of the observed data. The research delved into Bayesian estimation method of regression models to reduce the impact of some of the problems, posed by convectional method of estimating regression models, such as handling complex models, availability of small sample sizes and inclusion of background information in the estimation procedure. Posterior distributions are based on prior distributions and the data accuracy, which is the fundamental principles of Bayesian statistics to produce accurate final model estimates. Sensitivity analysis is an essential part of mathematical model validation in obtaining …

Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

Journal of Nonprofit Innovation

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

Ohio Recovery Housing: Resident Risk And Outcomes Assessment, Elyjiah Potter, Bivin Sadler

SMU Data Science Review

Addiction and substance abuse disorder is a significant problem in the United States. Over the past two decades, the United States has faced a boom in substance abuse, which has resulted in an increase in death and disruption of families across the nation. The State of Ohio has been particularly hard hit by the crisis, with overdose rates nearly doubling the national average. Established in the mid 1970’s Sober Living Housing is an alcohol and substance use recovery model emphasizing personal responsibility, sober living, and community support. This model has been adopted by the Ohio Recovery Housing organization, which seeks …

(R2060) An M/G/1 Retrial Queue With Recurrent Customers, General Retrial Times And Working Vacation, S. Pazhani Bala Murugan, A. Jeba Pauline Veronica

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we analyze an M/G/1 retrial queue with recurrent customers, general retrial times and working vacation. In this model, we use two types of customers. They are recurrent customers who rejoin the retrial queue after completion of their service and transit customers who leaves the system once their service complete. All the service times and retrial times for transit customers follows general distribution, retrial time for recurrent customers and working vacation time of the server are assumed to have an exponential distribution and also service time of the recurrent customers follows general distribution. We have used working vacation …