Allahabad Mathematical Society

Reducing Customer Abandonment Through the Characteristic Function of Two Queueing Games with Exponential Reneging

Author: D. Dutta, M. K. Patel

Pages: 45–67

Abstract:

Customer abandonment, also known as reneging, is a major problem for queueing systems. A high amount of reneging in a queueing system leads to reduced goodwill for the system. Queueing systems therefore aim to reduce reneging. Reneging is therefore modeled as a cost on the queueing system. In this paper, we propose a cooperative game theoretic approach to solve the problem of reneging in queueing systems with Poisson arrival, exponential service times and exponential reneging. We define two aggregation principles. These two aggregation principles give us two types of cooperative games. We consider the characteristic function as the average reneging rate in both games. We establish an inequality between the values of the characteristic functions of the two games. Subadditivity of the games is explored using random sampling. Furthermore, a differential evolution algorithm is used to find the non-emptiness of the core in both games. Finally, we use numerical illustrations to demonstrate the cost reduction that can be achieved using the two different aggregation principles. The nucleolus is used for distributing the cost of the grand coalition. The cost of reneging for each player in game one is compared with the cost for that player in game two. Read more

A Preliminary Note on a mixed finite difference approach for a singularly perturbed Gait model

Author: Shubhangini Gupta, Sourav Banerjee, Tamal Pramanick

Pages: 1–18

Abstract:

In this research, we introduce a simulation based numerical technique for addressing a class of linear second-order ordinary differential equations derived from a simplified biologically inspired model of human gait. Although the governing equation is posed as a time-dependent ODE, its inner rescaled formulation exhibits singular perturbation structure characteristic of boundary layer type behavior. The model incorporates significant physical aspects like gravity, damping, and leg stiffness, while also illustrating the vertical motion of the body's center of mass during ambulation or running. Standard numerical schemes may struggle to accurately resolve steep solution gradients arising for small perturbation parameters. To address this, we employ an asymptotic inner-outer decomposition combined with a time rescaling transformation to capture boundary layer behavior effectively. The Thomas approach is used to quickly solve the resulting tridiagonal problems within the mixed finite difference framework. The numerical experiments are also presented in order to validate the theoretical findings. Read more

Generalized Fuzzy Ideals in Ordered Ternary Semigroups

Author: Ravi Srivastava, Arvind Yadav, Neha Ahuja

Pages: 19–44

Abstract:

This work explores and defines generalized fuzzy left ideals, fuzzy right (lateral) ideals, fuzzy fuzzy bi-ideals, and quasiideals within the framework of ordered ternary semigroups, using a set-theoretic approach. Generalization of these fuzzy ideals is developed by employing Tom Head’s metatheorem as a foundational tool. Head’s metatheorem, which provides a unifying framework for transferring results from crisp algebraic systems to their fuzzy counterparts, serves as the foundational tool for our analysis. It is demonstrated that the classes of generalized fuzzy left (right, lateral) ideals, generalized fuzzy bi-ideals (quasi-ideals) and various types of fuzzy ideals are projection closed within an ordered ternary semigroup. Characterizations of generalized fuzzy bi-ideals and fuzzy quasi-ideals in terms of generalized fuzzy left (right, lateral) ideals are also provided. Tom Head’s metatheorem is applied to derive proofs for numerous propositions related to these various types of generalized fuzzy ideals, effectively simplifying the proof process by avoiding intricate calculations. Several important properties of fuzzy substructures in ordered ternary semigroups have been established. The intersection of a fuzzy subsemigroup and a generalized fuzzy bi-ideal is shown to be a generalized fuzzy bi-ideal. Similarly, the product of three generalized fuzzy bi-ideals, as well as the product of three generalized fuzzy quasi-ideals, results in a generalized fuzzy bi-ideal. Furthermore, every generalized fuzzy quasi-ideal is a generalized fuzzy bi-ideal, and every generalized fuzzy bi-ideal of a regular ordered ternary semigroup is a generalized fuzzy quasi-ideal. Additionally, the intersection of a generalized fuzzy left ideal, generalized fuzzy lateral ideal, and generalized fuzzy right ideal is a generalized fuzzy quasiideal. By situating fuzzy ideal theory in the richer setting of ordered ternary operations and applying Head’s metatheorem, this work advances the development of fuzzy algebraic systems and lays the groundwork for applications in logic, artificial intelligence, and multi-agent systems where uncertainty, order, and multi-arity operations coexist. Read more