Mathematical models are widely used to describe and control the dissemination of diseases in the population. Stochasticity is introduced to bring new insight into the modelling of population dynamics of diseases. Many systems in nature divulge stochasticity and are subject to stochastic perturbations. In this talk, we present differential equations with stochastic perturbations. We present briefly the existence and uniqueness of the global positive solution to the stochastic system. We present the stability conditions and illustrate the simulation results for the proposed model. The model parameters are estimated using the updated data estimation method, and the proposed model is validated. Finally, we present a computational experiment and validate the model by performing statistical analysis for the actual data for some diseases like COVID-19 and acute respiratory diseases.

We consider a single server retrial queue (RTQ) with no waiting space at the service station, A finite number M of orbits, with the first M-1 having finite capacities K, 2K, ...., (M-1)K in the increasing order, and the Mth one having infinite capacity are provided for retrial customers, depending on the number of retrials they have completed-- orbit one seats tho primary customers who encounter a busy server on their arrival, ... the (M-1)th accommodates those who have completed M-2 retrials and the last one is for those who is offered either the last retrial/direct access to the server through a SEARCH by the service system at the EPOCH OF COMPLETION OF A SERVICE. This mechanism provides exact information of the number of retrials an arbitrarily chosen customer in the system has completed. The increasing nature of the capacities of the orbits together with the assumption that retrial rates from orbits increase in a sequential manner, ensure minimum loss of retrial customers. This system is analysed under the assumption that the primary arrivals follow a Poisson process; inter-retrial times in each orbit are exponentially distributed. Customers in the last orbit are endowed with a privilege of being either taken for service at a customer service completion epoch through a SEARCH MECHANISM requiring negligible time or through the retrial. Two cases are considered: 1. Retrial customers return to orbit with probability one; 2. such customers abandon the system with probability p. A comparison of the performance of these two is carried out by introducing separate cost functions.

Often in several fields of real world applications future developments are functions of past events. Hence, the processes describing these situations cannot be Markovian. But a Hawkes process has an intensity that is stochastic and depends on the past history of the process itself, while an inverse Hawkes process is a process that has constant intensity and stochastic jump size, depending on its past history. That is why, this paper considers processes that are obtained by combining Hawkes processes and inverse Hawkes processes. In this context the asymptotic results studied are generalized versions of the Law of Large Numbers and of the Central Limit Theorem.

The rapidly expanding population and the race for urbanization have intensified the demand for efficient public transport. However, due to improper planning and a lack of precision in several critical areas, most of the existing urban road transport organizations (URTO) are unable to fulfill this travel demand. Further, efficient bus service is necessary for a country like India, where the population is expected to increase from 121.1 crores to 152.2 crores from 2011-2036 as per the Census of India, and most of them cannot afford private transport. So, people living in urban cities rely heavily on the Indian URTO (IURTO) for their daily commute. Going by the yearly financial performances of these IURTO, it is observed that almost every IURTO in India is facing huge losses. Despite their immediate rescue to increase the bus fares, the government puts them under contrary pressure to keep the fares at the current level to provide equitable service to the public. As the input prices rise, these IURTO come under huge financial pressure. Hence, the IURTO requires strategic interventions to minimize costs without changing the existing fare level. Further, road transport, particularly city buses, contributes higher CO2 emissions than other vehicles. According to a report published by the C40 Cities, considering a city like Bengaluru in India, even though buses constitute only 6% of the city vehicles, they are responsible for 25% of PM2.5 emissions. Hence, considering both huge monetary losses and higher CO2 emissions in operating buses, there is a need to identify all potential operational to strategic level decision problems to minimize the operational cost. In this presentation, one such prescriptive decision problem related to the strategic/ tactical level will be described. There can be several reasons behind the continuous losses faced by any IURTO, one of which is that the city buses travel various types of non-revenue generating distances. One such prominent non-revenue generating distance is the ‘Dead Kilometers (DK),’ which is the total distance traveled by bus between the depot and the starting point of the first trip and from the ending point of the last trip back to the depot without carrying passengers. According to the Central Institute of Road Transport (CIRT) statistics, all the buses in India collectively covered approximately 25.2 crore DK in 2019-2020. Further, the cost associated with DK (called DK Cost (DKC)) varies for every bus since mileage per unit liter of fuel for every bus varies. So, optimal/efficient matching of buses to depots is essential to minimize the DKC, thereby reducing the overall operational cost. This tactical-level decision problem is known in the literature as the Bus-Depot Matching Problem (BDMP). Moreover, the depot operating cost (DOC) for each bus differs across depots, making the proper matching of buses to depots critical for reducing DOC. Additionally, the choice of depot impacts the cost related to CO2 emissions, as each bus emits varying levels of CO2. Essentially, minimizing all three costs, DKC, DOC, and cost associated with CO2, will help IURTO minimize their cost. A quick review of the literature reveals that while the issue of minimizing DK has been ex- tensively studied, there is limited focus on minimizing DKC, particularly in the Indian context. Moreover, no existing research appears to address the BDMP by simultaneously minimizing eco- nomic costs (DKC and DOC) and environmental costs (associated with CO2 emissions) in bus operations, especially for TURTOs. This study aims to fill this gap by (i) proposing a mathemat- ical model that minimizes all three costs and demonstrating its applicability from a tiny example to a large-scale, real-world problem, (ii) introducing 52 simple heuristic methods to achieve near- optimal solutions, and (iii) conducting performance analyses using a triangulation approach with 360 randomly generated real-life-sized problems to identify the most efficient heuristic methods.

Reliability is defined to be the ability of a system or component to perform its required functions under stated conditions for a specified period of time. In the world of broadband network, different types of communication networks are facing a large demand due to its high- speed wireless data transmission, extensive coverage from base stations, and a luxury of upgrading protocol software. In this talk, an analytical model is discussed to determine reliability attributes of third generation and beyond Universal Mobile Telecommunication Systems (UMTS), Voice over Internet Protocol (VoIP), Long-Term Evolution Advanced (LTE-A), Vehicular Ad hoc Network (VANET), High-Altitude Platform (HAP) networks. These networks are modeled using stochastic models such as Markov chains, semi-Markov process, reliability block diagrams and Markov reward models to obtain these attributes. Numerical results illustrate the applicability of the proposed analytical model. In addition, it can help the guarantee of network connectivity after any failure, without over dimensioning the networks.

It is not well known that a nonsingular matrix with the so-called triangle property has a tridiagonal inverse. In this talk, I will give an overview of this matrix class. I will also touch upon some very recent results that we have obtained, for a related class of matrices whose inverses are (what we refer to as) bidiagonal matrices with the south-west structure.

Boundary value problems for PDEs employ subspace of Lp spaces. Boundaries of open sets have measure zero. Hence restriction of a function to a boundary is not well defined. Fractional order Sobolev spaces provide the required functional spaces using which Trace maps can be defined. They give the appropriate generalization of restriction operators.

Several queueing inventory models allow a product form solution for their steady-state distri- bution. The fundamental assumption in many of the above models is not to allow customers to join the system when the inventory level is zero. Under this assumption, with exponentially distributed inter-arrival and service times, the steady-state distribution of the queueing inventory system is the Kronecker product of the steady-state distribution of the classicM/M/1 queue and that of an inventory system with instantaneous service. Hence, despite the strong dependence between the queue and inventory, a product form for the steady state exists. However, the transient distribution of the above models doesn’t have a product form unless their initial probability distribution has a product form structure. In the present study, we apply the homotopy perturbation method for the transient analysis of an (s,S) inventory system with customer loss when the inventory level is zero.