Rasaki Olawale

Non-linear time series and linear models were not designed to detect probabilistic process that are depict by velocity and drift associated to returns the way Ornstein-Uhlenbeck stochastic process describes diffusion and velocity associated to series or waves influenced by Brownian motion or Levy process.  In this research, Brownian motion and Levy process were conflated as driving force for Ornstein-Uhlenbeck process with its solution applied to Naira-Dollar exchange rates from 2009-2019.The drift and diffusion estimates for the Ornstein-Uhlenbeck process driven by Brownian motion and Levy process are realization of AR (1) with 2.991 and 0.1672 respectively. The AR(1) realization for the Ornstein-Uhlenbeck process was stationary with estimate  that lies outside the unit circle. The AIC, BIC, RMSE, and MSE for the Ornstein-Uhlenbeck process were estimated to be 483.7572, 483.4782, 0.00101, and 8.395 respectively, compare to estimates of the same indexes for AR (1) of 767.5, 634.09, ...

Non-linear time series and linear models were not designed to detect probabilistic process that are depict by velocity and drift associated to returns the way Ornstein-Uhlenbeck stochastic process describes diffusion and velocity associated to series or waves influenced by Brownian motion or Lévy process. In this research, Brownian motion and Lévy process were conflated as driving force for Ornstein-Uhlenbeck process with its solution applied to Naira-Dollar exchange rates from 2009-2019.The drift and diffusion estimates for the Ornstein-Uhlenbeck process driven by Brownian motion and Lévy process are realization of AR (1) with 2.991 and 0.1672 respectively. The AR(1) realization for the Ornstein-Uhlenbeck process was stationary with estimate 0.7204  that lies outside the unit circle. The AIC, BIC, RMSE, and MSE for the Ornstein-Uhlenbeck process were estimated to be 483.7572, 483.4782, 0.00101, and 8.395 respectively, compare to estimates of the same indexes for AR (1) of 767.5, 634.09, 0.3819, and 23.48. The criterion via the residuals from the Ornstein-Uhlenbeck process was smaller, which connotes that the errors approximated in using drift, Brownian motion and 1 t x  to estimate t x is relatively small via the Ornstein-Uhlenbeck process.

In queuing theory one deals with the mathematical analysis of the performance of queuing systems. In our daily lives customers encounter queues while seeking services in institutions. The increase in the number of customers has resulted to congestion at revenue collection points in Kenyan towns. There is therefore need to study the queuing systems to identify possible remedies. This study sought to fit a queuing model to bus park revenue collection point as a preliminary action in studying the congestion problem in Kisii town, Kenya. The study considered and collected data on the inter-arrival times, service times and the number of servers at Kisii Bus Park Revenue Collection Point. The inter-arrival and service times were plotted and compared to a plot of a theoretical exponential distribution. The inter-arrival times resembled a theoretical exponential distribution with a parameter 1.022 and the service times resembled a theoretical exponential distribution with a parameter 1.209. Further, Kolmogorov Smirnov and Anderson Darling goodness of fit tests were conducted to determine if the inter-arrival and service times were exponentially distributed. In both cases, the test statistics were less than the critical value. The study therefore established that the inter-arrival and the service times could be modeled as exponential hence Markovian. The revenue collection point used two servers. This study assumed that the servers followed the same service distribution. This study concluded that the inter-arrival and the service times had an exponential distribution and the queuing model used was M/M/2. 1. Introduction Queues are experienced in our daily lives in business situations where customers have to wait for services to be delivered to them for example: in telephone exchange, in banks, in public transportation, in a supermarket, at the county revenue collection points, at a petrol station, waiting to use an automated teller machine (ATM) and people queuing to wait for their turn to vote [1, 2]. When using phones to conduct daily businesses, sometimes one may be put on hold and wait for their turn to receive the services. In modern life, queues are not experienced by humans only. Modern communication systems transmit messages, like emails, from one device to another by queuing them up inside the network. Modern communication systems maintain queues called inventories of raw materials, partly finished goods, and finished goods throughout the manufacturing process of a business institution. The supply chain management in businesses is nothing but the management of queues [3]. Queuing models help in the design process by predicting system performance in organizations. For example queuing models might be used to evaluate the costs and benefits of adding a server (s) to an existing system of an institution. The models enable institutions/organizations to compute the system performance measures in terms of more basic quantities like the waiting times, traffic intensity and the queue lengths in the systems. Some crucial measures of system (s) performance are: the mean queue length, probability of a customer waiting for service in the system, the probability of finding the system being idle, the probability distribution of the number of customers in the system, the utilization of the server (s) and customer waiting time [2]. Queuing systems have a wide range of applications in modern times [3]. The first queuing theory problem was found in the telephone exchange congestion studied by A.K. Erlang (1878-1929).