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Event Simulation Sample Essay & Outline

Introduction

Maggie’s Stop and Go fast food restaurant is one of the prominent drive-by fast food restaurants in Connecticut. Over the years, Maggie’s Stop and Go has relied extensively on the use of human labor to run and determine their operations and processes. This has resulted in a decline in customer numbers primarily due to poor service, and lack of proper coordination.

The manager of this restaurant resulted in applying a Discrete Event Simulation model that would enable them to coordinate their activities in a repeated fashion with the arrival of every customer. This is bound to bolster their position as one of the best drive-bys in Connecticut.

Entities

The entities that will be applied in the model adopted by Maggie’s Stop and Go are of both permanent and temporary nature. The permanent entities include the building, the drive-by lane, the ordering machine and the collection point. These are entities that this fast food joint has spent a small fortune improving, as part of their make-over program. The temporary entities in play will include the customers, the payment methods, the customer queue, and the employees.

The employees are grouped as temporary entities due to the fact that they work on a shift basis. The employees are shuffled between shifts hence no employee is assured of a regular time slot on a daily basis. The ordering machine uses smart technology and is touch sensitive. This ensures that the customers place their orders in a more efficient manner in comparison to the previous method where they had to speak into the microphone to place their order. The collection point has been separated from the payment area to ensure all customers pay their dues before they collect their orders.

Events

The events at Maggie’s Stop and Go have been narrowed down to a sequence of events that occur repeatedly with the arrival of each customer. In order to ensure a more reliable event queue, the microphone was replaced with the touch-sensitive ordering machine. This was to prevent situations in which the customer was either misinterpreted, or not heard clearly when placing their order.

This reduces to very low rates, the possibilities of complaints from customers that their needs have not been met by the customer service team. The simulation model will be multi-threaded due to the fact that many of the events taking place at Maggie’s Stop and Go are of concurrent nature. However, all events are designed to be sorted in a chronological order, and the same applies to the preparation of orders placed by the customers.

The event queue for the simulation model starts at the arrival of a customer and ends with their departure after collecting their order at the collection point. The first event on the event queue is the arrival of the customer at the drive-by lane where they slow down their car and approach the ordering machine. This is in the event that no other customer is using the ordering machine at the time of their arrival. In the event that another customer is using the machine, then the new customer is forced to wait in line until the first customer finishes placing their order.

The second event is the placing of the order. The ordering machine is updated automatically and provides the customer with the full menu of available food for purchase. The customer selects their desired food and states the quantity of the food either in packets or bottles or any measure that is provided. The customer then confirms their order and moves their car to the payment area. Upon confirmation of the order, an electronic receipt is created by the ordering machine as the customer simultaneously moves to the payment area.

The electronic receipt is used for the order preparation by the kitchen staff. At the payment area, Maggie’s Stop and Go receives payment for food and prints out a hard copy receipt that is handed to the customer as proof of their purchase. Payment is acceptable in terms of cash or credit card. The payment time taken ensures that the kitchen staff complete packing the customer’s order, and they place it in the collection point. The customer then collects their order and leaves the premises.

The event list for this model is as follows:

CUSTOMER ARRIVAL-CUSTOMER WAITS IN QUEUE-CUSTOMER PLACES ORDER-ORDER MACHINE PRINTS ELECTRONIC RECEIPT-CUSTOMER MOVES TO PAYMENT AREA-ORDER IS PREPARED-CUSTOMER COMPLETES PAYMENT-ORDER PLACED IN COLLECTION POINT-CUSTOMER COLLECTS ORDER-CUSTOMER DEPARTURE.

This event list is completed in chronological order, with exceptions in the event that one of the events in the list is not necessary. For example, the absence of a waiting queue would prompt the model to skip the event and hence the customer would proceed to place their order. This will also apply to the clock time.

The model developed for Maggie’s Stop and Go warrants the availability of event handlers that ensure all events proceed in the intended order. The first event handler is the guard at the gate who directs the vehicles into the drive-by lane. He ensures that all customers are served in chronological order, and that they also join the queue in the event that there is one.

The most important event handlers are the kitchen staff that ensures the order is prepared to customer satisfaction and specification, and placed in the collection point. The next event handler is the payment clerk who ensures that the payment transaction for the purchase is completed. All other events are automated and do not require event handlers to guarantee their successful completion. The manager is the overall event handler who oversees the successful completion of the sequence of events from start to finish.

State

This model makes use of a number of system states. These states are directly influenced by the events on the list, and they change in accordance to this list. The states that are affected by the events list include the status of the payment clerk, status of the kitchen staff, and the number of customers waiting in the queue. These states directly influence the simulation of events as their run-time is essential in the collection of statistical data that are necessary in understanding the simulation model.

The states also operate in a domino effect. A change in any of the states offsets a change in another state, such as is the case with a change in the status of the payment clerk signals a change in the number of customers waiting in the queue.

The variables in the system state can be modified in the event that change in one state does not necessarily warrant a change in another state. A good example is the change in payment clerk status. After serving the only customer at the premises, the change of payment clerk status to ‘idle’ does not warrant a change in corresponding states such as the number of customers waiting in the queue, or status of the kitchen staff. Such temporary modifications provide a clear picture of the efficiency of the simulation model. By providing data that are irregular, the simulation model can be understood from all perspectives.

Random Variables

The random variables in play in this simulation model include inter-customer-arrival time, the payment clerk service time, and the kitchen staff packaging time. These variables are greatly affected by the customer in question. The inter-customer arrival time affects the working conditions of the kitchen staff and the payment clerk, seeing that it increases the pressure under which they work.

Being the two most important parts of this simulation model, many customers arriving in between short periods of time translates into the need for these two components of this simulation model to operate faster than they would with fewer customers. The method of payment chosen by the customer also has great influence over the simulation model. A customer paying cash takes less time at the payment clerk as compared to a customer paying with their credit card. The order placed by the customer also determines the packaging time required by the kitchen staff. All these variables provide essential data that are useful in determining the right number of employees required to address any scenario that may arise at Maggie’s Stop and Go.

Ending Condition

The ending condition is set based on the shifts that the employees work. Many employees work 6-hour shifts, and hence the ending condition is set to be when the clock reaches the 6 hour mark. This provides a complete loop of the simulation. The simulation model then generates a statistical report on the average time for each event in the events list between customer arrival and customer departure. The data collected over the four 6 hour loops in a day will enable the management to determine the adequate staff and measures to be taken at different time periods.

References

Dagpunar, J. S. Discrete Event Simulation. Simulation and Monte Carlo: With Applications in Finance and MCMC, 135-156.
Fishman, G. S. (2001). Discrete-event simulation: Modeling, programming, and analysis. New York, NY [u.a.: Springer.
Robinson, S. (2010). Conceptual modeling for discrete-event simulation.

Mathematics Relation to Exercise Essay & Outline

Mathematics can be applied to virtually any human activity. It is especially important to realize that indeed mathematics is integral when it comes to physical exercises. It is through mathematics that persons exercising can be able to maintain exercise routines and develop new ones. Mathematics has become an important factor in the gymnasium, and it is helping a lot of people to conduct their exercises in an effective and efficient manner (Jacob & Andersson, 1998). This research paper is going to examine how mathematics relates to exercises.

Most of the exercise routines often involve repetitions of several numbers. Therefore, it can be seen that this is a major application of mathematics. There are repetitions of 15’s, 10’s or even 5’s. It is with these repetitions that one can understand whether they have completed a certain routine and whether they can go to the next one (Jacob & Andersson, 1998). There are certain muscles such as biceps and triceps that need a certain number of reps to grow, therefore, through mathematic calculations it is possible to ensure that indeed this happens.

Mathematics is used in the calculation of short term and long term goals (Williams & Wilson, 2008). Most at times when one is conducting the exercise, there is the aspect of what he or she wants to achieve (Jacob & Andersson, 1998). It is, therefore, through mathematics that one can be able to ensure that they can understand the true aspect of the progress that they have undergone and helped in the calculation of their short-term as well as long-term goals (Safrit & Wood, 2010).

Numbers and formulae can be described as being integral when it comes to exercise. For example, Body Mass Index commonly referred to as BMI is a formula that is used when conducting exercise (Williams & Wilson, 2008). There are certain exercises that are particular to people with a certain BMI. The BMI is often used to measure how the relation of a person’s height to his or weight. BMI is often an estimate of body fat, and it has been described as a good gauge of the risk of diseases which occur when one has more body fat.

In mathematics, there are functions that are used to describe motion. This is sometimes the focal point of the exercise, and there is a need to ensure the calculations are done in the right manner. This is more especially the case when it comes to professional athletes. Those that are in athletics, for example, have to be able to calculate their top speed when they are running to know which pace to take at the beginning of the race (Williams & Wilson, 2008). Further, it is not unusual for athletes such as those in golf to use angles to ensure that the ball goes towards the desired direction. The three-dimensional kinematic is also important when it comes to football players as they have to know which angle to throw the ball and which speed to run. Although, at times, there are no exact calculations that are done on a sheet of paper, it is through their knowledge of angles that they can execute several exercises (Jacob & Andersson, 1998).

Mathematics can be used to perfect exercise form. One can be able to understand different exercise routines and after how many reps one starts to feel pain the muscle. It is through this calculation of numbers that one can be able to effectively and efficiently improve his or her exercise form. It is also mathematics that tells one when to take a water break during an exercise. This is important in any exercise, further; it is also mathematics that helps in the calculation of the desired rest period (Williams & Wilson, 2008). It is during this rest period that muscles can develop, and one body gets back to its normal state of metabolism.

In the normal day to day exercise such as jogging, mathematics plays a very important role. For example, it is through mathematics that one calculates the distance that one should run to burn a certain number of calories (Williams & Wilson, 2008). Equipment such as treadmills is integrated with mathematical functions that help to calculate the slope of the belt, the speed and the weight of the person running to calculate the calories being burned. This is applied mathematics in physical exercise. This is also the same case when it comes to the weights section; the weights are numbered in a mathematical fashion for one to know which weight applies to his BMI and muscle strength.

In conclusion, Mathematics is hugely related to science in many different ways. It is through mathematics that persons can be able to maintain their exercise routines. It is also through mathematics that one can be able to calculate his or her short term and long term goals (Safrit & Wood, 2010). Mathematics is critical when it comes to ensuring people perfect their exercise form and in the measurement of growth regarding fitness levels. Lastly, gym equipment meant for exercise is hugely calibrated using mathematical terms and functions to ensure optimal exercise.

Reference

Williams, C. A., James, D. V. B., & Wilson, C. (2008). Mathematics and science for exercise and sport: The basics. London: Routledge.
Jacob, M., & Andersson, S. (1998). The nature of mathematics and the mathematics of nature. Amsterdam: Elsevier.
Safrit, M. J., & Wood, T. M. (2010). Measurement concepts in physical education and exercise science. Champaign, Ill: Human Kinetics Books.

Francois Viete's Contribution to Algebra Essay & Outline

Abstract

One of the leading lights in the science of mathematics is Francois Viete. The Frenchman is accredited with contributing to mathematics especially in the field of modern algebra as it is today. Ironically, Francois Viete was a lawyer by profession and only did minor mathematical assignments during his free time and during which time friends say he would ponder over mathematical problems for extraordinarily long hours. Algebraic notations using words is one of his most notable contributions to mathematics and in some quarters he is regarded as the father of Algebra.

Keywords: algebra, Francois Viete.

Francois Viete's Contribution to Algebra

Francois Viete is regarded in mathematical circles as the father of Algebra. He particularly revolutionized the way problems were solved by changing the methods of analysis and introduced the concept of representing variables using letters (Stedall, 2002). Viete concentrated on two key mathematical aspects that were commonly being used during his era – analysis and synthesis (Bashmakova & Smirnova, 2000). His work in mathematics begun in earnest with the publication of his work The Art of Analysis or New Algebra as it became known in some circles (Viete, 1983).

Viete's interest in mathematics, it is believed, stemmed from his passion for cosmology and astronomy. He had earlier been introduced to these subjects by Catherine, daughter of Antoinette d'Aubeterre and Jean de Parthenay-l'Archeveque, notable names in the 16th century France (O'Connor & Robertson, 2008). Viete would give lectures to Catherine about the mathematical relationship between celestial bodies, and one of his recovered manuscripts indicated that there seemed to be some mathematical foundation in the arrangement of heavenly bodies (O'Connor & Robertson, 2008). This manuscript known as the Harmonium celeste translated as "Harmonic Construction of the Heavens" appeared in the 1737 French publication Principes de comsographie which showed indications that Viete's mathematical works had a close relationship with cosmology and astronomy (O'Connor & Robertson, 2008)y.

To curve his niche, Viete had to remodel some famous mathematical concepts existing at the time. For example, in this document the Almagest he is seen challenging longstanding principles of Ptolemy but at the same time lending credence to Copernicus' theories (Bashmakova & Smirnova, 2000). In 1579, Viete's first work in trigonometry was published. In this work, he introduces computational methods to construct the cannon and explains how to use trigonometry to calculate plane angles and spherical triangles (Bashmakova & Smirnova, 2000).
Mathematicians presently can calculate properties of triangles and other figures if there exist other properties which are known. For example, one of the most common properties of triangles in modern algebra today is that all angles of a triangle add up to 360°. Viete called these features "determining components,” and the data presented in his works illuminates his understanding of magnitudes and ratios (Bos, 2001). He came up with a table of trigonometric ratios where he computed every minute of an arc to one part in 10,000,000. In one instance, he uses an inscribed polygon 6.144 sides and a circumscribed polygon with 12,288 sides to achieve a value of 29.083,819,59 to base 10,000,000 (Bos, 2001).

Throughout his Canon Mathematicus, Viete discovers theories that he regards can help in getting solutions to problems regarding right angle triangles and oblique triangles and in the long run, comes up with his invention – the law of tangents.

He describes which equations can be used to determine the unknown properties of a spherical triangle given that the other two variables are known – this application later became known as the Napier rule. Viete also made his contributions to the law of cosines and also expanded on Ptolemy's trigonometric identities. One of the identities he helped develop is the cosine rule which could be used calculate angles in spherical triangles. The formula they developed looked like this:

In as much as Viete was a genius in his own right, he also got inspiration from other mathematicians that preceded him. For example, most of his works seem to develop from the works of Pappus and Diophantus. Diophantus is credited with introducing short algebraic connotations and widening the application of rational numbers, equation operations, polynomial rules, and literal symbolism (Pycior, 1997). Similarly, his early works as depicted in his Arithmetica shows the application of negative numbers especially when he wanted to get solutions in the real rational domain (Pycior, 1997).

Viete's version of analysis is seen to borrow much from Diophantus' concept of unknown variables with powers and in particular, Euclid's theory as a foundation for solving equations. His concept of zetetics (analysis) encompassed two fundamental theoretical ideologies – zetetikon and poristikon where the former implied "the search for truth" and the latter, "the search for solutions." (Bashmakova & Smirnova, 2000). Using this approach, Viete attempted to show that once proof had been found, it could be turned into a synthesis and become a template for solving problems of similar structure. The poristkon ideology posited that a construction could be formed from a solution to a problem. By combining these two methods, Viete transformed the whole methodology behind algebra (Viete, 1983).

Reference

Bashmakova, I., & Smirnova, G. (2000). The Beginnings and Evolution of Algebra. (A. Shenitzer, Trans.) Washington, DC: MAA.
Bos, H. J. ( 2001). Redefining Geometrical Exactness. New York, NY: Springer.
O'Connor, J., & Robertson, E. (2008, 23 March). Francois Viete. Retrieved September 21, 2017, from MacTutor History of Mathematics: http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Viete.html
Pycior, H. (1997). Symbols, Impossible Numbers, and Geometric Entanglements. Cambridge, UK: Cambridge University Press.
Stedall, J. (2002). A Discourse Concerning Algebra: English Algebra to 1685. Oxford: Oxford University Press.
Viete, F. (1983). The Analytic Art. Kent, Ohio: Kent State .