INTRODUCTION

A system is a set of interrelated elements working together towards achieving a certain goal or objective (Merriam Webster). Systems theory is a way to view the world and it has influenced scientific, engineering as well as decision systems. Systems theory is used in this report as a paradigm for showing how systems work and how their elements relate together Fredrick Taylor (2001). System behavior is expressed in formulations and characteristics where the analysis is based on its simplicity and the number of elements among other factors R. Flood and M. Jackson (1991).

1.0 Task 1.

1.1 General Scenario:

Unity Auto Garage was established in 1974 by Mr. Rashid and a group of other directors driven by their ambitions to start up a user centered garage that would offer 24 hour service and at very friendly prices. The garage is located along Kampala Road within Nairobiâs busy industrial area and has introduced a very challenging competition line for Accidental and Auto repairs in Kenyaâs service industry. In the garage cars arrive for general services involving engine check, wheel alignment, and general car servicing auto body repairs, engine repairs, and safety inspections among others. The cars arrive for service in a single line and serviced by a team of garage attendants mostly comprising of 3 service team members. It is expected that a car will arrive at least every 30 minutes. The estimated average service time is 20 minutes per car.

1.2 1.2: Description and representation of basic concepts

Mr. Rashid, the garage founder is in dilemma over various decision making issues and he therefore wishes to solve the following decision based problems regarding the garage using the systems theory:

1. Know the average number of cars in the garage system.

2. Know the average time for a car in the system

3. Know the average waiting time

4. Know the average number of cars in queue

5. Know the utilization factor

6. Find out the percentage idle time.

7. The garage has a challenge of customers leaving if they find more than three cars on queue. The management wishes to know the probability that there will be more than three cars in the queue at a certain time.

From the above scenario two key problems can be formulated for the garage. These are;

1) Establishing the utilization factor for staff members and service channels in the garage

2) Determination of the queuing discipline to deploy for customer service.

Application of the systems theory is recommendable to Mr. Rashid in making the stated decisions. To do this, two systems theory approaches methods will be applied as follows; the analytical model to find out a solution for the first problem and a simulation model to find out a solution to the second problem.

1.3 1.3 Rationale of System Dynamics to the Problem:

Since a system is majorly composed of interrelated elements working towards a common objective, the garage system has a number of elements that foster its success. The major elements in the system are the garage human resource, technological resources available, the number of cars entering the garage queue for service and the number of service channels available at the garage. There are various events that result into state changes in the garage system. These are; a car entering the queue system for service, a car entering a service channel or a car leaving the service channel after service. An increase in the number of cars in the queue will result in the demand for more staff and the need for more resources such as finances and time. On the other hand, this may affect the degree of customer satisfaction as customers may be forced to wait for long in the service queues.

1.4 1.4 Representation of the diagram

(Fig. 1.1 The garage systemâs dynamic diagram)

1.5 1.5 An illustration of the diagram and critical discussion

Fig1.1 above shows the various relationships portrayed by the elements of the garage system. As indicated in the diagram an increase in cars on the queue will result in an increase in the garage staff team and overall this will call for more financial budgets and allocations. On the other hand, increase in service resources will lead to a reduction in the service time though the increase in resources means more financial expenditure for the garage. The diagram therefore suggests a negative feedback system for the garage.

Task 2: Use of Model building process

2.1 Identification of the decision variables

The garage system is dynamic in nature. Some of the systemâs variables are: cars arriving, time, and the size of the garage staff team, the available resources such as power and technology. According to the systems dynamics theory the systemâs variables will change in different situations. For instance it is expected that the arrival time will vary from a car to another. The systemâs behavior is expected to be of the exponential growth look. However, the pattern can be combined with oscillation to form a complicated system behavior.

2.2: Formulation of objective

Over and above answering the questions formulated in task 1, the system should also provide some loopback feedback to the garage management. Through the system the garage management will be informed on the cause and effect of either small or significant change in the systemâs variables.

2.3: Formulation of constraints

There are a number of constraints anticipated in the deployment of the system. The constraints include the fact that the number of cars reporting to the garage is just but an estimate and therefore there may be more than or fewer cars joining the queue that stated. Finances in the system can be termed exogenous as they seem to affect other variables in the system yet they are affected by no other variable.

2.4: Formulation of the assumption set

There are various assumptions in the system. They include the assumption that the car arrival will be uniform (after every 30 minutes). Another assumption in the system is that the service queue opens immediately for the next car after the car in service leaves. This means no slack of time as an event commences soon after the other. This assumption may lead to weak decisions as a small change in some variables may affect the results of the system.

Task 3: The simple analytical model

Construction of the analytical Model of the System:

In the systemâs analysis the following are the key considerations since it is a queuing system, the size of calling for service, the pattern of arrival and behaviour of arrival are to be considered. Other considerations to make are queue length and distribution and the service lines available. The analysis will be based on Kendallâs notation on queuing systems which consists of basic three symbol form. These are:

Arrival distribution | Service time distribution | Number of service channels open

Where;

M = Poison distribution for number of occurrences (Exponential times)

D = Constant deterministic rates

G = General distribution with mean and variance known

According to the system in proposal the following information and solutions can be deduced as follows.

â¢ The average number of cars in the system L =

â¢ The average car service time in system W=

â¢ Average number of cars in queue Lq=

â¢ Average service waiting time in the garage Wq=

â¢ Percent idle time for the service facility P0=

â¢ Probability the number of customers is k Pn>k=

Task 4: Analytical model application

The analytical solutions

Using the analytical model solutions to Mr. Rashidâs dilemma can be acquired as follows;

2 cars arriving every hour (since 1 car is arriving at least every 30 min)

3 cars serviced per hour (since the average service time given is 15 minutes per car)

â¢ The average number of cars the system will be;

L = = 2/ (3-2) = 2/1 which means there will be 2 cars in the system on average.

â¢ What is the average time that a car will spend in the system?

W= = 1/ (3-2) = 1 this means that on average a car will spend 1 hour in the service line.

â¢ What is the average number of cars waiting in queue?

Lq= = 22/ 3(3-2) = 1.33, this means that on average 1.33 (rounded off to the nearest whole car is 1 car) in the waiting time at a given time.

â¢ What is the average waiting time for a car in the system?

Wq = 2/ 3(3-2) = 0.67, this means 0.67 hours waiting time for a car in the system

â¢ The percentage of time that the facility is busy is also going to be the same as the waiting time, which is 0.67.

â¢ What is the probability that the facility is idle? i.e. (no car)

Pn>k =

P (0) =1-(2/3) = 0.33 this means that the probability that there are 0 cars in the system is 0.33

Task 5:

A Critical Evaluation of the Analytical Model:

Through the use of Kendallâs notation for queuing systems (Dr Conor McArdle EE414 – Performance Evaluation of queuing Systems) answers to the above system problems have been approached. The average number of cars in the system in an hour has been identified as 2 per hour. Other answers provided for the problem include the service average time that the service station is busy, the probability that the facility is idle, as well as the average waiting time for service. The utilization factor analysis helps the management make a decision on whether to add another service station or not. The queuing discipline determines which customer is selected from the queue for processing or service when a server becomes available. Examples of different queuing disciplines are: FCFS – First-Come-First-Served – If no queuing discipline is stated in the Kendall description, this one is assumed. All the systems we consider have a FCFS queuing discipline, LCFS-Last-Come-First-Served

PS – Processor-Sharing-All customers are served simultaneously where processing of all customers is equally spread across all servers.

Task 6:

6.1 Constructing the revised analytical model

Mr. Rashid wishes to increase the number of service channels in the garage by 100%. He therefore wishes to carry out a âwhat ifâ analysis for this in form of a revised analytical model. This will help him understand the impact of changing a certain variable in the system. For example what if the number of service stations was increased by 100% as he thinks? The result would be addition of another service line, while the number of cars visiting the garage for service still remains the same. From an analytical point of view the number of cars in the queue at a given time in the system would reduce and on the other hand the financial requirements would increase as a new staff team would be hired in place for the new service channel.

6.2: The analytical solution for the revised solution

With the âwhat ifâ analysis in consideration the new model would be as follows;

(Fig 6.2: revised solutionâs dynamic diagram)

There will be a significant reduction in the number of cars in queue for service as indicated in fig 6.2. This will mean faster service delivery to customers though the garage has to procure more finances for the new channel.

Task 7:

A critical evaluation of the revised system

The revised system helps the management in identifying critical decisions to be made. Through the revision of a certain system, various variables can be changed and therefore the results known in time. For instance Mr. Rashid is able to tell what if the number of service channels was increased up to a certain level. Therefore it is very important to factor analytical models in real life systems from a revision perspective as there will always crop a new system and solutions to more questions.

Task 8: Constructing the Simulated system

The system described in the scenario can be simulated as a discrete system. This means that the system constitutes of variables that change instantaneously at separated points in time. The systemâs state variables are;

â¢ The number of cars in queue

â¢ The number of cars arriving

â¢ The number of busy/serving stations.

The system has various crucial events that govern the change of state and these are arrival of a car, a car entering service and a car departure.

The system is discrete because it only changes when a car arrives for service, car enters for service or a car departs from service station.

Time simulation

To simulate time the next event time advance mechanism also known as (event to event time advance) will be used. In this case the simulation time will be initialized at zero, with the occurrence times for other events being determined by using probability methods since the occurrences anticipated will be random. The assumption is that at least one car will arrive in every 30 minutes. Inactivity periods will not be factored in this simulation as events follow each other therefore skipping inactivity time periods encountered by the system.

The systems event graphs

An event can be defined as an instantaneous occurrence that may change the state of a system. Determining the number of events in simulation sometimes may be challenging as well as specifying the state variables needed to keep the simulation running. Therefore for this system an event graph will help overcome these problems in my simulation as follows.

(Fig 1.2 the system simulation graph)

Task 9:

A Critical Evaluation of the Simulation:

As indicated in the above figure the arrival event can be scheduled initially and therefore schedules itself in the system. The other two events are scheduled by some preceding events somewhere. However the departure event may reschedule itself should the queue be very log for the customer to wait. Therefore there are strongly connected components in the simulation and these are;

Car arrival Car arrival

Car departure Car departure

The events car arrival, Car entering the service point and car departing are not strongly connected however.

Task 10: Conclusion:

The application of systems theory in business decision making is very crucial for approaching specific problems. The application of this enables a business know the cause and effect analysis of key elements of a system at work as well as the utilization factors associated. Through this an expression of the system overview is achieved and. Analytical methods involved help a business get the exact parameter of deployment for progressive performance. Simulation helps imitate the system and frame it in real world scenario perspectives. There are various queuing systems that can be applied at different levels and case scenarios. The choice of a good queuing system will help streamline operations and ensure quick service to customers in a business. Recommendation for further research could be analysis of queuing systems in multi service stations.

References

Robert B. Cooper. Introduction to Queuing Theory (2nd edition). 1981. 347 pp.

Borge Tolt. The solution manual (182 pp, 1981) available online at

http://www.cse.fau.edu/%7Ebob/publications/QueueingTheory_solns.pdf

Eitan Altman. NS simulator course for beginners. 2002. Lecture Notes 146 pp. Available at http://www-sop.inria.fr/mistral/personnel/Eitan.Altman/ns.htm

Moshe Zukerman Introduction to Queuing Theory and Stochastic Teletraffic Models

Donald Gross, John F. Shortle, James M. Thompson, and Carl M. Harris. Fundamentals of Queuing Theory, Solutions.

Banks, J., J.S. Carson, B.L. Nelson, and D.M. Nicol (2005), Discrete-Event System Simulation, Fourth Edition, Prentice-Hall, Upper Saddle River, NJ.

Sterman, J.D. (2000), Business Dynamics: Systems Thinking and Modeling for a Complex World, Irwin McGraw-Hill, Boston, MA.

Modeling and simulation in engineering available at http://www.hindawi.com/journals/mse/contents/

Ronald E Giachetti Design of enterprise systems

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