Study of Fuzziefied Single Server Queueing Systems with Vacation
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AREPORT ONSTUDY OF FUZZIEFIED SINGLE SERVER QUEUEING SYSTEMS WITH VACATIONSubmitted in partial fulfilment of the requirements ofSTUDY PROJECT(MATH F266)BYCHANKIT GOYAL2014B4TS950PM.Sc. (Hons.) MathematicsUnder the supervision ofDr. Chandra ShekharAssociate Professor[pic 1]BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI ABSTRACTQueueing theory is the mathematical study of waiting lines or queues. In queueing theory a model is constructed, so that performance indices like queue lengths and waiting lines can be evaluated and predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.The majority of the early queueing optimization problems were static, where the system parameters would not change over time. To overcome this, a new approach using fuzzy logic to choose the service rate dynamically based on the state of the system so as to minimize the average cost over an infinite horizon for a single server queueing system with vacation will be studied. A system with infinite queueing capacity and a single exponential server whose rate can be adjusted to finite service rate will be considered whose customers arrive according to Poisson fashion. A rule based fuzzy controller will be studied to minimize the average cost over an infinite horizon.ACKNOWLEDGEMENTI take this opportunity to thank our Course in-charge, Dr. Chandra Shekhar, Department of Mathematics, for his constant guidance and support. Without his help and assistance, this report would never have been possible. It has been a great learning experience. I want to thank Professor C B Gupta, Dr Rakhee, Dr Shivi Agrawal for providing me knowledge of Optimization, Operations Research and Applications of Fuzzy Logic in their respective courses.The guidance and support received from everyone is worth mentioning. Without their help, this project wouldn’t have existed in its present shape.ContentsABSTRACT        2ACKNOWLEDGEMENT        3INTRODUCTION        5Single Server with Vacations        6Figure 1. Queuing system with vacations        6Heyman Queuing Theory        7Figure 2. Heyman Queuing Equation        7State Evaluation        7Derivation of Decision Criteria        8Rule Base        9Figure 3. Single Server queueing model with vacation rule base        9Fuzzy Inference System        10Figure 4. Fuzzy Inference System GUI Diagram        10Figure 5. Fuzzy Inference Rule Editor        11

Figure 6. Fuzzy Inference System Rule Viewer¬†¬†¬†¬†¬†¬†¬†¬†12Membership functions¬†¬†¬†¬†¬†¬†¬†¬†13Figure 7. Membership functions of Accumulated Holding Cost¬†¬†¬†¬†¬†¬†¬†¬†13Figure 8. Membership functions of holding cost¬†¬†¬†¬†¬†¬†¬†¬†14Figure 9. Membership functions of traffic intensity¬†¬†¬†¬†¬†¬†¬†¬†15Figure 10. Membership functions of Output¬†¬†¬†¬†¬†¬†¬†¬†16Queueing simulation to obtain a sample performance result¬†¬†¬†¬†¬†¬†¬†¬†17Simulation Experiment with Sample data and Results¬†¬†¬†¬†¬†¬†¬†¬†18Figure 11. Number of customers and ¬†versus time¬†¬†¬†¬†¬†¬†¬†¬†19[pic 2]Figure 12. Number of customers versus time¬†¬†¬†¬†¬†¬†¬†¬†20Figure 13. Expected Number of Customers versus Poisson Arrival Rates¬†¬†¬†¬†¬†¬†¬†¬†21Conclusion¬†¬†¬†¬†¬†¬†¬†¬†22Future Scope¬†¬†¬†¬†¬†¬†¬†¬†23Bibliography¬†¬†¬†¬†¬†¬†¬†¬†24Appendixes¬†¬†¬†¬†¬†¬†¬†¬†25INTRODUCTIONIn queuing theory, a model is constructed so that queue lengths and waiting time can be predicted. Fuzzy approach helps in doing queuing problems with efficient and promising manner, in cases where analytical solutions do not exist.Although some optimization problems were introduced in queueing models early on, the majority of them are static or design problems in which the system characteristics do not change over time. Clearly, this type of models cannot meet the requirements of the majority of practical queueing applications, such as those related to the management of large scale systems in various domains: distribution, transportation, administration, informatics, etc. It is particularly so in many communication and computer applications, in which the performance of the studied system may be improved if some system parameters are adjusted as the system state changes. Hence, one has a dynamic or control problem in which the system characteristics are allowed to change over time.Queueing models have on occasion been classified into two general types descriptive or prescriptive. Descriptive models are models which describe some current ‚Äúreal-world‚ÄĚ situation, while prescriptive models (sometimes also known as normative) are models which prescribe what the real-world situation should be, that is, the optimal behaviour at which to aim.Most of the queueing models presented thus far are descriptive in that for given types of arrival and service patterns, and specified queue discipline and configuration, the state probabilities and expected-value measures of effectiveness which describe the system are obtained. This type of model does not attempt to prescribe any action (such as put on another server, change from FCFS to priority, etc.), and merely represents the current state of affairs.In this project, I will use Mamdani Inference System to solve problems by defining a valid rule base involving fuzzification (crisp inputs are transformed in fuzzy inputs) and defuzzification.

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Fuzzy Logic And Queue Lengths. (June 12, 2021). Retrieved from https://www.freeessays.education/fuzzy-logic-and-queue-lengths-essay/