Study of Fuzziefied Single Server Queueing Systems with Vacation

A
REPORT ON

STUDY OF FUZZIEFIED SINGLE SERVER QUEUEING SYSTEMS WITH VACATION

Submitted in partial fulfilment of the requirements of

STUDY PROJECT

(MATH F266)

BY

CHANKIT GOYAL

2014B4TS950P

M.Sc. (Hons.) Mathematics

Under the supervision of

Dr. Chandra Shekhar

Associate Professor

[pic 1]

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

ABSTRACT

Queueing 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.

ACKNOWLEDGEMENT

I 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.

Contents

ABSTRACT        2

ACKNOWLEDGEMENT        3

INTRODUCTION        5

Single Server with Vacations        6

Figure 1. Queuing system with vacations        6

Heyman Queuing Theory        7

Figure 2. Heyman Queuing Equation        7

State Evaluation        7

Derivation of Decision Criteria        8

Rule Base        9

Figure 3. Single Server queueing model with vacation rule base        9

Fuzzy Inference System        10

Figure 4. Fuzzy Inference System GUI Diagram        10

Figure 5. Fuzzy Inference Rule Editor        11

Figure 6. Fuzzy Inference System Rule Viewer        12

Membership functions        13

Figure 7. Membership functions of Accumulated Holding Cost        13

Figure 8. Membership functions of holding cost        14

Figure 9. Membership functions of traffic intensity        15

Figure 10. Membership functions of Output        16

Queueing simulation to obtain a sample performance result        17

Simulation Experiment with Sample data and Results        18

Figure 11. Number of customers and  versus time        19[pic 2]

Figure 12. Number of customers versus time        20

Figure 13. Expected Number of Customers versus Poisson Arrival Rates        21

Conclusion        22

Future Scope        23

Bibliography        24

Appendixes        25

INTRODUCTION

In 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 “real-world” 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.