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University Microfilms International 8612388 L ee, H ow oo A DIFFUSION APPROXIMATION FOR MULTI-SERVER FINITE-CAPACITY BULK QUEUES The Ohio State University Ph.D. 1986 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1986 by Lee, Howoo All Rights Reserved A DIFFUSION APPROXIMATION FOR MULTI-SERVER FINITE-CAPACITY BULK QUEUES Presented in partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Howoo Lee, B.S., M.S. ***** The Ohio State University 1 986 Reading Committee: Approved by Dr. Gordon M. Clark Dr. Walter C. Giffin Dr. Ramesh C. Srivastava Adviser Department of Industrial and Systems Engineering Copyright by Howoo Lee 1986 This dissertation is dedicated to my parents. - ii - ACKNOWLEDGEMENTS I wish to express my deepest appreciation to my mother and the deceased father. Without their care, enthusiasm for higher education and financial support, my graduate study would have been unsuccessful. I also wish to express my appreciation to my academic advisor, Dr. Gordon Clark, for his understanding and patience. His insight and guidance have greatly influenced me throughout my graduate education. I wish to acknowledge Dr. Walter Giffin who opened the door for me to the world of queueing theory. His constructive criticism always brought my poor engineering sense to my attention. Dr. John Neuhardt deserves my sincere thanks for haveing provided me with the opportunity to work as a research associate and the financial support when it was most needed. I also wish to thank Dr. Ramesh Srivastava for serving on my graduate study committee. Dr. Barry Nelson deserves my appreciation for his help with the variance reduction techniques. To my beloved wife, Yoosook, I owe my success. She has encouraged me everytime I needed it and endured those years with patience and understanding. My children, Sahngyool and Youngsun, deserve my special thanks. I always felt like a super man when they rushed to kiss me after a hard day's work. Sahngyool's most frequent question, "Are you going to school this evening, too?", was always more difficult to answer than the - iii - General Exam, questions. He always made me aware of the fact that besides a student, I am a father. - iv - VITA January 26, 1954 .......... Born: Wonju, Korea 1976 ....................... B.S., Industrial Engineering, Seoul National University, Seoul, Korea 1982 ....................... M.S., Industrial and Systems Engineering, The Ohio State University, Columbus, Ohio 1982 - 1983................ Research Associate, The Ohio state University 1983 - 1985................ Teaching Associate, The Ohio State University FIELDS OF STUDY Studies in Applied Stochastic Processes Dr. Gordon M. Clark, Department of Industrial and Systems Engineering, The Ohio State University Dr. Water C. Giffin, Department of Industrial and Systems Engineering, The Ohio State University Studies in Mathematical theory of Stochastic Processes Dr. Ramesh Srivastava, Department of Statistics, The Ohio State University Studies in Applied Statistics Dr. John B. Neuhardt, Department of Industrial and Systems Engineering, The Ohio State University - v - TABLE OF CONTENTS ACKNOWLEDGEMENTS .............................................. iii VITA .......................................................... iv NOTATIONS .................................................... viii LIST OF TAB L E S................................................ X LIST OF F I G U R E S .............................................. xi Chapter Page I. INTRODUCTION ................................................... 1 1.1 Introduction ................................................. 1 1.2 Motivation................................................... 2 1.3 Research objectives ......................................... 5 1.4 Examples of bulk queueing systems............................ 6 1.5 Outline and scope of this rese a r c h .......................... 9 1.6 Organization of the dissertation...............................11 II. LITERATURE REVIEW................................................. 12 2.1 Approximation techniques in queueing theory.................... 12 2.2 Literature review on diffusion approximation ............... 18 2.2.1 Approximation by solving Kolmogorov equations .......... 20 2.2.2 Diffusion approximation by weak convergence .......... 30 2.2.3 Diffusion approximation by stochastic differential equations................................................... 32 2.3 Conclusion of literature review.................................33 III. DEVELOPMENT OF DIFFUSION APPROXIMATION ....................... 35 3.1 Diffusion process .......................................... 35 3.1.1 D e f i n i t i o n ................................................. 35 3.1.2 Forward Kolmogorov Equation (Fokker-Planck equation). 37 3.2 Elementary Return Process (ERP)........... .................. 38 3.3 Interpretation in the queueing context .................. 43 3.4 Expressing infinitesimal mean and variance by queueing parameters..................................................... 44 3.5 Solutions of the ERP equations.................................51 3.6 Discretization................................................. 65 - vi - IV. ANALYSIS OF SYSTEM SIZE DISTRIBUTION .......................... 68 4.1 Proof of the legitimacy of the system size distribution. 68 4.2 Mean and variance of system s i z e .............................. 78 4.3 Derivation of infinite capacity distribution ............... 87 4.4 M/M/1 /k queue as a special case................................ 91 4.5 M /M/1 queue as a special c a s e ................................. 94 V. ACCURACY EVALUATION ......................................... 96 5.1 Performance measures ........................................ 96 5.2 Distributions of interarrival and service times.................99 5.3 Distributions of arrival and service size..................... 103 5.4 Comparison of MX/MX/l/k queue..................................103 5.5 Comparison with simulation estimates for multiple server cases.......................................................... 115 5.5.1 Accuracy of simulation estimates ....................... 115 5.5.2 Comparison of MX/MY/2/k queues.......................... 118 5.5.3 Comparison of MX/GY/2/k queues.......................... 124 5.5.4 Comparison of GI /M /2/k q u e u e s ........................ 128 5.5.5 Comparison of GIX/GY/2/k queues ...................... 132 5.5.6 Comparison of queues with different variabilities .... 136 5.5.7 Comparison of queues with ultra-heavy traffic ....... 138 5.5.8 Comparison of queues with many s e r v e r s ............... 140 5.6 Bulk queue vs. non-bulk queue.................................. 141 VI. CONCLUSIONS .................................................... 146 6.1 Research summary ............................................. 146 6.2 Suggestions for further study .............................. 149 6.2.1 infinitesimal mean and va r i a n c e........................... 149 6.2.2 Mean holding t i m e ......................................... 150 6.2.3 Waiting time................................................151 APPENDIX ............................................................... 152 Little's formula for bulk queues ........................... 152 Simulation program coding..................................... 159 FORTRAN program coding ...................................... 166 BIBLIOGRAPHY .......................................................... 171 - vii - NOTATIONS Pr(.) : Probability of an event X(t) : Diffusion process, or state of the diffusion process at time t f(.) : Density of the diffusion process Ca : Variability of the