STOCHASTIC DELAYS IN TRANSPORTATION TERMINALS: NEW RESULTS IN THE THEORY AND APPLICATION OF BULK QUEUES by WARREN BUCKLER POWELL B.S.E., Princeton University (1977) S.M., Massachusetts Institute of Technology (1979) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY January, 1981 o WARREN BUCKLER POWELL 1981 Signature of author.................-............................... Department of Civil Engineering Tonuary 27, 1981 Certified by.. -0 6-- ge A*@ Yjsef Sh4i T1si super"s Accepted by......................... Chairman Departmenta Committee on Graduate Students of the Department of Civil Engineering Archivas MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUL 1 1981 LIBRARIES 2 STOCHASTIC DELAYS IN TRANSPORTATION TERMINALS: NEW RESULTS IN THE THEORY AND APPLICATION OF BULK QUEUES by WARREN BUCKLER POWELL Submitted to the Department of Civil Engineering on January 27, 1981 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Civil Engineering ABSTRACT The incorporation of stochastic effects into models of terminal performance is traditionally accomplished through the use of simulation. In an effort to reduce the computational overhead of an analysis based on simulation, this research lays out an approach for studying stochastic delays in terminal operations based on theoretical results from the theory of bulk queues. After formulating different queueing problems as bulk queues, transforms of the queue length and waiting time distributions are derived for bulk arrival queues with service and random batch capacities. Also, the concept of a scheduled departure queue with cancellations is introduced and explored. In most cases, the relevant moment formulas for the mean and variance of both the length of the queue and the waiting time are derived. Following the theoretical work, several numerical problems associated with the application of transforms are addressed. First, it is shown that the task of finding the zeroes of a particular function, required to solve the transforms, does not pose any significant numerical problems, contrary to several recent papers, and can in fact be performed very quickly. Next, it is shown that these zeroes can be used to perform a partial inversion of the queue length transform, but that methods for performing complete numerical inversions are extremely sensitive to roundoff errors. Finally, a method for approximating moments which eliminates the need of solving for zeroes is developed and shown to yield accurate results. The last problem addressed is the validity of the assumption of a Poisson arrival process. It is shown that the Poisson arrival process is not a good approximation for general arrival processes which may in fact appear Poisson. With this in mind, a methodology is outlined for approximating general arrival processes. Thesis supervisor: Yosef Sheffi Title: Professor of Civil Engineering 3 Acknowledgements The successful completion of any substantial piece of work inevitably involves the support of many individuals, and this thesis is no exception. While it is difficult to name veryone who has influenced the research, I would like to acknowledge the efforts of those who helped mold the final product. My first thanks go to my committee, Amedeo Odoni, Dick Larson, Paul Roberts, Pierre Humblet and in particular my chairman, Yossi Sheffi. Outside of a yeoman's job of reading and critiquing some terrible earlier drafts, Yossi has provided an atmosphere of encouragement throughout the research which helped me get through several difficult stages of this research. I will be forever grateful of the many afternoons I spent in his office discussing anything but my thesis. I would like to thank Amedeo for his many comments on several earlier drafts, and Pierre, for his meticulous checking of the theoretical work and for several observations which both simplified and extended the results. Dick Larson also provided valuable comments on the organization of the material in an earlier draft of the thesis. The research presented in this thesis was intentionally confined to the development of analysis techniques from the esoteric theory of bulk queues. The motivation of the research, however, originated in the context of describing large scale, less-than-truckload trucking networks. In this respect, I would like to thank Paul Roberts for providing the "trucking" perspective of the research. I am also indebted to Mike McGee and John Terry of IU International for funding the first year of this research under the IU International Fellowship. This fellowship provided the much needed independence required to initiate such basic research. I would in particular like to thank Mike for setting up several interviews with terminal managers in Ryder and P.I.E., allowing me to see terminal operations first hand and to talk to shippers who use LTL services. No list of acknowledgements would be complete without the list of people who have contributed indirectly, though no less significantly, by providing a congenial working environment. In this respect I would like to thank Hani and Millie, Mike, Cliff, Frank, Brendon, Eve, Vic, Fred, Oscar, Jim and Josue. I would also like to thank Woody Richardson for introducing me to the topic in the first place. Finally, I would like to dedicate this thesis to my wife, Shari, whose enduring support and encouragement through the long haul has given this thesis purpose. 4 TABLE OF CONTENTS TITLE.............................................................. ABSTRACT.............................................................2 ACKNOWLEDGEMENT.......................................................3 TABLE OF CONTENTS.................................................... 4 LIST OF FIGURES............................ .......................... 7 LIST OF TABLES....................................................... 10 CHAPTER 1: Modeling Stochastic Delays in Transportation Terminals.... 11 1.1 Modeling delays in terminals.................................... 14 1.2 Formulating terminals as queues................................. 19 1.2.1 The unloading queue...................................019 1.2.2 The storage queue...................................... 20 1.2.3 The loading queue...................................... 20 1.2.4 Other aspects of loading and unloading queues.......... 23 1.3 Modeling demand.................................................28 1.4 Summary.........................................................30 CHAPTER 2: Review of the Literature on Bulk Queues and Queueing Networks................33 2.1 The bulk queueing literature-....................................34 2.1.1 Single arrival, bulk service queues....................36 2.1.2 Bulk arrival, single service queues.................... 40 2.1.3 Bulk arrival, bulk service queues...................... 42 2.1.4 Numerical methods in bulk queues....................... 43 2.1.5 Summary................................................46 2.2 The queueing network literature................................. 48 2.3 Summary............--------..----...-.... ..................... 51 5 CHAPTER 3: The #4/G/l Scheduled Departure Queue.......... 52 3.1 Notation............................................. 55 3.2 Queue length transforms for the lI/G /l queue..,... ..... 58 3.2.1 The MX/Gc/1 queue without cancellations....... 59 3.2.2 The Mx/GC /l queue with cancellations.........., 70 3.2.3 Scheduled and real-time queues under a go-when-filled policy........................ 77 3.2.4 Vehicles with random capacities.............. 80 3.3 Extensions of the scheduled departure queue........... 88 3.3.1 A light traffic approximation for scheduled departure queues with cancellations........... 89 3.3.2 Scheduled departures with correlated headways. 90 3.3.3 The transform of the size of an arriving group for a queue imbedded in a network..........................93 3.4 Waiting time transforms for bulk arrival, bulk service queues... 98 3.5 Relationship between the length of the queue from different perspectives.........................................107 3.6 The 1#/G/1 service queue.......................................108 3.7 Summary........................................................112 CHAPTER 4: Numerical Analysis of Transforms....... .114 4.1 Finding the zeroes........................... ... 115 4.1.1 Numerical stability................. ...119 4.1.2 Accuracy............................. .. 125 4.1.3 Computational efficiency............ ... 131 4.2 Inverting the transform...................... .. 136 4.3 Approximations for queue length distributions. ... 146 4. 4 Summary. ......... a...... ..........0.0 ..4..0...... .. .158 6 CHAPTER 5: Approximating the Arrival Process for G/Gc/l Queuese. 160 '5.1 Analyzing general arrival processes............................161 5.1.1 The microscopic view.................................164 5.1.2 The macroscopic view.................................177 5.2 A general arrival process in the context of a bulk queue.......187 5.3 Approximations for G/Gc/1 queues..............................191 5.4 Summary........................................................199 CHAPTER 6: Summary and Directions for Further Research...... ......200 6.1 Summary of major results................................ ....... 201 6.2 Directions for further research......................... ......205 REFERENCES. ............................. 207 APPENDIX A Neuts' algorithm.......................................214