Recent Title s i n This Serie s 137 F . I . Karpelevic h an d A . Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues . 199 4 136 Masayosh i Miyanishi , Algebrai c geometry . 199 4 135 Masar u Takeuchi , Moder n spherica l functions . 199 4 134 V . V . Prasolov , Problem s an d theorem s i n linea r algebra . 199 4 133 P . I . Naumki n an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n th e theor y o f waves , 199 4 132 Hajim e Urakawa , Calculu s o f variation s an d harmoni c maps . 199 3 131 V . V . Sharko, Function s o n manifolds : Algebrai c an d topologica l aspects . 199 3 130 V . V . Vershinin, Cobordisms an d spectra l sequences . 199 3 129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions . 199 3 128 V . P . Orevkov, Complexit y o f proof s an d thei r transformation s i n axiomati c theories . 199 3 127 F . L. Zak , Tangent s an d secant s o f algebrai c varieties . 199 3 126 M . L . Agranovskil , Invarian t functio n space s o n homogeneou s manifold s o f Li e group s an d applications. 199 3 125 Masayosh i Nagata , Theor y o f commutativ e fields. 199 3 124 Masahis a Adachi , Embedding s an d immersions . 199 3 123 M . A . Akivi s an d B . A . Rosenfeld , Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou , Theory o f entir e an d meromorphi c functions : Deficien t an d asymptoti c values and singula r directions . 199 3 121 LB . Fesenk o an d S. V . Vostokov, Loca l fields and thei r extensions : A constructive approach , 199 3 120 Takeyuk i Hid a an d Masuyuki Hitsuda , Gaussia n processes , 199 3 119 M . V . Karasev an d V . P . Maslov, Nonlinea r Poisso n brackets . Geometr y an d quantization , 199 3 118 Kenkich i Iwasawa , Algebrai c functions , 199 3 117 Bori s Zilber , Uncountabl y categorica l theories , 199 3 116 G . M . Feldman , Arithmeti c o f probabilit y distributions , an d characterizatio n problem s o n abelia n groups. 199 3 115 Nikola i V . Ivanov, Subgroup s o f Teichmiiller modula r groups , 199 2 114 Seiz o ltd , Diffusio n equations . 199 2 113 Michai l Zhitomirskil , Typica l singularitie s o f differentia l 1-form s an d Pfaffia n equations , 199 2 112 S . A . Lomov , Introductio n t o th e genera l theor y o f singula r perturbations , 199 2 111 Simo n Gindikin , Tub e domain s an d th e Cauch y problem , 199 2 110 B . V . Shabat, Introductio n t o comple x analysi s Par t II . Function s o f severa l variables , 199 2 109 Isa o Miyadera , Nonlinea r semigroups . 199 2 108 Take o Yokonuma , Tenso r space s an d exterio r algebra , 199 2 107 B . M. Makarov , M . G . Goluzina , A . A . Lodkin . an d A. N . Podkorytov , Selecte d problem s i n rea l analysis. 199 2 106 G.-C . Wen , Conformal mapping s an d boundar y valu e problems . 199 2 105 D . R. Yafaev , Mathematica l scatterin g theory : Genera l theory . 199 2 104 R . L . Dobrushin , R . Kotecky , an d S . Shlosman , Wulf f construction : A globa l shap e fro m loca l interaction. 199 2 103 A . K . Tsikh , Multidimensiona l residue s an d thei r applications . 199 2 102 A . M. Il'in , Matching o f asymptoti c expansion s o f solution s o f boundar y valu e problems , 199 2 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zhen-xi , Qualitativ e theory o f differentia l equations . 199 2 100 V . L. Popov , Groups, generators , syzygies . an d orbit s i n invarian t theory , 199 2 99 Nori o Shimakura , Partia l differentia l operator s o f ellipti c type . 199 2 98 V . A. Vassiliev , Complement s o f discriminant s o f smoot h maps : Topolog y an d applications , 199 2 (revised edition . 1994 ) 97 Itir o Tamura , Topology o f foliations : A n introduction . 199 2 {Continued in the back of this publication) This page intentionally left blank Heavy Traffic Limit s for Multiphase Queue s This page intentionally left blank 10.1090/mmono/137 Translations o f MATHEMATICAL MONOGRAPHS Volume 13 7 Heavy Traffic Limit s for Multiphase Queue s F. I. Karpelevich A. Ya. Kreinin American Mathematical Societ y 5 Providence , Rhode Island ''IVDB D OpHApnx H3paHjieBH H KapnejieBH H AjieKCaHAP ^KOBJieBH H KpettHH H MHOrOOA30BbIE CHCTEMM MACCOBOrO OBCJiyaCHBAHHf l B PE3KHME BOJIBIHOtt HArPY3KH Translated b y A. Kreinin an d A . Vainstein fro m a n origina l Russia n manuscrip t Translation edite d by Simeo n Ivano v 1991 Mathematics Subject Classification. Primar y 60K25 , 90B17 . ABSTRACT. Thi s boo k i s devote d t o th e analysi s o f severa l type s o f queuein g system s arisin g i n network theor y an d communicatio n theory . Numerou s method s an d result s fro m th e theor y o f stochastic processes are used. Th e main emphasis is made to problems of diffusion approximatio n of stochasti c processe s i n queuein g system s an d t o th e result s base d o n th e application s o f th e hydrodynamic limit method . This book will be useful to researchers working in the theory and applications of queueing theory and the theory o f stochastic processes. Library o f Congress Cataloging-in-Publication Dat a Karpelevich, Fridrikh Izrailevich . [Mnogofazovye sistem y massovogo obsluzhivaniia v rezhime bol'shoi nagruzki . English ] Heavy traffic limit s for multiphase queues/F. I. Karpelevich, A . Ya. Kreinin . p. cm . — (Translations of mathematical monographs, ISSN 0065-9282; v. 137 ) Includes bibliographical references . ISBN 0-8218-4597-7 (acid-free ) 1. Queuing theory. 2 . Stochastic processes. I . Kreinin, A . Ya. II . Title. III . Series. QA274.8.K3713 199 4 94-1241 8 515/.353—dc20 CI P Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g fo r them, are permitted to make fair us e of the material, such as to copy a chapter fo r us e in teaching or research. Permissio n is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment o f the source is given. Republication, systemati c copying, or multiple reproduction o f an y material in this publication (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests for suc h permission shoul d be addressed to the Manager of Editorial Services , American Mathematical Society , P.O . Bo x 6248, Providence, Rhod e Islan d 02940-6248 . Request s ca n als o be made by e-mail to reprint-permissionQmath.ams .org . The owner consents to copying beyond that permitted by Sections 10 7 or 10 8 of the U.S. Copy- right Law , provided tha t a fe e o f $1.0 0 plus $.2 5 pe r pag e fo r eac h cop y be paid directl y t o th e Copyright Clearanc e Center , Inc. , 22 2 Rosewoo d Drive , Danvers , Massachusett s 01923 . Whe n paying this fe e pleas e us e the code 0065-9282/94 t o refe r t o this publication . Thi s consen t doe s not exten d to othe r kind s o f copying , suc h a s copyin g fo r genera l distribution , fo r advertisin g o r promotional purposes , for creatin g new collective works, or for resale. © Copyrigh t 199 4 by the American Mathematical Society . Al l rights reserved. Printed in the United State s of America. The American Mathematical Societ y retains all rights except those granted to the United State s Government . © Th e paper used in this book is acid-free an d fall s within the guidelines established to ensure permanence and durability . O Printe d o n recycled paper. This volume was typeset using A/^-T^i, the American Mathematical Society' s Tg X macro system . 10 9 8 7 6 5 4 3 2 1 9 8 97 96 95 94 CONTENTS Preface i x Frequently Used Notation x i Introduction 1 Chapter I. Processe s with Reflection 1 1 § 1. Th e process X 1 1 §2. Th e process< J 1 5 §3. Th e infinitesimal operato r of the process £ 2 0 §4. Th e dependence of the processes X an d £ on the point x 0 2 2 §5. Th e convergence to the process £ 2 5 §6. Coordinat e supremum fo r the process £ 3 1 §7. Modification s o f the processes X an d £ 3 5 Chapter II. Multiphas e System s 4 3 § 1. Genera l multiphase systems 4 3 §2. Multiphas e queueing systems satisfying th e Kleinrock condition 4 7 §3. Multiphas e Kleinrock systems in heavy traffic 5 0 §4. Multiphas e Kleinrock systems in heavy traffic.