Tai Lieu Chat Luong BOUNDARY VALUE PROBLEMS IN QUEUEING SYSTEM ANALYSIS This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICSSTUDIES BOUNDARY VALUE PROBLEMS IN QUEUEING SYSTEM ANALYSIS J W COHEN 0.J BOXMA Department ofhlathematics State University of Utrecht Utrecht, The Netherlands 1983 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK * OXFORD 79 0North-Holland Publishing Company 1983 All rights reserved No part of this publication may he reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner ISBN: 444 86567 Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 VANDERBILT AVENUE NEW YORK, N.Y 10017 Library of Congress Cataloging in Publication Data Cohen, Jacob Willem Boundary value problems in queueing system analysis (North-Holland mathematics studies ; 79) Includes bibliographical references and index Boundary value problems Random w U k s (Mathematics) Queuing theory I Boxma, J., 195211 Title 111 Series QA379.c63 1983 519.8' 82-24589 ISBN 0-444-86567-5 (U S ) PRINTED IN THE NETHERLANDS to Annette and Jopie This Page Intentionally Left Blank Vii PREFACE The present monograph is the outcome of a research project concerning the analysis of random walks and queueing systems with a two-dimensional state space It started around 1978 At that time only a few studies concerning such models were available in literature, and a general approach did not yet exist The authors have succeeded in developing an analytic technique which seems to be very promising for the analysis of a large class of two-dimensional models, and the numerical evaluation of the analytic results so obtained can be effectuated rather easily The authors are very much indebted to F.M Elbertsen for his careful reading of the manuscript and his contributions to the numerical calculations Many thanks are also due to P van de Caste1 and G.J.K Regterschot for their assistance in some of the calculations in part IV, and to Mrs Jacqueline Vermey for her help in typing the manuscript Utrecht, 1982 J.W.Cohen 0.J Boxma Viii NOTE ON NOTATIONS AND REFERENCING Throughout the text, all symbols indicating stochastic variables are underlined The symbol ":= " stands for the defining equality sign References to formulas are given according to the following rule A reference to, say, relation (3.1) (the first numbered relation of section 3) in chapter of part I is denoted by (3.1) in that chapter, by (2.3.1) in another chapter of part I and by (1.2.3.1) in another part A similar rule applies for references to sections, theorems, etc CONTENTS ‘Preface Note on Notations and Referencing GENERAL INTRODUCTION Vii viii PART I INTRODUCTION TO BOUNDARY VALUE PROBLEMS I SINGULAR INTEGRALS Introduction Smooth arcs and contours The Holder condition The Cauchy integral The singular Cauchy integral 1.1.6 Limiting values of the Cauchy integral 1.1.7 The basic boundary value problem 1.1.8 The basic singular integral equation 1.1.9 Conditions for analytic continuation of a function given on the boundary 1.1 lo Derivatives of singular integrals 1.1.1 1.1.2 I 1.3 I 1.4 1.1.5 1.2 THE RIEMANN BOUNDARY VALUE PROBLEM 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.3 Formulation of the problem The index of G(t), t E L The homogeneous problem The nonhomogeneous problem Avariant of the boundary value problem (1.2) 19 19 22 25 26 27 31 34 35 36 38 39 39 40 41 45 48 THE RIEMANN-HILBERT BOUNDARY VALUE PROBLEM 50 1.3.1 1.3.2 1.3.3 1.3.4 50 52 55 56 60 1.3.5, Formulation of the problem The Dirichlet problem Boundary value problem with a pole Regularizing factor Solution of the Riemann-Hilbert problem Aspects of numerical analysis 392 IV.2.4 IV.2.4 Numerical results For the sake of comparison with the results of chapter we have again chosen, cf (1.6.1), for the LST’s of the service time distributions of the two types of customers As observed in remark 1.1, only three of the six examples of section 1.6 have been considered, i.e., those corresponding to tables 1.6.1, 1.6.3, 1.6.6 The contour L has been plotted in figure 14 for the first of these three cases (see also figure 13 for plots of the corresponding S1 and S2) Our experiments with the three cases suggest that L differs more from the unit circle (the exchangeable case) as r1-r2 grows (and also the iteration procedure appears to converge slower) The computer programs were written in Fortran and the calculations were performed on a Cyber 175 computer The values in the three cases obtained for oi(0), ai(l), oi(l) and E{x(~)} under consideration are listed in table 4.1, together with the values obtained for these cases in chapter The agreement is generally good However, a drawback of the method of the present chapter appears to be a sensitivity of a (11, u,(l) and in particular of o;(l) for the correct values of the p ( ) function determining L (see also the discussion concerning the behaviour of H(5) near 5-1 in (2.12) and (2.131)- This effect could explain the differences between the values found here and those in the preceding chapter The agreement between oi(1) and mean queue length values is generally still excellent, due to the insensitivity of these quantities for small values of r2 (see 1V.2.4 The alternating sentee discipline - A random walk approach 393 the conclusions in section 1.6; see also the case with r2=4/9) Remark 4.1 The total CPU time involved in calculating the exact results f o r one particular example did not exceed seconds Each iteration of p(.) and ( ) demanded 0.6 seconds CPU time Summarizing, the numerical experiments o f this chapter show that it is indeed possible to obtain accurate numerical values f o r quantities related to the two-dimensional random walk stud- ied in chapter 11.3 For the mere purpose Gf studying the alter- nating service model the direct approach o f chapter is better suited In particular the numerical analysis of the complex singular integral equation (11.3.6.15) is considerably more complicated than that of Theodorsen's integral equation (compare also the statements in remark 1.6,l and remark 4.1) I Figure 14 The contour L ( ) f o r the-case of table 1.6.1: X = 0.44, r = lO/ll, B unit circle) ( - denotes the Aspects of numerical analysis 394 IV.2.4 TABLE Comparison o f results obtained via the methods of the present chapter and of chapter , for the cases: A X 0.44, r1 X 0.44, rl = 5/6 C X = , r1 = 5/9 B lO/ll, , , 8, = (table ) , R2 = (table , , = (table ) ~ Case A Case B Case C chap chap chap chap chap chap 0.04911 0,04925 0.09117 0.09128 0.16323 0.16325 0.51089 0.51075 0.46883 0.46872 0.38677 0.38675 0.10523 0.10424 0.20045 0.19960 0.59097 0.59094 1.74159 1.74060 1.53378 1.53293 0.81319 0,81316 0.05563 0.05509 0.10951 lo948 0.49974 0.50012 1.49610 1.49621 1.27042 1,27043 0,46110 0.46079 0.05047 0.05059 0.09542 0.09543 0.60032 0.60029 0.56763 0.56750 0.52704 0.52703 0.24909 0.24939 395 REFERENCES EVGRAFOV, M.A., Analytic Functions (Dover Publ., New York, 1966) TITCHMARSH, E.C., The Theory of Functions (Oxford Univ Press, London, 1952) NEHARI, Z., Conformal Mapping Dover Publ., New York, 1975) ZABREYKO, P.P., KOSHELEV, A.I., KRASNOSEL'SKII, M.A., MICHLIN, S.G., RAKOVSHCHIK, L.S , STET'SENKO, V.Ya., Integral Equations - A Reference Text (Noordhoff Int Publ., Leyden, 1975) , s , Singu1;ire Integraloperatoren MICHLIN , s G & P R ~ S S D O R F (Akademie Verlag, Berlin, 1980) GAKHOV, F.D., Boundary Value Problems (Pergamon Press, Oxford, 1966) MUSKHELISHVILI, N.I., Singular Integral Equations (Noordhoff, Groningen, 1953) GOLUSIN, G.M., Geometrische Funktionentheorie (V.E.B Deutscher Verlag der Wissensch., Berlin, 1957) GAIER, D., Konstruktive Methoden der Konformen Abbildung (Springer Verlag, Berlin, 1964) 10 KOBER, H., Dictionary of Conformal Representations (Dover Publ., New York, 1957) 11 BECKENBACH, E.F (ed.), Construction and Application of Conformal'Maps (Nat Bureau of Standards, Appl Math Series 18, Washington D.C., 1952) 12 SPITZER, F., Principles of Random Walk (Van Nostrand, Princeton, 1964) 13 LUKACS, E., Characteristic Functions (Griffin, London, 1960) 14 LOEVE, M., Probability Theory (Van Nostrand, Princeton, 1960) 15 COHEN, J.W & BOXMA, O.J., The M/G/l queue with alternating service formulated as a Riemann-Hilbert problem In: Performance '81, Proc of the 8th Intern, Symp., ed F J Kylstra, p 181-199, North-Holland Publ Co , Amsterdam, 1981 For more details see also Report 177, Math Inst Univ of Utrecht, 1980 References 396 16 BLANC, J.P.C., Application of the Theory of Boundary Value Problems i n the Analysis of a Queueing Model with Paired Services (Mathematical Centre Tract 153, Amsterdam, 1982) 17 KINGMAN, J.F.C., Two similar queues in parallel Ann.Math Statist (19611 1314-1323 18 FAYOLLE, G & IASNOGORODSKI, R., Two coupled processors: the reduction to a Riemann-Hilhert problem Wahrsch Verw Gebiete 47 (1979) 325-351 19 FLATTO, L & McKEAN, H.P., Two queues in parallel Corn Pure Appl Math 30 (1977) 255-263 20 FAYOLLE, G., Me'thodes Analytiques pour les Files d'Attente Couple'es (Thesis, Univ de Paris VI, Paris, 1979) 21 IASNOGORODSKI, R., Problsme-Frontibes dans les Files d'Attente (Thesis, Univ de Paris VI, Paris, 1979) 22 COHEN, J.W., The Single Server Queue (North-Holland Publ Co., Amsterdam, 1982 - n d edition) 23 VEKUA, N.P., Systems of Singular Integral Equations (Noordhoff, Groningen, 1967) 24 IVANOV, V.V., The Theory of Approximate Methods and their Applications to the Numerical Solution of Singular Integral Equations (Noordhoff Int Publ., Leyden, 1976) 25 ERDELYI, A., MAGNUS, W., OBERHETTINGER, F., TRICOMI, F.G., Higher Transcendental Functions Vol I1 (McGraw-Hill, New York, 1953) 26 HANCOCK, H., Theory of Elliptic Functions (Dover Publ., New York, 1958) 27 KONHEIM, A.G., MEILIJSON, I., MELKMAN, A., Processor sharing of two parallel lines J Appl Probab 18 (1981) 952-956 28 NOBLE, B., Methods Based on the Wiener-Hopf Technique (Pergamon Press, London, 1958) 29 POLLACZEK, F The'orie Analytique des Problsmes Stochastiques Relatifs un Groupe de Lignes Tglkphoniques avec Dispositif d'Attente (Gauthier Villars, Paris, 1961) References 391 30 DE SMIT, J.H.A., The queue GI/M/s with customers of different types or the queue GI/Hm/s To appear in Adv in Appl Probab 31 NEUTS, M.F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (Johns Hopkins Univ Press, Baltimore, 1981) 32 KIEFER, J & WOLFOWITZ, J., On the theory of queues with many servers Trans Amer Math SOC (1955) 1-18 33 MARCUS, M & MINC, H., A Survey of Matrix Theory and Matrix Inequalities (Allyn and Bacon, Boston, 1964) 34 HOKSTAD, P., On the steady-state solution of the M/G/2 queue Adv in Appl Probab 21 (1979) 240-255 35 COHEN, J.W., On the M/G/2 queueing model Stoch Process Appl 12 (1982) 231-248 - 36 EISENBERG, M., Two queues with alternating service SIAM J Appl Math 36 (1979) 287-303 37 GOHBERG, I & GOLDBERG, S., Basic Operator Theory (Birkh:user Verlag, Basel, 1981) 38 HCBNER, 0., Zur Numerik der Theodorsenschen Integralgleichung in der Konformen Abbildung (Mitteilungen Math Seminar Giessen, Heft 140, Giessen, 1979) 39 MELAMED, B., On Markov jump processes imbedded at jump epochs and their queueing-theoretic applications Math Oper Res (1982) 111-128 40 WHITTAKER, E.T & WATSON, G.N., A Course of Modern Analysis (Cambridge Univ Press, London, 1946) 41 DELVES, L.M & WALSH, J (eds.), Numerical Solution of Integral Equations (Clarendon Press, Oxford, 1974) 42 SMITH, N.M.H., On the Distribution of Queueing Time for Queues with Two Servers (Thesis, Austr Nat Univ., 1969) 43 KUHN, P.J , Private communication This Page Intentionally Left Blank 399 SUBJECT INDEX A Alternating service discipline, 12, 271, 345, 377 Analytic, 26 - arc, 23 - continuation, 36, 110 - contour, 23 Aperiodicity, s e e strong aperiodicity Arc length, 22 Argument principle, 132 B Birth- and death process, 242 Boundary value problem, 7, 34 - with a pole, 5 - with a shift, 164 simultaneous -, 250 Dirichlet -, s e e Dirichlet Riemann -, s e e Riemann Riemann-Hilbert -, s e e Riemann-Hilbert Wiener-Hopf -, s e e Wiener-Hop€ Busy period, Tpl , - of an M / G / I queue, 300, 328 - of an M/M/1 queue, 246 residual - of an M/G/I queue, 299 400 Subject index C Caratheodory's theorem, 135 Cauchy integral, singular - , 27 Cauchy principal value, 19, 27 Cauchy-Riemann conditions, 52 Complex integral equation, s e e integral equation (simultaneous) Component random walk, s e e random walk Conformal mapping, 8, 63, 65 Conformally equivalent, 66 Connected set, , 111 Corresponding boundaries principle, 67 Corresponding boundaries theorem, 66 Counterclockwise, 22 Coupled processors , 13, 288 D Derivative of singular integral, s e e singular integral Dirichlet problem, 52, 225, 278 Domain, 64 Exchangeable, 85 F Fan's theorem, 341 Fredholm integral equation, s e e integral equation Function element, 110 Sabject index H Harmonic, Holder c o n d i t i o n , , H6lder c o n s t a n t , , Holder i n d e x , , 25 Homogeneous random w a l k , s e e random walk I m p l i c i t f u n c t i o n t h e o r e m , 109 Index, 40 Indicator function, 79 Inhomogeneous random w a l k , s e e random walk I n t e g r a l equation Fredholm -, 165 -, 176, 197, 209, 382 T h e o d o r s e n ' s , 72, 96, 220, 236, 346 simultaneous J J a c o b i e l l i p t i c f u n c t i o n , [ s n ] , 251 J o r d a n a r c , 22 Jordan contour, K K e l l o g g Is t h e o r e m , Kernel, 3¶ 81, 87, 132, 153, 158, , 274, 297 L L i o u v i l l e ' s theorem, 34 401 402 Subject index Lower s h e e t , s e e s h e e t Lyapounov c o n t o u r , 68 M Many s e r v e r q u e u e , 319 Maximum modulus t h e o r e m , Maximum p r i n c i p l e , Meromorphic, M / G / q u e u e , , 296, 317 M/G/2 q u e u e , 14, 319 N N e a r l y c i r c u l a r a p p r o x i m a t i o n , , 279, 354 Non-recurrent, Null-recurrent, 83 83 Number o f c u s t o m e r s p r e s e n t i n t h e s y s t e m , s e e q u e u e l e n g t h Numerical e v a l u a t i o n , - of c o n f o r m a l mapping, 345 of s i n g u l a r i n t e g r a l e q u a t i o n s , 345 P P h a s e t y p e method, 320 P i e c e w i s e s m o o t h , 23 P l e m e l j - S o k h o t s k i f o r m u l a ( s ) , , 33 Poisson formula, 54 P o i s s o n k e r n e l , , 214, 275 P o s i t i v e r e c u r r e n t , 83, P r i n c i p a l v a l u e , s e e Cauchy p r i n c i p a l v a l u e P r i n c i p l e of c o r r e s p o n d i n g b o u n d a r i e s , s e e c o r r e s p o n d i n g boundaries P r i n c i p l e of p e r m a n e n c e , 112 Subject index Priwalow’s theorem, 208 PS formula(s), s e e Plemelj-Sokhotski formula(s) Q Queue length - at time t, alternating service discipline, 347 queues in parallel, 242 - immediately after departure, [-n z I, alternating service discipline, 271, 347 mean -, alternating service discipline, 347 Queues in parallel, 11, 241 [xtl, R Random walk, component -, 82 homogeneous -, 378 inhomogeneous - , 379 two-dimensional -, 77, 377 symmetric two-dimensional -, 85 general two-dimensional -, 151 Rectifiable, 22 Regular at a point, 26, 64 Regular in a domain, 64 Regularizing factor, 56 Return time, 129 Riemann boundary value problem, 4, 39, 102, 148, 181, 197, 200, 208, 210, 384 homogeneous -, 41 inhomogeneous -, 45 Riemann-Hilbert boundary value problem, 50, 225, 252 homogeneous - , 60 403 404 Subject index inhomogeneous -, Riemann mapping t h e o r e m , 6 Rouch6's t h e o r e m , S Schwarz f o r m u l a , 53 Sheet, l o w e r s h e e t , 256 u p p e r s h e e t , 256 Simultaneous i n t e g r a l e q u a t i o n s , see i n t e g r a l e q u a t i o n Simply c o n n e c t e d c u r v e , 23 Simply c o n n e c t e d s e t , 6 S i n g u l a r Cauchy i n t e g r a l , s e e Cauchy i n t e g r a l Singular integral, 19 d e r i v a t i v e of -, 38 S i n g u l a r i n t e g r a l e q u a t i o n , 35 Skipfree, S l i t , 218, 275 Smooth a r c , 2 Smooth c o n t o u r , 23 Sn-function, 251 Starshaped c o n t o u r , 72, 157 Strong a p e r i o d i c i t y , 86 T Theodorsen's i n t e g r a l e q u a t i o n , s e e i n t e g r a l e q u a t i o n numerical a n a l y s i s o f -, 349 Theodorsen's procedure, 70 Two-dimensional random w a l k , s e e random walk 405 U Uniformisation, 11 Univalent a t a p o i n t , U n i v a l e n t i n a domain, Upper s h e e t , s e e s h e e t W W a i t i n g t i m e i n M/G/2, [w], 337 Wiener-Hopf boundary v a l u e p r o b l e m , 302 Wiener-Hopf decomposition, Workload, 9 , 326 [xt , - o f c o u p l e d p r o c e s s o r model, - o f M/G/2 q u e u e , This Page Intentionally Left Blank