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continuation on page 411
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Kiinzi
Operations Research
171
G. F. Newell
Approximate Behavior
of Tandem Queues
Spri nger-Verlag
Berlin Heidelberg New York 1979
Editorial Board
H. Albach' A. V. Balakrishnan' M. Beckmann (Managing Editor)
P. Dhrymes . J. Green' W. Hildenbrand' W. Krelle
H. P. Kunzi (Managing Editor) . K. Ritter' R. Sato . H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. Kunzi
Brown University Universitat Zurich
Providence, RI 02912/USA 8090 Zurich/Schweiz
Author
Gordon F. Newell
Institute of Transportation Studies
University of California
Berkeley, CA 94720/USA
ISBN-13:978-3-540-09552-1 e-ISBN-13:978-3-642-46410-2
001: 10.1007/978-3-642-46410-2
AMS Subject Classifications (1980): 60K25, 90 B22, 90-02
Library of Congress Cataloging in Publication Data
Newell, Gordon Frank, 1925-
Approximate behavior of tandem queues.
(Lecture notes in economics and mathematical systems; 171 : Operations research)
Bibliography: p.
Includes index.
1. Queuing theory. I. Title. II. Series: Lecture notes in economics and mathematical
systems; 171.T57.9.N486 519.8'279-20953
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© by Springer-Verlag Berlin Heidelberg 1979
2142/3140-543210
Preface
The following monograph deals with the approximate stochastic behavior
of a system consisting of a sequence of servers in series with finite storage
between consecutive servers. The methods employ deterministic queueing and
diffusion approximations which are valid under conditions in which the storages
and the queue lengths are typically large compared with 1. One can disregard
the fact that the customer counts must be integer valued and treat the queue
as if it were a (stochastic) continuous fluid. In these approximations, it
is not necessary to describe the detailed probability distribution of service
times; it suffices simply to specify the rate of service and the variance rate
(the variance of the number served per unit time).
Specifically, customers are considered to originate from an infinite
reservoir. They first pass through a server with service rate ~O' vari
ance rate ~O' into a storage of finite capacity cl . They then pass
through a server with service rate ~l' variance rate ~l' into a storage
of capacity c2 ' etc., until finally, after passing through an nth server,
they go into an infinite reservoir (disappear). If any jth storage become
full j = 1, 2, , n , the service at the j-lth server is interrupted
and, of course, if a jth storage becomes empty the jth server is inter
rupted; otherwise, services work at their maximum rate.
Equivalently one could have a system of servers 1, 2, ••. ,n fed
by an arrival process of rate ~O ' variance rate ~O. If the storage cl
is full, the arrival process is considered either to be interrupted or to
throw any excess arrivals that cannot enter the full storage out of the system
("lost call") or back into the infinite source.
IV
The properties of the system are described in terms of the random
in which D. (t) is the cumulative
J
number of customers to pass the jth server by time t starting from some
initial state at time 0 •
Chapter I first describes the general formulation of the problem. The
deterministic approximation (~j 0) is then analysed leading to an ex-
plicit evaluation of the D. (t) starting from an arbitrary initial state
J
and for arbitrary choices of the ~j and n. For sufficiently large t
all servers will serve at the rate ~ .. min~. of the "bottleneck." For
J
~j > 0, the stochastic properties of the Dj(t) are described in terms
of their time-dependent joint probability density. This probability den-
sity is shown to satisfy (approximately) a diffusion equation in n+l space
variables. plus time. The density must, in addition, satisfy certain bound-
ary conditions when one or more of the storages is either empty or full.
Various general properties of the system of equations, such as overall
service rate, marginal queue length distributions, etc., are described.
Subsequent chapters will deal with solutions of the equations in whatever
special cases one can obtain solutions in some ~nageable form.
Chapter II will deal with the case n = I a single queue. The
analysis is for arbitrary choices of the ~O' ~O' cI ' ~l' ~l and
describes the time-dependent behavior of the joint distribution of DO(t) ,
DI(t) from an arbitrary initial state. In contrast with previous treat
ments of a single server system, this analysis describes both the input
and the output or equivalently the queue length and the output. In partic-
ular, the results give an explicit formula for the equilibrium service rate
~ and the equilibrium variance rate of the output as a function of the
storage capacity cI (and the ~O' ~O' ~l' ~l)'
v
Chapters III and IV will deal with equilibrium queue distributions for
n = Z in the special cases ~O = ~l = ~Z ' but arbitrary cl ' Cz ' ~O ' ~l
and ~Z. The joint probability density of the two queue lengths satisfies
a diffusion equation inside a rectangle (sides cl ' cZ) in a two-dimensional
space, but a linear transformation of coordinates will map the equilibrium
distribution into a solution of Laplace's equation in a parallelogram (sub-
ject to an unconventional type of boundary condition). These equations are
solved through a series of conformal mappings which eventually yield a solu-
tion in parametric form. Chapter III describes the formal solution. Chapter
IV gives numerical evaluations of the marginal queue length distributions and
the dependence of service rate ~ and
Chapter V deals with the time-dependent properties of the joint proba-
bility distributions of the cumulative departures DO(t) , DI(t) ,DZ(t) past
servers 0, 1, and Z for a two-server system with infinite storage (c = c =(0)
I Z
and equal variance coefficients ~O = ~l = ~Z· It gives general solutions for
this joint distribution starting from any initial state DO(O) , Dl (0) , DZ(O)
or initial distribtuion of states. This is derived by image methods, but the
solution requires multiple reflections over several boundaries and gives a
rather unwieldy formula containing six terms, each of which involves some
multiple integrals. Although the methods used can be generalized to more
servers, the conclusion of this chapter is little more than "it can be done."
The general results seem to be too clumsy to be of much practical use.
Chapters VI and VII employ Laplace Transform and techniques similar
to the Wiener-Hopf factorization to derive the joint equilibrium queue
distribution for the two-server system with very large (00) storages cl '
Cz but general service rates ~O' ~l ,~Z and variance coefficients
VI
Chapter VI discusses the case ~O < ~l ' ~2' i.e., an
input rate ~O less than the service rates of servers I and 2. The
joint distribution of QI' Q2 and particularly the marginal distribution
of Q2 are analyzed in some detail for the special case ~O = ~2 = 0 ,
~l > 0 (regular input, regular server at 2). Properties of the queue
distribution for general ~j are described but explicit solutions are
obtained only for a few other special choices of the ~. , including
J
~l = 0 , ~O' ~2 > 0; ~O = ~l' ~2 > 0; and ~O 0 ~l = ~2 . Com-
parisons are made between the distributions of Q2 for ~O < ~l' ~2
and corresponding systems from Chapters III, IV with ci < 00 c2 = 00
= = but with the chosen so that the input rate to server 2
~o ~l ~2
in the latter system is equal to the ~O of the former.
Chapter VII deals with the case of arbitrarily large but
~l < ~O' ~2' i.e., server I is a bottleneck. One is concerned here
with the joint distribution of the number of vacant storage spaces Q'
I
ci - QI upstream of server I and the queue Q2 downstream. The mar-
ginal distributions of QI' and Q2 for cI' c2 + 00 are known to be
exponential (for arbitrary ~O' ~2 > ~l and , ~.J IS) so the main emphasis
is on the statistical dependences between particularly the
QI' Q2'
asymptotic properties for large These distributions are then
used to estimate the effect of finite but large values of and
The main conclusion is that the reductions in the overall service rate
due to fini te and to finite are nearly additive.
Chapters III, IV, VI, and VII all deal with equilibrium queue dis-
tributions for a system with an input server followed by two other servers
in tandem. The queues QI' Q2 behind servers I and 2 depend upon the
service rates ~O' ~l' ~2' variance rates ~O' ~l' ~2 and the storages
VII
cl ' c2 • No practical analytic method was found for evaluating these dis-
tributions accurately if are finite and the are different.
Chapters III, IV deal with ~O = ~l = ~2 and cl ' c2 < 00 giving special
attention to the analytic singularities at corners of the state space, and
the blocking effects. Chapters VI and VII deal with ~O # ~l # ~2 but
cl = c2 = 00 , giving special attention to the effect of different ~j on
the shape of the queue distribution. By comparing the effects of different
~j and the effects of finite cl ' c2 ' one can, however, infer how the queue
distributions vary qualitatively with all the parameters. This is probably
all that one would want from an analytic formulation anyway, since, for any
specific choice of the parameters, one could evaluate the distribution by
simulation.
Chapter VIII, Epilogue, is a commentary on how one can (usually) analyze
a real tandem queueing system with many servers by identifying a critical
server or a critical pair of interacting servers with finite storage. There
is also a discussion of where the problem now stands and what techniques
are likely to produce further advances.
The research described in this monograph was supported in part by the
National Science Foundation under a series of grants entitled, "Application
of Mathematics to Transportation Studies." The work was done over a time
span from 1974 to 1978 and was previously distributed as Research Report
UCB-ITS-RR-78-3 and UCB-ITS-RR-77-l9. The typing of the manuscript was done
by Inta Vodopals.
Contents
I. General Theory
1. Introduction 1
2. Graphical Representations and
Deterministic Approximation 5
3. Motion of Holes 11
4. Diffusion Equation 15
5. Queue Length Distribution 27
6. Soft Boundaries 31
7. Moments 37
References 42
II. A Single Server
1. Diffusion Equation 43
2. Queue Dis tribution 44
3. Service Rates 47
4. Longtime Behavior of the Joint Dis tributions 49
5. Service Variances 62
6. Image Solution cl 00 67
7. Longtime Behavior cl = 00 78
8. Discussion 86
III. Equilibrium Queue Distributions
Two Servers, ~O = ~l = ~2' Theory
1. Introduction 90
2. Formulation 93
3. Conformal Mappings 106
4. Marginal Distributions 110
5. Symmetry 121
x
6. Saddle Points and Singularities 124
7. One Large Storage 134
8. Expansions of the Marginal Distributions 139
References 148
IV. Equilibrium Queue Distributions,
Two Servers ~O = ~l = ~2' Numerical Results
1. Introduction 149
2. Marginal Distributions for c2 = 00 150
3. Relation between c*l ' c*2 and wI' w3 171
4. Marginal Distributions c*l ' c*2 < 00 178
5. The Service Rate 189
6. Joint Distributions 208
v.
Time-dependent Solutions ~O ~l
1. Introduction 218
2. Image Solution 219
3. Time-dependent Queue Distribution 229
VI. Laplace Transform Methods,
Equilibrium Queue Distributions for n
1. Analysis of Transforms 231
2. Equilibrium Distributions cl c2 00 , ~O ~2 0 241
3. Numerical Evaluations 248
4. Equilibrium Distributions cl c2 00 262
5. Other Special Cases 283
6. Interpretation 304
VII. Equilibrium Queue Distributions; n 2·,
~l < ~O' ~2; cl ' c2 ->-00
1. In troduc tion 313
2. Joint Distribution for ~O = ~2 = 0 316
3. Joint Distribution for ~O' ~2 > 0 331
