The M/M/∞Service System with Ranked Servers in Heavy Traffic

141 Pages·1984·3.912 MB·English

by  G. F. Newell (auth.)

Lecture Notes in Economics and Mathematical Systems
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Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
231
G. F. Newell
The MIMloo Service System
with Ranked Servers
in Heavy Traffic
With a Preface by Franz Ferschl
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Dhrymes
G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor) H.P. Kunzi
G.L. Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fUr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
G. F. Newell
Professor of Transportation Engineering and Operations Reserach
University of California, 416C McLaughlin, Berkeley, CA 94720, USA
ISBN-13: 978-3-540-13377-3 e-ISBN-13: 978-3-642-45576-6
001: 10.1007/978-3-642-45576-6
Library of Congress Cataloging in Publication Data. Newell, G. F. (Gordon Frank), 1925-The
MIMI 00 service system with ranked servers in heavy traffic. (Lecture notes in economics and
mathematical systems; 231) Includes index. 1. Queuing theory. I. Title. II. Series. T57.9.N494
1984 519.8'2 84-14091
ISBN-13: 978-3-540-13377-3
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© by Springer-Verlag Berlin Heidelberg 1984
2142/3140-543210
PREFJl.CE
by Franz Ferschl
In trying to place and appreciate this contribution of
Gordon F. Newell, one must first settle the question: What can
be considered a "solution" to a problem in queuing theory. Even
a casual survey of the literature suggests a variety of demands
that have been made and--more rarely--have been satisfied. A few
examples may be given to indicate the range of answers.
At one end of the scale and as a methodological starting
point one finds the existence theorems for equilibrium solutions
e.g., by Lindley (1'952) for GI/G/l, by Kiefer and Wolfowitz
(1955) for GI/G/S and by Loynes (1964) for tandem queues, to
mention just three examples. Especially the last two. examples
show that contributions which to mathematicians are indeed
fundamental, may have little to offer that is useful to a
practitioner. He "knows" anyway that equilibrium distributions
must exist--but what do they look like? One step further in the
direction of greater concreteness are solutions in the form of
Laplace transforms or of other generating functions, sometimes
given only as solutions to certain equations. Even though these
solutions are still behind the "Laplacian curtain" they can be
very useful in practice particularly when one is interested only
in moments of low order, such as expected value and variance.
From other quarters one can hear, however, "down with Rouche"
where we remind the reader that Rouche's theorem is the principal
tool in the theory of functions that is used to pin down the
initial conditions for solutions in terms of generating
functions. The next step is the search for algorithms for
obtaining numerical solutions, of the type that is contained in
the voluminous tables of Hillier and Oliver (1981) for the models
Em/En/S. One can next look for rapidly converging algorithms, a
specialty of the school of ~.F. Neuts.
Of course, the ideal of a solution is and remains a closed
form solution in terms of an explicit formula containing the
interesting variables, preferably as elementary functions of the
IV
model parameters. By the way it is just such a formula which
forms the basis of Chapter 2 of this book. Incidentally this is
one of the oldest successful modelsj it was treated already by
A.K. Erlang (1912) in the context of the fundamental system
M/M/s/s which is basic for the planning of telephone systems.
There is no way to surpass the simplicity and elegance of
this solution: the probability that j out of s channels are
occupied is given by the jth term of the Poisson distribution
truncated at s
-a'-j ,
.
p(j,s) ]
as
1 + a + • •• + s.I
"Despite the formal simplicity of "p(j,s) ••• " (p. 7) Newell
perceives the need for and possibility of approximations. Thus
for large a and s this formula is rather cumbersome (" ••• it
is not in a form which can be easily interpreted" p. 7).
The key word "approximation methods" opens one's eyes to a
variety of methods, approaches, and ambitions. One knows that
exact and at the same time practical solutions are rare in
queuing theory and therefore one feels the need for approxima
tions. Approximations can be brought about through modification
of the assumptions in the model. This is the oldest way, first
tried by A.K. Erlangj as everyone knows this consists in the use
of phases of exponentially distributed length, in order to
describe service times or times between successive arrivals, a
method which has been further refined in the hands of M.F. Neuts
and his school. A second approach championed, among others, by
J.F.C. Kingman attempts approximations by going to the limit
within certain classes of models. These are the well-known heavy
traffic and light traffic approximations. They bring to the
surface the particular role of the exponential and geometric
distributions under more general conditions.
v
A.third method of approach which is particularly well
adapted to the needs of traffic scientists are approximations
which are generated by a particularly large number of customers
(travellers). This is the proper domain of Professor Newell.
His work on deterministic flow models as well as approximations
of the second order which rest on diffusion approximations have
been fundamental. As far as the description of the behavior of
systems with large number of customers, or for that matter with
large dimensions, is concerned, diffusion models are among the
strongest tools suitable for obtaining practical results. One
attempt to use his method to solve the otherwise untractable
tandem problem has been published by Newell as a monograph in
this series (Volume 171: Approximate Behavior of Tandem Queues).
There are also approximation methods aimed at evaluation of
known formulae in order to render them more transparent and
applicable in practice. This last category includes the present
research report. The M/M/s/s model which was mentioned above
is here refined and generalized 1) by allowing the possibility of
infinitely many parallel channels, and 2) by introducing a
preference order among these channels according to which the most
preferred are occupied first by arriving customers.
In 1937 Kosten solved the question of the common
distribution of the number of occupied channels when two groups
are formed, namely, the group of primary and that of secondary
service channels, by developing expressions which are obtained by
evaluating certain generating function. The above mentioned
probability p(j,s) is a simple special case. Here, once more,
the question of a practically useful well-interpretable
approximations is posed and solved.
It cannot be the purpose of this preface to discuss a
detailed list of the solutions that are offered by the author.
All we can do is to give a fortaste of what is to come by means
of one example which I might characterize as a qualitative but
substantive description. Suppose that a very large (practically
infinite) number of parallel service lines is given which may be
VI
interpreted as parking spaces. These carry a preference order
since parking spaces "close" to the destination--for instance,
the entrance of a supermarket--are preferred. How can we
describe the pattern of the free parking spaces in the primary
group, e.g., among the s best spaces? That depends, of course,
on the traffic intensity a, that is, the average number of
arrivals per unit time, a~. For a» I a first approximation
shows "that it is (nearly) equally likely that the best available
space is anywhere in the sea of mostly occupied sites" (po 9)
and" there is an average of only about one idle server among
the first a/2 servers." Suppose now that s is fixed (at a
sufficiently large value) and consider different traffic
intensities or rather their standardized deviations from s,
that is, K = {s-a)/Ia between K» I and -K» 1. The
behavior of the system is now described by a variety of
qualitative statements--derived exactly from the implicit
formulae by sometimes rather lengthy derivations. This is the
stuff that this book is made of, no short summary can give an
adequate idea of all aspects of this work.
I can promise interesting results to the practitioner and a
fascinating book to the expert. My hope is that G.F. Newell will
some day pick up some other "unsolved" problem of queuing theory,
treat it in his own inimitable way and thereby give us further
stimulation through practically meaningful and at the same time
theoretically sound research.
(Translated by Martin J. Beckmann)
PREFACE
We are concerned here with a service facility consisting of a large (in-
finite) number of servers in parallel. The service times for all servers are
identical, but there is a preferential ordering of the servers. Each newly
arriving customer enters the lowest ranked available server and remains there
until his service is completed.
It is assumed that customers arrive according to a Poisson process of rate
A , that all servers have exponentially distributed service times with rate ~
and that a = A/~ is large compared with 1. Generally, we are concerned with
the stochastic properties of the random function N(s ,t) describing the number
of busy servers among the first s ordered servers at time t.
Most of the analysis is motivated by special applications of this model to
telephone traffic. If one has a brunk line with s primary channels, but a
large number (00) of secondary (overflow) channels, each newly arriving customer
is assigned to one of the primary channels if any are free; otherwise, he is
assigned to a secondary channel. The primary and secondary channels themselves
could have a preferential ordering. For some purposes, it is convenient to
imagine that they did even if an ordering is irrelevant.
The evaluation of the (marginal) distribution of N(s ,t) for a single
value of s is the classic "lost call" problem. Any customer who arrives
and finds the first s servers busy is "lost" to the primary servers. Whether
he is routed to other servers or goes away is irrelevant to the behavior of
N(s ,t) itself. The equilibrium distribution of N(s ,t) is the famous
Erlang distribution (1912).
The more challenging problem is to analyze the behavior of the secondary
(overflow) channels, i.e., the properties of N (t) = N(oo , t) - N(s , t) and/or
s
the joint properties of this and N(S , t); or, more generally, the joint proper
ties of N(s1 ' t) , N(s2 ' t) ••• for arbitrary values of s1 • s2 ' ••.
VIII
"Exact" solutions for the equilibrium distribution of Ns(t) and the
joint distribution of Ns(t) , N(s ,t) were obtained by Kosten (1937) using
rather complicated generating function techniques, but the solution is in
such a form as to be virtually useless for numerical evaluation with a» 1
(the ~ase of most practical interest). His solution actually gives the factorial
moments of Ns (t) , or the conditional factorial moments of Ns (t) given N(s, t) ,
but even these are rather awkward to evaluate.
The bulk of the following analysis deals with the asymptotic properties of
these distributions, particularly in the limit a, s + 00 with K = (s - a)/IS
finite (of order 1).
Section 1, Introduction, reviews the known properties of this system and
various applications. Section 2 discusses the limit properties for a» 1
of the Erlang and related distributions. Section 3 gives a qualitative de-
scription of the time-dependent behavior of the number of busy secondary servers
NS(t) and the number Ni(t) of the idle primary servers, particularly for
K »1 or -K» 1
Section 4, the longest section, describes various analytic procedures that
can be used to obtain asymptotic formulas for the equilibrium factorial moments
of Ns(t) , from Kosten's formulas, and how these moments can be inverted to
obtain approximations to the distribution of N (t) Cases with K» 1,
s
JKJ « 1, and -K» 1 must be treated separately. Much of the complication
here arises from the fact that the continuum approximation to the distribution
of Ns(t) has a singularity at the origin.
Section 5 extends the methods of Section 4 to evaluate asymptotic equilibrium
joint distributions of Ni(t) and Ns(t) . One can see qualitatively the approxi-
mate shape of these distributions,although detailed numerical calculations are
tedious (and uninteresting).
For K of order 1 the continuum approMimation to the joint distributions
of Ni (t). Ns (t) satisfies a (time-dependent) diffusion equation. General
IX
properties of the diffusion equation and its boundary conditions are discussed
in section 6. Some time-dependent solutions are described in section 7, in-
eluding the complete solution for K = o. Section 8 describes some of the
mathematical complexities of the equilibrium equation that lead to the un-
pleasant solutions analyzed in section 4 and 5.
Finally, section 9 discusses the application of the formulas of section 4
to one of the simplest network type problems: a single group of secondary
channels serving the overflow traffic from several independent groups of primary
servers. It is shown here that one of the commonly used methods of approximation,
the so-called "equivalent random method:' may, in some cases, give poor estimates
of the overflow distribution.
The research reported here was supported in part by the National Science
Foundation under a sequence of grants entitled, "Application of Mathematics
to Transportation Studies," Mes 80-07393, 81-02457, and 82-05607. It started
in the summer of 1980 with some quick successes, including the simple solution
for the equilibrium distribution for K = 0 and for -K» 1. The anticipated
quick analysis of the whole range of K, however, took three more summers to
complete.
It was my intent that the analysis of the problems considered here would
be followed by an analysis of other joint distributions of N(s. , t) and of
J
some (simple) applications to a variety of telephone network problems. After
four years, however, most of my enthusiasm has waned. There are some simple
and intuitive results buried in the jungle of formulas; indeed most of the
complication arises in the transition between K» 1 and -K» 1 which,
for IS» 1 , represents a relatively narrow range of traffic intensity
a - s + o(~) • That the details of the transition may be of minor practical
importance was overshadowed by my determination to meet the challenge. The