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\noindent{\bf F\'elix POLLACZEK}\\
b. 1 December 1892 - d. 29 April 1981
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\noindent{\bf Summary.}
Pollaczek was a pioneer in mathematical modelling of queueing
phenomena. He was a fine scientist and made fundamental contributions to
queueing theory.
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F\'elix Pollaczek was born in 1892 in Vienna and passed
away in at his home in Boulange-Billancourt, France.  In
Vienna he attended the ``Akademie Gymnasium'' (latin grammar
school). In 1910 he enrolled as a student in electrical engineering
at the Technological University Vienna. During the first World War
he served in the Austrian army and reached the rank of lieutenant.
In 1921 he married Hilda Geiringer.  His only child Magda, a
daughter, was born in 1922.  They soon separated.  In 1944 Hilda
became the wife of von Mises (q.v.) [1].  Felix remarried in 1934 to Vera
Jacobowitz. It was a happy marriage, and Vera was a great support
for him.

In 1916 Pollaczek wrote his first paper.  It was on Fermat's (q.v.) Last
Theorem and published in the 1917 proceedings of the Imperial
Vienna Academy of Sciences.  In 1918 he resumed his studes and
obtained his master's degree in 1920 at the Technological
University of Brno, Czechoslovakia.  He then moved to Berlin and
obtained his doctor's degree in 1922 from the University of Berlin
with a thesis on number theory under supervision of I. Shur.

In 1921 he was employed by the A.E.G., a large German electrical
company.  In 1923 he joined the research institute of the German
Post Office.  The Nazi rules forced him, being a Jew, to leave in
1933.  He went to Paris.  Vaulot, also a pioneer in telephone
congestion theory, got him a consultant position with the
``Soci\'et\'e d'\'Etudes pour Liasons T\'el\'ephoniques et
T\'el\'egraphiques". Visa problems caused him to leave in 1936.  He
and Vera returned to Brno, where his mother's family lived.
Khintchine invited him to Russia in 1937.  He spent three months
there, and visited several universities; a professorship was
offered to him by the University of Tiflis.  Difficulties with a
visa and the unattractive political climate in the Soviet Union
made him renounce this position.  When the Germans invaded
Czechoslovakia in 1938, a last minute escape brought F\'elix and
Vera to Paris again.  During 1939/1940 he was consulting with
``Center National de la Recherche Scientique" (CNRS) as ``Ma\^itre
de Recherches".  The war time was hard for them, but they succeeded
in escaping the Nazi persecution of the Jews, see [3].  After the
liberation of France in 1944 Pollaczek again became affiliated with
CNRS. Although already in his fifties, Pollaczek was unbroken.  He
resumed his research, published, visited congresses and accepted
invitations to lecture.  His economic position was rather
difficult, but in the mid fifties a pension from his former German
employer made the life of F\'elix and Vera more relaxed.  In 1947
they obtained French nationality.

Pollaczek's publications are listed in [4].  The list, as composed
by him, is divided into five groups: Number Theory, Mathematical
Analysis, Mathematical Physics, Rational Mechanics and Probability
Theory.  Number Theory contains three papers, the last one dates
from 1929.  Eleven papers on Mathematical Analysis, all published
after 1944, for the greater part they concern orthogonal
polynomials, see also Pollaczek polynomials in [11].  The fourteen
ones on Mathematical Physics concern mainly electro-magnetic
problems arising in the design of transmission lines and cables.
Almost all of them date from Pollaczek's time with the German Post
Office.  The paper in Mechanics is on wave propagation.  Those
listed under Probability Theory form the greater part of his
publications among them two monographs, one on the single server
waiting time model [9] the other on the many server model [10].

Pollaczek is best known for his mean waiting time formula for the
stable M/G/1 queueing model derived in 1930 [5].  This famous
formula is known as the Pollaczek-Khintchine formula.  Khintchine
[6] derived it independently a few years later.

The paper [5] of Pollaczek was the second study in queueing
literature of a queueing model with an arbitrary service time
distribution.  The first one stems from Vaulot [7].  In [7] it is
shown that the stationary distribution of the number of busy
servers for the classical Erlang loss model depends only on the
first moment of the service time distribution and not on any other
characteristic of this distribution; the so called ``insensitivity
property".  Forty years later insensitivity phenomena became an
important research subject in the performance analysis of queueing
networks.  In his paper [7] Vaulot introduced the so-called method
of ``the supplementary variable"; it became an important tool in
analytic Queueing Theory.  Former modelling of queueing situations
used negative exponential distributions for the description of
interarrival- and service time distributions.  With this assumption
an analytic approach, presently known as the birth-and-death
technique, could be applied.  This technique has been extensively
used by E.C. Molina from the Bell System Laboratories (see
[8] for details.

The analysis of the one- and many server waiting time model with
unspecified interarrival- and service time distributions, the
GI/G/s model, became the main theme of Pollazcek's research.  His
monograph [9] discussed the GI/G/1 model, and it is a masterpiece
of classical hard analysis.  The starting point is the relation

$$W_{n+1} = max(0, w_n + \tau _n - \sigma _{n+1})\;.$$

\noindent Here $w_n$ is the waiting time of the nth arriving customer,
 $\tau_n$ his service time and $\sigma_{n+1}$ the time between the
 arrivals of the nth and $(n+1)$th customer.  The equation is a
 specimen of relations encountered in waiting time- and storage models.
 The relaion is transformed into a singular integral equation for the
 generating function

$$\Phi_\omega(\tau,\rho):=\sum^\infty_{n=1}\tau^n{\rm E}\{e^{-\rho
{w}_n}|{w}_1 = \omega\},\;|\tau| < 1,\; {\rm Re}\;\rho\geq 0\;,$$

\noindent of the sequence of Laplace-Stieltjes transforms of the
distributions of $w_n$. The explicit solution of the integral
equation for $\Phi(\tau,\:\rho)$ by means of an integral expression
was a remarkable achievement.  The only assumption introduced is
that of the regularity of the L.S.-transform $\beta(\rho)$ of the
service time distribution $B(t)$ at $\rho = 0$.  It implies
that $B(t)$ should not have too heavy a tail.  The assumption,
however, can be removed and the result can be used for heavy-tailed
distributions $B(.)$ which arise in current modelling.

The technique developed by Pollaczek also leads to explicit results
for busy period and idle time distributions, queue length
distributions and the covariance of the sequence $w_n$ for $n
\rightarrow \infty$ is another important result.  Pollaczek's approach
became a powerful technique for the analysis of a large class of
models in Queueing Theory, Storage, Risk and Dam Theory.  Around
1960 Fluctuation Theory was developed.  It soon appeared that the
probabilistic structure of the GI/G/1 model was closely related to
those studied in Fluctuation Theory, and that many of Pollazcek's
results could be rephrased in terms  of stochastic characteristics
of partial sums of independent random variables.

Pollaczek seems to have been little interested in the stochastic
structure of the processes which he investigated.  His goal was to
provide the designers of telephone systems with sharp tools to
judge the system performance which was measured in terms of
congestion- and delay probabilities, average waiting times and
queue lengths.  His approach was the classical one of Applied
Mathematics.  Make a model which incorporates all the essential
features of the actual engineering problem.  Once an adequate model
had been set-up its analysis is a mathematical problem, which was
solved only if the solution is amenable to numerical evaluation. It
is a point he mentioned in several conversations.  He considered
the analysis of the GI/G/1 model as trivial.  The GI/G/s, $s> 1$,
is the really hard one.  He was never fully satisfied with his
analysis of it, see the monograph [10], and still worked on it in
the late seventies.  Numerical accessibility was the problem, but
rather recently the tools became available.

Pollaczek was a sharp and original mathematician, whose work
strongly influenced analytical Queueing Theory.  In the eighties it
turned out that his approach fits well into the class of
Riemann-Hilbert Boundary Value Problems, a chapter in the theory of
integral equations.  In this context the GI/G/1 is indeed almost
trivial.  However, not so the many server model.

Pollaczek was a kind and modest man, maybe too modest; at
conferences he usually remained in the background.  He was a
learned man, who fostered scientific standards and integrity.  In
1977 he was awarded the John van Neumann Theory Prize by the
Operations Research Society of America.

The reference [3] contains a detailed biography of Pollaczek
composed by Schreiber and Le Gall.  The details about his course of
life are borrowed from their paper.
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\begin{thebibliography}{3}
\bibitem{1}  [1] Siegmund-Schultze, R. (1993). Hilda Geiringer-von Mises, Charlier Series,
Ideology, and the Human Side of the Emancipation of Applied
Mathematics of the University of Berlin during the 1920s. {\it Historia
Mathematica}, {\bf 20}, 364-381.

\bibitem{2}  [2] Special Issue on Teletraffic Theory and Engineering in Memory
of F\'elix Pollaczek, ed. J.W. Cohen, P.J. K\"uhn, F. Schreiber, R.
Syski, AE\"U, {\it International Journal of Electronics and Communication}, {\bf 47}
(1993), issues 5, 6.

\bibitem{3} [3] Schreiber, F., LeGall, P., In Memoriam F\'elix Pollaczek,
see [2] above, 275-281.

\bibitem{4}  [4] Cohen, J.W. (1981). Obituary: F\'elix Pollaczek.
{\it Journal of Applied Probability},
{\bf 18}, 958-963.

\bibitem{5}  [5] Pollaczek, F. (1930). \"Uber eine Aufgabe der Wahrscheinlichkeitstheorie.
{\it Mathematische Zeitschrift}, {\bf 32}, 64-100, 729-750.

\bibitem{6}  [6] Khintchine, A.Y. (1932). Mathematical theory of a stationary queue.
{\it Matematicheski
Sbornik}, {\bf 39}, 73-84.

\bibitem{7}  [7] Vaulot, E. (1927). Extension des formules d'Erlang au cas
o\`u les dur\'ees des conversations suivent une loie quelconque,
{\it R\'evue G\'en\'erale de l'Electricit\'e}, {\bf 22}, 1164-1171.

\bibitem{8}  [8] Cohen, J.W. (1986). Some samples of modelling. In: {\it
The Craft of
Probabilistic Modelling}, ed. J. Gani, Springer-Verlag, New York,
89-108.

\bibitem{9}  [9] Pollaczek, F. (1957). {\it Probl\`emes Stochastiques pos\'es par le
ph\'enom\`ene de formation d'une queue d'attente \`a un guichet et
par des ph\'enom\`enes apparant\'es}. Gauthier Villars, Paris.

\bibitem{10}  [10] Pollaczek, F. (1961). {\it Th\'eorie Analytique des 
Probl\`emes Stochastiques
relatif \`a un group de Lignes t\'el\'ephoniques avec dispositif
d'attente}. Gauthier Villars, Paris.

\bibitem{11}  [11] Erd\'elyi, A. et.al. (1953). {\it Higher 
Transcedental Functions}, McGraw-Hill
Book Company, New York.
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\hfill{J.W. Cohen}

\end{thebibliography}

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