**Félix Pollaczek**'s parents were Alfred Pollaczek (1862-1923) and Marie Gomperz (1867-1923). It was a well-respected Jewish family with Alfred being a local councillor and an inspector on the Austrian railways. In fact the Pollaczek family had a tradition of working for the Austrian railways with Alfred's father, Samuel Pollaczek, being General Inspector on the Austrian railways. Alfred and Marie were married in 1891 and Félix was the eldest of their two children; he had a younger brother Gustav Pollaczek (1895-1966). We shall say a little more about Gustav later in this biography but let us record at this point that his first occupation was in the family tradition of the Austrian railways.

Félix attended the Akademie Gymnasium, the Latin High School, in Vienna. This school, the oldest and most famous in Vienna, was founded in 1553 and had been situated in the Beethovenplatz since 1866. Erwin Schrödinger graduated from the school four years before Pollaczek who graduated at the age of seventeen in 1910. In that year he entered the Technical University of Vienna, enrolling as a student of electrical engineering. The political situation in 1914 rapidly deteriorated and by August of that year Austria-Hungary was allied with Germany and at war with Russia, France and Great Britain. Pollaczek served in the Austrian army during World War I. By the end of the war he had reached the rank of lieutenant. However, he had time to think deeply about mathematics and he wrote his first paper in 1916 while serving in the army. This paper on Fermat's Last Theorem, *Über den grossen Fermatschen Satz* Ⓣ, was his first contribution to number theory and was published in 1917.

After World War I ended in 1918, Pollaczek went to Brno, Czechoslovakia, where he studied for a Master's Degree in electrical engineering. He was awarded the degree in 1920 and, following this, he went to Berlin where he undertook research for his doctorate in mathematics. In 1921 he married Hilda Geiringer who, like Pollaczek, was born in Vienna. She had obtained her doctorate in mathematics in that city working under Wilhelm Wirtinger and had moved to Berlin in 1921 to become Richard Von Mises' assistant in the Institute of Applied Mathematics. Pollaczek's thesis advisor was Issai Schur and he undertook research on number theory, working in an area close to the work he had undertaken on Fermat's Last Theorem during the war. He submitted his thesis *Über die Kreiskörper der l-ten und l ^{2}-ten Einheitswurzeln* Ⓣ in 1922 and was awarded his doctorate. In this thesis he studied fields generated by the

*p*th and

*p*

^{2}th roots of one, where

*p*is an irregular prime number. He published a paper based on the work of his thesis in 1924. The year 1922 when he was awarded his doctorate was also the year that Félix and Hilda's daughter Magda was born. However, the marriage was not successful and they soon separated, and eventually they were divorced. Before his thesis was submitted, Pollaczek had taken a job in 1921 as an engineer with A.E.G. [Allgemeine Elektricitäts Gesellschaft] in Berlin. He only worked for this major German electrical company, which was responsible for installing Germany's first electrical power system, for two years before being appointed as a scientific adviser to the German Postal, Telephone and Telegraph Services at Reichspost-Zentralamt Berlin-Tempelhof.

Pollaczek began publishing papers on mathematical physics, particularly on electrical engineering, the first two being *Das Einschaltproblem für das homogene Kabel bei beliebiger Einschaltung* Ⓣ (1924) and *Theorie der Einschaltvorgänge des vielgliedrigen künstlichen Kabels* Ⓣ (1925). He published three further papers in this area in 1926, and continued publishing on this topic over the following years. However, his interests were broad and in 1929 he published* Über die Fortpflanzung mechanischer Vorgänge in einen linearen Gitter* Ⓣ on rational mechanics, and *Über die Einheiten relativ-abelscher Zahlkörper* Ⓣ on number theory. In the following year he published two papers on probability theory. It is one of these papers *Über eine Aufgabe der Wahrscheinlichkeitstheorie* Ⓣ published in *Mathematische Zeitschrift* which contains the result for which he is best known today, the Pollaczek-Khinchin formula. This formula for the mean waiting time of a queueing model with an arbitrary service time distribution was also derived by the Russian mathematician Aleksandr Yakovlevich Khinchin in 1932.

If political events in Germany had turned out differently, it seems likely that Pollaczek would have continued to work for the German Postal, Telephone and Telegraph Services, and publish research articles on a wide range of mathematical topics. However, when Hitler became Chancellor of Germany in 1933, he immediately announced legal actions against Germany's Jews. On 7^{th} April 1933, Hitler introduced a law for the "Restoration of the civil service". This meant that all non-Aryans and Jewish civil servants were dismissed from their positions with the exception of those who either had fought in the Great War or had been in office since August 1914. Pollaczek was dismissed from his position in the German Postal, Telephone and Telegraph Services as a consequence of these laws. He realised that he would have to get out of Germany if he was to survive so he went to Paris. Pollaczek's last paper written in German appeared in 1934. He wrote all his subsequent papers and books in French (except for two papers written very late in his life which he wrote in English).

Émile Vaulot, chief engineer at the Poste, Téléphone et Télécommunications (the French Post Office) and a lecturer at the École Polytechnique, was an expert on telephone engineering who had also worked on queueing models with an arbitrary service time distribution. He was able to obtain a position for Pollaczek as a consultant for the Société d'Études pour Liasons Téléphoniques et Télégraphiques. Pollaczek married Vera Jacobowitz in 1934; unlike his first marriage, this marriage was very successful and Vera proved a great support to her husband through the following difficult years. As tensions rose between the countries of Europe, Pollaczek's position in France became more difficult and by 1936 he could not get his visa renewed to continue to work there. Accompanied by his wife, he went to Brno, Czechoslovakia, where members of his mother's family were living.** **Aleksandr Yakovlevich Khinchin, who was working on the theory of stationary random processes, had interests similar to those of Pollaczek and had found the same Pollaczek-Khinchin formula a couple of years after Pollaczek. Khinchin invited Pollaczek to the Soviet Union. After working in Moscow, Khinchin spent the two years 1935-37 at Saratov University before returning to Moscow. Pollaczek spent three months in the Soviet Union in 1937, visiting Khinchin and making trips to a number of universities. He made a very favourable impression and was offered a professorship at the University of Tiflis in Georgia. This was a tempting offer as Tiflis was a major cultural and educational centre. In addition to the university there were several other higher education institutions, and many research establishments. However, working there presented visa difficulties similar to those he had experienced in France and, in addition, the Soviet political system did not appeal to him. He therefore returned to Brno.

The situation in Europe deteriorated further during 1938. In March of that year German troops marched into Austria which was declared part of Germany. Even more worrying for Pollaczek was Hitler's announcement in May that he would destroy Czechoslovakia. Attempts to appease Hitler over the following months led to the Sudetenland becoming part of Germany and German troops entered areas close to Brno in October. Pollaczek was one of 12,000 Jews in Brno who heard of the increasing anti-Semitic acts throughout Germany with Jewish businesses smashed in November. On 15 March 1939 German tanks entered Brno and two days later Hitler visited Brno to emphasise his conquest. Pollaczek had seen what was coming and he had escaped with his wife just before the German troops arrived. They made their way to Paris and at first it seemed that he had made a good move for he was appointed Maître de Recherches at the Centre National de la Recherche Scientifique. Let us note here that the Pollaczeks did well in leaving Brno for less that 1,000 of the 12,000 Jews there survived the war.

World War II began on 1 September 1939 when German forces entered Poland. On the following day, Britain, France and several other countries, declared war on Germany but over the following months France was not involved in any fighting, but spent time trying to build defences to protect the country from an invasion by Germany. During this time Pollaczek continued to work in Paris. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders and, by June, France had surrendered and fighting had ended. Pollaczek and his wife, as Jews, were again in great danger and there was no way he could continue working for the Centre National de la Recherche Scientifique. The Pollaczeks spent the war years avoiding persecution by the Nazis. They succeeded in surviving the war but suffered years of hardship. In August 1944 Paris was liberated and Pollaczek was able to begin working again for the Centre National de la Recherche Scientifique. He had lost nearly twelve of what should have been the most productive years of his life. However, he launched himself back into research and in 1946 published the number theory paper *Relations entre les dérivées logarithmiques de Kummer et les logarithmes π-adiques* Ⓣ as well as three papers on probability *Sur I'application de la théorie de fonctions au calcul de certaines probabilités continues utilisées dans la theorie des réseaux téléphoniques* Ⓣ, *La loi d'attente des appels téléphoniques* Ⓣ, and *Sur un problème du calcul des probabilités qui se rapporte a la téléphonie* Ⓣ. An application for French citizenship from Pollaczek and his wife was granted in 1947. He did not have a good income over the following ten years but, after that, he received a pension from the German Postal, Telephone and Telegraph Services which made him financially more comfortable.

We promised at the start of this article to give some further details of Félix Pollaczek's brother Gustav. Gustav published works such as *Grundzüge des französischen Eisenbahn-Frachtrechts* Ⓣ (1935), *International Legislation in the Field of Transportation* (1944), *Rebuilding the European transportation system* (1945), and *German Transportation and Communication Systems* (1946). He had moved to the United States and was involved in the American labour conference on international affairs which considered postwar reconstruction. He was described in a 1946 publication as follows: Doctor of laws, University of Vienna; former official, Austrian state railways; formerly consultant, Foreign Economic Administration; now in Office of Economic Security Policy, Department of State.

During the ten years from 1945 on, Félix Pollaczek worked on yet another mathematical topic, this time mathematical analysis. In this area his main interests were in orthogonal polynomials and as well as working on Legendre polynomials, Hermite polynomials and Laguerre polynomials, he introduced polynomials named by Arthur Erdélyi as the 'Pollaczek polynomials'. He lectured during these years at the Institut de Statistique at the Faculty of Science in Paris. For example a course he gave in the Institute in 1956 was published as *Application de la théorie des probabilités à des problèmes posés par l'encombrement des réseaux téléphoniques* Ⓣ (1959). The published material contains a:-

In 1957 Pollaczek published the monograph... treatment in telephony of the simplest many-server system - the "fully accessible" case in which any idle server may attend any service demand. Both loss(busy signal)and delay systems are examined.

*Problèmes stochastiques posés par le phénomène de formation d'une queue d'attente à un guichet et par des phénomènes apparentés*Ⓣ. John Riordan writes in a review:-

Pollaczek also published the bookThis is a thorough study of problems associated with a single server stochastic system. These problems are mainly those of delay when a waiting line is permitted, service being in order of arrival without defections from the waiting line. The case, arising in air traffic, where a delayed arrival in joining service suffers an additional(random)delay due to this very fact, as well as the usual case where joining service is instantaneous, is considered and is modified later to permit only quantized delays, integral multiples of some appropriate time unit. But also the problem of the distribution of the "busy period", and the problem of loss when arrivals finding the server busy are dismissed(without effect on future arrivals), appear in their proper places. ... The treatment throughout employs complex variable theory and to an expert like the author may be regarded, as he says, as a set of exercises in the elementary part of that theory,(which should test the skill of most novices). This of course conditions the generality of the various distribution functions in question but, as the author says, in a way not serious for any of the known applications.

*Théorie analytique des problèmes stochastiques relatifs à un groupe de lignes téléphoniques avec dispositif d'attente*Ⓣ (1961). He described his methods in queueing theory in a letter to Ryszard Syski which is reproduced in [7]:-

Pollaczek received a major award in 1977 from the Operations Research Society of America and The Institute of Management Sciences. This is described by the probabilist Jacob Willem Cohen (1923-2000) in [1]:-Some of the guiding ideas of the theory outlined in my paper may be approximately stated as follows. Under the rule of strict order servicing of an s server system with given initial conditions, all quantities we are interested in, like waiting times, busy periods, etc., are definite functions of a great number of stochastic variables of given distributions, and of some parameters signifying given initial values. Therefore all unknown expressions for probabilities and distribution functions are in fact Lebesgue-Stieltjes integrals, the integrands of which are formed by means of the aforesaid definite functions. In my theory the task of carrying out these integrations is reduced to the problem of resolving one or several systems of s linear non-homogeneous integro-functional equations of a new kind. The circumstances that this theory employs uniquely analytic methods and dispenses with all resources of classical Probability Calculus probably accounts for the fact that hitherto my methods have been employed by nobody save myself.

Let us end by quoting again from [1]:-In1977Pollaczek was awarded the John von Neumann Theory Prize by the ORSA-TIMS prize committee. Although he had always enjoyed good health, his age then prevented him from travelling to the United States to receive this prize in person. At the country house of the hospitable Le Gall family in Bous-le-Roi, France, the prize was presented to him during a short ceremony in the presence of Pierre Le Gall, Ryszard Syski and myself. Pollaczek felt very honoured by this award from the American Operations Research community.

Pollaczek made basic contributions in pointing to the direction in which the analytic approach must develop. In him queueing theory had a fine and sharp scientist. His modest and kind personality will be missed by his friends.

**Article by:** *J J O'Connor* and *E F Robertson*