UCN@Sophia Labex Seminar - Uri Yechiali : "Tandem Jackson networks asymmetric inclusion processes and catalan numbers"

Uri Yechiali - Tel Aviv University
Communication systems

Date: May 4th 2017
Location: Eurecom - Eurecom

Abstract: The Tandem Jackson Network is a system of n sites (queues) in series, where single particles (customers, jobs, packets, etc.) move, one by one and uni-directionally, from one site to the next until they leave the system (think, for example, on a production line, or on a line in a cafeteria). When each site is an M/M/1 queue, the Tandem Jackson Network is famous for its product-form solution of the multi-dimensional distribution function of the sites' queue-sizes. In contrast, the Asymmetric Inclusion Process (ASIP) (introduced recently by Reuveni, Eliazar and Yechiali) is a series of n Markovian queues (sites), each with unbounded capacity, but with unlimited-size batch service. That is, when service is completed at site k, all particles present there move simultaneously to site k+1, and form a cluster with the particles present in the latter site (think, for example, on cars travelling on a single-lane road, forming sets of convoys, where a faster convoy joins the tail of its preceding slower one, and the two convoys continue moving as a single larger convoy). We analyze the ASIP and show that its multi-dimensional Probability Generating Function (PGF) does not posses a product-form solution. We then present a method to calculate this PGF. We further show that homogeneous systems are ‘optimal’ and derive limit laws (when the number of sites becomes large) for various variables (e.g. busy period, draining time, etc.). Considering the occupancies of the sites (‘queue sizes’) we show that occupation probabilities in the ASIP obey a discrete two-dimensional boundary-value problem. Solving this problem we find explicit expressions for the probability that site k is occupied by m particles (m=0,1,2,…). Catalan's numbers are shown to naturally arise in the context of these occupation probabilities. Speaker's bio: Uri Yechiali received his Ph.D. in Operations Research from Columbia University, New York, in 1969, and was appointed an Assistant Professor at NYU. He joined the Department of Statistics and Operation Research of Tel Aviv University as a Senior Lecturer in 1971 and promoted to Full Professor in 1981. Over the years he has held visiting positions in New York and Columbia Universities, INRIA (France) and EURANDOM (The Netherlands). Professor Yechiali serves on the editorial boards of three international scientific journals and has given numerous invited lectures at scientific meetings on various topics, mainly in the areas of Queueing Theory, Performance Evaluation of Communication Networks, and Reliability. During his academic career he has supervised 38 M.Sc. (with thesis) and 9 Ph.D. students, most of whom now hold prestigious positions in leading academic institutions. Prof. Yechiali served as a consultant to AT&T Bell Labs and to various Israeli companies and government ministries, including BEZEQ and the Ministry of Defense, on problems related to Teletraffic Analysis, Reliability issues, Statistics and Optimization. He has published close to 100 research papers, mostly in the areas of Queueing Theory, Performance Evaluation, Reliability and Operations Research, and received various grants for joint research projects with European institutes. Professor Yechiali served as the President of TIMS Israeli Chapter during 1976-1981 and as the President of ORSIS (Operations Research Society of Israel) during 1983-1985. He held the position of Chairman of the Department of Statistics and Operations Research in the School of Mathematical Sciences of Tel Aviv University during 1981-1983 and during 1996-1999. In 2004 he received the ORSIS LIFE ACHIEVEMENT AWARD for his excellent scientific achievements and broad activities in Operations Research.

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