## Multi-class queues and stochastic networks – LNMB Fall 2016

Course information on LNMB website

### Course description:

Complex stochastic systems, like communication systems, computer networks and manufacturing systems, may often be modeled as queueing networks with multiple nodes and/or multiple classes. The performance of these systems may be evaluated in terms of queue lengths, sojourn times or blocking probabilities. This course focuses on basic queueing networks for which performance measures can be obtained in closed form. First, the course focuses on a class of networks where the equilibrium distribution has a so-called product-form solution. Topics include the output theorem, reversibility, partial balance, quasi reversibility and product-form. Examples include Jackson networks, Kelly-Whittle networks, BCMP networks, loss networks and processor sharing networks. Second, the course considers the sojourn time distribution in simple networks. Third, computation of performance measures often requires effcient algorithms. To this end, Mean Value Analysis and approximation techniques will be studied. Finally, fluid queues will be addressed.

### Detailed content:

reversibility, stationarity, basic queues, output theorem, feedforward networks - partial balance, Jackson network, Kelly-Whittle netwerk, arrival theorem - quasi-reversibility, customer types, BCMP networks, bandwidth sharing networks - blocking, aggregation, decomposition - loss networks, insensitivity via supplementary variables - sojourn time distribution in networks - MVA, AMVA, QNA - fluid queues, basic models - feedback fluid queues, networks of fluid queues

### Literature:

- R. Nelson, Probability, Stochastic Processes and Queueing Theory, 1995
- F.P. Kelly, Reversibility and Stochastic Networks, Wiley, 1979 (available on-line)
- R.W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, 1989.
- R.J. Boucherie, N.M. van Dijk (editors), Queueing Networks – A Fundamental Approach, International Series in Operations Research and Management Science Vol 154, Springer, 2011 [link]
- Handouts, slides and references to relevant additional literature will be made available at the lectures.

*Prerequisites:*
The participants should have followed courses in probability theory, stochastic processes and queueing theory.*Examination:*
Take home problems.

*Address of the lectureres:*

Prof.dr. R.J. Boucherie
Stochastic Operations Research; Department of Applied Mathematics; Faculty of Electrical Engineering, Mathematics, and Computer Science; University of Twente,
P.O. Box 217 NL-7500 AE Enschede
Phone: 053-4893432 Email: r.j.boucherie@utwente.nl

Dr.ir. W.R.W. Scheinhardt Stochastic Operations Research; Department of Applied Mathematics; Faculty of Electrical Engineering, Mathematics, and Computer Science; University of Twente, P.O. Box 217 NL-7500 AE Enschede Phone: 053-4893832 Email: w.r.w.scheinhardt@utwente.nl

### Slides:

- slides lecture 1 Boucherie
- slides lecture 2 Boucherie
- slides lecture 3 Boucherie
- slides lecture 4 Boucherie
- slides lecture 5 Boucherie
- slides lecture 6 Boucherie
- slides lecture FQ-1 Scheinhardt
- slides lecture FQ-2 Scheinhardt

### Exercises:

- exercise set 1 Boucherie
- exercise set 2 Boucherie
- exercise set Scheinhardt-6
- exercise set Scheinhardt-7
- exercise set Scheinhardt-8+9

Due date exercises part Boucherie:

- provide explicit proofs of the results, i.e., you cannot state that the result is “by analogy with” result in book.
- hand in as single pdf file: December 2, 2016
- marked by: January 6, 2017

Due date exercises part Scheinhardt:

- hand in as single pdf file: December 16, 2016
- marked by: January 20, 2017