Landelijk Netwerk Mathematische Besliskunde
Course AsQT: Asymptotic Methods in Queueing Theory
Time: 
Monday 11.00  12.45 (September 9  November 11).

Location: 
Hans Freudenthalgebouw, Room 611AB, Budapestlaan, Utrecht (De Uithof) 
Lecturer : 
Prof.dr. A.P. Zwart (CWI / TU/e) 
Course description:
Exact analysis of complex queueing systems is often out of scope. For many queueing systems it is all but impossible to obtain exact expressions for expected values of performance measures such as queue lengths, waiting times and sojourn time. Also, average values may not even be the most informative measures to describe a system's performace, but one may rather be interested in performance quantiles for example. For such cases a wide range of asymptotic techniques are available that may serve to develop suitable approximations and provide valuable insights. In this course we will discuss several such techniques and illustrate them on more advanced queueing models such as GPS queues, DPS queues, and bandwidthsharing networks. The following techniques and topics will be discussed:
 Large deviations and tail asymptotics: We discuss several techniques to estimate tail probabilities in queueing systems. We distinguish two intrinsically different scenarios: one in which performance characteristics have light tailed distributions and one with heavy tails. We will explain the fundamental differences between these two scenarios ("conspiracy" versus "disaster" scenarios) and illustrate several analysis techniques that one may resort to in obtaining asymptotically accurate estimates, including analytic asymptotics, probabilistic bounds and coupling arguments.
 Fluid and diffusion limits: For optimization of complex stochastic processes, one may search for simpler versions of the processes that are still accurate enough to design meaningful optimizing control strategies. Fluid and diffusion limits are particularly useful in this context. For the fluid limit, one starts off the stochastic process (for example a queue length process) at an exceptional high level x and monitors it over a long period of time (order x). As the scaling parameter x tends to infinity, the stochastic process can often be shown to satisfy a functional strong law of large numbers, which is commonly referred to as the fluid limit. In applications, the fluid limit may not give sufficient information to design optimal control strategies and one will typically be interested in deviations from the fluid limit. The diffusion limit describes these deviations.
 Perturbation analysis and timescale separation: analyzing Markovian queueing networks as multidimensional Markov processes may be notoriously difficult. One abstraction is to isolate the behavior of a single queue, and capture the influence of other queues in what is called the random environment. The state of the random environment determines the transition laws of the queueing system at hand. As the random environment changes state, the queue can move from one mode of operation to another (for example from lightly loaded conditions to overloaded conditions and back). When the state changes of the random environment occur on a much faster time scale than the queueing dynamics, one obtains a socalled fluid approximation (this is a somewhat different notion than the earlier mentioned fluid limits). On the contrary, if the state changes are extremely slow the limiting process is called a quasistationary approximation. This concept of timescale separation can be formalized using perturbation analysis for Markov processes.
 Heavy traffic: For efficiency, in practice service systems are aimed at being deployed at fairly high loads. As the load on a (queueing) system approaches the critical capacity, typical performance characteristics such as queue lengths and sojourn times grow beyond limits. In the 1960s, Kingman showed that for singleserver queues, the queue length process can be scaled such that a meaningful limit is obtained as the critical capacity is approached. In the past half a century, this concept has been extended to much more complex systems and successfully applied in practice, particularly in inventory systems, production facilities, call centers and communication networks. In the course we will discuss the founding principles of heavy traffic theory.
Literature:
Handouts, slides and references to relevant 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 lecturer:
Prof.dr. A.P. Zwart
CWI, Science Park 123
1098 XG Amsterdam
Phone: 0205924018
Email: Bert.Zwart@cwi.nl
