Section: New Results

Advances in Methodological Tools

Participants : Eitan Altman, Konstantin Avrachenkov, Ilaria Brunetti, Ioannis Dimitriou, Mahmoud El Chamie, Majed Haddad, Alain Jean-Marie, Philippe Nain, Giovanni Neglia.

Queueing theory

In [21] K. Avrachenkov and P. Nain in collaboration with U. Yechiali (Tel Aviv Univ., Israel) study a retrial queueing system with two independent Poisson streams of jobs flowing into a single-server service system, having a limited common buffer that can hold at most one job. If a type-$i$ job ($i=1,2$) finds the server busy, it is blocked and routed to a separate type-$i$ retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. They study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann-Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts.

In [19] K. Avrachenkov in collaboration with E. Morozov (Petrozavodsk State Univ., Russia) consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has c identical servers and can accommodate up to K jobs (including c jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. They establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that they obtained have clear probabilistic interpretation.

In [20] K. Avrachenkov in collaboration with E. Morozov and R. Nekrasova (Petrozavodsk State Univ., Russia) and B. Steyaert (Ghent Univ., Belgium) study a retrial queueing system with $N$ classes of customers, where a class-$i$ blocked customer joins orbit $i$. Orbit $i$ works like a single-server queueing system with (exponential) constant retrial time (with rate ${\mu }_{0i}$) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, they present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, they deduce a set of necessary stability conditions for such a system. They will show that these conditions have a very clear probabilistic interpretation. They also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that they focus on.

In [75] , I. Dimitriou investigates a single server system accepting two types of retrial customers and paired services. The service station can handle at most one customer, and if upon arrival a customer finds the server busy it is routed to an infinite capacity orbit queue according to its type. Upon a service completion epoch, if at least one orbit queue is non-empty, the server seeks to find customers from the orbits. If both orbit queues are non-empty, the seeking process will bring to the service area a pair of customers, one from each orbit. If only one is non-empty, then a customer from this orbit queue will be brought to the service area. However, if a primary customer arrives during the seeking process it will occupy the server immediately. It is shown that the joint stationary orbit queue length distribution at service completion epochs, can be determined via transformation to a Riemann boundary value problem. Stability condition is investigated, while an extension of the model is also discussed and analyzed. Numerical results are obtained and yield insight into the behavior of the system. The theoretical system can be used to model a relay node for two connections in wireless communication, where network coding is used.

When individuals have to take a decision on whether or not to join a queue, one may expect to have threshold equilibria in which customers join the queue if its size is smaller than a threshold and do not join if it exceeds the threshold. In [74] , P. Wiecek (Wroclaw Univ. of Technology, Poland), E. Altman and A. Ghosh (Univ. of Pennsylvania, USA) have studied queueing in which the congestion cost per user decreases in the queue size. An example for such a situation is multicast communication where all individuals that participate in the multicast session share the transmission cost. They showed that many equilibria exist and computed the asymptotic system behavior as the arrival rate of individuals grows.

Markov processes

In [16] K. Avrachenkov in collaboration with A. Eshragh (Univ. of Adelaide, Australia) and J. Filar (Flinders Univ., Australia) present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix $P$ in this class corresponds to a Hamiltonian cycle in a given graph $G$ on $n$ nodes and to an irreducible, periodic, Markov chain. They show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first $n-1$ powers of $P$, whose coefficients can be explicitly derived. They also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices. As an illustration of these analytical results, they exploit them to develop a new heuristic algorithm to determine a non-Hamiltonicity of a given graph.

Control theory

In [17] K. Avrachenkov and O. Habachi (former post-doc in Maestro ) in collaboration with A. Piunovskiy and Y. Zhang (both from the Univ. of Liverpool, UK) investigate infinite-horizon deterministic optimal control problems with both gradual and impulsive controls, where any finitely many impulses are allowed simultaneously. Both discounted and long-run time-average criteria are considered. They establish very general and at the same time natural conditions, under which the dynamic programming approach results in an optimal feedback policy. The established theoretical results are applied to the Internet congestion control, and by solving analytically and non-trivially the underlying optimal control problems, they obtain a simple threshold-based active queue management scheme, which takes into account the main parameters of the transmission control protocols, and improves the fairness among the connections in a given network.

Game theory

Estimating the Shapley-Shubik index

In [15] K. Avrachenkov in collaboration with L. Cottatellucci (EURECOM) and L. Maggi (Create-Net , Italy) consider simple Markovian games, in which several states succeed each other over time, following an exogenous discrete-time Markov chain. In each state, a different simple static game is played by the same set of players. They investigate the approximation of the Shapley-Shubik power index in simple Markovian games (SSM). They prove that an exponential number of queries on coalition values is necessary for any deterministic algorithm even to approximate SSM with polynomial accuracy. Motivated by this, they propose and study three randomized approaches to compute a confidence interval for SSM. They rest upon two different assumptions, static and dynamic, about the process through which the estimator agent learns the coalition values. Such approaches can also be utilized to compute confidence intervals for the Shapley value in any Markovian game. The proposed methods require a number of queries, which is polynomial in the number of players in order to achieve a polynomial accuracy.

Evolutionary games

Evolutionary games attempt to explain the evolution of species and the dynamics of competition. The player's utility is called “fitness” and a larger fitness indicates a larger rate of reproducibility. In standard evolutionary games, one studies interactions between individuals each of which is consider as a player. In [49] , I. Brunetti, E. Altman, and R. El-Azouzi (Univ. of Avignon) argue that in many situations both in biology as well as in networking, one cannot attribute a fitness to an individual but rather to a group of individuals that behaves as an altruistic entity. For example, in a hive of bees it is only the queen that reproduces and thus one cannot model a single bee as a selfish player. They present new definitions for evolutionary games for such situations and study their equilibrium.

This, as well as other considerations in multi-population evolutionary games, is applied in [56] by H. Gaiech and R. El-Azouzi (Univ. of Avignon), M. Haddad, E. Altman and I. Mabrouki (Univ. of Manouba, Tunisia) to Multiple Access Control for which the equilibrium is explicitly computed.

In [84] E. Altman presents a summary of the foundations of classical evolutionary games addressed to a wide public. Both the equilibrium notion of ESS (Evolutionary Stable Strategy) as well as the replicator dynamics (which describes the non-equilibrium behavior) are presented.

Sequential Anonymous Games

Stationary anonymous sequential games are a special class of games that combines features from both population games (infinitely many players) with stochastic games. It allows studying competition in complex systems where each individual belongs to a community (which we call individual state) which may change in time as a result of actions taken by the individual. Unlike standard evolutionary games, a player does not just optimize its immediate reward (fitness) but some long term reward over the time. P. Wiecek (Wroclaw Univ. of Technology, Poland) and E. Altman proved in [42] the existence of an equilibrium for the general model and studied the two applications. The first one is described in § 6.5.1 .

The second application is a maintenance repair problem: each of a large number of cars can decide whether to behave gently or to drive fast. By driving fast it takes larger risks for having an accident. The probability of an accident depends on the fraction of drivers that drive fast. An internal state of the car is either good (g) or bad (b). A car gets to a state b as a result of an accident and then it has some penalty and costs for repair. The advantage of driving fast is reducing delay costs. This problem is formulated as a sequential anonymous game and its equilibrium is computed. They computation makes use of the linear structure of both the transition probabilities and the immediate fitness in the global state.

Optimization

In [55] M. El Chamie and G. Neglia provide a methodology for solving smooth norm optimization problems under some linear constraints using the Newton's method. This problem arises in many machine learning and graph optimization applications. They show how Newton's method significantly outperforms gradient methods both in terms of convergence speed and in term of robustness to the step size selection.