An Intelligent Swarm of Markovian Agents

We define a Markovian Agent Model (MAM) as an analytical model formed by a spatial collection of interacting Markovian Agents (MAs), whose properties and behavior can be evaluated by numerical techniques. MAMs have been introduced with the aim of providing a flexible and scalable framework for distributed systems of interacting objects, where both the local properties and the interactions may depend on the geographical position. MAMs can be proposed to model biological inspired systems since are suited to cope with the four common principles that govern swarm intelligence: positive feedback, negative feedback, randomness, multiple interactions. In the present work, we report some results of a MAM model for WSN routing protocol based on swarm intelligence, and some preliminary results in utilizing MAs for very basic ACO benchmarks. 1 Swarm Intelligence: a modeling perspective Swarm intelligent (SI) algorithms are variously inspired from the way in which colonies of biological organisms self organize to produce a wide diversity of functions [1, 2]. Individuals of the colony have a limited knowledge of the overall behavior of the system and follow a small set of rules that may be influenced by the interaction with other individuals or by modifications produced in the environment. The collective behavior of large groups of relatively simple individuals, interacting only locally with few neighboring elements, produces global patterns. Even if many approaches have been proposed that differentiate in many respects, four basic common principles have been isolated that govern SI. ⋄ Positive feedback ⋄ Negative feedback ⋄ Randomness ⋄ Multiple interactions The same four principles govern also a class of algorithms inspired by the expansion dynamics of slime molds in the search for food [3, 4], that have been utilized as the base for the generation of routing protocols in Wireless Sensor Networks (WSN). Through the adoption of the above four principles, it is possible to design distributed, self-organizing, and fault tolerant algorithms able to self-adapt to the environmental changes, that present the following main properties [1]: i) Single individuals are assumed to be simple with low computational intelligence and communication capabilities; ii) individuals communicate indirectly, through modification of the environment (this property is known as stigmergy [2]); iii) The range of the interaction may be very short, nevertheless a robust global behavior emerges from the interaction of the nodes; iv) The global behavior adapts to topological and environmental changes. The usual way to study such systems is through simulation, due to the large number of involved individuals that lead to the well-known state explosion problem. Analytical techniques are preferable if, starting from the peculiarities of SI systems, allow to realize effective and scalable models. Along this line, new stochastic entities, called Markovian Agents (MAs) [5, 6], have been introduced with the aim of providing a flexible, powerful, and scalable technique for modeling complex systems of distributed interacting objects, for which feasible analytical and numerical solution algorithms can be implemented. Each object has its own local behavior that can be modified by the mutual interdependencies with the other objects. MAs are scattered over a geographical area and retain their spatial position so that the local behavior and the mutual interdependencies may be related to their geographical positions and other features like the transmittance characteristics of the interposed medium. MAs are modeled by a discrete-state continuous-time finite CTMC whose infinitesimal generator is influenced by the interaction with other MAs. The interaction among agents is represented by a message passing model combined with a perception function. When resident in a state or during a transition, an MA is allowed to send messages that are perceived by the other MAs, according to a spatial dependent perception function, modifying their behavior. Messages may model real physical messages (as in WSN) or simply the mutual influences of an MA over the other ones. The flexibility of the MA representation, the spatial dependency, and the mutual interaction through message passing and perception function, make MA models suited to cope with various biological inspired mechanisms governed by the four aforementioned principles. In fact, the Markovian Agent Model (MAM), whose constituent elements are the MAs, was specifically studied to cope with the following needs [6]: i Provide analytical models that can be solved by numerical techniques, thus avoiding the need of long and expensive simulation runs; ii Provide a flexible and scalable modeling framework for distributed systems of interacting objects; iii Provide a framework in which local properties can be coupled with global properties; iv Local and global properties and interactions may depend on the position of the objects in the space (space-sensitive models); v The solution algorithm self-adapts to variations in the system topology and in the interaction mechanisms. Interactive Markovian Agents have been first introduced in [7, 5] for single class MAs and then extended to Multi-class Multi-message Markovian Agent Model in successive works [8–10]. In [9, 11, 12] MAs have been applied to routing algorithms in WSN, adopting SI principles [13]. The present work describes the structure of MAMs and the numerical solution algorithms in Section 2. Then, applications derived from biological models are presented: a swarm intelligent algorithm for routing protocols in WSN (Section 3) and a simple Ant Colony Optimization (ACO) example (Section 4). 2 Markovian Agents Models The structure of a single MA is represented in Figure 1. States i, j, . . . , k are the states of the CTMC representing the MA. The transitions among the states are of two possible types that are drawn differently: Solid lines (like the transition from i to j or the self-loops in i or in j) indicate the fixed component of the infinitesimal generator and represent the local or autonomous behavior of the object that is independent of the interaction with the other MAs (like, for instance, the time to failure distribution, or the reaction to an external stimulus). Note that we include in the representation also self-loop transitions that require a particular notation since are not visible in the infinitesimal generator of the CTMC [14]. Dashed lines (like the transition from i to k or the transitions into i or j) represent the transitions induced by the interaction with the other MAs. The way in which the rates of the induced transitions are computed is explained in the following section.