The increased complexity of modern networks and the increasingly dynamic access patterns in multimedia consumption have led to new challenges for content delivery.

Dynamic networks and dynamic access patterns result in a complex system. To deliver content efficiently we introduced  an artificial hormone system that is capable of handling the dynamics, is self-organizing, robust and adaptive. The content placement problem is NP complete and is closely related to several hard problems including edge-disjoint path routing, scheduling and the bin packing problem.

The evaluation of self-organizing algorithms brings also a real challenge. For a first evaluation we created and ILP model of the problem. It is applied to get the exact optimum that serves as a bound in the evaluation of the solution algorithms.

In this paper, we examine the convergence of the algorithm and found that the hormone levels converge to a limit at each node in the typical cases.

We form a series of theorems on convergence with different conditions by starting with a specific base case and then we consider more general, practically relevant cases.

The theorems can be proved by exploiting the analogy between the Markov chains and the artificial hormone system

Böszörmenyi, L

Elmenreich, W

Sobe, A

Szkaliczki, Tibor

SZTAKI Publication Repository

Analysis of an Artificial Hormone System(Extended abstract)Tibor Szkaliczki∗eLearning DepartmentComputer and Automation Research Instituteof the Hungarian Academy of SciencesKende u. 13-17.1111 BudapestHungaryszkaliczki.tibor@sztaki.mta.huAnita SobeDepartment of Computer ScienceUniversity of NeuchatelRue Emile-Argand 11CH-2000 NeuchtelSwitzerlandanita.sobe@unine.chWilfried ElmenreichInstitute of Networked and Embedded Systems/ Lakeside LabsAlpen-Adria-Universita¨t Klagenfurt, AustriaComplex Systems Engineering,Universita¨t Passau, Germanywilfried.elmenreich@aau.atLaszlo Bo¨szo¨rmenyiInstitute of Information TechnologyAlpen-Adria-Universita¨t KlagenfurtUniversita¨tsstrasse 65-67, 9020 Klagenfurt,Austrialaszlo@itec.uni-klu.ac.atAbstract:The increased complexity of modern networks and the increasingly dynamic access patterns inmultimedia consumption have led to new challenges for content delivery. Dynamic networksand dynamic access patterns result in a complex system. To deliver content efficiently weintroduced an artificial hormone system that is capable of handling the dynamics, is self-organizing, robust and adaptive. The content placement problem is NP complete and is closelyrelated to several hard problems including edge-disjoint path routing, scheduling and the binpacking problem. The evaluation of self-organizing algorithms brings also a real challenge.For a first evaluation we created and ILP model of the problem. It is applied to get the exactoptimum that serves as a bound in the evaluation of the solution algorithms. In this paper,we examine the convergence of the algorithm and found that the hormone levels converge toa limit at each node in the typical cases. We form a series of theorems on convergence withdifferent conditions by starting with a specific base case and then we consider more general,practically relevant cases. The theorems can be proved by exploiting the analogy between theMarkov chains and the artificial hormone system.Keywords: self-organizing algorithm, convergence, routing, content delivery1 IntroductionArticial hormone systems [1] are bio-inspired self-organizing algorithms that promise a robust and adap-tive behavior [2] to cope with the problem of content delivery in dynamic networks [3]. The algorithmintroduced in[4] is inspired by the endocrine system of higher mammals and provides the possibility ofsharing small multimedia units in a self-organizing manner during and after a social event. In case of the∗Research is supported by the Hungarian National Development Agency under grant HUMAN MB08-1-2011-0010.1artificial hormone algorithm, nodes are glands that create, consume and forward hormones through thenetwork (the blood stream). Hormones indicate interest in a specific content unit. Content units reacton hormones, by moving from lower towards higher hormone concentration. Therefore, the algorithm cansuccessfully deliver the units on the requesting node if the hormone level increases strictly monotonicallytowards the requesting node. The hormone levels at the time of decisions on the directions of forwardingthe units are formed after many iterations of the algorithm. We examine the convergence of the hormonelevels in order to approximate the hormone levels with their limits. Our main result is that the hormonelevels converge to a limit in the general cases. Furthermore, we are interested in whether the hormonedistribution is monotonic towards the requesting node.Our talk gives an overview of our results on the analysis of the content placement problem and thehormone-based algorithm. Section 2 introduces the problem and refers to the NP-completeness results.Section 3 describes the hormone based algorithm and shows some results on the monotonicity of thehormone levels. Section 4 includes theorems on the convergence of the hormone-based algorithms.2 Optimal Content Placement ProblemEach network node can store and forward content objects and generate requests (hormones) for them.The atomic building element of the content is called content unit that cannot be further split and eachcontent unit is routed along a single path. The units may form compositions which include a sequence ofunits with possible timing constraints. If a unit is requested by several users at the same time, then theirdelivery paths may share common segments. The problem input includes the network graph, the initiallocation of the units, the storage capacities of the nodes, the link bandwidths, the unit sizes and theseries of the requests. Several requests generated at the same time are processed simultaneously. Queuesare formed for each link where the incoming content units can wait before forwarding if the link is busy.The content placement problem is to route all requested content objects simultaneously to the requestingnodes. The problem is a multi-objective optimization problem in the general case. The optimizationgoals to be minimised include the average delay and the number of units that do not arrive in time (ifvideos are transmitted, the subsequent video frames have to arrive within a given deadline to provideproper quality).The problem is computationally hard. If storage capacities are given for the nodes, the Bin Packingproblem can be easily reduced to this problem in order to prove the NP-completeness. Even if storagecapacities are not considered (i.e., infinite storage is assumed), the problem still remains computationallydifficult. In this case, the NP-completeness can be shown with the help of the Edge-Disjoint Pathproblem. It follows from the proof that the content placement problem remains computationally hard,even if each request demands only one single unit. If compositions are formed from the units, the Time-Path Scheduling Problem that is a special Multiprocessor Scheduling Problem can be reduced to thecontent placement problem even if only one request is created in the system.We developed the Integer Linear Program model (ILP) of the problem as well. Our idea was touse ILP for the evaluation of other algorithms [5]. Although ILP offers an inefficient solution in thepractice it produces absolute bounds for the evaluation of practical methods. Based on the ILP model,an optimization tool has been introduced that includes preprocessing, generating ILP model for a probleminstance, solving the generated ILP and generating statistics. We applied the tool to evaluate the self-organizing algorithm presented in the next section.3 Hormone-based algorithmThe hormone-based algorithm is a self-organizing algorithm introduced in [4]. The algorithm creates anartificial hormone system where requests for units are mapped to hormones. The hormone level can berepresented by a real number and it may vary on the different nodes of the network. Since there areseveral paths on which hormones can spread, an evaporation mechanism is introduced. The hormones canbe created by the network nodes and diffused over the network. Since there are several paths on which2hormones can spread, an evaporation mechanism is introduced. The hormone-based algorithm includessearch for the requested unit and then the delivery of the units to the requesting clients. In the searchphase, the hormone is spreading in the network. If the hormone reaches a node storing the requestedunit then the increasing hormone levels attract the required unit and guide it on an appropriate path tothe requester.3.1 The steps of the algorithmThe main steps of the hormone-based algorithm are as follows: handling incoming requests, diffusinghormones, moving units, evaporating known hormones. These steps are continuously repeated in eachnode of the distributed content delivery system.In the algorithmic step of handling incoming requests, the requesting node starts the presentation ofthe unit if the requested unit is present on it. Otherwise, it generates hormones in order to indicate thedemand for the unit. In the step of diffusing hormones, a part of the hormone on the node is distributedamong the neighbors according to their provided link quality. If the hormone reaches a node storing therequested unit then the unit is forwarded from the storing node to its neighbor with the highest hormonelevel in the step of moving units and the unit gets towards the place of the request. The evaporationreduces the hormones by a constant value and then deletes the hormone if its value is below a specifiedthreshold. This step ensures that the hormones on alternative paths will disappear from the system afterdelivering the unit at the destination. The changes in the hormone level for a whole iteration can beformulated as follows:h(t+ 1) = (h(t) + b(t+ 1))AWhere• h(t) = (h1(t), h2(t), ..., hn(t)): the vector containing the hormone levels on each node at iteration t• A: The diffusion matrix. Matrix item aij gives how much of the hormone on node i is forwarded tonode j in the step of diffusion (or preserved if i = j). Network edges with better quality get morehormones and the corresponding matrix element becomes larger.• b(t) = (b1(t), b2(t), ..., bn(t)): the vector containing the additive terms for each node at iteration t.bi(t) may be negative as well and vary in each iteration.3.2 Some definitionsThe elements of A are all non-negative, less than or equal to one and the sum of the elements for each rowis equal to one. Therefore, the elements can be the transition probabilities between the states of a Markovchain and A corresponds to a transition probability matrix. Therefore, we introduced the regularity forthe hormone update function, which is one of the basic terms of the Markov chains. We call a hormonesystem regular if some power of the diffusion matrix has only positive elements. A necessary conditionfor the regularity is that the network is strongly connected. The connectivity is sufficient as well if thereare loop edges at each node which is typical in case of the hormone-based algorithm and therefore theresults on regular Markov chains can be applied to the hormone update in most cases.For simplicity, A is assumed to be constant, i.e., independent from the time. In reality, the elementsof the matrix may vary depending on the changes in the load of the corresponding links. Copying/movingunits in the network causes also changes in the diffusion. However, the duration of forwarding a unittakes typically much longer time than updating the hormone levels, therefore, the matrix is assumed tobe constant during the hormone-update process. Let us call the diffusion time-homogeneous if both thediffusion matrix and the location of the units are the same after each iteration.The normalised fixed vector w of the diffusion matrix has the property wA = w and the sum of itscomponents is 1. The normalized fixed vector is unique for a regular hormone system similarly to thefixed probability vector for the regular Markov chains.3Some nodes may delete the hormone (e.g., at evaporation if the hormone level is low). Let us callthe nodes deleting the hormone deleting nodes. The other nodes that do not delete hormones are calledpreserving nodes. For deleting nodes, the additive term depends on the current hormone level (bi(t) =−hi(t)). The deleting nodes receive hormones but do not forward them to any neighbors.3.3 Monotonicity in timeThe series of the hormone levels in subsequent iterations on the same node have some nice monotonicfeatures. For simplicity, we assume that only one request is generated in the first iteration. If the hormonelevels are increasing in an iteration then the hormone update steps provide almost always greater or equalhormone levels for the next iteration. Based on the monotonic features of the algorithmic steps, we provedthat if the hormone level monotonically increases with time at each node in iteration t0(t0 > 1) then itdoes in each later iteration as well. The condition on monotonicity for each node seems to be restrictive,but we found the following simple condition for it: If the hormone level increases at the requesting nodein the second iteration then the hormone level at each node monotonically increases in each iteration.A simple corollary is that if a hormone level is larger than 0 on a node in iteration t0(t0 > 1) and thehormone level monotonically increases with time at each node in the same iteration then the hormonelevel remains always positive on the node. As a consequence, the set of preserving nodes is monotoneincreasing in each iteration t ≥ t0.4 Convergence resultsThe hormone update function is iterated for several times before the decision is made on the directionof forwarding the units based on the current hormone levels. The time scale of copying a unit is muchlonger than the one of the hormone update because the size of the content unit is much larger than theone of the messages containing the hormones. Since the number of iterations between the decisions isusually large, the hormone levels at the decisions can be approximated with their limits. We examinethe convergence of the algorithm on a series of cases by starting with a basic, restrictive condition andthen we consider more common, practically relevant conditions. We give formulas for the limits of thehormone values in each case. The results are based on the analogy of the artificial hormone system withthe Markov-chains. All of our results refer to regular systems and time-homogeneous diffusion.4.1 Zero additive termWe found that if the additive term b(t) is zero for iterations t > 1 then the hormone level converges at eachnode. The limit of the hormone level vector is η0w, where w is the normalised fixed vector of the diffusionmatrix and η0 is the hormone level generated by the requesting node in the first iteration. For this reason,the relative values of the hormone in the network depends only on the normalised fixed vector (w) of thediffusion matrix and is independent from the algorithm parameters and the location of the requestingnode. In these systems, the units are forwarded always to the same direction independently where therequesting node is located. This result is not surprizing if we think on the analogy with Markov chains.The regular Markov chains converge to the stationary distribution (also called as equilibrium distribution)independently from the starting distribution.4.2 Constant additive termIf the additive term is constant for iterations t > 1 and∑ni=1 bi > 0 (bi is the ith component of b) thenthe hormone levels are asymptotically equivalent with the linear function t · (∑ni=1 bi) ·w where w is thenormalised fixed vector of the diffusion matrix. According to this, the hormone levels diverge and areunbounded. However, if the algorithm is stopped after a specific time the hormone levels can be wellapproximated by a linear function. Similarly to the case of the zero additive term, the relative values of4the hormone levels also depend on the fixed vector (w) of the diffusion matrix. Therefore, the requestedunit would be always guided to the same node independently from the location of the requesting node.4.3 Fixed set of deleting nodesThe condition on the constant additive term is not typical because the network usually contains deletingnodes for which the additive term varies in time. The deleting nodes correspond to the absorbing statesof the Markov chains while the preserving nodes correspond to the transient states if the network isconnected. The elements of Q denoting the diffusion matrix restricted to the columns and rows ofpreserving nodes correspond to the transition probabilities between the transient states of an absorbingMarkov chain. For this reason, we can apply the theorems on the absorbing Markov chains to provethe convergence of the hormone system with deleting nodes. If the set of preserving (or deleting) nodesis fixed and nonempty, the convergence results can be reformulated as follows: if the additive term iszero for the preserving nodes for iterations t > 1 then the hormone levels converge to zero at each node.Furthermore, if the additive term is constant nonzero for the preserving nodes then the hormone levelsconverge at each node. According to this, the convergence of the hormone levels is changed and thealgorithm becomes convergent in the case of the constant additive term.4.4 General regular systemNow, let us turn to the general case and drop the condition of the fixed set of preserving nodes. Ifthe hormone system monotonically increases with time, there is a maximal set of preserving nodes thatnever decreases (see Subsection 3.3). The convergence results on the fixed set of preserving nodes can beapplied to the maximal set and we get that the hormone levels converge at each node in the general caseas well.References[1] U. Brinkschulte, M. Pacher, and A. von Renteln, Towards an Artificial Hormone Systemfor Self-organizing Real-Time Task Allocation. In The 5th IFIP Workshop on Software Technologiesfor Future Embedded & Ubiquitous Systems, pages 339–347, 2007.[2] W. Elmenreich, R. D’Souza, C. Bettstetter, and H. de Meer, A survey of models anddesign methods for self-organizing networked systems. In Proceedings of the Fourth InternationalWorkshop on Self-Organizing Systems, volume LNCS 5918, pages 37–49. Springer Verlag, 2009.[3] K. Schelfthout and T. Holvoet, A Pheromone-Based Coordination Mechanism Applied inPeer-to-Peer. In Agents and Peer-to-Peer Computing, volume 2872 of Lecture Notes in ComputerScience, pages 109–132. Springer Berlin / Heidelberg, 2005.[4] A. Sobe, W. Elmenreich, and L. Bo¨szo¨rmenyi, Towards a Self-organizing Replication Modelfor Non-sequential Media Access, Proceedings of the 2010 ACM workshop on Social, adaptive andpersonalized multimedia interaction and access (SAPMIA 2010), Florence, Italy, (2010) pp. 3-8,ACM[5] T. Szkaliczki, A. Sobe, and L. Bo¨szo¨rmenyi, Discovering Bounds of Performance of Self-Organizing Content Delivery Systems, Pre-Proceedings of 1st International Workshop on Evaluationof Self-Adaptive and Self-Organizing Systems Tools, Techniques and Case studies, Sixth IEEE In-ternational Conference on Self-Adaptive and Self-Organizing Systems (SASO 2012), Lyon, France;10-14 September 2012, pp. 20-275

Analysis of an Artificial Hormone System

Abstract

Similar works

Full text

Available Versions

SZTAKI Publication Repository