INTERNATIONAL JOURNAL OF NETWORK MANAGEMENT Int. J. Network Mgmt 2008; 18: 465–466 Published online 19 September 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/nem.711
Special Issue on Mathematical Methods in Network Management Computer science has its traditions in logic and discrete mathematics, not really in the full range of mathematical disciplines, such as physics and the other natural sciences. Management, on the other hand, is more like physics or engineering than it is like mathematics. It deals with the real-world issue of approximation and uncertainty. It uses idealizations to model its complexity and this in turn requires a broader repertoire of modeling methods. Few systems are simple enough to be described by discrete means alone. In this issue we wanted to open up this repertoire now that there is a more widespread realization of the need for such methods. We received many submissions, some in the formative stages and some more rehearsed. As Machiavelli noted, there will always be resistance when one attempts to bring about a change in the establishment, but we believe that the introduction of mathematical methods was inevitable all along in computer science. We have now simply reached the point at which it is an imperative. Natural scientists met the issue of complexity much earlier in their respective fields and have long since resigned themselves to the art of ‘suitably idealized approximation’. Now computer science too must succumb to what it actually already knows—namely that as the scale of something grows one needs to hide irrelevant details and find simpler modes of description at a higher level. The goal of this special issue is show new applications of mathematical methods to the complex problems in the area of network management. The availability of new quantitative method applications and algorithms, in turn, should facilitate the construction of next-generation systems for the management of networks and services. Both researchers and software designers should find utility in this special issue as a range of methods is covered. There are two papers using game-theoretic approaches along with one each based on mathematical programming, stochastic modeling and one using an algorithmic approach. In total 21 papers were submitted, each undergoing an average of three reviews, leading to a competitive 24% of the papers being accepted. In the following a short overview of the respective papers is given. The first paper in our collection is by Zhu et al. of HP Labs on ‘Automated Application Component Placement in Data Centers using Mathematical Programming’. In this paper a generalized yet finegrained model of data center applications and resources is developed. The optimization problem of placing each application in the data center on the right server is formulated in a mixed integer program. The objective function is to minimize the utilization of links between switches. Constraints including link capacities were modeled linearly; those that were quadratic were linearized. The utilization of fiber channel storage area network links was not included in the objective function based on topology considerations; instead directional link capacity was modeled as a feasibility constraint. A standard LP/MIP solver was used for the numerics. Using a real-world and a synthetic dataset, the MIP model was evaluated against a random placement algorithm in which the former achieved better link utilization rates. Furthermore, the authors explore sequential versus simultaneous placement of applications, whereby sequential placement was found to provide a preferable solution quality to computation cost ratio for the case study performed. The paper ‘A Game-Theoretic Model for Capacity-Constrained Fair Bandwidth Allocation’ by Yan, El-Atawy and Al-Shaer present a distributed bandwidth allocation and admission control model with minimum bandwidth allotment based on a non-cooperative game framework. The optimization is on fair bandwidth allotment with a fairness criterion termed ‘residual capacity fairness’. They define a Harsanyi-type social welfare function as an aggregate of the users’ utility functions with pricing of bandwidth *Correspondence to: Michael Alexander, Koellnerhofgasse 3/15A, 1010 Vienna, Austria. Email:
[email protected]
Copyright © 2008 John Wiley & Sons, Ltd.
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at the edge of the network. Bandwidth is modeled as a vector of minimum and maximum data rate. It is shown that the main solution to the defined game is its max–min strategy subject to the constraint of a fairness condition for the case in which there is at least one link experiencing a bottleneck. The paper furthermore looks at the model’s price determinants of bandwidth allocation, which is shown to be sensitive to the ratio of link prices and a Lagrange multiplier linked to two constraints in the main model. Finally, a distributed admission control algorithm is derived from the model and pricing considerations. The full model is evaluated using simulation contrasting global and distributed bandwidth allocation. Focusing on the latter it is shown that it performs within 10–20% of the global allocation approach and a ratio of congested to total flows is less than 50%. The paper ‘End-to-End DWDM Optical Link Power Control via a Stackelberg Revenue-Maximizing Mode’ by Zhu and Pavel explores optimizing the signal-to-noise ratio in optical transport systems with a Stackelberg game and consecutively added power capacity constraints. The model extends a prior noncooperative game approach by the authors with a Stackelberg leader. Both uniform and differentiated pricing for power allotment of network users is considered. In addition, a distributed allotment algorithm based on geometric programming is presented. It is evaluated in a two-user simulation in which the algorithm is shown to converge for both within a few iterations. ‘Delay Management in Delay-Tolerant Network’ by Zheng et al. focuses on Markov chains to examine a series of location-based node mobility scenarios and their QoS properties: encounter probability and encounter delay over time. The Markov paths are based on the movement of nodes between locations, whereby messages are exchanged when two nodes meet at the same location. Locations are fitted with a number of properties such as attraction. The movement pattern between location sets are random, random current location-dependent or pseudo-random—‘agenda-based’. The study looks at constant dwell time with discrete Markov chains and variable dwell time at a given location with continuous time Markov chains. Message life time in terms of ‘steps’ in the model is constrained by a time-to-live counter. Analyzing two- and three-location settings, results show that short-location dwell time for random movement with constrained 'steps' leads to a steep concave encounter probability curve. The results furthermore suggest that long dwell time, in turn, is preferable for 'agenda-based' movement. ‘Traffic Matrix Estimation Based on a Square Root Kalman Filtering Algorithm’ by Zhou et al. introduces a refined Kalman filter method to estimate traffic matrices that improves on ill-conditioning problems of prior approaches, facilitating implementation in software systems. Their square root Kalman filtering traffic matrix estimation algorithm is based on matrix decomposition and ensures positive definiteness of error covariance matrices. The validation looks at the algorithm’s numerical performance, with algorithm stability shown to improve on the prior main Kalman filter approach. In closing, we would like to express our thanks to all contributors and especially the reviewers for their dedication and time. Michael Alexander Mark Burgess
Copyright © 2008 John Wiley & Sons, Ltd.
Int. J. Network Mgmt 2008; 18: 465–466 DOI: 10.1002/nem
