A distributed approach to capacity allocation in logical networks

Abstract

We consider a dynamic capacity reallocation scheme in a logically fully-connected telecommunications network. We show that the problem of optimal capacity allocation can be solved in a distributed manner, an essential feature of such a scheme. Our continuous-capacity reallocation scheme can be used as a foundation for a discrete system. This is useful from the perspective of practical implementation.

Introduction

Modern telecommunications networks are expected to carry many different types of traffic. There is a specific need for algorithms that can allocate capacity across routes in a network in an effective manner; for example by optimising quality of service, or maximising revenue earned. To operate in networks that are continuously evolving, such algorithms must be scalable and robust. In order to achieve scalability, an algorithm is likely to have to be distributed in nature.

In this paper, we consider a physical network that is already dimensioned (see [5] for more details on dimensioning), where links have finite capacity. Requests for connection between given origin-destination pairs are satisfied by allocating capacity on a set of physical links that join the origin to the destination. We call this set of links a route. Since the capacity of each physical link is constrained, the traversing routes are essentially competing for resources. We are interested in designing a scheme for resolving this competition in a way that benefits the network as a whole.

The construction of such a scheme essentially depends on the admission policy that is applied to requests for connection. One alternative is a complete sharing strategy in which requests for connection are admitted if there is sufficient capacity on all of the links of the route that the connection wishes to access. We would expect such a scheme to have a low average blocking probability over the whole network, at the expense of needing complex admission strategies that can monitor the state at all links along a route.

Alternatively, a network operator could use a complete partitioning strategy in which end-to-end logical links, such as virtual paths in ATM networks or label-switched paths in IP networks, are created for each route and allocated a specific amount of dedicated capacity. A connection is admitted only if there is sufficient capacity on the logical link corresponding to the connection’s route. In such a scheme, the logical links act as intermediaries, obtaining capacity from the network and then making it available to the connections as they require it. We would expect that the network average blocking probability of a complete partitioning scheme would be higher than for complete sharing, but admission control would be more straightforward, since the decision to admit or reject a request for connection can be based upon on the current occupancy of the logical link alone.

The question of whether complete partitioning can be worthwhile was considered by Arvidsson et al. [3], who investigated the trade-off between multiplexing gains and overhead costs. They found that, in general, complete sharing is preferable when overhead costs are low, and complete partitioning is preferable when overhead costs are high. In fact, complete partitioning was found to be cost-effective in many cases using various realistic revenue rates and overhead cost assumptions.

Intermediate between the two schemes mentioned above is a scheme in which end-to-end logical links are created with an allocated capacity, but where capacity can be transferred between logical links on a slower timescale than the connection arrival and departure process. With such a facility, the network would be able to transfer capacity from logical links that are temporarily lightly-loaded to logical links that are heavily-loaded. In this way, capacity can be made to track stochastic fluctuations in traffic.

A capacity reallocation scheme along the lines discussed above was proposed in [2], where simulation studies showed that it could significantly reduce blocking probabilities in comparison with a complete partitioning scheme with optimally allocated capacity. This scheme used the capacity value function developed for the $M/M/k/k$ loss model by Chiera and Taylor [8]. Using a knowledge of current occupancy, this function values an extra unit of capacity in an $M/M/k/k$ loss system according to the reduction in expected loss of revenue that would occur over a specified time horizon T. As the occupancies on the various routes change, the scheme moves capacity to those routes which, possibly temporarily, value it most highly.

Although it operates physically in a different context, the problem that we consider is mathematically similar to optimisation-based flow control, which is well established in the literature. Kelly [12] posed a global optimisation problem for rate control schemes for broadband multiservice networks. The need for a distributed approach motivated a decomposition of the original formulation into a user problem and a network problem. Via Lagrangian arguments, Kelly showed how to tie these problems together in such a way that the resulting vector of capacities is the unique solution to the original global problem. This was further developed by Kelly et al. [13] through the construction of a rate control algorithm, which had the feature of additive increase/multiplicative decrease, prevalent in IP networks. Low and Lapsley [14] also used an optimisation approach for transmission rate control where sources adjust rates in response to variations in the network. They used a gradient projection method to solve the dual problem, resulting in a distributed algorithm.

Both of these algorithms envisaged a situation where the number and the parameters of the different flows are constantly changing and the goal is to track the ‘optimal’ flow rate. However, the authors of [12], [13], [14] presented their results in a context where the number of flows was fixed, establishing that the algorithms converged to the optimal solution in this case.

In our physical model, the number and parameters of the logical links are fixed, but the utility of the allocated capacity is constantly changing, since it is a function of the randomly-varying occupancy. Although simulation results are promising [2], analytical results regarding such a system are difficult to obtain, since they involve transient properties of the whole network. To make a start on understanding the properties of the proposed scheme, we adopt the same approach as [12], [13], [14] and study whether it can solve a static optimisation problem. Specifically, we consider the case where the time horizon T becomes large and so logical links value their capacity according to their long-term average traffic parameters, rather than their current occupancy. Under this scenario, the utility of capacity is a deterministic function for each route and standard non-linear optimisation techniques can be used.

We consider primal penalty and Lagrangian-based methods, which have a stronger focus on maintaining primal feasibility than the dual algorithm of [14], and which are tailored for the capacity partitioning problem considered here. We propose a distributed algorithm for solving the partitioning problem, and we investigate the behaviour of the algorithm when capacity is continuous, followed by the case where capacity is discrete.

This paper is organised as follows: Section 2 introduces the global framework for the problem, in particular the optimisation formulation. We also motivate the need for a decentralised scheme and outline what is meant by “local” or “distributed”. Section 3 contains a discussion and examples of a pure penalty method and a multiplier method. Section 4 uses the continuous-capacity framework to devise two corresponding discrete systems that are practically implementable. We end with concluding remarks and possible directions for future research.