Jasper Vanlerberghe

A hybrid analytical/simulation optimization of Generalized Processor Sharing

With Generalized Processor Sharing (GPS), packets of different applications are backlogged in different queues and the different queues are served according to predetermined weights. It is well-established that GPS is a viable approach to provide different QoS for different applications. However, since the analysis of systems with GPS is a notoriously hard problem, it is not easy to find the weights that optimize GPS for some given objective function. The latter is important from a practical point of view. In this paper, we assume the objective function to be some weighted combination of (non-linear) increasing functions of the mean delays. We use results from strict priority scheduling (which can be regarded as a special case of GPS) to establish some exact theoretical bounds on when GPS is more optimal than strict priority. Some important case studies are included, thereby resorting to Monte-Carlo estimation to find the optimal weights for GPS systems.

On generalized processor sharing and objective functions: analytical framework

Today, telecommunication networks host a wide range of heterogeneous services. Some demand strict delay minima, while others only need a best-effort kind of service. To achieve service differentiation, network traffic is partitioned in several classes which is then transmitted according to a flexible and fair scheduling mechanism. Telecommunication networks can, for instance, use an implementation of Generalized Processor Sharing (GPS) in its internal nodes to supply an adequate Quality of Service to each class. GPS is flexible and fair, but also notoriously hard to study analytically. As a result, one has to resort to simulation or approximation techniques to optimize GPS for some given objective function. In this paper, we set up an analytical framework for two-class discrete-time probabilistic GPS which allows to optimize the scheduling for a generic objective function in terms of the mean unfinished work of both classes without the need for exact results or estimations/approximations for these performance characteristics. This framework is based on results of strict priority scheduling, which can be regarded as a special case of GPS, and some specific unfinished-work properties in two-class GPS. We also apply our framework on a popular type of objective functions, i.e., convex combinations of functions of the mean unfinished work. Lastly, we incorporate the framework in an algorithm to yield a faster and less computation-intensive result for the optimum of an objective function.

Strict monotonicity and continuity of mean unfinished work in two queues sharing a server

We consider a system where two queues share one server. In case of conflict, the first (second) queue is served with probability (1 respectively). We prove strict monotonicity and continuity w.r.t. of the mean unfinished work in queues 1 and 2. Restrictive assumptions are avoided as much as possible, by only assuming that the total unfinished work is a regenerative process. Finiteness of the second moment of the length of a regeneration cycle is generally required for continuity.

Approximating the optimal weights for discrete-time generalized processor sharing

Generalized Processor Sharing (GPS) is a simple, flexible and fair scheduling mechanism to achieve delay differentiation between several customer classes. The amount of delay differentiation is regulated by the weights given to the classes. In this paper we assume a discrete-time, two-class GPS queueing system. Our goal is to derive the optimal weights in order to minimize a weighted sum of functions of the mean delays of both classes. As analytical results are scarce we use an approximation method. The approximation is based on power series expansions of the mean queue length of each of the queues for certain weights. Padé approximants are used to extrapolate the approximation to the whole domain of possible weights, resulting in a set of approximations. An algorithm is proposed to filter out the infeasible solutions (with regard to monotonicity and other characteristics of the system) and aggregate the others, resulting in a single approximation. The result proves to be an accurate approximation of the optimal weights w.r.t. the cost function. For a load of 90% we have a maximum misprediction of 1% of the cost, in the case of a weighted sum of squares of the mean delays. The main contribution of this article is that power series approximations can be used effectively for optimization purposes.

Calculation of the performance region of an easy-to-optimize alternative for generalized processor sharing

Abstract Service differentiation is a basic requirement in every modern queueing system with multiple classes of customers. In this paper, we look at Hierarchical Generalized Processor Sharing (H-GPS), which is a discrete-time hierarchically-structured implementation of the well-known idealized Generalized Processor Sharing (GPS) scheduling discipline. We prove that, for three classes, H-GPS can be configured to obtain any performance possible by other scheduling mechanisms, such as priority queueing or GPS. The hierarchical nature of a H-GPS system, however, has the major advantage that optimization is easier and more intuitive. To this end, we also present an algorithm to calculate the configuration parameters for H-GPS given a certain performance objective.

On the optimization of two-class work-conserving parameterized scheduling policies

Numerous scheduling policies are designed to differentiate quality of service for different applications. Service differentiation can in fact be formulated as a generalized resource allocation optimization towards the minimization of some important system characteristics. For complex scheduling policies, however, optimization can be a demanding task, due to the difficult analytical analysis of the system at hand. In this paper, we study the optimization problem in a queueing system with two traffic classes, a work-conserving parameterized scheduling policy, and an objective function that is a convex combination of either linear, convex or concave increasing functions of given performance measures of both classes. In case of linear and concave functions, we show that the optimum is always in an extreme value of the parameter. Furthermore, we prove that this is not necessarily the case for convex functions; in this case, a unique local minimum exists. This information greatly simplifies the optimization problem. We apply the framework to some interesting scheduling policies, such as Generalized Processor Sharing and semi-preemptive priority scheduling. We also show that the well-documented

On the benefits of a hierarchical version of generalized processor sharing

c\mu

Determining optimal weights for Generalized Processor sharing

cμ-rule is a special case of our framework.