A reduced-load equivalence for generalised processor sharing networks with heavy-tailed input flows

We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPS-based scheduling algorithms are considered important for providing differentiated quality of service in integrated-services networks. We are interested in the workload of a particular flow~$i$ at the bottleneck node on its path. Flow $i$ is assumed to have long-tailed traffic characteristics. We distinguish between two traffic scenarios, (i) flow~$i$ generates instantaneous traffic bursts and (ii) flow $i$ generates traffic according to an on/off process. In addition, we consider two configurations of feed-forward networks. First we focus on the situation where other flows join the path of flow~$i$. Then we extend the model by adding flows which can branch off at any node, with cross traffic as a special case. We prove that under certain conditions the tail behaviour of the workload distribution of flow~$i$ is equivalent to that in a {em two-node tandem network where flow~$i$ is served in isolation at {em constant rates. These rates only depend on the traffic characteristics of the other flows through their average rates. This means that the results do not rely on any specific assumptions regarding the traffic processes of the other flows. In particular, flow~$i$ is not affected by excessive activity of flows with `heavier-tailed' traffic characteristics. This confirms that GPS has the potential to protect individual flows against extreme behaviour of other flows, while obtaining substantial multiplexing gains

Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs

In this paper we derive large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model. We assume both classes to have Gaussian characteristics. We distinguish three cases depending on whether the GPS weights are above or below the average rate at which traffic is sent. First, we calculate exact asymptotic upper and lower bounds, then we calculate the logarithmic asymptotics, and finally we show that the decay rates of the upper and lower bound match. We apply our results to two special Gaussian models: the integrated Gaussian process and the fractional Brownian motion. Finally we derive the logarithmic large-buffer asymptotics for the case where a Gaussian flow interacts with an on-off flo

A reduced-load equivalence for generalised processor sharing networks with heavy-tailed input flows

We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPS-based scheduling algorithms are considered important for providing differentiated quality of service in integrated-services networks. We are interested in the workload of a particular flow~$i$ at the bottleneck node on its path. Flow $i$ is assumed to have long-tailed traffic characteristics. We distinguish between two traffic scenarios, (i) flow~$i$ generates instantaneous traffic bursts and (ii) flow $i$ generates traffic according to an on/off process. In addition, we consider two configurations of feed-forward networks. First we focus on the situation where other flows join the path of flow~$i$. Then we extend the model by adding flows which can branch off at any node, with cross traffic as a special case. We prove that under certain conditions the tail behaviour of the workload distribution of flow~$i$ is equivalent to that in a {em two-node tandem network where flow~$i$ is served in isolation at {em constant rates. These rates only depend on the traffic characteristics of the other flows through their average rates. This means that the results do not rely on any specific assumptions regarding the traffic processes of the other flows. In particular, flow~$i$ is not affected by excessive activity of flows with `heavier-tailed' traffic characteristics. This confirms that GPS has the potential to protect individual flows against extreme behaviour of other flows, while obtaining substantial multiplexing gains

Sample-path large deviations for tandem and priority queues with Gaussian inputs

This paper considers Gaussian flows multiplexed in a queueing network. A single node being a useful but often incomplete setting, we examine more advanced models. We focus on a (two-node) tandem queue, fed by a large number of Gaussian inputs. With service rates and buffer sizes at both nodes scaled appropriately, Schilder's sample-path large deviations theorem can be applied to calculate the asymptotics of the overflow probability of the second queue. More specifically, we derive a lower bound on the exponential decay rate of this overflow probability and present an explicit condition for the lower bound to match the exact decay rate. Examples show that this condition holds for a broad range of frequently-used Gaussian inputs. The last part of the paper concentrates on a model for a single node, equipped with a priority scheduling policy. We show that the analysis of the tandem queue directly carries over to this priority queueing system. iffalse {it Perhaps:} We conclude by presenting a number of motivated conjectures for the analysis of a queue operating under the generalized processor sharing discipline

Sample-path large deviations for generalized processor sharing queues with Gaussian inputs

In this paper we consider the Generalized Processor Sharing (GPS) mechanism serving two traffic classes. These classes consist of a large number of independent identically distributed Gaussian flows with stationary increments. We are interested in the logarithmic asymptotics or exponential decay rates of the overflow probabilities. We first derive both an upper and a lower bound on the overflow probability. Scaling both the buffer sizes of the queues and the service rate with the number of sources, we apply Schilder's sample-path large deviations theorem to calculate the logarithmic asymptotics of the upper and lower bound. We discuss in detail the conditions under which the upper and lower bound match. Finally we show that our results can be used to choose the values of the GPS weights. The results are illustrated by numerical examples

A tandem queue with Lévy input: a new representation of the downstream queue length.

In this paper we present a new representation for the steady state distribution of the workload of the second queue in a two-node tandem network. It involves the difference of two suprema over two adjacent intervals. In case of spectrally-positive

Generalized processor sharing queues with heterogenous traffic classes

We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic flows are served in accordance with the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behavior of the light-tailed traffic flow under the assumption that its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed flow served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is in fact asymptotically equivalent to that in the isolated system, multiplied with a certain pre-factor, which accounts for the interaction with the heavy-tailed flow. Specifically, the pre-factor represents the probability that the heavy-tailed flow is backlogged long enough for the light-tailed flow to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario

Large deviations of infinite intersections of events in Gaussian processes

The large deviations principle for Gaussian measures in Banach space is given by the generalized Schilder's theorem. After assigning a norm ||f|| to paths f in the reproducing kernel Hilbert space of the underlying Gaussian process, the probability of an event A can be studied by minimizing the norm over all paths in A. The minimizing path f*, if it exists, is called the most probable path and it determines the corresponding exponential decay rate. The main objective of our paper is to identify the most probable path for the class of sets A that are such that the minimization is over a closed convex set in an infinite-dimensional Hilbert space. The `smoothness' (i.e., mean-square differentiability) of the Gaussian process involved has a crucial impact on the structure of the solution. Notably, as an example of a non-smooth process, we analyze the special case of fractional Brownian motion, and the set A consisting of paths f at or above the line t in [0,1]. For H>1/2, we prove that there is an s such that

User-level performance of elastic traffic in a differentiated-services environment

We consider a system with two service classes, one of which supports elastic traffic. The traffic characteristics of the other class can be completely general, allowing streaming applications as an important special case. The link capacity is shared between the two traffic classes in accordance with the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, provide a flexible mechanism for service differentiation and prioritization. We examine the user-level performance of the elastic traffic. The elastic traffic users randomly initiate file transfers with a heavy-tailed distribution. Within the elastic traffic class, the active flows share the available bandwidth in an ordinary Processor-Sharing (PS) fashion. The PS discipline has emerged as a natural paradigm for evaluating the user-perceived performance of bandwidth sharing algorithms like TCP. For a certain parameter range, we establish that the transfer delay incurred by the elastic traffic flows is asymptotically equivalent to that in an isolated PS system with constant service rate. This service rate is only affected by the streaming traffic through its average rate. Specifically, the elastic traffic is largely immune from possible adverse traffic characteristics or performance degradation due to prioritization of the streaming traffic. This confirms that GPS-based multiplexing mechanisms achieve significantly better performance for bot

Two coupled queues with heterogeneous traffic

We consider a system with two heterogeneous traffic classes, one having light-tailed characteristics, the other one exhibiting heavy-tailed properties. When both classes are backlogged, the two corresponding queues are each served at a certain nominal rate. However, when one queue empties, the service rate for the other class increases. This dynamic sharing of surplus service capacity is reminiscent of the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, provide a candidate implementation mechanism for achieving differentiated Quality-of-Service in a DiffServ architecture. We characterize the asymptotic workload behavior of both traffic classes. The tail of the workload distribution of the {em heavy-tailed/ class is asymptotically equivalent to that of the heavy-tailed class in isolation -- but with its nominal service rate inflated by the slack capacity of the light-tailed class. For the {em light-tailed/ class, we show a sharp dichotomy in the qualitative behavior, depending on whether its load exceeds its nominal service rate or not. In underload scenarios, the tail of its workload distribution is equivalent to that of the light-tailed class in isolation, multiplied with a certain pre-factor. The pre-factor represents the probability that the heavy-tailed class is backlogged long enough for the light-tailed class to build up a large workload. This provides a measure for the extent to which the light-tailed class benefits from sharing surplus capacity with the heavy-tailed class. In contrast, in overload situations, the light-tailed class is adversely affected by the heavy-tailed class, and inherits its traffic characteristics