Michel Mandjes

Generalized processor sharing with light-tailed and heavy-tailed input

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, 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 for Small Buffers: An Insensitivity Result

This article focuses on a queue fed by a large number of “semi-Markov modulated fluid sources”, e.g., on/off sources with on and off-times that have general distributions. The asymptotic regime is considered in which the number of sources grows large, and the buffer and link rate are scaled accordingly. We aim at characterizing the exponential decay rate of the buffer overflow probability for the regime of small buffers. An insensitivity result is proven: the decay rate depends on the distributions of the on and off-times only through their means. The efficiency gain to be achieved by using small buffers is significant, as the decay rate grows fast: proportionally to the square root of the buffer size.

Delay performance analysis of the new Internet services with guaranteed QoS

In addition to the traditional best effort Internet protocol (IP) service of the Internet, the Internet Engineering Task Force has defined two new services that provide quality of service guarantees on throughput and delay. In this paper, we analyze the delay performance that can be achieved with the service that provides the most firm guarantees-the guaranteed quality of service. Using a voice and a video application as examples, the end-to-end delay is calculated. Three different scenarios are presented: local, regional, and long distance, differing in the distance covered and the number of IP routers traversed. It is shown that even in the long-distance case, the achievable end-to-end delay for the guaranteed service can be kept sufficiently low to support interactive applications with strict delay requirements. It is concluded that for low-rate applications, the efficiency cannot be higher than about 60-75%. For connections traversing a large number of routers, the variable delay as advertised by the routers is much higher than the expected delay. The second new service, called the controlled-load network element service, provides less firm delay guarantees. The end-to-end delay for this service is estimated using similar techniques. The results suggest that this service is suitable for interactive applications in the local and regional scenarios only. For the long distance scenario, the end-to-end delay is expected to be below 300 ms.

Exact asymptotics for fluid queues fed by multiple heavy-tailed on–off flows

We consider a fluid queue fed by multiple On–Off flows with heavy-tailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a “dominant” subset of the flows, with the original service rate subtracted by the mean rate of the other flows. We describe how a dominant set may be determined from a simple knapsack formulation. The dominant set consists of a “minimally critical” set of On–Off flows with regularly varying On periods. In case the dominant set contains just a single On–Off flow, the exact asymptotics for the reduced system follow from known results. For the case of several On–Off flows, we exploit a powerful intuitive argument to obtain the exact asymptotics. Combined with the reduced-load equivalence, the results for the reduced system provide a characterization of the tail of the workload distribution for a wide range of traffic scenarios.

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.

Load characterization and anomaly detection for voice over IP traffic

We consider the problem of traffic anomaly detection in IP networks. Traffic anomalies typically arise when there is focused overload or when a network element fails and it is desired to infer these purely from the measured traffic. We derive new general formulae for the variance of the cumulative traffic over a fixed time interval and show how the derived analytical expression simplifies for the case of voice over IP traffic, the focus of this paper. To detect load anomalies, we show it is sufficient to consider cumulative traffic over relatively long intervals such as 5 min. We also propose simple anomaly detection tests including detection of over/underload. This approach substantially extends the current practice in IP network management where only the first-order statistics and fixed thresholds are used to identify abnormal behavior. We conclude with the application of the scheme to field data from an operational network.

ON SPECTRAL SIMULATION OF FRACTIONAL BROWNIAN MOTION

This article focuses on simulating fractional Brownian motion (fBm). Despite the availability of several exact simulation methods, attention has been paid to approximate simulation (i.e., the output is approximately fBm), particularly because of possible time savings. In this article, we study the class of approximate methods that are based on the spectral properties of fBms stationary incremental process, usually called fractional Gaussian noise (fGn). The main contribution is a proof of asymptotical exactness (in a sense that is made precise) of these spectral methods. Moreover, we establish the connection between the spectral simulation approach and a widely used method, originally proposed by Paxson, that lacked a formal mathematical justification. The insights enable us to evaluate the Paxson method in more detail. It is also shown that spectral simulation is related to the fastest known exact method.

Estimation of performance measures for product form cellular mobile communications networks

This paper investigates cellular mobile communications networks. The purpose of the paper is twofold. First, it is shown that the restrictive assumption of reversible routing is not required for the network population distribution to be of product form. Different protocols with their specific ways of handling congestion, all of them yielding product form, are discussed. Second, the notoriously difficult task of obtaining performance measures derived from product form expressions is attacked by an efficient method based on importance sampling. This algorithm substantially speeds up the computational time required to estimate, for example, the probability that a call attempting a hand‐over is blocked. In addition, qualitative insight is gained into the network conditional on blocking in a specific cell: are neighbouring cells overloaded as well? The examples include networks with capacity constraints due to effective interference between cells, and a reasonably sized network containing 49 cells and 7 cell reuse groups.

Optimal trajectory to overflow in a queue fed by a large number of sources

We analyse the deviant behavior of a queue fed by a large number of traffic streams. In particular, we explicitly give the most likely trajectory (or ‘optimal path’) to buffer overflow, by applying large deviations techniques. This is done for a broad class of sources, consisting of Markov fluid sources and periodic sources. Apart from a number of ramifications of this result, we present guidelines for the numerical evaluation of the optimal path.

A tandem queue with server slow-down and blocking

Abstract We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a ‘blocking threshold.’ In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a ‘slow-down threshold, ’ which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρ i is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results.