Overflow Probability Research Articles

The Buffer-Bandwidth Trade-off Curve is Convex

To achieve a constant overflow probability, the two queueing resources, viz. buffer and bandwidth, can be traded off. In this paper we prove that, under general circumstances, the corresponding trade-off curve is convex in the ‘many-sources scaling’. This convexity enables optimal resource partitioning in a queueing system supporting heterogeneous traffic, with heterogeneous quality-of-service requirements.

The significance of finite buffer edge effects in histogram measurements of queues

The empirically oberved "fractal" or "self-similar" nature of packet traffic implies heavy tailed queue processes for such traffic. However, based on our simulation analysis using real network data as well as standard models, we have found that the actual losses sustained are remarkably lower than those suggested by the heavy tail distribution. This can be explained by an effect observed in the tail of the histogram of a finite buffer queue process, which we call "tail-raising", which contains information pertinent to performance estimation. This effect is also responsible for a significant reduction in packet losses for finite buffer systems, than would be otherwise predicted by the buffer overflow probability for heavy-tailed queues. We define a new parameter X B on the histogram of a queue process for a finite buffer system, to calculate the tail of the queue process based on the information available in the histogram on the finite buffer. We propose an estimator that approximates X B , namely, X min?, which is measurable because of the tail-raising effect and has a robust measurement method. The proposed estimator shows promise as a good predictor for performance metrics of queueing systems. We propose an innovative packet loss ratio estimation technique which uses histogram measurements combined with a virtual buffer scheme to find and extrapolate the objective packet loss rate using a binning strategy for histogram measurement, namely, Symmetric Logarithmic Binning (SLB).

Cell loss probability for M/G/1 and time-slotted queues

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

Cell loss probability for M/G/1 and time-slotted queues

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

Overflow behavior in queues with many long-tailed inputs

We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B and link rate C are scaled by n, so that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where the on periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull.The results show a sharp dichotomy in the qualitative behavior, depending on the shape of the function v(t) := - logP(A* > t) for large t, A* representing the residual on period. If v(.) is regularly varying of index 0 (e.g., Pareto, Lognormal), then, during the path to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will be ‘more than linear’ in the buffer size. In contrast, if v(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is roughly proportional to the buffer size.In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribute at their mean rates. These observations lead to approximations for the overflow probability. The approximations may be extended to the case of heterogeneous sources. The results provide further insight into the so-called reduced-load approximation.

Modeling and simulation of a readout architecture for pixel detectors

This paper analyzes in detail some theoretical aspects in the modeling of a proposed readout architecture for pixel detectors. The readout architecture is designed for a chip containing about 3000 pixels of 50 /spl mu/m/spl times/400 /spl mu/m. The main objective is to get the maximum pixel hit readout with the minimum probability of hit loss. The readout architecture is modeled as a Markov stochastic process. The pixel front-end and readout are simulated and tested with Monte Carlo data. The simulations allow to optimize the communication channel bandwidths and local buffering. The probability of system overflow of the simulated system is confronted with the one obtained by modeling.

Analysis of local internet/enterprise networkingmultiplexerfor ON-OFF VoIP sources

The authors analyse the multiplexing of a small number of independent and homogenous ON-OFF traffic sources into a single packet switched buffer: a possible model of VoIP in a local internet. This work complements a previous paper in which a large number of sources was considered. It is shown how a previously developed formula for queues can be applied to this system, allowing accurate calculation of the buffer overflow probability.

Design rules and equivalent capacity for bufferingof Pareto source

Formulae are developed that act as design rules when buffering an ON-OFF source with long range dependent (LRD) characteristics. These can be used to predict the buffer overflow probability, or the equivalent capacity required to ensure a specified buffer overflow probability. The traffic source has Pareto distributed ON and OFF periods, and therefore this analysis can be seen as an extension of earlier results for exponential ON and OFF periods.

Traffic overflow in loss systems with selective trunk reservation

We consider two traffic streams competing for service at an n-server queuing system. Jobs from stream 1, the protected stream, are blocked only if all n servers are busy. Jobs from stream 2, the best effort stream, are blocked if n− r, r≥1, servers are busy. Blocked customers are diverted to a secondary group of c− n servers with, possibly, a different service rate. For the case r=1, we calculate the joint probabilities of the number of primary and secondary busy servers. For r>1, we describe a procedure for deriving the joint probabilities. These probabilities allow for the calculation of various performance measures including the overflow probabilities of the primary server and secondary server group. Our model is applicable to traffic control in communication networks that use the selective trunk reservation method.

Modelling the self-similar behaviour of network traffic

Traditional Poisson or Markovian traffic models have been proven, by several studies, inappropriate for predicting the performance of networks with long-range dependent traffic. Thus, the use of conventional call admission control algorithms based on short-range dependent models cannot achieve the quality of service required by self-similar sources, since network performance is actually worse in terms of buffer overflow probability and delay. The variety of traffic types that depict long-range dependence creates the need for new models suitable for network design and congestion control mechanisms. In this paper, we present a call admission control (CAC) scheme for traffic with self-similar behaviour based on the notion of effective bandwidth. Our solution satisfies the quality of service requirements of the source, expressed in terms of buffer overflow probability, while taking into account the long-range dependence property of the traffic through the Hurst parameter. We base our CAC mechanism on a Gaussian model initially studied by Norros. We show that this model performs well in the case of one source as well as in the case of many aggregated sources. Our scheme assumes homogenous sources, each one represented by a fractional Gaussian noise process characterised by the mean bit rate and the self-similarity parameters (H, α). We study the estimation of ( H, α) from the real traffic and show the connection between their estimates. Finally, we present a way to calculate the equivalent capacity, which is necessary so that the aggregated stream will achieve the desired loss rate. In our simulation experiments, we used data collected from an IP over ATM high-speed link, connecting the National Research Network of Greece (GRnet) to the Trans-European network backbone (TEN-34), a part of the global Internet.

Design rules for buffering overlapping Pareto processesin packetised networks

A formula is presented that acts as a design rule when multiplexing many homogeneous long range dependent (LRD) ON-OFF processes through a buffer. The formula enables the buffer overflow probability, or the required buffer length, to be predicted. Sources have Pareto distributed ON periods and exponentially distributed OFF periods. From the results (formulas and numerical), valuable tools and insight can be developed which will be applicable to the design of future packetised communication networks, switches and routers.

Queueing analysis of scheduling policies in copy networks of space-based multicast packet switches

Space-based multicast switches use copy networks to generate the copies requested by the input packets. In this paper our interest is in the multicast switch proposed by Lee (1988). The order in which the copy requests of the input ports are served is determined by the copy scheduling policy and this plays a major part in defining the performance characteristics of a multicast switch. In any slot, the sum of the number of copies requested by the active inputs of the copy network may exceed the number of output ports and some of the copy requests may need to be dropped or buffered. We first propose an exact model to calculate the overflow probabilities in an unbuffered Lee's copy network. Our exact results improve upon the Chernoff bounds on the overflow probability given by Lee by a factor of more than 10. Next, we consider buffered inputs and propose queueing models for the copy network for three scheduling policies: cyclic service of the input ports with and without fanout splitting of copy requests and acyclic service without fanout splitting. These queueing models obtain the average delay experienced by the copy requests. We also obtain the sustainable throughput of a copy network, the maximum load that can be applied to all the input ports without causing an unstable queue at any of the inputs, for the scheduling policies mentioned above.

STATIC AND DYNAMIC EVALUATION OF QoS PROPERTIES

Efficient utilization of modern high bandwidth communication networks relies on statistical multiplexing of many logical channels through one physical channel. Communication requests typically include some statistical characterization of the requested connection (such as pick value, mean values, etc). The task of the network management procedure is to accommodate as many communication requests as possible while keeping the failure (e.g. overflow) probability bounded by a pre-specified parameter. When the network consists of one link, the task reduces to evaluating the probability that a sum of random variables does not exceed a given bound. Techniques such as the method of effective bandwidth give a practical solution for the one link problem. In this paper we address the more realistic setting of estimating QoS properties of multi-link networks with arbitrary patterns. The related optimization problem for that setting is #P-complete even for the most simple communication characteristics. Our main result is an efficient Monte-Carlo method for estimating the failure probability of a general network. Our method is particularly useful in a dynamic setting in which communication requests are dynamically added and eliminated from the system. The amortized cost in our solution of updating the estimate after each change is proportional to the fraction of links involved in the change rather than to the total number of links in the network.

A Prediction System for the Reliability Improvement of the Vehicle-Follower Control on the Automated Highway

This paper describes a prediction approach which allows the evaluation of collision and unhooking risks in the automatic control of truck platoons on highways. In a real time context, this study aims at predicting the time that remains to a follower truck to overflow one of the collision or the unhooking limits. On the basis of this predicted time, and of the truck dynamic and temporal characteristics, this prediction system must deliver an alarm when the overflow probability exceeds a predefined threshold, and thus, the transition to the manually operated mode will be activated. The prediction system is based on the linear regression model of the inter-truck spacing signal trend.

Overflow probability in shared cell switched buffers

The ubiquity of the cell switching technique is a clear indication of its continued importance for the future of telecommunications worldwide. The performance of shared buffers in cell switches provides substantial improvement over switches with separate output buffers. The conventional approach to shared buffer analysis is numerical convolution; however, previous work provides an analytical solution based on the Chernoff bound. In this paper we develop a new closed-form expression for the cell loss probability in the shared buffer that offers superior accuracy, enhanced simplicity, and better generality.

Buffer overflow probability for multiplexed on-offVoIP sources

The authors analyse the multiplexing of a large number of independent and homogenous on-off traffic sources into a single packet switched buffer: a possible model of VoIP. It is shown how a previously developed formula for queues can be applied to this system, allowing accurate calculation of the buffer overflow probability via the burst-scale decay rate of the buffer state probabilities. Numerical results comparing the expression with simulation and with other approximations shows that the approach gives superior accuracy, with equivalent complexity.

Overflow behavior in queues with many long-tailed inputs

We consider a fluid queue fed by a superposition of $n$ homogeneous on-off sources with generally distributed on- and off-periods. We scale buffer space $B$ and link rate $C$ by $n$, such that we get $nb$ and $nc$, respectively. Then we let $n$ grow large. In this regime, the overflow probability decays exponentially in the number of sources $n$; we specifically examine the situation in which also $b$ is large. We explicitly compute asymptotics for the case in which the on-periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull. We provide a detailed interpretation of our results. Crucial is the shape of the function $v(t) := - log Pr( A^* < t )$ for large~$t$, $A^*$ being the residual on-period. If $v(.)$ is slowly varying (e.g., Pareto, Lognormal), then, during the trajectory to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill `slowly'', and the typical time to overflow will be `more than linear'' in the buffer size. In contrast, if $v(.)$ is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is essentially proportional to the buffer size. In both cases there is a substantial fraction of the sources that remain in the on-state during the entire path to overflow, while the other sources contribute at mean rate. These observations lead to straightforward approximations for the overflow probability. These approximations can be extended to the case of heterogeneous sources; the results provide further insight into the so-called reduced-load approximation.

Overflow probability for a discrete-time queue with non-stationary multiplexed input

Recently it has been reported that various traffic exhibit long?range dependence and/or self?similarity. However, there exist some reports which suggest that traffic may be non?stationary. In this paper we discuss the average overflow probability JN L(Ly)?E[N?1Σn=1 NI{Qn L>Ly}] in a finite period [1,N] for a discrete?time queue Qn L with a non?stationary multiplexed input from L sources. By using the large deviation principle, we analyze the asymptotic behavior of JN L(Ly) as L increases and obtain an approximation formula of the form Jn L(Ly)?C(L,y)?e??(y). We apply the approximation formula to two examples. One is a non?stationary process with deterministic level shifts and the other is a Gaussian process with spectrum of the form f??. The latter is non?stationary if ??1.

Allocating Bandwidth for Bursty Connections

In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probability p are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most $p$. In bin-packing we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many connections as possible. For the stochastic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variables. For the other two problems we have algorithms restricted to on-off sources (the most common special case studied in the statistical multiplexing literature), with a somewhat weaker range of performance guarantees. A standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth---this is a means of associating a fixed demand with a bursty connection that represents its distribution as closely as possible. Our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce; the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution.

On the decay rates of buffers in continuous flow lines

Consider a tandem system of machines separated by infinitely large buffers. The machines process a continuous flow of products, possibly at different speeds. The life and repair times of the machines are assumed to be exponential. We claim that the overflow probability of each buffer has an exponential decay, and provide an algorithm to determine the exact decay rates in terms of the speeds and the failure and repair rates of the machines. These decay rates provide useful qualitative insight into the behavior of the flow line. In the derivation of the algorithm we use the theory of Large Deviations.