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Consider a network multiplexer with a finite buffer fed by a superposition of independent heterogeneous On-Off sources. An On-Off source consists of a sequence of alternating independent activity and silence periods. During its activity period a source produces fluid with constant rate. For this system, under the assumption that the residual activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary overflow probability and loss rate. The derived asymptotic formulas, in addition to their analytical tractability, exhibit excellent quantitative accuracy, which is illustrated by a number of simulation experiments. We demonstrate through examples how these results can be used for efficient computing of capacity regions for network switching elements. Furthermore, the results provide important insight into qualitative tradeoffs between the overflow probability, offered traffic load, available capacity, and buffer space. Overall, they provide a new set of tools for designing and provisioning of networks with heavytailed traffic streams. Keywords---Network multiplexer, Finite buffer fluid queue, On-Off process, Heavy-tailed distributions, Subexponential distributions, Long-range dependence I.

. We consider a fluid queue with ON periods arriving according to a Poisson process and having a long--tailed distribution. This queue has long range dependence, and we compute the asymptotic behavior of the steady state distribution of the buffer content. The tail of this distribution is much heavier than the tail of the buffer content distribution of a queue which does not possess long range dependence and which has light tailed ON periods and the same traffic intensity. 1. Introduction and preliminaries We consider a model of a network server (multiplexer) defined as follows. Sessions arrive to the server according to a Poisson process with rate ? 0. Each session lasts a random length of time with distribution F that has a finite mean . The lengths of different sessions are independent of each other and of the Poisson arrival process. A session generates work or traffic or fluid at unit rate, commonly measured in some units of network traffic, e.g. packets; the work that cannot be ...

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 P(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(·) ...

Consider a fluid queue with a finite buffer B and capacity c fed by a superposition of N independent On-Off processes. An On-Off process consists of a sequence of alternating independent activity and silence periods. Successive activity, as well as silence, periods are identically distributed. The process is active with probability p on and during its activity period produces fluid with constant rate r. For this queueing system, under the assumption that the residual activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary loss probability and loss rate. In the case of homogeneous sources with residual activity periods equal in distribution to on r , the queue overflow probability is asymptotically, as B !1, equal to P[Q B = B] = ` N k 0 ' p k 0 on P on r ? B k 0 (r \Gamma ae) +N ae \Gamma c k 0 (1 + o(1)); where ae = rp on , N ae ! c and k 0 is the smallest integer greater than (c...

We derive the sojourn time asymptotics for a multi-class G/G/1 queue with regularly varying service requirements operating under the Discriminatory Processor-Sharing (DPS) discipline. DPS provides a natural approach for modelling the flowlevel performance of differentiated bandwidth-sharing mechanisms. Under certain assumptions, we prove that the service requirement and sojourn time of a given class have similar tail behaviour, independent of the specific values of the DPS weights. As a by-product, we obtain an extension of the tail equivalence for ordinary Processor-Sharing (PS) queues to non-Poisson arrivals. The results suggest that DPS offers a potential instrument for effectuating preferential treatment to high-priority classes, without inflicting excessive delays on low-priority classes. To obtain the asymptotics, we develop a novel method which only involves information of the workload process and does not require any knowledge of the steady-state queue length distribution. In particular, the proof method brings sufficient strength to extend the results to scenarios with a time-varing service capacity.

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 two-node tandem network where flow i is served in is...

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 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 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.

Abstract. Motivated by the problem of the coexistence on transmission links of telecommunication networks of elastic and unresponsive traffic, we study in this paper the impact on the busy period of an M/M/1 queue of a small perturbation in the server rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε of the mean value of the busy period duration. This allows us to study the validity of the Reduced Service Rate (RSR) approximation, which consists in comparing the perturbated M/M/1 queue with the M/M/1 queue where the service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of

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. We exploit a powerful intuitive argument to obtain the exact asymptotics for the reduced system. Combined with the reduced-load equivalence, the results for the reduced system provide an asymptotic characterization of the buffer behavior. 2000 Mathematics Subject Classification: 60K25 (primary), 60F10, 90B18, 90B22 (secondary). Keywords and Phrases: fluid models, heavy-tailed distributions, knapsack problem, large deviations, queueing theory, reduced-load equivalence. I.