Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases

Sojourn times in a multiclass processor sharing queue

We consider a processor sharing queue with several customer classes. For an arbitrary customer of class i we show that the sojourn time distribution is regularly varying of index -\nu_i iff the service time distribution is regularly varying of index -\nu_i, and derive an explicit asymptotic formula. Furthermore, the tail of the sojourn time distribution of customer class i is shown to be unaffected by the tails of the service time distributions of other customer classes, even if some of the latter tails are heavier. This result implies that, when the sojourn time of a customer is large, this is not due to long service requirements of other customer types. In particular, short-range dependent traffic does not suffer from longe-range dependent traffic if processor sharing is used as a service discipline

A fluid queue with a finite buffer and subexponential input

We consider a fluid model similar to that of Kella and Whitt [33], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established in [33] are extended to the finite buffer case. It is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N on-off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue

Queues with regular variation

X+173hlm.;24c

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Heavy Loads and Heavy Tails

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic limit in infinite-variance scenarios. This serves as a tribute to some of the pioneering results of J.W. Cohen in this domain. We specifically demonstrate that reduced-load equivalence properties established for the tail asymptotics of the workload naturally extend to the heavy-traffic limit

Long range dependence in network traffic and the closed loop behaviour of buffers under adaptive window control

We consider an Internet link carrying http-like traffic, i.e., transfers of finite volume files arriving at random time instants. These file transfers are controlled by an adaptive window protocol (AWP); an example of such a protocol is TCP. We provide analysis for the auto-covariance function of the AWP-controlled traffic into the link's buffer; this traffic, in general, cannot be represented by an on-off process. The analysis establishes that, for TCP-controlled transfer of Pareto-distributed file sizes with infinite second moment, the traffic into the link buffer is long range-dependent (LRD). We also develop an analysis for obtaining the stationary distribution of the link buffer occupancy under an AWP-controlled transfer of files sampled from some distribution. For any AWP, the analysis provides us with the Laplace-Stieltjes transform (LST) of the distribution of the link buffer occupancy process in terms of the functions defining the AWP and the file size distribution. The analysis also provides a necessary and a sufficient condition for the finiteness of the mean link buffer content; these conditions again have explicit dependence on the AWP used and the file size distribution. This establishes the sensitivity of the buffer occupancy process to the file size distribution. Combining the results from the above analyses, we provide various examples in which the closed loop control of an AWP results in finite mean link buffer occupancy even though the file sizes are Pareto-distributed (with infinite second moment), and the traffic into the link buffer is long range-dependent (with Hurst parameters which would suggest an infinite mean queue occupancy under open loop analysis). We also study the effect of window reductions due to active queue management and find that window reductions lead to further lightening of the tail of buffer occupancy distribution. The significance of this work is three-fold: (i) by looking at the window evolution as a function of the amount of data served and not as a function of time, this work provides a new framework for analysing various processes related to the link buffer under AWP-controlled transfer of files with a general file size distribution; (ii) it indicates that the buffer behaviour in the Internet may not be as poor as predicted from an open loop analysis of a queue fed with LRD traffic; and (iii) it shows that the buffer behaviour (and hence the throughput performance for finite buffers) is sensitive to the distribution of file sizes