He is the author of numerous scientific papers on stochastic processes and their applications and the coauthor of the influential book on Modelling of Extremal Events for Insurance and Finance. He has published extensively on selfsimilarity and stable processes. Home Selfsimilar Processes. Add to Cart.

More about this book. Selfsimilar processes crop up in a wide range of subjects from finance to physics, so this book will have a correspondingly wide readership. Everybody is talking about scaling, and selfsimilar stochastic processes are the basic and the clearest examples of models with scaling. In applications from finance to communication networks, selfsimilar processes are believed to be important.

Yet much of what is known about them is folklore; this book fills the void and gives reader access to some hard facts. And because this book requires only modest mathematical sophistication, it is accessible to a wide audience. Paul Embrechts.

## Self-Similarity of Network Data Analysis

Subject Areas. Continuous-time arrival processes are useful in performance evaluation involving continuous-time intervals as in the case of certain random access tree algorithms. For example, see, J. Longo, Ed. New York: Springer, , pp. Gallager, Conflict resolution in random access broadcast networks, in Proc. Theory and Applications, Provincetown, Mass. Clearly, N t is a wide-sense stationary count process. The unit of time may be chosen to be a slot, a frame, and the like. To this end, it suffices to have each customer, starting from its arrival, generate a new packet every unit-time interval during its life time i.

The relation between r k and p k is governed by Eq. In view of Eq. Thus, an exactly second-order self-similar traffic stream is obtained.

### Kundrecensioner

Conversely, known forms for the service time distribution may be utilized for approximating certain autocorrelation structures, if desired asymptotic autocorrelation properties can be established via Eq. This approach leads to the construction of asymptotically second-order self-similar traffic. Having established heavy-tailed distributed service time to be a necessary condition for exact self-similarity, it can be shown that it is a sufficient condition for asymptotic self-similarity.

Without loss of the asymptotic nature of F n , the conventional form of the denominator has been revised in Eq. Given Eq. Only a generic time unit is required for defining the basic count process. Mean aggregate processes are obtained by averaging over increasing sizes of time intervals. To facilitate comparison, and in consistence with the definition cited previously for N m t , the ordinate is normalized to represent the number of packets per time unit for all aggregate count processes. It should also be noted that all traffic stream examples shown in FIGS. Other packet arrival rates have been found to give rise to self-similar patterns and bursty features as well.

Recalling the normalization process applied to FIG. The fluctuation, or burstiness, of the traffic count process of the present invention, however, gradually decreases with increasing order of time averaging processes.

### More about this book

Such smoothing behavior also occurs when traffic patterns generated from different arrival rates are compared with the ordinates scaled proportionally to the respective arrival rates, thereby indicating smoothing over source multiplexing. Such smoothing properties suggest that time or source multiplexing may suppress the effects of self-similarity on certain network performance, as confirmed below.

One of the aggregate processes for both exactly and asymptotically second-order self-similar traffic shown in FIG. In that regard, FIGS.

## USB1 - Portable self-similar traffic generation models - Google Patents

Specifically, FIGS. As anticipated, the smaller the H value, the less bursty the traffic. As H decreases to 0. It is interesting to note that asymptotic self-similarity, as modeled by a Pareto service time distribution, appears to be more bursty than its exact counterpart having the same H value. Accordingly, the model provided by the present invention can be conveniently incorporated into a trace-driven simulation for providing in-flight traffic arrival generation, thereby precluding a separate traffic generation and input procedure, and saving memory.

The popular slotted Aloha protocol and its corresponding multichannel version are now considered for illustrating the effects of input traffic patterns and of statistical multiplexing on random access protocols. A brief description of the subject is provided before the simulation results are presented.

## Self-Similar Processes in Telecommunications

The slotted Aloha protocol that is considered below has been stabilized by pseudo-Bayesian control, such as disclosed by R. Theory, vol. IT, pp. Consequently, delay-throughput simulation was carried out.

An infinite source population model has been assumed with each source terminal having at most one packet to transmit at any given time. Moreover, a packet is successfully transmitted if and only if there is one terminal transmitting during a time slot, while collided packets resulting from simultaneous transmissions from multiple terminals are retransmitted later on.

Thus, the stabilized Aloha operates as follows:. At the beginning of each time slot, say slot k, backlogged terminals i. The multichannel slotted Aloha protocol considered herein is also defined by extending random access to a certain number c of slotted Aloha channels.

By pooling multiple channels together, individual traffic streams that would otherwise transmit on separate channels were multiplexed into a single pool of channels, while the capacity of the server is accordingly increased from that of a single channel to that of a pool of channels. According to queuing theory, for unslotted contention-free transmissions, such pooling decreases the average delay by a factor of c, such as disclosed by D. Bertsekas et al. Upper Saddle River, N. Note that the average delay includes both waiting time, i.

Furthermore, under a Poisson offered load, the maximum achievable throughput of the multichannel slotted Aloha remains the same as for the unichannel slotted Aloha, as can be readily established.

The multichannel slotted Aloha can be stabilized by a simple extension of the above pseudo-Bayesian rule. The effect of self-similarity on unichannel slotted Aloha was tested first.

- Almost uniqueness result for reversed variational inequalities.
- US6526259B1 - Portable self-similar traffic generation models - Google Patents.
- J.K. Lassers Homeowners Tax Breaks 2006: Your Complete Guide to Finding Hidden Gold in Your Home (J.K. Lasser);
- Product description;
- EUREKA: Physics and Engineering.
- Self-Similar Processes in Communications Networks.
- Sight Unseen.

Similar behavior has been noted in the literature for a common channel under a tree algorithm having a self-similar input. It is here conjectured that this is caused by the self-similar nature of the input traffic, which generates packet arrivals correlated over any time scale. In particular, a traffic stream with a given average arrival rate has in a realization a certain number of packets input to the channel over some time period. Most of these packets go through the channel with some average delay and delay variation over that time period.

The pattern of fluctuations along the run time changes not only with average arrival rate, but also with each realization i. Consequently, an open issue is raised whether statistical averages are still meaningful performance measures in characterizing some random access systems. On the other hand, a pseudo-Bayesian controlled channel slotted Aloha is adaptable to traffic self-similarity, as shown by FIG. Again, similar results were obtained when self-stable tree algorithms were used in place of stabilized Aloha algorithms.

In a separate study by J. It may be observed that pooling frequency-division channels is essentially equivalent to pooling time-division channels. Thus, the detrimental impact of self-similarity is drastically magnified on single-channel protocol performance, but greatly subdued on multi-channel protocol performance. While the present invention has been described in connection with the illustrated embodiments, it will be appreciated and understood that modifications may be made without departing from the true spirit and scope of the invention. Effective date : Year of fee payment : 4.

According to one aspect of the method, the predetermined service time distribution is selected based on the selected predetermined autocorrelation function, thereby the step of generating the simulation of self-similar traffic generates self-similar traffic having an exact second-order self-similarity. According to another aspect of the method, the predetermined service time distribution is selected based on a desired predetermined heavy-tailed distribution function; thereby the step of generating the simulation of self-similar traffic generates self-similar traffic having an asymptotic second-order self-similarity.

Field of the Invention The present invention relates to the field of telecommunications. Description of the Related Art Traditional traffic models used for characterizing the behavior of a telecommunications network, such as traffic models that are based on a Poisson process, do not sufficiently model the bursty nature of broadband telecommunications. What is claimed is: 1. A method for mimicking a stream of self-similar traffic in a telecommunications network, the method comprising steps of:.

The method according to claim 1 , wherein the predetermined service time distribution is selected based on the selected predetermined autocorrelation function, and. The method according to claim 1 , wherein the predetermined service time distribution is selected based on a desired predetermined heavy-tailed distribution function, and.

The method according to claim 1 , further comprising a step of applying the generated stream of self-similar traffic to a portion of a telecommunications network. The method according to claim 4 , further comprising the step of simulating a behavior of the portion of the telecommunications network based on the applied stream of self-similar traffic. USP true USB1 en. Method and apparatus for improving performance in a network using a virtual queue and a switched poisson process traffic model.