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Delay analysis plays a role in real-time systems in computer communication networks. This paper gives our results in the aspect of delay analysis of fractal traffic passing through servers. There are three contributions presented in this paper. First, we will explain the reasons why conventional theory of queuing systems ceases in the general sense when arrival traffic is fractal. Then, we will propose a concise method of delay computation for hard real-time systems as shown in this paper. Finally, the delay computation of fractal traffic passing through severs is presented.

There are two categories of communications to perform the delivery of a message M from the source A to the destination B. One is in the sense of best effort. By best effort, one means that the computer communication system, which is denoted by S, does not guarantee the connection of sending M from A to B, and accordingly, the quantity of the time delay

In the case of guaranteed connections, there are two types of communication systems. One is in the type of real-time systems. The other is in the type of nonreal-time ones. By real-time system, one implies that the predetermined time delay should be guaranteed (Natarajan and Zhao [

In the field of computer communications, there are two categories of real-time systems. One is for hard real-time systems, and the other is for soft ones. By hard real-time systems, we mean that the time constraint, more precisely, the predetermined time delay, has to be assured. Otherwise, the communication is regarded as a failure ([

Recall that the time constraint mentioned above is the message delay suffering from S from A to B (Sandmann [

While we mentioned above that delay serves as a key parameter in the aspect of traffic passing through servers in the field of computer networks, one may say that the delay denoted by

Queuing system for single server.

Note that traffic of the Markovian type implies that it is light tailed. By light tail, we mean that its autocorrelation function (ACF) is exponentially decayed and so are its power spectrum density (PSD) function and probability density function (PDF) (Li and Zhao [

Possible applications of conventional queuing theory to delay analysis are in the case of fractal traffic models with finite variance, such as fractional Brownian motion (fBm), fractional Gaussian noise (fGn); see, for example, Norros [

The previous discussions imply that the key reason that makes the conventional queuing theory very difficult, if not impossible, to be used in the delay analysis of communication systems with fractal arrivals is the fractal properties of traffic, namely, self-similarity and LRD. Thus, fractal arrival traffic substantially challenges queuing theory of real-time systems.

As known, performance analysis of conventional queuing systems has to assume that statistical means and variances of arrival traffic exist (Cooper et al. [

Note that variance analysis of random functions or time series plays a key role in statistics (Bendat and Piersol [

There are two categories with respect to the theory of network calculus. One is for deterministic delay analysis of queuing systems (Le Boudec and Thiran [

This paper aims at presenting novel computation methods of delay of fractal traffic passing through servers without relating to the concepts of means and variances of arrival traffic.

The rest of the paper is organized as follows. We will give the brief of fractal traffic in Section

Denote by

Denote by

Let

Traffic

Expressing

We consider the local behavior of traffic

Taqqu’s law says that the PDF of a random function

Denote by

Previous discussions imply the following remarks.

Traffic follows power laws.

It is LRD.

It is approximately self-similar.

It is a type of

It is heavy tailed.

LRD is a global property of traffic, which is measured by

Fractal dimension

In general, we do not talk about means and variances of traffic. Instead, we are interested in other two, namely, local self-similarity and LRD in the theory of fractal traffic.

Network calculus may be applied to the delay analysis with respect to quality of service (QoS) in computer communication networks ([

Traffic passing through single server.

Question 1: how to model arrival traffic

Question 2: how to design a service scheme, which is denoted by

Question 3: in order to guarantee the predetermined delay when

The answer to question 1 is about traffic modeling. The one to question 2 is about system modeling. That to the third is the relationship among the arrival

In order to assure a predetermined delay

The literature regarding statistical envelopes of light-tailed random functions is rich, as they are needed in many fields of sciences and technologies, ranging from electronics engineering to ocean one; see, for example, Rice [

In the society of computer science, people are interested in a type of envelopes of traffic, called bounding models of traffic (Michiel and Laevens [

There are two parameters in the above expression. One is

As a matter of fact, on one hand, we have

The deterministic envelop of traffic, namely,

Remark

Denote a service curve of a server by

As previously mentioned,

Let

The reports regarding delay computation are rich; see, for example, [

Denote by

According to (

Suppose that

Note that

Denote the inverse of

Let (

Consider arrival traffic

Arrival traffic

One way to find the end-to-end delay of

Denote

The discussions in the previous subsections produce the following remarks.

The above delay analysis and its computations do not need any information of the statistics of arrival traffic

The delay can be deterministically guaranteed. Hence, the deterministic queuing systems as Le Boudec and Thiran stated in [

The advantage described by Remarks

We previously reported our bound of arrival traffic by taking into account its fractal dimension

Denote by

According to (

The bandwidth regarding

Remark

Theorem

Note that (

We previously mentioned several times that we are studying queuing systems irrelevant to statistical means and variances of arrival traffic because variances and or means of traffic may not exist [

We have explained the reasons why conventional theory of queuing systems is inappropriate to be used in the delay analysis of queuing systems when arrival traffic is fractal. Then, we have given concise method of delay computation of deterministic queuing systems. Finally, we have derived the computation method of delay when arrival traffic is fractal.

This work was supported in part by the National Natural Science Foundation of China under the project Grant nos. 61272402, 61070214, and 60873264 and by the 973 Plan under the project Grant nos. 2011CB302800.

^{α}power law noise generation

^{γ}power spectrum noise sequence generator