- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Computational and Mathematical Methods in Medicine

Volume 2012 (2012), Article ID 154038, 11 pages

http://dx.doi.org/10.1155/2012/154038

## Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

^{1}School of Information Science & Technology, East China Normal University, No. 500 Dong-Chuan Road, Shanghai 200241, China^{2}Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa, Macau

Received 23 March 2011; Accepted 20 April 2011

Academic Editor: Carlo Cattani

Copyright © 2012 Ming Li and Wei Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

#### 1. Introduction

We use the term cyber-physical networking systems (CPNS) instead of cyber-physical systems (CPS) as that in Song et al. [1] for the meaning of Internet of Things (IoT) that was stated by Commission of the European Communities [2] or Networks of Things (NoT) as discussed by Ferscha et al. [3], intending to emphasize the point that we are interested in the networking theory in CPS. Communication networks in CPNS include, but are never limited to, the Internet. Physical systems considered in CPNS are heterogeneous, ranging from telemedicine systems to geophysical ones, see, for example, Clifton et al. [4], Traynor [5], Chang [6]. Obviously, data in various physical systems are heterogeneous, see, for example, Chang [6], Goodchild [7], Lai and Xing [8], Mandelbrot [9–11], Hainaut and Devolder [12], Cattani [13–17], Chen et al. [18–22], Mikhael and Yang [23], Bakhoum and Toma [24–26], Li [27–32], Li et al. [33–39], Messina et al. [40], Humi [41], Dong [42], Liu [43], Toma [44], Abuzeid et al. [45], [46–49], Werner [50], and West [51], just naming a few.

There are two challenge issues in CPNS. On the one hand, data models that are irrelevant of statistics of a random function are greatly desired. On the other hand, theory that may be used to linearize nonlinear data transmission systems but irrelevant of their nonlinearity is particularly expected, because communication systems, including the Internet, are, in nature, nonlinear due to queuing, see, for example, Akimaru and Kawashima [52], Yue et al. [53], Gibson [54], Cooper [55], Pitts and Schormans [56], McDysan [57], and Stalling [58]. In short, we are interested in *data models that are irrelevant of their statistics and system theory that is irrelevant of the nonlinearity of systems*.

The early work regarding the above in italic may refer to Cruz [59–61], Zhao and Ramamritham [62], Raha et al. [63], Chang [64, 65], Boudec [66], Boudec and Patrick [67], Firoiu et al. [68], and Agrawal et al. [69]. Following Cruz [59, 60], the theory for the above in italic is called network calculus, see, for example, [66, 67], Jiang and Liu [70]. Chang [71] uses the term calculus, which is taken as the synonym of network calculus of Cruz in this paper.

The main application area of network calculus is conventionally to computer science, the Internet in particular, see, for example, Wang et al. [72, 73], Li and Zhao [74, 75], Fidler [76], Jiang [77], Jiang et al. [78], Liu et al. [79], Li et al. [80], Li and Kinghtly [81], Burchard et al. [82], Ng et al. [83], Raha et al. [84, 85], Starobinski and Sidi [86], Fukś et al. [87], Jia et al. [88], Golestani [89], and Lenzini et al. [90]. However, we have to emphasize the point that its applications are never limited to computer science. Rather, it is a theory to model data irrelevant of their statistics and to deal with data transmission without the necessity in principle to consider the nonlinearity of transmission systems, as we shall explain in the next section. Therefore, it may be a promising tool to deal with data and systems in CPNS.

Basically, the fundamental theory of network calculus consists of three parts as described below.(i) model of arrival data ,(ii)relationship between , single system (or node or server) that is usually called service curve, and departure data ,(iii)departure data of a series of systems (nodes or servers) , driven by arrival data ,

where min-plus algebra plays a role, see, for example, [66, 67, 70, 71, 76].

The contributions of this paper are in the following three aspects: (i)the problem statement,(ii)the proof of the existence of the identity in the min-plus algebra in the domain of generalized functions,(iii)the asymptotic expression of the identity.

The rest of paper is organized as follows. Research background is discussed in Section 2. In Section 3, we will brief the min-plus algebra and state the problem regarding the identity in this algebra system. In Section 4, we shall address the existence of the identity in the min-plus algebra. The asymptotic expression of the identity is presented in Section 5. Discussions are given in Section 6, which is followed by conclusions.

#### 2. Research Background

Data in CPNS are heterogeneous. They may be from sensors like radio-frequency identification (RFID), see, for example, [91], Ilie-Zudor et al. [92], Ahuja and Potti [93], data traffic in the Internet [38], transportation traffic (see [94–98]), ocean waves (see [31]), sea level (see [36, 99]), medical signals (see [14]), hydrological data (see [100]), financial data (see [101]), and so on. They may be Gaussian (see [29, 31]) or non-Gaussian (see [12, 102]). They may be in fractional order or integer order. In the case of fractional order, they may be unifractal or multifractal. The sample size of data of interest may be long enough for statistical analysis or very short, for example, a short conversation in mobile phone networks. On the other side, systems are also heterogeneous. Therefore, CPNS challenges us two tough issues. One is in data modeling and the other system modeling. We shall exhibit that the min-plus algebra in network calculus may yet serve as a tool in this regard.

##### 2.1. Network Model

We first explain a single node in CPNS. Then, a model of tandem network is mentioned.

###### 2.1.1. Nonlinearity of Node in CPNS

Denote by a node in CPNS, see Figure 1. Suppose there are clients arriving at the input of at time , see, for example, Starobinski et al. [103].

Without confusions, we use to represent the operator of node such that Recall that queuing is a phenomenon often occurring in CPNS. For instance, cars in highways are often queued. Clients in a library for borrowing or returning books need queuing. Suppose client suffers from delay . Then, Note that is a random variable in two senses. One is The other is Therefore, we have the following remark.

*Remark 1 (nonlinearity). *A node in CPNS is usually nonlinear. That is,

###### 2.1.2. Number of Arrivals is Random

The number of arrivals, denoted by in Figure 1, is random.

*Note 1. *We need theory to deal with a nonlinear node with arrival clients, where is a random variable.

###### 2.1.3. Tandem Network Model

A single node previously described is not enough in CPNS since a client may be served by a series of nodes, which we call tandem network, see Figure 2.

According to Remark 1, each node in Figure 2 is nonlinear. In addition, considering Note 1, we see that the number of arrival clients at the input of each node is random. Some clients may go through from to while others may not. For instance, client leaves the tandem network when it passes through . Further more, some clients, for example, , arrive at this tandem network at the input of . In general, how many clients leave the tandem network at the output of a specific node and how many clients arrive at the input of another specific node are uncertain.

*Note 2. *We need theory to handle a nonlinear system that is a tandem network as that in Figure 2 to assure the quality of service (QoS) of a specific client or of a specific class of clients within a given period of time.

The above Note 1 and Note 2 propose two challenge tasks in system theory. We shall explain how min-plus algebra is capable of dealing with those tasks late.

##### 2.2. Data Modeling

We consider two classes of data flow. One is arrival data in the aggregated case, or aggregated clients, and the other arrival data of a specific client. In terms of network communications, the former is usually called aggregated arrival traffic while later arrival traffic at connection level. Without confusions, we use the term traffic rather than client.

One of radical properties of arrival traffic (traffic for short) is remarked below.

*Remark 2 (positive). *Traffic is positive. That is,
where is the set of real numbers.

Another radical property of traffic is that the maximum of is finite. More precisely, the value of may never be infinite. Thus, we have the following remark.

*Remark 3 (finite range). *The maximum of is finite. That is,

*Remark 4 (randomness). *The function is usually random. This implies that

###### 2.2.1. Traffic at Connection Level

At connection level, for instance, for the connection, traffic is . One particularity of is that for usually lasts within a finite time interval, say, . The width of the interval may be short, such as a short conversation like a word “hello” or long, such as a long speech over a network. In any case, it is finite. Modeling with short interval is particularly desired and challenging.

*Note 3. *In the discrete case, the length of may be too short to the proper statistical analysis of in practice.

*Note 4. *Without confusions, we use to represent the interval in both the continuous case and the discrete one. In the continuous case, . In the discrete case, , where is the set of integer numbers, implying . We use to represent an interval the starting point of which is nonzero.

###### 2.2.2. Aggregated Traffic

We adopt Figure 1 to discuss aggregated traffic. At time , aggregated traffic denoted by at a node is expressed by In contrary to , the particularity of is that for usually lasts within an interval longer than that of . As a matter of fact, if passes through a node, another arrival flow may arrive at the node. Consequently, in general, we should consider for .

##### 2.3. Accumulated Traffic

Traffic, either or , discussed previously is instantaneous one. Data modeling of instantaneous traffic is essential, as we need understanding what its behaviors are at instantaneous time at the input of a node. However, from the point of view of the service of a node, we also need data modeling of accumulated traffic within a time interval, say, , without loss of generality, because it is desired for us to understand what the service performance of the node is for the purpose of proper design of a buffer size as well as scheduling policy of the node.

###### 2.3.1. Accumulated Traffic at Connection Level

In the continuous case, the accumulated traffic of within the interval is denoted by . It is given by In the discrete case,

###### 2.3.2. Accumulated Traffic in the Aggregated Case

Denote by the accumulated traffic in the aggregated case within the interval . Then, in the continuous case, we have In the discrete case,

The mathematical expressions of and appear similar except the subscript . However, differs from substantially in analysis in methodology. On the one hand, for should be assumed to be short such that conventional methods in statistics fail to its statistical analysis. On the other hand, for may be large enough such that it may be sectioned for the statistical analysis, see, for example, Li et al. [104].

###### 2.3.3. A Basic Property of Accumulated Traffic

One property of accumulated traffic, either or , is the wide sense increasing. By wide sense increasing, we mean that or

Therefore, the data functions or series we face with are increasing ones in the wide sense.

###### 2.3.4. Model of Data

For and , the following is called the model of data ,

*Note 5. *The model expressed by (16) is irrelevant of any information of statistics of . The advantage of this model is at the cost of using inequality instead of equality.

*Note 6. *The model of (16) is simple in computation. Thus, it may be effective in practice, particularly in environments of CPNS, where simple computations are always expected.

For accumulated traffic , we have Due to sufficiently large , we may set the starting time by . In this case, we have Moreover, we are allowed to section the above integral such that Without loss of generality, we use (17) to explain and .

*Remark 5. *The parameter represents the bound of the burstness or local irregularity of , because

Note that the above integral does not make sense if for the continuous even in the field of the Lebesgue’s integrals, see Dudley [105], Bartle and Sherbert [106], and Trench [107] for the contents of the Lebesgue’s integrals. However, it makes sense when it is considered in the domain of generalized functions, which we shall brief in the following section. A simple way to explain (20) is where and is the Dirac- function.

*Remark 6. *The parameter represents the bound of the average rate of , because

*Remark 7. *The parameter measures the local property of while is a measure of global property of .

#### 3. Min-Plus Algebra and Problem Statement

Min-plus convolution is essential in the min-plus algebra. In this section, we first briefly review the conventional convolution in linear systems. Then, we shall visit min-plus convolution. Finally, we shall state the problem in the aspect of identity in the min-plus algebra.

##### 3.1. Conventional Convolution

Denote by a real number that satisfies . If a function defined on , where is allowed to be and is allowed to be , is measurable and we say that .

Suppose that two functions . Then, one says that convolutes if where is the symbol implying the operation of convolution. We call it conventional convolution so as to distinguish it from the min-plus convolution we are discussing in this paper.

The conventional convolution is crucial for linear systems, see, for example, Gibson [54], Box et al. [108], Mitra and Kaiser [109], Papoulis [110], Harris [111], Mikusinski [112], Fuller [113], and Bendat and Piersol [114], just naming a few. It has the properties described by the following lemmas.

Lemma 1. *In the algebra system , the conventional convolution is commutative. *

Lemma 2 (closure of *). *If , then .*

Lemma 3. *In the algebra system , where + implies the ordinary addition, * with respect to + is distributive.*

Lemma 4. *For , .*

Lemma 5. *The identity in is the Dirac- function that is defined by
**
where is continuous at .*

In fact, in the domain of generalized functions, we have Thus, in the sense of generalize functions. Consequently, is taken as the asymptotic identity in in the domain of generalized functions. Accordingly, the inverse of the conventional convolution discussed by, for instance, Mikusinski [112], Bracewell [115], Huang and Qiu [116], Abutaleb et al. [117], Rhoads and Ekstrom [118], Todoeschuck and Jensen [119], and Moreau et al. [120], exists because the necessary and sufficient condition that the inverse of an operation exists is that there exists the identity in that system, see, for example, Korn and Korn [121], Zhang [122], Riley et al. [123], Bronshtein et al. [124], and Stillwell [125], but it should be in the sense of generalized functions. As a matter of fact, the conventional convolution itself is in that sense, see, for example, Smith [126].

Theorem 1. *The algebra system is a group.*

*Proof. *First, the operation * is closed in . Second, * is commutative because, for any ,

Finally, there exists the left identity denoted by and the right one again denoted by in such that

Thus, is a group.

##### 3.2. Min-Plus Convolution

Considering the property of wide sense increasing of accumulated traffic mentioned in Section 2.3, we denote by the set that contains all functions that are greater than or equal to zero and that are wide sense increasing.

*Definition 1. *Let . Then, the following operation is called min-plus convolution:
where represents the operation of the min-plus convolution.

*Example 1. *Let for and 0 elsewhere. Then, .

Lemma 6 . (closure of ). *Let . Then, .*

Lemma 7. *The operation is commutative. That is,
*

Define another operation that is denoted by such that

Then, we have an algebra system denoted by that follows the distributive law.

Lemma 8. *The operation with respect to is distributive. That is, for , one has
*

The following rule useful in this research is stated as follows.

Lemma 9. *Suppose . Then, for , one has
**
where + is the ordinary addition.*

Denote by the conventional identity in the min-plus algebra, which is defined by see [66–70].

It seems quite obvious when one takes as the identity in the min-plus algebra since However, we shall soon point the contradictions of below.

##### 3.3. Problem Statement

Denote by the Heavyside unit step function. That is, Then, for , we have Using (34), we have The above is an obvious contradiction regarding the conventional identity defined by (34).

In addition to the above contradiction, we now state another problem regarding (34). As a matter of fact, if we let and in Lemma 9, then, on the left side of (33) in Lemma 9, we have On the other side, on the right side of (33) in Lemma 9, we have Comparing the right sides of (39) with that of (40) yields another contradiction expressed by

The above discussions imply that the definition of the identity of (34) in the min-plus algebra, which is commonly used in literature, see, for example, [66–70], may not be rigorous at least. Therefore, the conventional representation of the identity, that is, (34), may be inappropriate since it may mislead computation results like those in (39) and (40). Consequently, rigorous definition of the identity needs studying.

#### 4. Existence of Identity in Min-Plus Algebra

The problems regarding the definition of the conventional identity, which we stated in Section 3.3, give rise to a question whether or not the identity in the min-plus algebra exists. The answer to this question is rarely seen, to the best of our knowledge. Another question resulted from Section 3.3 is what the rigorous representation of the identity is. We shall provide the answer to the first question in this section. The answer to the second will be explained in the next section.

##### 4.1. Preliminaries

We brief some results in generalized functions [127–129] for the purpose of discussing the existence of identity.

*Definition 2. *Let be the support of a function . It implies . The function is said to have a bounded support if there exist such that .

*Definition 3. *A function is said to have time continuous derivatives if its first derivatives exist and are continuous. If its derivatives of all orders exist and are continuous, is said to be infinitely differentiable. In this case, is said to be smooth.

*Definition 4. *A test function is a smooth with . The set of all test functions is denoted by .

*Definition 5. *A linear functional on is a map such that, for and , .

*Definition 6. *Denote by a sequence of test functions and another test function. We say that if the following holds:(1)there is an interval that contains and for all ,(2) uniformly for .

*Definition 7. *A functional on is continuous if it maps every convergent sequence in into a convergent sequence in . A continuous linear functional on is termed a generalized function. It is often called a distribution in the sense of Schwartz.

*Definition 8. *A function is locally integrable if for all .

Lemma 10. *Any continuous, including piecewise continuous, function is locally integrable.*

Lemma 11 (regular). *Any locally integrable function is a generalized function defined by
**
In this case, is called regular.*

Lemma 12. *Any generalized function has derivatives of all orders.*

Lemma 13. *There exists the Fourier transform of any generalized function.*

*Definition 9 (rapid function). *A function of rapid decay is a smooth function such that as for all , where is the space of complex numbers. The set of all functions of rapid decay is denoted by .

Lemma 14. *Every function belonging to is absolutely integrable.*

##### 4.2. Proof of Existence

Define the norm and inner product of by where . Combining any with its limit yields a Hilbert space that we denote again by without confusions.

Let be a system function such that it transforms its input to the output by

Denote the system by the operator . Then, we purposely force the functionality of such that it maps an element to another element . Note that is a linear operator. In fact, according to Lemma 8, we have In addition, from Lemma 9, we have Therefore, is a linear mapping from to .

Denote by the space consisting of all such operators by Then, from Lemmas 8 and 9, one can easily see that is a linear space.

Lemma 15 (archimedes criterion). *For any positive real numbers and , there exists positive integer such that (see [130]).*

Lemma 16 (archimedes). *If , there exists such that (see [106]).*

Lemma 17. *An operator is invertible if and only if there exists constant such that for all , , where and are linear normed spaces (see [131]).*

From the above discussions, we obtain the following theorem.

Theorem 2 (existence). *For and and , if or , then both and are invertible. Consequently, the identity in the min-plus algebra exists.*

*Proof. *Consider
Since
and , we have
According to Lemmas 15 and 16, there exists such that
Therefore,
Similarly, if is such that , we have since , where is a constant. Thus, according to Lemma 17, Theorem 2 holds.

*Note 7. *In Theorem 2, we need the conditions of and . Since and are wide sense increasing, we need in fact and .

#### 5. Representation of Identity in Min-Plus Algebra

Express the Dirac- function by

For the purpose of distinguishing the identity we present from the conventional one, we denote as the identity in what follows instead of as used in Section 3.

Theorem 3 (representation). *The identity in the min-plus algebra is expressed by
*

*Proof. *Take the following into account
Then, the identity in the discrete case is given by
The identity in the continuous case is taken as the limit expressed by
Considering the Poisson’s summation formula, we have
In the limit case,
This completes the proof.

*Remark 8. *If one uses the representation in Theorem 3, the contradictions given in (38) and (41) vanish.

*Note 8. *The identity expressed by (59) is an asymptotic one.

#### 6. Discussions

We mention an application of min-plus algebra to CPNS. Denote by the accumulated function characterizing the output of the node (Figure 3). Then, the min-plus convolution can be used to establish the relationship between , , and by

Suppose a traffic function passes through tandem nodes from the first node with the service curve to the th node with the service curve to reach the destination as indicated in Figure 4. Denote the departure traffic of the th node by . Then, where (see [132])

*Note 9. *Min-plus algebra can be used to linearize a nonlinear system as can be seen from (62). Thus, it may yet be used as a theory in the aspect of data transmission systems in CPNS.

#### 7. Conclusions

We have proposed the problem regarding the conventional identity in the min-plus algebra. In addition, we have presented the proof that the identity in the min-plus algebra exists in the domain of generalized function. Moreover, we have given the asymptotic expression of the identity in the system of min-plus algebra.

#### Acknowledgment

This work was partly supported by the National Natural Science Foundation of China (NSFC) under the project Grant nos. 60873264, 61070214, and the 973 plan under the Project no. 2011CB302801/2011CB302802.

#### References

- Z. Song, Y. Q. Chen, C. R. Sastry, and N. C. Tas,
*Optimal Observation for Cyber-Physical Systems*, Springer, 2009. - Commission of the European Communities, “Internet of things—an action plan for Europe,”
*Journal of International Wildlife Law and Policy*, vol. 12, no. 1-2, pp. 108–120, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Ferscha, M. Hechinger, A. Riener et al., “Peer-it: stick-on solutions for networks of things,”
*Pervasive and Mobile Computing*, vol. 43, no. 3, pp. 448–479, 2008. View at Publisher · View at Google Scholar · View at Scopus - G. D. Clifton, H. Byer, K. Heaton, D. J. Haberman, and H. Gill, “Provision of pharmacy services to underserved populations via remote dispensing and two-way videoconferencing,”
*American Journal of Health-System Pharmacy*, vol. 60, no. 24, pp. 2577–2282, 2003. View at Google Scholar · View at Scopus - K. Traynor, “Navy takes telepharmacy worldwide,”
*American Journal of Health-System Pharmacy*, vol. 67, no. 14, pp. 1134–1136, 2010. View at Google Scholar - K. T. Chang,
*Introduction to Geographical Information Systems*, McGraw–Hill, New York, NY, USA, 2008. - M. F. Goodchild, “Twenty years of progress: GIScience in 2010,”
*Journal of Spatial Information Science*, vol. 2010, no. 1, pp. 3–20, 2010. View at Google Scholar - T. L. Lai and H. Xing,
*Statistical Models and Methods for Financial Markets*, Springer, 2008. - B. B. Mandelbrot,
*Gaussian Self-Affinity and Fractals*, Springer, 2001. - B. B. Mandelbrot,
*Multifractals and 1/f Noise*, Springer, 1998. - B. B. Mandelbrot,
*The Fractal Geometry of Nature*, W. H. Freeman, New York, NY, USA, 1982. - D. Hainaut and P. Devolder, “Mortality modelling with Lévy processes,”
*Insurance*, vol. 42, no. 1, pp. 409–418, 2008. View at Publisher · View at Google Scholar · View at Scopus - C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 207–217, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. Cattani, “Fractals and hidden symmetries in DNA,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 507056, 31 pages, 2010. View at Publisher · View at Google Scholar - G. Mattioli, M. Scalia, and C. Cattani, “Analysis of large-amplitude pulses in short time intervals: application to neuron interactions,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 895785, 15 pages, 2010. View at Publisher · View at Google Scholar - C. Cattani, “Shannon wavelets for the solution of integrodifferential equations,”
*Mathematical Problems in Engineering*, vol. 2010, 22 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. Cattani, “Shannon wavelets theory,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 164808, 24 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,”
*IEEE Transactions on Image Processing*, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation for vision processing,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 416963, 12 pages, 2011. View at Publisher · View at Google Scholar - S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,”
*IEEE Sensors Journal*, vol. 11, no. 2, pp. 389–390, 2011. View at Publisher · View at Google Scholar - S. Y. Chen and Q. Guan, “Parametric shape representation by a deformable NURBS model for cardiac functional measurements,”
*IEEE Transactions on Biomedical Engineering*, vol. 58, no. 3, pp. 480–487, 2011. View at Publisher · View at Google Scholar - S. Chen, J. Zhang, H. Zhang et al., “Myocardial motion analysis for determination of tei-index of human heart,”
*Sensors*, vol. 10, no. 12, pp. 11428–11439, 2010. View at Publisher · View at Google Scholar - W. Mikhael and T. Yang, “A gradient-based optimum block adaptation ICA technique for interference suppression in highly dynamic communication channels,”
*EURASIP Journal on Advances in Signal Processing*, vol. 2006, Article ID 84057, 10 pages, 2006. View at Publisher · View at Google Scholar - E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 428903, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 695208, 15 pages, 2010. View at Publisher · View at Google Scholar - E. G. Bakhoum and C. Toma, “Relativistic short range phenomena and space-time aspects of pulse measurements,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 410156, 20 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “Change trend of averaged Hurst parameter of traffic under DDOS flood attacks,”
*Computers and Security*, vol. 25, no. 3, pp. 213–220, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “A class of negatively fractal dimensional gaussian random functions,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 291028, 18 pages, 2011. View at Publisher · View at Google Scholar - M. Li, “Generation of teletraffic of generalized cauchy type,”
*Physica Scripta*, vol. 81, no. 2, Article ID 025007, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “Fractal time series-a tutorial review,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar - M. Li, “A method for requiring block size for spectrum measurement of ocean surface waves,”
*IEEE Transactions on Instrumentation and Measurement*, vol. 55, no. 6, pp. 2207–2215, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in hilbert space-a further study,”
*Applied Mathematical Modelling*, vol. 31, no. 3, pp. 625–631, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Li, W. Zhao, and S. Chen, “MBm-based scalings of traffic propagated in internet,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 389803, 21 pages, 2011. View at Publisher · View at Google Scholar - M. Li and W. Zhao, “Variance bound of ACF estimation of one block of fGn with LRD,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 560429, 14 pages, 2010. View at Publisher · View at Google Scholar - M. Li and W. Zhao, “Detection of variations of local irregularity of traffic under DDOS flood attack,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 475878, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. Li, C. Cattani, and S.-Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 654284, 13 pages, 2011. View at Publisher · View at Google Scholar - M. Li and J. Y. Li, “On the predictability of long-range dependent series,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 397454, 9 pages, 2010. View at Publisher · View at Google Scholar - M. Li and S. C. Lim, “Modeling network traffic using generalized cauchy process,”
*Physica A*, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. Li, S. C. Lim, and Sy. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 657839, 9 pages, 2011. View at Publisher · View at Google Scholar - A. R. Messina, P. Esquivel, and F. Lezama, “Time-dependent statistical analysis of wide-area time-synchronized data,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 751659, 17 pages, 2010. View at Publisher · View at Google Scholar - M. Humi, “Assessing local turbulence strength from a time series,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 316841, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Dong, “A tutorial on nonlinear time-series data mining in engineering asset health and reliability prediction: concepts, models, and algorithms,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 175936, 22 pages, 2010. View at Publisher · View at Google Scholar - Z. Liu, “Chaotic time series analysis,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 720190, 31 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - G. Toma, “Specific differential equations for generating pulse sequences,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 324818, 11 pages, 2010. View at Publisher · View at Google Scholar - O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 652306, 22 pages, 2010. View at Publisher · View at Google Scholar - M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 302786, p. 13, 2012. View at Google Scholar - M. Li and S. C. Lim, “Power spectrum of generalized cauchy process,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 219–222, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. He, H. Qian, Y. Zhou, and Z. Li, “Cryptanalysis and improvement of a block cipher based on multiple chaotic systems,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 590590, 14 pages, 2010. View at Publisher · View at Google Scholar - Z. Liao, S. Hu, and W. Chen, “Determining neighborhoods of image pixels automatically for adaptive image denoising using nonlinear time series analysis,”
*Mathematical Problems in Engineering*, vol. 2010, 2010. View at Publisher · View at Google Scholar · View at Scopus - G. Werner, “Fractals in the nervous system: conceptual implications for theoretical neuroscience,”
*Frontiers in Fractal Physiology*, vol. 1, p. 28, 2010. View at Google Scholar - B. J. West, “Fractal physiology and the fractional calculus: a perspective,”
*Frontiers in Fractal Physiology*, vol. 1, 2010. View at Google Scholar - H. Akimaru and K. Kawashima,
*Teletraffic: Theory and Applications*, Springer, 1993. - W. Yue, H. Takagi, and Y. Takahashi,
*Advances in Queueing Theory and Network Applications*, Springer, 2009. - J. D. Gibson,
*The Communications Handbook*, IEEE Press, 1997. - R. B. Cooper,
*Introduction to Queueing Tehory*, Elsevier, 2nd edition, 1981. - J. M. Pitts and J. A. Schormans,
*Introduction to ATM Design and Performance: With Applications Analysis Software*, John Wiley & Sons, New York, NY, USA, 2nd edition, 2000. - D. McDysan,
*QoS & Traffic Management in IP & ATM Networks*, McGraw-Hill, New York, NY, USA, 2000. - W. Stalling,
*High-Speed Networks: TCP/IP and ATM Design Principles*, Prentice Hall, 2nd edition, 2002. - R. L. Cruz, “A calculus for network delay—part 1: network elements in isolation part 2,”
*IEEE Transactions on Information Theory*, vol. 37, no. 1, pp. 114–141, 1991. View at Publisher · View at Google Scholar · View at Scopus - R. L. Cruz, “A calculus for network delay—part 2: network analysis,”
*IEEE Transactions on Information Theory*, vol. 37, no. 1, pp. 132–141, 1991. View at Publisher · View at Google Scholar · View at Scopus - R. L. Cruz, “Quality of service guarantees in virtual circuit switched networks,”
*IEEE Journal on Selected Areas in Communications*, vol. 13, no. 6, pp. 1048–1056, 1995. View at Publisher · View at Google Scholar · View at Scopus - W. Zhao and K. Ramamritham, “Virtual time csma protocols for hard real-time communication,”
*IEEE Transactions on Software Engineering*, vol. 13, no. 8, pp. 938–952, 1987. View at Google Scholar · View at Scopus - A. Raha, S. Kamat, and W. Zhao, “Guaranteeing end-to-end deadlines in ATM networks,” in
*Proceedings of the 15th International Conference on Distributed Computing Systems*, pp. 60–68, June 1995. View at Scopus - C. S. Chang, “On deterministic traffic regulation and service guarantees: a systematic approach by filtering,”
*IEEE Transactions on Information Theory*, vol. 44, no. 3, pp. 1097–1110, 1998. View at Google Scholar · View at Scopus - C. S. Chang, “Stability, queue length, and delay of deterministic and stochastic queueing networks,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 5, pp. 913–931, 1994. View at Publisher · View at Google Scholar · View at Scopus - J. Y. Le Boudec, “Application of network calculus to guaranteed service networks,”
*IEEE Transactions on Information Theory*, vol. 44, no. 3, pp. 1087–1096, 1998. View at Google Scholar · View at Scopus - J. Y. Le Boudec and T. Patrick,
*Network Calculus, A Theory of Deterministic Queuing Systems for the Internet*, Springer, 2001. - V. Firoiu, J. Y. Le Boudec, D. Towsley, and Z. L. I. Zhang, “Theories and models for internet quality of service,”
*Proceedings of IEEE*, vol. 90, no. 9, pp. 1565–1591, 2002. View at Publisher · View at Google Scholar · View at Scopus - R. Agrawal, F. Baccelli, and R. Rajan,
*An Algebra for Queueing Networks with Time Varying Service and Its Application to the Analysis of Integrated Service Networks*, INRIA, 1998, RR-3435. - Y. M. Jiang and Y. Liu,
*Stochastic Network Calculus*, Springer, 2008. - C. S. Chang,
*Performance Guarantees in Communication Networks*, Springer, 2000. - H. Wang, J. B. Schmitt, and I. Martinovic, “Dynamic demultiplexing in network calculus—theory and application,”
*Performance Evaluation*, vol. 68, no. 2, pp. 201–219, 2011. View at Publisher · View at Google Scholar - S. Q. Wang, D. Xuan, R. Bettati, and W. Zhao, “Toward statistical QoS guarantees in a differentiated services network,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 253–263, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. Z. Li and W. Zhao, “Stochastic performance analysis of non-feedforward networks,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 237–252, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Li and W. Zhao, “Representation of a stochastic traffic bound,”
*IEEE Transactions on Parallel and Distributed Systems*, vol. 21, no. 9, pp. 1368–1372, 2010. View at Google Scholar - M. Fidler, “Survey of deterministic and stochastic service curve models in the network calculus,”
*IEEE Communications Surveys and Tutorials*, vol. 12, no. 1, pp. 59–86, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. M. Jiang, “Per-domain packet scale rate guarantee for expedited forwarding,”
*IEEE /ACM Transactions on Networking*, vol. 14, no. 3, pp. 630–643, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. M. Jiang, Q. Yin, Y. Liu, and S. Jiang, “Fundamental calculus on generalized stochastically bounded bursty traffic for communication networks,”
*Computer Networks*, vol. 53, no. 12, pp. 2011–2019, 2009. View at Publisher · View at Google Scholar · View at Scopus - Y. Liu, C. K. Tham, and Y. M. Jiang, “A calculus for stochastic QoS analysis,”
*Performance Evaluation*, vol. 64, no. 6, pp. 547–572, 2007. View at Publisher · View at Google Scholar · View at Scopus - C. Li, A. Burchard, and J. Liebeherr, “A network calculus with effective bandwidth,”
*IEEE/ACM Transactions on Networking*, vol. 15, no. 6, pp. 1442–1453, 2007. View at Publisher · View at Google Scholar · View at Scopus - C. Li and E. Knightly, “Schedulability criterion and performance analysis of coordinated schedulers,”
*IEEE/ACM Transactions on Networking*, vol. 13, no. 2, pp. 276–287, 2005. View at Publisher · View at Google Scholar · View at Scopus - A. Burchard, J. Liebeherr, and S. D. Patek, “A min-plus calculus for end-to-end statistical service guarantees,”
*IEEE Transactions on Information Theory*, vol. 52, no. 9, pp. 4105–4114, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. K. Y. Ng, S. Song, and W. Zhao, “Integrated end-to-end delay analysis for regulated ATM networks,”
*Real-Time Systems*, vol. 25, no. 1, pp. 93–124, 2003. View at Publisher · View at Google Scholar · View at Scopus - A. Raha, S. Kamat, X. Jia, and W. Zhao, “Using traffic regulation to meet end-to-end deadlines in ATM networks,”
*IEEE Transactions on Computers*, vol. 48, no. 9, pp. 917–935, 1999. View at Publisher · View at Google Scholar · View at Scopus - A. Raha, W. Zhao, S. Kamat, and W. Jia, “Admission control for hard real-time connections in ATM LANs,”
*IEE Proceedings*, vol. 148, no. 4, pp. 1–12, 2001. View at Publisher · View at Google Scholar · View at Scopus - D. Starobinski and M. Sidi, “Stochastically bounded burstiness for communication networks,”
*IEEE Transactions on Information Theory*, vol. 46, no. 1, pp. 206–212, 2000. View at Publisher · View at Google Scholar · View at Scopus - H. Fukś, A. T. Lawniczak, and S. Volkov, “Packet delay in models of data networks,”
*ACM Transactions on Modeling and Computer Simulation*, vol. 11, no. 3, pp. 233–250, 2001. View at Publisher · View at Google Scholar · View at Scopus - X. Jia, W. Zhao, and J. Li, “An integrated routing and admission control mechanism for real-time multicast connections in ATM networks,”
*IEEE Transactions on Communications*, vol. 49, no. 9, pp. 1515–1519, 2001. View at Publisher · View at Google Scholar · View at Scopus - S. J. Golestani, “Network delay analysis of a class of fair queueing algorithms,”
*IEEE Journal on Selected Areas in Communications*, vol. 13, no. 6, pp. 1057–1070, 1995. View at Publisher · View at Google Scholar · View at Scopus - L. Lenzini, E. Mingozzi, and G. Stea, “A methodology for computing end-to-end delay bounds in FIFO-multiplexing tandems,”
*Performance Evaluation*, vol. 65, no. 11-12, pp. 922–943, 2008. View at Publisher · View at Google Scholar · View at Scopus - INFSO D.4 Networked Enterprise & RFID INFSO G.2 Micro & Nanosystems, in co-operation with the Working Group RFID of the ETP EPOSS, Internet of Things in 2020, Roadmap for the Future[R]. Version 1.1, May 2008.
- E. Ilie-Zudor, Z. Kemény, F. van Blommestein, L. Monostori, and A. van Der Meulen, “A survey of applications and requirements of unique identification systems and RFID techniques,”
*Computers in Industry*, vol. 62, no. 3, pp. 227–252, 2011. View at Publisher · View at Google Scholar - S. Ahuja and P. Potti, “An introduction to RFID technology,”
*Communications and Network*, vol. 2, no. 3, pp. 183–186, 2010. View at Google Scholar - R. Wang, L. Zhang, R. Sun, J. Gong, and L. Cui, “A pervasive traffic information acquisition system based on wireless sensor networks,”
*IEEE Transactions on Intelligent Transportation Systems*, vol. 12, no. 2, pp. 615–621, 2011. View at Publisher · View at Google Scholar - W. H. K. Lam, S. C. Wong, and H. K. Lo, “Emerging theories in traffic and transportation; and emerging methods for transportation planning and operations,”
*Transportation Research Part C*, vol. 19, no. 2, pp. 169–171, 2011. View at Google Scholar - L. N. Emmanuel, R. T. Antonio, and L. M. Ernesto, “A modeling framework for urban traffic systems microscopic simulation,”
*Simulation Modelling Practice and Theory*, vol. 18, no. 8, pp. 1145–1161, 2010. View at Google Scholar - J. A. Laval, “Hysteresis in traffic flow revisited: an improved measurement method,”
*Transportation Research Part B*, vol. 45, no. 2, pp. 385–391, 2011. View at Publisher · View at Google Scholar - B. G. Heydecker and J. D. Addison, “Analysis and modelling of traffic flow under variable speed limits,”
*Transportation Research Part C*, vol. 19, no. 2, pp. 206–217, 2011. View at Google Scholar - P. Santoro, M. Fernández, M. Fossati, G. Cazes, R. Terra, and I. Piedra-Cueva, “Pre-operational forecasting of sea level height for the Río de la Plata,”
*Applied Mathematical Modelling*, vol. 35, no. 5, pp. 2264–2278, 2011. View at Publisher · View at Google Scholar - D. She and X. Yang, “A new adaptive local linear prediction method and its application in hydrological time series,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 205438, 15 pages, 2010. View at Publisher · View at Google Scholar - S. V. Muniandy, S. C. Lim, and R. Murugan, “Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates,”
*Physica A*, vol. 301, no. 1–4, pp. 407–428, 2001. View at Publisher · View at Google Scholar · View at Scopus - G. Samorodnitsky and M. S. Taqqu,
*Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance*, Chapman and Hall, New York, NY, USA, 1994. - D. Starobinski, M. Karpovsky, and L. A. Zakrevski, “Application of network calculus to general topologies using turn-prohibition,”
*IEEE/ACM Transactions on Networking*, vol. 11, no. 3, pp. 411–421, 2003. View at Publisher · View at Google Scholar - M. Li, S. Wang, and W. Zhao, “A real-time and reliable approach to detecting traffic variations at abnormally high and low rates,”
*Springer Lecture Notes in Computer Science*, vol. 4158, pp. 541–550, 2006. View at Google Scholar · View at Scopus - R. M Dudley,
*Real Analysis and Probability*, Cambridge University Press, 2002. - R. G. Bartle and D. R. Sherbert,
*Introduction to Real Analysis*, John Wiley & Sons, 3rd edition, 2000. - W. F. Trench,
*Introduction to Real Analysis*, Pearson Education, 2003. - G. E. P. Box, G. M. Jenkins, and G. C. Reinsel,
*Time Series Analysis: Forecasting and Control*, Prentice Hall, 1994. - S. K. Mitra and J. F. Kaiser,
*Handbook for Digital Signal Processing*, John Wiley & Sons, 1993. - A. Papoulis,
*The Fourier Integral and Its Applications*, McGraw-Hill, 1962. - C. M. Harris,
*Shock and Vibration Handbook*, McGraw-Hill, 4th edition, 1995. - J. Mikusinski,
*Operational Calculus*, Pergamon Press, 1959. - W. A. Fuller,
*Introduction to Statistical Time Series*, Wiley, 2nd edition, 1996. - J. S. Bendat and A. G. Piersol,
*Random Data: Analysis and Measurement Procedure*, John Wiley & Sons, 3rd edition, 2000. - R. N. Bracewell,
*The Fourier Transform and Its Applications*, McGraw-Hill, New York, NY, USA, 2nd edition, 1978. - X. Huang and P. H. Qiu, “Blind deconvolution for jump-preserving curve estimation,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 350849, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. S. Abutaleb, N. M. Elhamy, and M. E. S. Waheed, “Blind deconvolution of the aortic pressure waveform using the malliavin calculus,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 102581, 27 pages, 2010. View at Publisher · View at Google Scholar - R. L. Rhoads and M. P. Ekstrom, “Removal of interfering system distortion by deconvolution,”
*IEEE Transactions, Instrumentation and Measurement*, vol. 17, no. 4, pp. 333–337, 1968. View at Google Scholar - J. P. Todoeschuck and O. G. Jensen, “Scaling geology and seismic deconvolution,”
*Pure and Applied Geophysics*, vol. 131, no. 1-2, pp. 273–287, 1989. View at Publisher · View at Google Scholar · View at Scopus - S. Moreau, G. Plantier, J. C. Valière, H. Bailliet, and L. Simon, “Estimation of power spectral density from laser Doppler data via linear interpolation and deconvolution,”
*Experiments in Fluids*, vol. 50, no. 1, pp. 179–188, 2011. View at Publisher · View at Google Scholar · View at Scopus - G. A. Korn and T. M. Korn,
*Mathematical Handbook for Scientists and Engineers*, McGraw-Hill, 1961. - H. R. Zhang,
*Elementary of Modern Algebra*, People's Education Press, China, 1978. - K. F. Riley, M. P. Hobson, and S. J. Bence,
*Mathematical Methods for Physics and Engineering*, Cambridge Press, 2006. - I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig,
*Handbook of Mathematics*, Springer, 2007. - J. Stillwell,
*Mathematics and Its History*, Springer, 3rd edition, 2010. - D. C. Smith, “An introduction to distribution theory for signals analysis—part 2. The convolution,”
*Digital Signal Processing*, vol. 16, no. 4, pp. 419–444, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. Lighthill,
*An Introduction to Fourier Analysis and Generalised Functions*, Cambridge University Press, 1958. - R. P. Kanwal,
*Generalized Functions: Theory and Applications*, Birkhauser, 3rd edition, 2004. - I. M. Gelfand and K. Vilenkin,
*Generalized Functions*, vol. 1, Academic Press, New York, NY, USA, 1964. - A. D. Aleksandro,
*Mathematics, Its Essence, Methods and Role*, vol. 3, USSR Academy of Sciences, 1952. - V. I. Istratescu,
*Introduction to Linear Operator Theory*, Marcel Dekker, New York, NY, USA, 1981. - M. Li and W. Zhao, “Sufficient condition for min-plus deconvolution to be closed in the service-curve set in computer networks,”
*International Journal of Computers*, vol. 1, no. 3, pp. 163–166, 2007. View at Google Scholar