A Weighted Ranking Algorithm For Facet-Based Component Retrieval System Facet-based component retrieval techniques have been proved to be an effective way for retrieving . These Techniques are widely adopted by component library systems , but they usually simply list out all the retrieval results without any kind of ranking . In our work , we focus on the problem that how to determine the ranks of the components retrieved by user . Factors which can influence the ranking are extracted and identified through the analysis of ER-Diagram of facet-based component library system . In this paper , a mathematical model of weighted ranking algorithm is proposed and the timing of ranks calculation is discussed . Experiment results show that this algorithm greatly improves the efficiency of component retrieval system . A high efficiency retrieval system for software component library is important for the reuse of software components . The point of high efficiency is not that the time performance in one matching or retrieving process which can be measured by how many seconds or how many milliseconds elapsed , but that the efficiency to make the component consumers be able to find what they need as soon as possible , even though the former is the basis of the latter . No matter accuracy matching or fuzzy matching , our component retrieval system usually simply lists out all the retrieval results without any kind of ranking , or at least without a systematic ranking . Users have to view the detail information of all the retrieval results one by one to find out which is the best to fit their requirements , or else they have to adjust their query conditions to retrieve again . If there are a large number of components retrieved from the component library , it could be a tough and torturous experience to find a proper component . However , it 's a fact that there 's a matching degree between the query conditions and retrieval results . The matching degree is just the similarity and relevancy between the query condition and its retrieval results . Only when we rank the retrieval results by the matching degree as the Web search engines can component consumers easily find what they need . They only have to compare the first several retrieval results but not all of them . According to the discussion above , it 's clear that a formula to calculate the matching degree and its corresponding ranking algorithm , which can greatly improve the retrieval efficiency for software component library , are needed . In this paper , we propose a weighted ranking algorithm for facet-based component retrieval system . This algorithm has been implemented in a software component library , called DLCL , and greatly improves the efficiency of the retrieval system . 2.1 Existing Retrieval Methods for Component Library Retrieval of software components is a core technique of component library . Today there are lots of retrieval methods for software component library . The main are as follows -LSB- 1 , 2 -RSB- : -LRB- 1 -RRB- Specification matching method ; -LRB- 2 -RRB- AI Based method ; -LRB- 3 -RRB- Information science method ; -LRB- 4 -RRB- Hypertext browsing method . As to the four methods , each has its own features and there 's no a general formula to calculate the matching degree . For example , specification matching method uses formal specifications to describe the behavior of software components and relies on theorem proving to determine match and mismatch . AI Based method relies on the use of AI planning techniques to automatically search software components in component library . So we have to use different calculating strategies to calculate the matching degree of each retrieval method . Among the retrieval methods discussed above , information science method is widely used in practice . Information science method usually comprises several different retrieval methods which are attribute-value , enumerated , faceted , and keyword method . Of the four methods , facet-based component retrieval method has been proved to be an effective way for retrieving and has been widely adopted by component library systems . In the following section , we 'll discuss the facet-based retrieval method . 505-049 274 2.2 Facet-based Retrieval Method A component classification is a set of -LCB- facet , facet term -RCB- pairs , also called descriptors -LSB- 3 -RSB- . Reusable software components -LRB- RSC -RRB- are classified by assigning appropriate facet terms for all applicable facets . The objective in classifying the RSC is to make it possible for all users who might use the RSC to retrieve it as a result of their requests . Faceted classification scheme is an effective way for classifying the components and widely adopted by component library systems . Correspondingly , there are several retrieving algorithms for faceted classification scheme . Some systems use the traditional database query techniques in facet-based retrieval . Wang YF proposed a tree matching algorithm in his PH.D dissertation -LSB- 4 -RSB- . This algorithm maps the component facets into a facet tree and maps the query conditions into a query tree . The matching algorithm deals with the match of the facet tree and query tree and calculates the matching cost . This algorithm bases on the tree matching theories , such as tree embedding , tree inclusion , and tree containment . These three tree matching methods are becoming more and more elastic in order to improve the retrieving recall while maintaining the precision to a certain extent . Matching cost of the tree matching will be calculated to measure the approximate degree between the facet trees of the components and the query tree . The data structure of a tree is represented by a three-tuple : T = -LRB- V , E , root -LRB- T -RRB- -RRB- , V represents a limited set of vectors , root -LRB- T -RRB- represents the root of the tree , E represents the set of edges . There 's no a general formula to calculate the matching degree due to the different feature of each retrieval method . Facet-based retrieval method has been widely adopted by existing component library systems , such as REBOOT , Proteus , Asset Library , and JBCL -LSB- 5 -RSB- . It has been proved to be an effective method to the retrieval of component library system . And therefore , it makes great sense to propose a component ranking algorithm for facet-based retrieval system . 3.1 ER-Diagram of Software Component Library The extraction and identification of the influential factors which are used to calculate the matching degree is the first step to establish a mathematical model . To Analyze the ER-Diagram of software component library is an effective way to extract the factors . An ER-diagram of facet-based component library was given below : Producer Provide Consume r Component Reuse Facet Term Describe Feedback Summary Feedback 1 n 1 1 1 1 n n n n m m Relate n m Include Describe Fig. 1 . ER-Diagram of Component Library Entities list : Component : component is the basic and primary entity in component library . Besides the attributes , there are facet-term pairs and information summary to describe a component . User Feedback : an opinion , a comment or a score provided by users after they have used a component . Component Summary : an information summary to describe a component which enables users to know well the component quickly . Facet : facet and its terms are used to classify and represent the components . 3.2 Factors of Weighted Ranking Algorithm As to a facet-based component library system , facet is the most important method to classify and represent the components . Correspondingly , facet-based retrieval methods , such as facet tree matching method , are important for the component retrieval system . The matching degree between facet tree and query tree is of much importance for ranking . However , matching degree of facet is not the only factor which is able to influence the ranking . Retrieval system of component library usually has two search modes : simple query and complex query . Simple query just simply uses the traditional database query method to match the Attribute-Valued pairs . In contrast , complex query is a much more effective way which combines several query methods together to match different kinds of component information . And therefore , 275 we should take other factors into account besides the facet for ranking the retrieval results . According to the analysis of ER-Diagram above , we can extract some other factors which are able to influence the ranking of component retrieval results while using the complex query . Attributes of component , such as component name , can be used to match the keywords in the query conditions . The matching degree of Attribute-Valued pairs should be an influential factor for ranking . Summary of component can also be used to match the query conditions . Query conditions usually consist of several keywords . The density , prominence , and position of keywords within the component summary will influence the ranking of components . The keyword density is just the number of occurrences of the keywords within the component summary divided by the total number of words . Keyword prominence is related to the location of keywords in the summary . For example , keywords placed at the beginning of the summary maybe carry more weight than those towards the end of it . User feedback of a component is very useful for other users who want to use it to evaluate the quality and other features of the component . They can acquire much more objective description and useful information about the component besides the component attributes and summary . How many times the component information has been visited and how many times the component has been downloaded for reusing should also be taken into account as the factors to calculate the matching degree for ranking . They reflect the popularity and reusability of the component from another aspect . 3.3 Mathematical Model Retrieval results consist of a collection of components matching the query conditions : Definition 1 : Components -LRB- C 1 , C 2 , ... , C i , ... , Cn -RRB- ; -LRB- n N , n1 -RRB- It makes no sense to discuss the circumstance of empty retrieval results , since that we are going to discuss the ranking of component lists . Accordingly , each component has a rank value : Definition 2 : Ranks -LRB- R 1 , R 2 , ... , R i , ... , Rn -RRB- ; -LRB- n N , n1 -RRB- The query condition consists of a collection of keywords : Definition 3 : Keywords -LRB- K 1 , K 2 , ... , K i , ... , K n0 -RRB- ; -LRB- n 0 N , n 0 1 -RRB- Each component is described by a set of Attribute-Valued pairs : Definition 4 : Attributes -LRB- A 1 , A 2 , ... , A i , ... , A n1 -RRB- ; -LRB- n 1 N , n 1 1 -RRB- Besides with Attribute-Valued pairs , components are also classified and represented by a set of facets and their terms : Definition 5 : Facets -LRB- F 1 , F 2 , ... , F i , ... , F n2 -RRB- ; -LRB- n 2 N , n 2 1 -RRB- Summary of component information differs from Attribute-Valued pairs . It provides a comprehensive description of a component in context . Definition 6 : Summary -LRB- S -RRB- ; User feedback includes all the comments and feedback to a specific component : Definition 7 : User Feedback -LRB- U 1 , U 2 , ... , U i , ... , U n3 -RRB- ; -LRB- n 3 N , n 3 1 -RRB- User feedback must be analyzed and evaluated to a relative number . We use E to represent the Evaluation number of user feedback . Definition 8 : E = Evaluate -LRB- User Feedback -RRB- . Definition 9 : Visited times of a component : Visited times -LRB- V -RRB- ; Definition 10 : Downloaded times of a component : Downloaded times -LRB- D -RRB- ; We have listed out all the influential factors above , which constitute a six-tuple : Factors -LRB- A , F , S , U , V , D -RRB- ; Their influential weights differ from each other according to their feature and importance : Definition 11 : Weights -LRB- W A , W F , W S , W U , W V , W D -RRB- ; -LRB- 0W A , W F , W S , W U , W V , W D 1 , W A + W F + W S + W U + W V + W D = 1 -RRB- W A represents the weight of Attributes ; W F represents the weight of Facets ; W S , W U , W V , and W D also represent the weight of corresponding factor discussed above . There are several functions for calculating the matching degree of some factors . The core calculating formula of each function relies on its corresponding matching algorithm . Functions Formulas Summary F S -LRB- Keywords , Summary -RRB- = no i S Ki match 1 -RRB- , -LRB- Facets F F -LRB- Keywords , Facets -RRB- = = no i n j Fj Ki match 1 2 1 -RRB- , -LRB- Attributes F A -LRB- Keywords , Attributes -RRB- = = no i n j Aj Ki match 1 1 1 -RRB- , -LRB- Match function of component summary uses the content-based similarity measurement algorithm of the search engine techniques . A Best-First algorithm was proposed by Cho -LSB- 6 -RSB- . This algorithm uses a vector space model to calculate the similarity between the keywords and the content . Its formula is given as following : = p k q k kq kp p q k kp kq W W W W p q sim 2 2 | -RRB- , -LRB- The variable q represents the collection of keywords , p represents the content , and W kp represents the importance of k to a specific topic . In our mathematical model , variable q represents the query condition , and the variable p represents the summary of the component information . 276 Facet-based retrieving method usually adopts the facet tree matching . And therefore , its match function calculates the matching degree between the facet tree of component and the query tree . A formula to calculate the matching cost of tree containment matching was given by Xu -LSB- 7 -RSB- : Q = -LRB- V , E , root -LRB- Q -RRB- -RRB- , D = -LRB- W , F , root -LRB- D -RRB- -RRB- are two unordered label tree , TCostM -LRB- Q , D -RRB- represents the tree containment matching cost from tree Q to tree D. -LRB- -RRB- -LCB- -RCB- D Q f f = : | min D -RRB- TCostM -LRB- Q , -LRB- -RRB- -LRB- -RRB- -LRB- -RRB- -LRB- + + = -RRB- -LRB- -RRB- -LRB- -RRB- -LRB- -RRB- -LRB- -RRB- -LRB- -RRB- -LRB- -RRB- -RRB- -LRB- -LRB- -RRB- -LRB- f Range f spectrum w f domain V v f domain v v label v label v f label v label f -RRB- If f is a tree containment matching from tree Q to tree D , and -LRB- f -RRB- = TCostM -LRB- Q , D -RRB- , then f is the tree containment matching which obtains the minimum matching cost from tree Q to tree D . This definition could be also applied to the containment matching between tree and forest or between forest and forest . As to the match function of component attributes , we just use the traditional database query methods to deal with it . E , V , and D are three ranking factors without any relation to query keywords . Even though they are numbers , we could not use them directly for ranking . Functions should be provided to transform them . Functions Evaluation of Feedback F E -LRB- E -RRB- Visited Times F V -LRB- V -RRB- Downloaded Times F D -LRB- D -RRB- According to the discussion above , we finally draw out a very simple formula to calculate the rank for each component : Rank = F A W A + F F W F + F S W S + F E W E + F V W V + F D W D We can use matrix operation to represent the calculation of rank value for each component in the retrieval results . There are n components and 6 influential factors . F 6 n W 1 6 = R 1 n : = n i D V E S F A Dn Vn En Sn Fn An Di Vi Ei Si Fi Ai D V E S F A D V E S F A R R R R W W W W W W F F F F F F F F F F F F F F F F F F F F F F F F M M M M M M M M M M M M M M 2 1 2 2 2 2 2 2 1 1 1 1 1 1 We specify the weights with experiential values at the very beginning , and then use the data mining technology to analyze the user logs to dynamically and iteratively adjust those values . 3.4 Timing of Ranks Calculation It 's time for us to determine when to deal with the calculation of ranks , since that we have designed how to deal with it . It 's just the matter of the timing of rank calculation . There are two probable times for calculation : calculating after the retrieving process has been finished ; calculating during the process of retrieving . Both of the probable times have their own advantages and disadvantages . If we calculate the rank after the retrieving process has been finished , we can deal with the retrieving and ranking separately . It will be much easier for us to design and maintain the system , since the retrieving and ranking process are independent . However , it costs much more time and space to calculate the rank . It needs a lot of memory space to store a large number of retrieval results temporally before they are ranked , and costs much more time to manage the transmission of data between storage devices and CPU . On the contrary , if we calculate the rank during the process of retrieving , we have to combine the retrieving with the ranking process completely or partly . Undoubtedly , it will be hard for us to implement and maintain the system , but it can greatly improve the time and the space performances . According to the discussion above , we can choose a proper time to calculate the ranks . Which solution we should choose depends on the requirements of the system . The solution which calculates the ranks during the retrieving process should be adopted if the time and the space performances are rigorously required . Our component library system , named DLCL , is implemented with J2EE platform . The mathematical model and its algorithm we discussed above have been implemented in this system with Java . Java is an object-oriented language . Its implementation consists of several core interfaces and classes . The core interfaces , classes and the relationship between them are demonstrated in the UML class diagram : 277 Fig. 2 . Class Diagram of Ranking Module We calculate the ranks during the retrieving process to improve the time and space performances , and therefore , we have to combine the ranking module partly with the retrieving module . In order to lower the coupled degree between these two modules , a callback mechanism was adopted . We define an interface Rank , which consists of only one method to calculate the ranks . RankComponentImpl is a class implementing the interface Rank to calculate the ranks of those components retrieved by users . The method of its concrete object can be executed by the searcher during the retrieving process . Class Component encapsulates those methods which provide the component information . ComponentMatchingDegree is a class providing those methods for calculating the matching degree between the query keywords and component . Each influential factor has its own strategy for calculation . There 's also a class Weight , which provides the methods to get the influential weight of each factor . In order to verify the efficiency of the component retrieving system which adoptes our weighted ranking algorithm , we design an experiment and carry out this experiment in our component library system , named DLCL . There are more than 1000 components in this system . Retrieving system of DLCL splits the retrieval results into several pages if there are too many components retrieved , and lists out 10 components per page . The experiment separates the users into two groups , group 1 and group 2 . Each group consists of 10 persons . All the users know about the knowledge of component reuse to a certain extent . Both groups use the facet-based component retrieving method to retrieve the components . Retrieval results of group 1 are listed out without any ranking , however , those of group 2 are ranked by our weighted ranking algorithm . There are several aspects to measure the efficiency of each group : how many pages they turned ; how many times they had to adjust the query condition ; and the most important , how many time was elapsed during the whole retrieving process . The experimental results are given in the following table : Group 1 Group 2 Average Turned Pages -LRB- pages -RRB- 2.7 1.4 Average Adjusted Times -LRB- times -RRB- 2.3 1.1 Average Time elapsed -LRB- minutes -RRB- 26.6 9.5 The experimental results obviously shows that the efficiency of group 2 is greatly higher than group 1 . By applying the weighted ranking algorithm into the retrieving system of DLCL , users need n't turn too many pages to view and compare the component information or to adjust the query condition to improve the query precision . Only to view the first page of retrieval results will be enough most of the time . And therefore , it greatly saves the time and retrieval costs . The idea of Component Rank comes from computing fair impact factors of published papers -LSB- 8 -RSB- . Google is a web search engine . Its method can be considered as an HTML extension of the method proposed for counting impact of publications , called influence weight in -LSB- 8 -RSB- . Google computes the ranks -LRB- called PageRanks -RRB- for HTML documents in the Internet -LSB- 9 , 10 -RSB- . In reference -LSB- 11 -RSB- , the authors present the Component Rank model for ranking software components , and show a system for computing Component Rank . In this model , a collection of software components is represented as a weighted directed graph whose nodes correspond to the components and edges correspond to the usage relations . Similar components are clustered into one node so that effect of simply duplicated nodes is removed . The nodes in the graph are ranked by their weights which are defined as the elements of the eigenvector of an adjacent matrix for the directed graph . A major distinction of Component Rank model in -LSB- 11 -RSB- from PageRank and the influence weight in -LSB- 9 , 10 -RSB- is that Component Rank model explores similarity between components before the weight computation . In this paper , we also propose a weighted ranking algorithm for component retrieval system . This weighted ranking algorithm uses different calculating strategies according to the feature of facet-based retrieval methods . While in -LSB- 11 -RSB- , the authors employed only statical use relations . In this paper , a mathematical model of weighted ranking algorithm is proposed and the timing of ranks calculation is discussed . We have applied this ranking algorithm into our component library system , named DLCL . The experiment we carried out shows that this algorithm greatly improves the efficiency of component retrieving system , saving the time and retrieval costs for component reusing . 278 This research is partially supported by the National High Technology Development 863 Program under Grant No . 2004AA116010 . -LSB- 1 -RSB- Frakes WB , Pole TP , An empirical study of representation methods for reusable software components , IEEE Transactions on Software Engineering , 1994 , l20 -LRB- 8 -RRB- , pp617-630 -LSB- 2 -RSB- H. Mili , R. Rada , W. Wang , K. Strickland , C. Boldyreff , L. Olsen , J. Witt , J. Heger , W. Scherr , and P. Elzer , Practitioner and SoftClass : A Comparative Study of Two Software Reuse Research Projects , J. Systems and Software , 1994 , 27 -LRB- 5 -RRB- -LSB- 3 -RSB- NEC Software Engineering Laboratory , NATO Standard for Management of a Reusable Software Component Library , NATO Communications and Information Systems Agency , 1991 -LSB- 4 -RSB- Wang YF . Research on retrieving reusable components classified in faceted scheme -LSB- Ph.D. . Thesis -RSB- . Shanghai : Fudan University , 2002 . -LSB- 5 -RSB- Chang JC , et al. . Representation and Retrieval of Reusable Software Components -LSB- J -RSB- . Computer Science , 1999 , 26 -LRB- 5 -RRB- : 41-48 . -LSB- 6 -RSB- Cho J , Garcia-Molina H , Page L. Efficient Crawling Through URL Ordering -LSB- J -RSB- . Computer Networks , 1998 , 30 -LRB- 1 ~ 7 -RRB- : 161-172 -LSB- 7 -RSB- Xu RZ , et al. . Research on Matching Algorithm for XML-Based Software Component Query , Journal of Software , 2003 , 14 -LRB- 7 -RRB- : 1195-1202 . -LSB- 8 -RSB- G. Pinski and F. Narin . `` Citation Influence for Journal Aggregates of Scientific Publications : Theory , with Application to the Literature of Physics '' . Information Processing and Management , 12 -LRB- 5 -RRB- : 297.312 , 1976 . -LSB- 9 -RSB- L. Page , S. Brin , R. Motwani , and T. Winograd . `` The PageRank Citation Ranking : Bringing Order to the Web '' . Technical Report of Stanford Digital Library Technologies Project , 1998 . `` http://www-db . stanford.edu/.backrub/ pageranksub.ps '' . -LSB- 10 -RSB- J. Kleinberg . `` Authoritative Sources in a Hyperlinked Environment '' . Journal of the ACM , 46 -LRB- 5 -RRB- : 604.632 , 1999 . -LSB- 11 -RSB- Katsuro Inoue , Reishi Yokomori , Hikaru Fujiwara , Tetsuo Yamamoto , Makoto Matsushita , Shinji Kusumoto : Component Rank : Relative Significance Rank for Software Component Search . ICSE 2003 : 14-24 279

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O

component rank
retrieval system
matching degree
facet
ranking algorithm
component retrieval
component library

and component library
facet-based component retrieval
weighted ranking algorithm


An

Analytical

Model

Based

On

With

Self-Similar

Input

To

Provide

End-To-End

Qos

In

Networks

The

dramatic

increase

in

demand

for

wireless

Internet

access

has

lead

to

the

introduction

of

new

wireless

architectures

and

systems

including

Wi-Fi

and

WiMAX

systems

such

as

UMTS

and

are

leaning

towards

an

all-IP

architecture

for

transporting

IP

multimedia

services

mainly

due

to

its

scalability

and

promising

capability

of

inter-working

heterogeneous

wireless

access

networks

During

the

last

ten

years

substantial

work

has

been

done

to

understand

the

nature

of

wired

IP

traffic

and

it

has

been

proven

that

IP

traffic

exhibits

self-similar

properties

and

burstiness

over

a

large

range

of

time

scales

Recently

because

of

the

large

deployment

of

new

wireless

architectures

researchers

have

focused

their

attention

towards

understanding

the

nature

of

traffic

carried

by

different

wireless

architecture

and

early

studies

have

shown

that

wireless

data

traffic

also

exhibits

strong

long-range

dependency

Thus

the

classical

tele-traffic

theory

based

on

a

simple

Markovian

process

can

not

be

used

to

evaluate

the

performance

of

wireless

networks

Unfortunately

the

area

of

understanding

and

modeling

of

different

kinds

of

wireless

traffic

is

still

immature

which

constitutes

a

problem

since

it

is

crucial

to

guarantee

tight

bound

QoS

parameters

to

heterogeneous

end

users

of

the

mobile

Internet

In

this

paper

we

make

several

contributions

to

the

accurate

modeling

of

wireless

IP

traffic

by

presenting

a

novel

analytical

model

that

takes

into

account

four

different

classes

of

self-similar

traffic

The

model

consists

of

four

queues

and

is

based

on

a

queueing

system

We

analyze

it

on

the

basis

of

priority

with

no

preemption

and

find

exact

packet

delays

To

date

no

closed

form

expressions

have

been

presented

for

with

priority

C.

-LSB-

Computer

Communication

Networks

-RSB-

Network

Management

G.

-LSB-

Mathematics

of

Computing

-RSB-

Queueing

Theory

Markov

Processes

Theory

Design

Performance

Reliability

During

the

past

decade

researchers

have

made

significant

efforts

to

understand

the

nature

of

Internet

traffic

and

it

has

been

proven

that

Internet

traffic

exhibits

self-similar

properties

The

first

study

which

stimulated

research

on

self-similar

traffic

was

based

on

measurements

of

Ethernet

traffic

at

Bellcore

-LSB-

-RSB-

Subsequently

the

self-similar

feature

has

been

discovered

in

many

other

types

of

Internet

traffic

including

studies

on

Transmission

Control

Protocol

-LRB-

TCP

-RRB-

-LSB-

-RSB-

WWW

traffic

-LSB-

-RSB-

VBR

video

-LSB-

-RSB-

and

Signaling

System

No

-LSB-

-RSB-

Deeper

studies

into

the

characteristics

of

Internet

traffic

has

discovered

and

investigated

various

properties

such

as

self-similarity

-LSB-

-RSB-

long-range

dependence

-LSB-

-RSB-

and

scaling

behavior

at

small

time-scale

-LSB-

-RSB-

The

references

-LSB-

-RSB-

provide

two

extensive

bibliographies

on

self-similarity

and

long-range

dependence

research

covering

both

theoretical

and

applied

papers

on

the

subject

Concurrently

over

the

past

few

years

we

have

witnessed

a

growing

popularity

of

Third

Generation

Systems

-LRB-

-RRB-

which

have

been

designed

to

provide

high-speed

data

services

and

multimedia

applications

over

mobile

personal

communication

networks

The

Universal

Mobile

Telecommunication

System

-LRB-

UMTS

-RRB-

is

the

predominant

global

standard

for

developed

by

Third

Generation

Partnership

Project

-LRB-

-RRB-

-LSB-

-RSB-

The

UMTS

architecture

is

shown

in

Fig.

It

consists

of

two

service

domains

a

Circuit

Switched

-LRB-

CS

-RRB-

service

domain

and

a

Packet

Switched

-LRB-

PS

-RRB-

service

domain

which

is

of

interest

in

this

paper

In

the

PS

service

domain

a

UMTS

network

connects

to

a

public

data

network

-LRB-

PDN

-RRB-

through

Serving

GPRS

Support

node

-LRB-

SGSN

-RRB-

and

Gateway

GPRS

support

node

-LRB-

GGSN

-RRB-

has

defined

four

different

QoS

classes

for

UMTS

-LRB-

-RRB-

Conversational

-LRB-

-RRB-

Interactive

-LRB-

-RRB-

Streaming

and

-LRB-

-RRB-

Background

conversational

being

the

most

delay-sensitive

and

background

the

least

delay

sensitive

class

-LSB-

-RSB-

With

the

increasing

demand

of

Internet

connectivity

and

the

flexibility

and

wide

deployment

of

IP

technologies

there

has

emerged

a

paradigm

shift

towards

IP-based

solutions

for

wireless

networking

-LSB-

-RSB-

Several

Wireless

IP

architectures

have

been

proposed

-LSB-

-RSB-

based

on

three

main

IP

QoS

models

IntServ

-LSB-

-RSB-

DiffServ

-LSB-

-RSB-

and

MPLS

-LSB-

-RSB-

has

also

recently

introduced

a

new

domain

called

IP

Multimedia

Subsystem

-LRB-

IMS

-RRB-

for

UMTS

The

main

objective

of

IMS

is

to

deliver

innovative

Permission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page

To

copy

otherwise

or

republish

to

post

on

servers

or

to

redistribute

to

lists

requires

prior

specific

permission

and/or

a

fee

MobiWAC

October

Torremolinos

Malaga

Spain

Copyright

ACM

Fig.

A

Simplified

UMTS

Network

Architecture

and

cost-effective

services

such

as

IP

telephony

media

streaming

and

multiparty

gaming

by

providing

IP

connectivity

to

every

mobile

device

-LSB-

-RSB-

In

the

light

of

this

researchers

have

recently

focused

on

understanding

the

nature

of

wireless

IP

traffic

and

early

studies

have

shown

that

wireless

data

traffic

also

exhibits

self-similarity

and

long-range

dependency

-LSB-

-RSB-

Much

of

the

current

understanding

of

wireless

IP

traffic

modeling

is

based

on

the

simplistic

Poisson

model

which

can

yield

misleading

results

and

hence

poor

wireless

network

planning

Since

the

properties

and

behavior

of

self-similar

traffic

is

very

different

from

traditional

Poisson

or

Markovian

traffic

several

issues

need

to

be

addressed

in

modeling

wireless

IP

traffic

to

provide

end-to-end

QoS

to

a

variety

of

heterogeneous

applications

We

begin

by

giving

an

overview

of

related

work

on

wired

and

wireless

IP

traffic

modeling

along

with

a

comparison

of

our

model

with

previous

work

In

this

section

we

first

discuss

the

related

work

which

has

been

done

in

the

area

of

performance

evaluation

of

wired

IP

and

Wireless

IP

networks

under

self-similar

input

and

then

we

compare

our

model

with

the

previous

ones

Previous

Work

on

IP

Traffic

Modeling

There

has

been

much

work

done

on

Internet

traffic

modeling

based

on

queueing

theory

in

the

presence

of

self-similar

traffic

-LSB-

-RSB-

In

-LSB-

-RSB-

a

Matrix

Geometric

-LRB-

analytical

-RRB-

method

is

used

to

compute

numerical

results

for

a

two

class

DiffServ

link

being

fed

by

a

Markovian

Modulated

Poisson

Process

-LRB-

MMPP

-RRB-

input

A

weakness

of

this

model

is

that

MMPP

may

require

an

estimation

of

a

large

number

of

parameters

An

OPNET

based

simulation

approach

was

adopted

in

-LSB-

-RSB-

to

see

the

impact

of

self-similarity

on

the

performance

evaluation

of

DiffServ

networks

As

a

result

an

idea

of

expected

queue

length

was

given

in

relation

to

the

Hurst

parameter

and

server

utilization

It

is

difficult

to

offer

guaranteed

QoS

parameters

on

the

basis

of

such

analysis

The

major

weakness

of

the

majority

of

available

queueing

based

results

is

that

only

the

FIFO

queueing

discipline

has

been

considered

for

serving

the

incoming

traffic

and

thus

differential

treatment

to

different

kinds

of

traffic

can

not

be

provided

In

addition

the

previous

results

are

asymptotic

We

also

refer

the

readers

to

-LSB-

-RSB-

for

an

overview

of

previous

work

that

has

been

carried

out

to

evaluate

the

performance

of

IP

networks

The

major

drawback

of

the

existing

work

is

that

the

queueing

models

considered

are

not

able

to

capture

the

self-similar

characteristics

of

Internet

traffic

Furthermore

it

is

important

to

note

that

most

of

the

previous

work

is

focused

on

the

analysis

of

one

type

of

traffic

only

without

discussing

its

affect

on

the

performance

of

other

kinds

of

network

traffic

Previous

Work

on

Wireless

IP

Traffic

Modeling

Few

studies

have

focused

on

wireless

traffic

modeling

and

here

we

discuss

the

most

relevant

work

As

shown

in

Fig.

the

principle

of

allocation

of

data

flows

between

end

users

and

GGSN

leads

to

increasing

load

on

the

network

elements

when

moving

closer

to

the

GGSN

Hence

GGSN

is

the

node

most

exposed

to

self-similar

influence

in

UMTS

-LSB-

-RSB-

The

influence

of

self-similar

input

on

GGSN

performance

in

the

UMTS

Release

IM-subsystem

has

been

analyzed

on

the

basis

of

a

W

queueing

system

-LRB-

FBM-Fractional

Brownian

Motion

-RRB-

in

-LSB-

-RSB-

In

this

work

different

probabilistic

parameters

of

GGSN

such

as

average

queue

length

and

average

service

rate

were

also

found

The

work

in

-LSB-

-RSB-

presents

modeling

and

a

simulation

study

of

the

Telus

Mobility

-LRB-

a

commercial

service

provider

-RRB-

Cellular

Digital

Packet

Data

-LRB-

CDPD

-RRB-

network

The

collected

results

on

average

queueing

delay

and

buffer

overflow

probability

indicated

that

genuine

traffic

traces

produce

longer

queues

as

compared

to

traditional

Poisson

based

traffic

models

To

get

an

overview

of

the

analysis

done

in

wireless

IP

traffic

modeling

with

self-similar

input

we

refer

the

readers

to

-LSB-

-RSB-

These

studies

are

merely

based

on

characterization

of

wireless

traffic

To

provide

differential

treatment

to

multiple

traffic

classes

with

different

QoS

demands

there

is

a

need

to

accurately

determine

end-to-end

QoS

parameters

such

as

delay

jitter

throughput

packet

loss

availability

and

per-flow

sequence

preservation

Comparison

of

our

Model

with

Prior

Work

To

overcome

the

limitations

of

the

previous

work

in

traffic

modeling

-LRB-

wired

and

wireless

IP

traffic

-RRB-

we

present

a

realistic

and

novel

analytical

model

by

considering

four

different

classes

of

traffic

that

exhibit

long-range

dependence

and

self-similarity

Our

model

implements

four

queues

based

on

a

queueing

system

and

we

analyze

it

on

the

basis

of

priority

with

no

preemption

The

traffic

model

considered

is

parsimonious

with

few

parameters

and

has

been

studied

in

-LSB-

-RSB-

The

model

is

furthermore

similar

to

on/off

processes

in

particular

to

its

variation

N-Burst

model

studied

in

-LSB-

-RSB-

where

packets

are

incorporated

However

only

a

single

type

of

traffic

is

considered

in

-LSB-

-RSB-

We

present

a

novel

analytical

approach

and

make

the

following

contributions

to

Wireless

IP

traffic

modeling

Interarrival

Time

Calculations

We

calculate

the

packet

interarrival

time

distributions

for

the

particular

self-similar

traffic

model

-LSB-

-RSB-

for

the

first

time

in

this

paper

The

distribution

of

cross

interarrival

time

between

different

types

of

packets

is

derived

on

the

basis

of

single

packet

results

Packet

Delays

for

Multiple

Self-Similar

Traffic

Classes

We

consider

a

queueing

system

which

takes

into

account

four

different

classes

of

self-similar

input

traffic

denoted

by

and

analyze

it

on

the

basis

of

non

preemptive

priority

and

find

exact

packet

delays

To

date

no

closed

form

expressions

have

been

presented

for

with

priority

Embedded

Markov

Chain

Formulation

We

also

formulate

the

embedded

Markov

chain

of

by

considering

all

possible

states

and

derive

the

corresponding

transition

probabilities

The

rest

of

the

paper

is

organized

as

follows

Section

and

are

devoted

to

explaining

the

self-similar

traffic

model

with

multiple

classes

and

the

calculation

of

interarrival

times

respectively

Section

explains

the

procedure

of

formulating

the

embedded

Markov

Chain

along

with

the

derivation

of

packet

delays

The

applications

of

the

model

are

discussed

in

section

Finally

conclusion

and

future

work

is

given

in

Section

The

traffic

model

considered

here

-LSB-

-RSB-

belongs

to

a

particular

class

of

self-similar

traffic

models

also

called

telecom

process

in

-LSB-

-RSB-

recently

The

model

captures

the

dynamics

of

packet

generation

while

accounting

for

the

scaling

properties

of

the

traffic

in

telecommunication

networks

Such

models

also

called

infinite

source

models

are

similar

to

on/off

processes

with

heavy

tailed

on

and/or

off

times

What

is

more

our

model

abstracts

the

packet

arrival

process

in

particular

and

facilitates

queueing

analysis

by

the

approaches

developed

in

the

sequel

In

the

framework

of

a

Poisson

point

process

the

model

represents

an

infinite

number

of

potential

sources

The

traffic

is

found

by

aggregating

the

number

of

packets

generated

by

such

sources

Each

source

initiates

a

session

with

a

heavy-tailed

distribution

in

particular

a

Pareto

distribution

whose

density

is

given

by

-RRB-

-LRB-

r

b

r

g

r

b

where

is

related

to

the

Hurst

parameter

by

-RRB-

-LRB-

H

The

sessions

arrive

according

to

a

Poisson

process

with

rate

The

packets

arrive

according

to

a

Poisson

process

with

rate

locally

over

each

session

For

each

class

the

traffic

Y

-LRB-

t

-RRB-

measured

as

the

total

number

of

packets

injected

in

-LSB-

t

-RSB-

is

found

by

-RRB-

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

t

S

i

i

i

i

S

t

R

U

t

Y

where

denote

the

local

Poisson

process

the

duration

and

the

arrival

time

of

session

i

respectively

Hence

Y

-LRB-

t

-RRB-

corresponds

to

the

sum

of

packets

generated

by

all

sessions

initiated

in

-LSB-

t

-RSB-

until

the

session

expires

if

that

happens

before

t

and

until

t

if

is

does

not

The

stationary

version

of

this

model

based

on

an

infinite

past

is

considered

in

calculations

below

The

packet

sizes

are

assumed

to

be

fixed

because

each

queue

corresponds

to

a

certain

type

of

application

where

the

packets

have

fixed

size

or

at

least

fixed

service

time

distribution

i

i

i

S

R

U

The

traffic

model

Y

is

long-range

dependent

and

almost

second-order

self-similar

the

auto

covariance

function

of

its

increments

is

that

of

fractional

Gaussian

noise

Three

different

heavy

traffic

limits

are

possible

depending

on

the

rate

of

increase

in

the

traffic

parameters

-LSB-

-RSB-

Two

of

these

limits

are

well

known

self-similar

processes

fractional

Brownian

motion

and

Levy

process

which

do

not

account

for

packet

dynamics

in

particular

Packet

interarrival

time

distributions

for

the

particular

self-similar

traffic

model

are

calculated

for

the

first

time

in

this

paper

We

consider

a

single

type

of

packet

first

The

distributions

of

cross

interarrival

time

between

different

types

of

packets

are

derived

on

the

basis

of

single

packet

results

Interarrival

Times

for

a

Single

Class

Although

the

packet

arrival

process

itself

is

long-range

dependent

and

shows

self-similarity

the

number

of

alive

sessions

at

a

period

of

time

say

of

length

t

has

a

stationary

distribution

and

is

Poisson

distributed

The

alive

sessions

at

any

time

can

be

further

split

into

independent

components

as

those

session

that

last

longer

than

t

and

those

that

expire

before

t

Such

results

are

well

known

-LSB-

pg

-RSB-

and

will

be

used

to

derive

the

interarrival

time

distribution

of

the

packets

Given

that

there

is

a

packet

arrival

at

an

instant

in

time

we

aim

to

find

the

distribution

of

the

time

until

next

arrival

denoted

by

T

We

will

find

-RRB-

-LRB-

t

F

for

When

the

event

is

considered

the

information

that

there

is

a

packet

arrival

is

equivalent

to

the

information

that

there

is

at

least

one

session

alive

at

the

given

instant

This

follows

from

the

assumption

that

local

packet

generation

process

is

Poisson

over

each

session

The

probability

that

next

interarrival

is

greater

than

t

-RCB-

-LCB-

t

T

P

t

-RCB-

-LCB-

t

T

on

a

particular

session

is

the

same

as

the

probability

that

the

remaining

time

until

next

arrival

is

greater

than

t

due

to

the

memoryless

property

of

exponential

distribution

That

is

-RCB-

-LCB-

t

T

P

P

-LCB-

Next

packet

interarrival

is

greater

than

t

there

is

a

packet

arrival

-RCB-

P

-LCB-

Next

packet

interarrival

is

greater

than

t

there

is

at

least

one

alive

session

-RCB-

where

is

the

probability

that

there

is

at

least

one

alive

session

in

other

words

the

utilization

of

an

system

The

event

that

next

packet

interarrival

is

greater

than

t

can

be

split

as

follows

G

M

The

active

sessions

that

expire

before

t

do

not

incur

any

new

arrivals

The

active

sessions

that

expire

after

t

do

not

incur

any

new

arrivals

No

new

session

arrivals

in

t

or

at

least

one

session

arrival

with

no

packet

arrival

in

t

We

find

the

probability

that

all

three

events

occur

at

the

same

time

by

using

the

independence

of

a

Poisson

point

process

over

disjoint

sets

The

result

is

-RCB-

-LCB-

t

T

P

-RRB-

-RSB-

-LRB-

exp

-LSB-

-LCB-

-RRB-

-LRB-

-RRB-

-LRB-

t

B

v

A

v

e

t

e

e

t

t

-RRB-

-RSB-

-RRB-

-RRB-

-LRB-

-LRB-

exp

-LSB-

-RRB-

-RSB-

-RRB-

-LRB-

-LRB-

exp

-LSB-

t

e

B

v

e

A

v

t

t

t

t

-RRB-

-RSB-

e

-RRB-

-LRB-

-RRB-

-RSB-

-LRB-

exp

-LSB-

-LRB-

exp

-LSB-

e

e

-RRB-

-LRB-

-RRB-

-LRB-

t

t

t

B

A

A

e

t

t

t

-RCB-

-RRB-

-RSB-

-RRB-

e

-RRB-

-LRB-

-LRB-

-RRB-

-RSB-

-LRB-

exp

-LSB-

-LRB-

exp

-LSB-

e

e

-RRB-

-LRB-

-RRB-

-LRB-

t

B

e

t

t

t

t

B

A

t

t

where

-RRB-

-LRB-

t

A

t

dy

y

g

t

y

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

t

t

t

G

t

dy

y

yg

B

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

and

-RRB-

-RSB-

duration

session

-LSB-

E

exp

-LRB-

-RSB-

-RRB-

-LRB-

exp

-LSB-

b

because

the

steady

state

number

in

the

system

in

G

M

queue

is

Poisson

distributed

with

mean

E

-LSB-

Session

duration

-RSB-

-LSB-

-RSB-

and

and

b

are

the

parameters

of

the

session

duration

with

complementary

distribution

function

G

and

density

-RRB-

-LRB-

r

b

r

g

b

r

which

is

Pareto

Interarrival

Times

for

Multiple

Classes

Here

we

explain

the

detailed

procedure

to

find

out

the

Interarrival

times

for

two

classes

the

Interarrival

times

for

more

than

two

classes

can

be

found

in

a

similar

way

Let

denote

the

interarrival

time

between

a

class

i

packet

that

comes

first

and

a

class

j

packet

that

follows

The

analysis

which

can

be

extended

to

provides

a

method

for

other

self-similar

models

as

well

provided

that

the

distribution

of

interarrivals

are

available

ij

T

j

i

j

i

i

T

For

the

consecutive

packet

arrival

time

we

have

T

-LCB-

-RCB-

-LCB-

t

T

P

t

T

P

no

arrivals

of

class

in

-RCB-

T

t

T

ds

s

f

P

-RRB-

-LRB-

-RCB-

s

in

class

of

arrivals

no

-LCB-

t

T

ds

s

f

s

F

-RRB-

-LRB-

-RRB-

-LRB-

where

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

t

A

v

B

v

e

e

t

F

-RSB-

-RRB-

-LRB-

exp

-LSB-

-RRB-

-RSB-

-LRB-

exp

-LSB-

t

t

t

e

A

v

e

t

-LRB-

-RRB-

-RSB-

-RRB-

-RRB-

-LRB-

-LRB-

exp

-LSB-

t

e

B

v

t

t

-RRB-

-LRB-

t

A

and

-RRB-

-LRB-

t

B

are

defined

analogously

as

in

-LRB-

-RRB-

and

-LRB-

-RRB-

and

we

used

the

independence

of

class

and

packet

inputs

Here

F

is

found

through

similar

arguments

used

for

P

-LCB-

T

t

-RCB-

in

the

last

subsection

without

assuming

that

there

is

an

alive

session

of

type

As

a

result

by

differentiation

we

find

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

F

t

f

t

f

T

T

Now

consider

the

interarrival

time

T

occurring

between

a

class

packet

followed

by

a

class

packet

For

T

we

get

-RCB-

-LCB-

t

T

P

t

ds

s

P

s

f

-RCB-

arrived

packet

a

in

class

of

arrivals

no

-LCB-

-RRB-

-LRB-

t

T

ds

s

F

s

f

-RRB-

-LRB-

-RRB-

-LRB-

where

is

the

density

function

corresponding

to

the

event

that

there

is

an

arrival

of

class

packet

at

time

s

and

we

used

independence

of

class

and

packet

streams

As

a

matter

of

fact

can

be

obtained

by

taking

the

derivative

of

the

complementary

distribution

function

-RRB-

-LRB-

s

f

-RRB-

-LRB-

s

f

F

given

in

-LRB-

-RRB-

As

a

result

we

get

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

F

t

f

t

f

T

T

Similarly

it

can

be

shown

that

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

F

t

f

t

f

T

T

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

F

t

f

t

f

T

T

We

consider

a

model

of

four

queues

based

on

by

considering

four

different

classes

of

self-similar

input

traffic

denoted

by

and

analyze

it

on

the

basis

of

priority

with

no

preemption

Let

the

service

time

distribution

have

rate

and

for

type

type

type

and

type

packets

respectively

and

let

type

packets

have

the

highest

priority

and

type

packets

have

the

lowest

priority

With

Four

Classes

M

SS

The

usual

embedded

Markov

chain

-LSB-

-RSB-

formulation

of

is

based

on

the

observation

of

the

queueing

system

at

the

time

of

arrival

instants

right

before

an

arrival

At

such

instants

the

number

in

the

system

is

the

number

of

packets

that

arriving

packet

sees

in

the

queue

plus

packets

in

service

if

any

excluding

the

arriving

packet

itself

We

specify

the

states

and

the

transition

probability

matrix

P

of

the

Markov

chain

with

the

self-similar

model

for

four

types

of

traffic

M

G

Let

denote

the

embedded

Markov

chain

at

the

time

of

arrival

instants

As

the

service

is

based

on

priority

the

type

of

packet

in

service

is

important

at

each

arrival

instant

of

a

given

type

of

packet

to

determine

the

queueing

time

Therefore

we

define

the

state

space

as

-RCB-

-LCB-

n

X

n

-RCB-

-RCB-

-LCB-

-RCB-

-LCB-

-RRB-

-LCB-

-LRB-

Z

i

i

i

i

I

s

s

s

s

s

a

a

a

a

a

s

a

i

i

i

i

S

-LRB-

-RRB-

where

are

labels

to

denote

the

type

of

arrival

are

labels

to

denote

the

type

of

packet

in

service

are

the

number

of

packets

in

each

queue

including

a

possible

packet

in

service

I

denotes

the

idle

state

in

which

no

packet

is

in

service

or

queued

and

is

the

set

of

nonnegative

integers

Some

of

the

states

in

the

state

space

S

given

in

-LRB-

-RRB-

have

zero

probability

For

example

is

impossible

The

particular

notation

in

-LRB-

-RRB-

for

S

is

chosen

for

simplicity

although

the

impossible

states

could

be

excluded

from

S.

Each

possible

state

the

reachable

states

from

each

and

the

corresponding

transition

probabilities

will

be

calculated

a

a

a

a

s

s

s

s

i

i

i

i

Z

-RRB-

-LRB-

s

a

i

i

i

States

of

the

Embedded

Markov

Chain

The

states

of

the

Markov

chain

and

the

possible

transitions

with

respective

probabilities

can

be

enumerated

by

considering

each

case

We

will

only

analyze

the

states

with

non-empty

queues

in

this

paper

States

with

-RRB-

-LRB-

s

a

i

i

i

i

i

i

i

i

We

can

divide

the

states

and

transitions

into

groups

Because

-LRB-

a

s

-RRB-

can

occur

different

ways

and

the

next

state

-LRB-

p

q

-RRB-

can

be

composed

similarly

in

different

ways

as

-RCB-

-LCB-

a

a

a

a

p

a

and

We

will

analyze

only

the

first

one

in

detail

the

others

follow

similarly

-RCB-

-LCB-

s

s

s

s

q

s

Transition

from

-RRB-

-LRB-

-RRB-

-LRB-

s

a

j

j

j

j

s

a

i

i

i

i

This

is

the

case

where

a

transition

occurs

from

an

arrival

of

type

to

an

arrival

of

type

such

that

the

first

arrival

has

seen

a

type

packet

in

service

packets

of

type

-LRB-

equivalently

total

of

queue

and

the

packet

in

service

-RRB-

and

packets

of

type

-LRB-

in

this

case

only

queue

-RRB-

packets

of

type

and

packets

of

type

in

the

system

The

transition

occurs

to

packets

of

type

packets

of

type

with

a

type

packet

in

service

packets

of

type

and

packets

of

type

in

the

system

Due

to

priority

scheduling

an

arrival

of

type

can

see

a

type

packet

in

service

in

the

next

state

only

if

all

type

packets

including

the

one

that

arrived

in

the

previous

state

are

exhausted

during

the

interarrival

time

That

is

why

can

take

only

the

value

and

exactly

i

i

i

i

j

j

j

j

j

i

packets

of

type

are

served

In

contrast

the

number

of

packets

served

from

queue

say

k

can

be

anywhere

between

and

i

as

at

least

one

type

packet

is

in

the

system

one

being

in

service

when

a

new

arrival

occurs

The

transition

probabilities

are

-RRB-

-RCB-

-LRB-

-RRB-

-LRB-

-LCB-

s

a

i

i

i

i

X

s

a

i

i

k

i

X

P

n

n

-LCB-

i

P

served

from

type

k

served

from

type

and

a

type

packet

remains

in

service

during

-RCB-

T

where

we

use

the

fact

that

the

remaining

service

time

of

a

type

packet

in

service

has

the

same

exponential

distribution

Exp

-LRB-

-RRB-

due

to

the

memory-less

property

of

a

Markovian

service

Therefore

for

i

k

K

-RRB-

-RCB-

-LRB-

-RRB-

-LRB-

-LCB-

s

a

i

i

i

i

X

s

a

i

i

k

i

X

P

n

n

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dt

dx

ds

t

f

x

f

s

f

k

i

where

sum

of

l

independent

service

times

of

type

m

packets

m

l

m

S

Z

l

Note

that

has

an

Erlang

distribution

with

parameters

l

m

S

-RRB-

-LRB-

m

l

as

each

service

time

has

an

exponential

distribution

and

the

sum

being

the

sum

of

several

exponentially

distributed

random

variables

has

a

hypoexponential

distribution

The

density

functions

of

all

these

distributions

can

easily

be

evaluated

numerically

Similarly

we

can

enumerate

all

cases

The

results

for

first

cases

are

given

in

Table

in

the

Appendix

l

l

S

S

Limiting

Distribution

and

Waiting

Times

Steady

state

distribution

as

seen

by

an

arrival

can

be

found

by

solving

P

using

the

transition

matrix

P

of

the

Markov

chain

analyzed

above

In

practice

the

queue

capacity

is

limited

in

a

router

So

the

steady

state

distribution

exists

To

the

best

of

our

knowledge

no

previous

analytical

expressions

are

available

for

the

waiting

time

of

a

queue

with

priority

Our

analysis

relies

on

the

limiting

distribution

of

the

state

of

the

queue

at

the

arrival

instances

which

can

be

computed

using

the

analysis

given

above

for

our

self-similar

traffic

model

In

general

the

following

analysis

is

valid

for

any

queueing

system

where

the

limiting

distribution

at

the

arrival

instances

can

be

computed

The

expected

waiting

time

for

the

highest

priority

queue

can

be

found

as

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RSB-

-LSB-

s

a

j

j

j

j

j

s

a

j

j

j

j

j

W

E

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

s

a

j

j

j

j

j

s

a

j

j

j

j

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

where

and

are

the

respective

capacities

of

each

queue

This

follows

clearly

from

the

fact

that

an

arriving

packet

of

higher

priority

will

wait

until

all

packets

of

the

same

priority

as

well

as

the

packet

in

service

are

served

Depending

on

the

type

of

the

packet

in

service

we

have

the

constituent

expressions

in

the

sum

J

J

J

J

On

the

other

hand

we

obtain

the

expected

waiting

time

for

the

low

priority

queues

by

analyzing

the

events

that

constitute

this

delay

The

amount

of

work

in

the

system

at

any

time

is

defined

as

the

-LRB-

random

-RRB-

sum

of

all

service

times

that

will

be

required

by

the

packets

in

the

system

at

that

instant

The

waiting

time

of

a

type

packet

-LRB-

which

is

nd

highest

priority

queue

-RRB-

can

be

written

as

Z

Z

Z

W

-LRB-

-RRB-

where

Z

is

the

amount

of

work

seen

by

the

arriving

packet

in

queue

and

queue

-LRB-

i.e

higher

priority

and

equal

priority

-RRB-

Z

is

the

amount

of

work

associated

with

higher

priority

-LRB-

i.e.type

-RRB-

packets

arriving

during

Z

Z

is

the

amount

of

work

associated

with

type

packets

arriving

during

Z

and

so

on

As

illustrated

in

Fig.

the

waiting

time

of

an

arriving

packet

of

type

is

indeed

given

by

the

total

workload

building

in

front

of

it

The

arrows

in

the

figure

denote

the

arrival

times

of

type

packets

and

all

the

oblique

lines

have

degrees

angle

with

the

time

axis

In

this

figure

the

waiting

time

is

Z

Z

Z

Z

W

for

example

Let

M

j

denote

the

number

of

type

j

arrivals

over

Z

i

j

Then

L

M

M

S

S

Z

W

where

denotes

the

random

sum

of

M

j

M

S

j

independent

service

times

of

type

packets

Then

L

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-LSB-

-RSB-

M

E

S

E

M

E

S

E

Z

E

W

E

since

the

service

times

and

the

arrival

process

are

independent

For

a

stationary

packet

arrival

process

we

get

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-LSB-

j

j

j

j

j

Z

E

c

Z

c

E

Z

M

E

E

M

E

due

to

mentioned

independence

where

is

a

constant

particular

to

the

arrival

process

That

is

expectation

of

the

number

of

arrivals

in

any

period

of

time

is

proportional

to

the

length

of

that

period

because

of

stationarity

in

time

and

linearity

of

expectation

In

our

stationary

self-similar

traffic

input

process

c

c

is

the

expected

number

of

arrivals

per

unit

time

which

can

be

called

the

arrival

rate

given

by

the

product

of

the

arrival

rate

of

session

arrivals

the

arrival

rate

of

packets

over

a

session

and

the

expected

session

length

-LSB-

-RSB-

Explicitly

Hence

the

expected

waiting

time

reduces

to

-RRB-

-LRB-

b

c

L

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-LSB-

-RSB-

Z

E

c

S

E

Z

E

c

S

E

Z

E

W

E

-RSB-

-LSB-

-RSB-

-LSB-

-RRB-

-RSB-

-LSB-

-RSB-

-LSB-

-LRB-

-RSB-

-LSB-

W

E

c

Z

E

Z

E

Z

E

c

Z

E

L

In

view

of

-LRB-

-RRB-

therefore

we

get

from

-RSB-

-LSB-

W

E

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RSB-

-LSB-

s

a

j

j

j

j

j

j

s

a

j

j

j

j

j

j

W

E

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

-RSB-

-LSB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

W

E

c

s

a

j

j

j

j

j

j

s

a

j

j

j

j

j

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

Z

Z

Z

Z

time

Work

-LRB-

time

-RRB-

Fig.

Waiting

time

of

a

type

packet

in

terms

of

Z

j

's

Similarly

we

can

directly

write

down

the

expected

waiting

time

for

a

packet

of

type

-LRB-

rd

priority

queue

-RRB-

and

type

-LRB-

lowest

priority

queue

-RRB-

The

expected

waiting

time

for

a

packet

of

type

can

be

found

from

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RSB-

-LSB-

s

a

j

j

j

j

j

j

j

s

a

j

j

j

j

j

j

j

W

E

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

-RSB-

-LSB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

W

E

c

c

s

a

j

j

j

j

j

j

j

s

a

j

j

j

j

j

j

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

and

can

be

determined

from

-RSB-

-LSB-

W

E

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RSB-

-LSB-

s

a

j

j

j

j

j

j

j

j

s

a

j

j

j

j

j

j

j

j

W

E

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

s

a

j

j

j

j

j

j

j

j

s

a

j

j

j

j

j

j

j

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

J

j

-RSB-

-LSB-

-RRB-

-LRB-

W

E

c

c

c

Here

we

give

an

overview

of

the

prime

application

of

the

model

systems

such

as

UMTS

and

are

leaning

towards

an

all-IP

network

architecture

for

transporting

IP

multimedia

services

-LSB-

-RSB-

An

all-IP

DiffServ

platform

is

the

currently

most

promising

architecture

to

interwork

the

heterogeneous

wireless

access

networks

and

the

Internet

to

provide

broadband

access

seamless

global

roaming

and

QoS

guarantees

for

various

IP

multimedia

services

-LSB-

-RSB-

To

transport

UMTS

services

through

IP

networks

without

loosing

end-to-end

QoS

provisioning

a

consistent

and

efficient

QoS

mapping

between

UMTS

services

and

IP

QoS

classes

is

required

According

to

UMTS-to-IP

QoS

mapping

is

performed

by

a

translation

function

in

the

GGSN

router

that

classifies

each

UMTS

packet

flow

and

maps

it

to

a

suitable

IP

QoS

class

-LSB-

-RSB-

In

order

to

make

accurate

mappings

and

to

ensure

guaranteed

QoS

parameters

to

the

end

user

of

mobile

Internet

it

is

essential

to

being

able

to

accurately

model

the

end-to

end

behavior

of

different

classes

of

wireless

IP

traffic

-LRB-

conversational

interactive

streaming

and

background

-RRB-

passing

through

a

DiffServ

domain

Several

queueing

tools

have

been

developed

that

can

be

implemented

in

IP

routers

within

different

QoS

domains

including

Priority

Queueing

-LRB-

PQ

-RRB-

Custom

Queueing

-LRB-

CQ

-RRB-

Weighted

Fair

Queueing

-LRB-

WFQ

-RRB-

Class

Based

Weighted

Fair

Queueing

-LRB-

CBWFQ

-RRB-

and

Low-Latency

Queueing

-LRB-

LLQ

-RRB-

-LSB-

-RSB-

Our

model

is

directly

applicable

to

the

problem

of

determining

the

end-to-end

queueing

behavior

of

IP

traffic

through

both

Wired

and

wireless

IP

domains

but

modeling

accuracy

is

more

crucial

in

resource

constrained

environments

such

as

wireless

networks

For

example

our

model

is

directly

able

to

analyze

the

behavior

of

four

different

QoS

classes

of

UMTS

traffic

passing

through

a

DiffServ

domain

in

which

routers

are

implemented

with

priority

queueing

Thus

the

model

enables

tighter

bounds

on

actual

behavior

so

that

over-provisioning

can

be

minimized

It

also

enables

translations

of

traffic

behavior

between

different

kinds

of

QoS

domains

so

that

it

is

possible

to

map

reservations

made

in

different

domains

to

provide

session

continuity

In

this

paper

we

have

presented

a

novel

analytical

model

based

on

queueing

system

for

accurate

modeling

of

wireless

IP

traffic

behavior

under

the

assumption

of

four

different

classes

of

self-similar

traffic

We

have

analyzed

it

on

the

basis

of

non-preemptive

priority

and

explicit

expressions

of

expected

waiting

time

for

the

corresponding

classes

have

been

derived

The

model

represents

an

important

step

towards

the

overall

aim

of

finding

realistic

-LRB-

under

self-similar

traffic

assumptions

-RRB-

end-to-end

QoS

behavior

-LRB-

in

terms

of

QoS

parameters

such

as

delay

jitter

and

throughput

-RRB-

of

multiple

traffic

classes

passing

through

heterogeneous

wireless

IP

domains

-LRB-

IntServ

DiffServ

and

MPLS

-RRB-

At

the

moment

we

are

working

on

the

numerical

analysis

to

find

solutions

to

the

suggested

imbedded

Markov

Chain

in

order

to

find

exact

QoS

parameter

bounds

for

a

given

system

Our

future

work

will

focus

to

analyze

the

performance

of

different

QoS

domains

implemented

with

different

queueing

disciplines

-LSB-

-RSB-

W.

Leland

M.

Taqqu

W.

Willinger

and

D.

Wilson

On

the

self-similar

nature

of

Ethernet

traffic

-LRB-

extended

version

-RRB-

IEEE/ACM

Transactions

on

Networking

vol

no.

pp.

Feb.

-LSB-

-RSB-

V.

Paxon

Empirically

derived

analytical

models

of

wide-area

TCP

connections

IEEE/ACM

Transactions

on

Networking

vol

pp

Aug.

-LSB-

-RSB-

V.

Paxon

and

S.

Floyd

Wide-area

traffic

the

failure

of

Poisson

modeling

in

Proc

ACM

SIGCOMM

London

U.K.

Aug.

pp.

-LSB-

-RSB-

M.

Crovella

and

A.

Bestavros

Explaining

World

Wide

Web

Traffic

Self-Similarity

Tech

Rep.

Boston

University

CS

Dept

Boston

MA

Aug.

-LSB-

-RSB-

M.

W.

Garrett

and

W.

Willinger

Analysis

Modeling

and

Generation

of

Self-Similar

VBR

Video

Traffic

ACM

Computer

Communication

Review

vol

Oct.

SIGCOMM

Symposium

-LSB-

-RSB-

W.

Willinger

et

al

Statistical

analysis

of

traffic

data

from

working

CCS

subnetworks

IEEE

Journal

on

Selected

Areas

of

Communication

vol

no.

pp.

Apr.

-LSB-

-RSB-

M.

Crovella

and

A.

Bestavros

Self-Similarity

in

World

Wide

Web

Traffic

Evidence

and

Possible

Causes

in

ACM

Sigmetrics

May

-LSB-

-RSB-

J.C

Bolot

and

M.

Grossglauser

On

the

Relevance

of

Long-Range

Dependence

in

Network

Traffic

Computer

Communication

Review

vol

no.

pp.

October

-LSB-

-RSB-

Z.

L.

Zhang

V.

Ribeiro

S.

Moon

and

C.

Diot

Small-Time

Scaling

behavior

of

internet

backbone

traffic

An

Empirical

Study

in

IEEE

INFOCOM

march

-LSB-

-RSB-

M.

S.

Taqqu

Self-Similar

processes

In

S.

Kotz

and

N.

Johnson

editors

Encyclopedia

of

Statistical

Sciences

vol

pp.

Wiley

New

York

-LSB-

-RSB-

W.

Willinger

M.S

Taqqu

and

A.

Erramilli

A

bibliographical

guide

to

self-similar

traffic

and

performance

modeling

for

modern

high

speed

networks

In

F.

P.

Kelly

S.

Zachary

and

I.

Ziedins

editors

Stochastic

Networks

Theory

and

Applications

pp.

Claredon

Press

Oxford

-LSB-

-RSB-

H.

Holma

and

A.

Taskala

WCDMA

for

UMTS

Radio

Access

for

Third

Generation

Mobile

Communications

nd

Edition

John

Wiley

Sons

Ltd.

pp.

-LSB-

-RSB-

J.

Yang

and

I.

Kriaras

Migration

to

all-IP

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York

APPENDIX

Table

The

States

of

the

Markov

Chain

and

Transition

Probabilities

Initial

State

Reachable

States

-LRB-

m

-RRB-

Transition

Probability

-RRB-

-LRB-

s

a

i

i

i

k

i

m

i

k

K

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

dt

dx

ds

t

f

x

f

s

f

m

k

-RRB-

-LRB-

s

a

i

i

k

i

m

i

k

K

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dt

dx

ds

t

f

x

f

s

f

m

k

i

-RRB-

-LRB-

i

k

s

a

i

k

i

m

dsdxdt

t

f

x

f

s

f

m

k

i

i

T

S

S

S

t

x

t

S

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

s

a

i

i

i

i

-RRB-

-LRB-

i

k

s

a

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

i

-RRB-

-LRB-

s

a

i

i

i

k

i

m

i

k

K

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dt

dx

ds

t

f

x

f

s

f

m

k

-RRB-

-LRB-

s

a

i

i

i

i

m

-RRB-

-LRB-

-RRB-

-LRB-

t

T

S

dt

ds

t

f

s

f

m

-RRB-

-LRB-

s

a

i

i

k

i

m

K

i

and

i

k

K

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dt

dx

ds

t

f

x

f

s

f

m

k

i

-RRB-

-LRB-

s

a

i

i

i

i

-RRB-

-LRB-

i

k

s

a

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

-RRB-

-LRB-

i

k

s

a

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

i

-RRB-

-LRB-

i

k

s

a

i

i

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

-RRB-

-LRB-

i

k

s

a

i

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

-RRB-

-LRB-

s

a

i

i

i

i

m

-RRB-

-LRB-

-RRB-

-LRB-

t

T

S

dt

ds

t

f

s

f

m

-RRB-

-LRB-

s

a

i

k

i

m

i

i

k

K

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

dt

dx

ds

t

f

x

f

s

f

m

k

i

i

-RRB-

-LRB-

s

a

i

i

i

i

-RRB-

-LRB-

i

k

s

a

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

i

-RRB-

-LRB-

i

k

s

a

i

i

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

-RRB-

-LRB-

i

k

s

a

i

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

-RRB-

-LRB-

i

k

s

a

i

k

i

m

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

-RRB-

-LRB-

s

a

i

i

i

i

m

-RRB-

-LRB-

-RRB-

-LRB-

t

T

S

dsdt

t

f

s

f

m

-RRB-

-LRB-

s

a

i

i

i

i

-RRB-

-LRB-

i

s

a

k

i

m

i

k

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

t

x

t

T

S

S

S

S

S

dsdxdt

t

f

x

f

s

f

m

k

i

i

i

traffic modelling
queuing model
self-similar
self-similar traffic
3g networks