A General Model of Hybrid Data Dissemination

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1 A General Model of Hyrid Data Dissemination Hao-Ping Hung and Ming-Syan Chen Graduate Institute of Communication Engineering National Taiwan University, Taiwan, ROC ABSTRACT Hyrid data dissemination, which comines the push-ased (i.e., roadcast) and pull-ased (i.e., on-demand) data delivery, is the most common technique to deliver information in a moile computing system. Most of the prior works in hirid dissemination are ased on the assumption that each delivered data item is of the same size. However, in the modern communication environment in which various information is delivered, the conventional dissemination schemes suffer from the efficiency issues. In this paper, we consider a general model of hyrid data dissemination, in which each data item is allowed to have an aritrary size. The analytical model MGBC (Model of General Broadcast Channels) and MGOD (Model of General On-demand Channels) are first proposed to descrie the roadcast and on-demand channels, respectively. In addition, the scheme GDS (General Dissemination Scheme) is adopted to perform the channel allocation and the data classification. Experimental results show that the proposed approach gives a near-optimal solution in achieving the minimun access time in the general dissemination environment. Keywords Moile Computing, Hyrid Data Dissemination Categories and Suject Descriptors H.2.4 [DATABASE MANAGEMENT]: Systems-Query processing General Terms Algorithms, Management 1. INTRODUCTION The advance in wireless communication enales users to access data anytime, anywhere, via laptops, PDAs and smart phones. There are several techniques for an information 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 distriuted for profitorcommercialadvantageandthat copies ear this notice and the full citation on the first page. To copy otherwise, to repulish, to post on servers or to redistriute to lists, requires prior specific permission and/or a fee. MDM Ayia Napa Cyprus (c) 2005 ACM /05/05...$5.00 system to disseminate data items to moile users. The roadcast mechanism [1], [5], [9] also known as roadcast disk, is adopted to disseminate the data items periodically and continuously in order to conserve the energy and the andwidth. Moreover, the hyrid dissemination [2], [4], [7],classifies the data items into popular and unpopular (respectively, hot and cold) ones. The popular data items are delivered via roadcast (i.e., push-ased) channels, while the unpopular data items are disseminated via on-demand (i.e., pull-ased) channels. In this paper, we focus on the topic of hyrid dissemination. Prior works in this field studied the channel allocation and data classification according to the access frequency of each data item. In [7], Lee uses the analytical model of roadcast and on-demand channel to derive the optimal channel assignment, and thus determines the popluarity of the data items. In [4], Hu proposes the scheme ABS to perform the channel assignment and data classification simultaneously. Since the previous works are ased on the assumption that the disseminated items are of the same size, in this paper, we consider the environment in which the item size varies from one another. In order to distinguish the environment from others, we name it a general environment. In a general environment, like the access frequency, the sizeofadataitemalsoaffects the performance of the roadcast and on-demand channels. In order to disseminate data more efficiently, an information system should e sensitive to the effect that oth the access frequency and the data item size cause. However, the conventional analytical models for oth roadcast and on-demand channels only consider the access frequency. Based on the conventional analytical models, the dissemination scheme which performs channel allocation and data classificationwillnotealetoecarriedout efficiently. Therefore, new analytical models MGBC (standing for Model of General BroadCast channels) and MGOD (standing for Model of General On-Demand channels) are proposed in this paper to descrie the characteristics of the roadcast and on-demand channels. In MGBC, a roadcast disk scheduling algorithm called MinError Scheduling is also devised together with the analytical roadcast model to reduce the error of estimation to the minimum. Algorithm MinError Scheduling is similar to the Deficit Round Roin (areviated as DRR) technique [10]. However, DRR considers the unlimited sequential input of network flows, while the MinError Scheduling is an algorithm to arrange a limited numer of data items in the roadcasting environment, in which data items will e roadcast periodically. In MGOD, the M/G/c queue rather than the con-

2 ventional M/M/c queue is used to descrie the on-demand channels. Based on MGBC and MGOD, a general dissemination scheme (areviated as GDS) is adopted to perform the channel allocation and data classification. In the eginning, the average access time of the roadcast and the on-demand channels are derived from MGBC and MGOD, respectively. After that, the channel allocation and the data classification are performed according to the derivation of MGBC and MGOD. In the conventional schemes [2], [4], [6], [7], the channel allocation and the data classification according to the access frequency of each data item ecause of the assumption that each item is of the same size. However, in the modern dissemination environment, the size of each data item is different from one another. To remedy this, GDS classifies the data according to the access frequency and the item size into the roadcast cluster and the on-demand cluster. In addition, in order to overcome the efficiency issues, a two-phase classification mechanism is adopted in GDS. The first phase performs the coarse classification according to the access frequency to achieve the coarse alance point. The second phase performs the fine classification according to the product value of the access frequency and the data item size to achieve the fine alance point. To measure the performances of the proposed models and scheme, several experiments are conducted. Since there is no previous work, a directly perceived dissemination scheme, named CS, is used to compare the efficiency of the proposed GDS. The experimental results show that not only can the proposed models estimate the channel characteristics accurately, ut also the proposed scheme achieves the alance point of access time efficiently. The rest of the paper is organized as follows. In Section 2, preliminaries of the conventional hyrid data dissemination environment and concepts of M/G/c queue are given. In Section 3, the dissemination scheme GDS is proposed with the analytical models MGBC and MGOD. The experimental results are shown in Section 4. Finally, this paper will e concluded in Section PRELIMINARIES 2.1 Hyrid Data Dissemination In Figure 1, the architecture of an information system with hyrid data dissemination is illustrated. By collecting the requests of the moile users, the information system classifies the requested data items into popular (i.e., hot) and unpopular (i.e., cold) according to their access frequencies. The popular items are scheduled y a roadcast program into roadcast channels, which can e viewed as roadcast disks. The unpopular items are put in an ondemand queue. The moile users receive the requested data items via the roadcast (i.e., push-ased) and on-demand (i.e., pull-ased) channels. The conventional analytical models of the roadcast and on-demand channels are descried as follows. Conventional Analytical Models : In roadcast channels, the data items are roadcast periodically in each roadcast cycle. Consider n data items with equal size z, andc push roadcast channels with individual andwidth. In a flat roadcast environment, the access time for retrieving data will e: Information System Broadcast Program On-demand Queue Up-link Channel Pushed and Pulled Channels Moile Clients Figure 1: The architecture of hyrid data dissemination W = nz + z 2c push. (1) As for the on-demand channels, the server will reply the data item directly. Since the arrival process of the requests is assumed to e Poisson with the arrival rate λ pull, the c pull on-demand channels with individual andwidth can e modeled as an M/M/c queue. The average access of an on-demand channel will thus e: r c pull W om/m/c = 1 µ +( )pe, where (2) c pull!(c pullµ)(1 ρ) 2 µ z, ρ = λ pull c pull µ,r = λ pull µ, and p e =( X c pull 1 r n /n!+r c pull / ((1 ρ)c pull!)) 1. n=0 2.2 M/G/c Queue Analysis It is in essence more difficult to analyze the M/G/c queue than the M/M/c queue numerically. Therefore, related research in M/G/c queue focuses on the ounds and approximations of performance measures [3], [11], [12], [13]. In this susection, we introduce the model adopted in [11], [12], and riefly descrie the approximation of waiting time and queue length in M/G/c queue. The approximating results will e adopted to analyze the channel characteristics in the next section. Before descriing the procedure of the approximation, we list the related theory and definitionsasfollows: Definition 1: Processors Sharing (PS) queueing discipline is defined as a queueing discipline in which all customers in the queue receive service simultaneously. In a single server or multi-server PS queue, the capacity C of the server(s) is equally shared etween the customers in the system. There are several properties for PS. If there are N customers in the system, each receives the service at the rate C/N. Customers do not have to wait at all; the service starts immediately upon arrival. Moreover, the remaining service times of the customers in system are i.i.d. Definition 2: The work in system, V(t) for any t 0, is defined as the sum of the service times of all customers

3 in queue and the remaining service times of all customers in service. Theory 1: In an M/G/1 queue, let a random variale S e represent the remaining service time of the server. The expected value of S e can e derived as: E[S e ]= E[S2 ], where 2E[S] E[S] represents the expected value of the service time, while E[S 2 ] represents the expected value of the second moment of the service time. The proof of this theory can e found in [12] for interested readers. Approximation Procedure of an M/G/c Queue: Let V M/G/c e the corresponding stationary work of an M/G/c queue under FIFO. We get an approximation on first-moment performance as E[V M/G/c ] E[V PS], where V PS is the stationary work of a processors sharing queue. V PS = P N PS j=1 Sej. NPS represents the numer of customers in the system. S ej represents the remaining service time of each customer. To e specific, N PS N M/M/c. Since S ej are i.i.d., we derive the following equations: NX PS E[V PS]=E[ S ej] =N PSE[S e]=n M/M/c E[S e]. j=1 On the other hand, E[V M/G/c ]=Q M/G/c E[S]+cρE[S e ], where ρ = λe[s]/c. Q M/G/c is the average numer in the M/G/c queue, and c represents the numer of the servers. The approximation will e Q M/G/c E[S]+cρE[S e] N M/M/c E[S e]. Since N M/M/c = Q M/M/c +cρ and E[S e]=e[s 2 ]/(2E[S]), the right-hand size of the approximation can e rewritten E[S as Q 2 ] M/M/c 2E[S] + cρ E[S2 ] 2E[S], while the left-hand size will e Q M/G/c E[S] +cρ E[S2 ]. Therefore, we derive the final approximation 2E[S] as: Q M/G/c Q M/M/c E[S 2 ] 2E 2 [S]. (3) 3. MODELS MGBC/MGOD AND SCHEME GDS In this section, the analytical model of the roadcast channels (MGBC) and the analytical model of the on-demand channels (MGOD) are first descried to derive the average access time. After that, ased on the analytical models, a disseminating scheme GDS is proposed. A list of the symols used in this section is given in Tale 1. Consider a server with a dataase containing D = M items. In the dataase, the size of each data item d i is denoted y z i. Assume that the request for each item d i forms a Poisson process with the arrival rate λ i. The data items d 1,d 2...d k U pull are disseminated via the on-demand channels, while the data items d k+1,d k+2,...d M U push are disseminated via roadcast channels, where U push and U pull,representing the roadcast and on-demand items, respectively, are two disjoint susets of D. Note that U push U pull = D. Let λ push = P M i=k+1 λi e the push request arrival rate and λ pull = P k i=1 λ i ethepullrequestarrivalrate. Thenumers of roadcast and on-demand channels are c push and c pull respectively. 3.1 MGBC for Broadcast Channels: Assume that the data items d k+1,d 2...d M are roadcast periodically via c push roadcast channels. The aggregate size of the data items is hence P M i=k+1 zi. Moreover, the aggregate andwidth for roadcast channels is c push,where represents the andwidth of a single roadcast channel. Since the roadcast cycle can e derived y ( P M i=k+1 z i)/c push, the average proe time is ( P M i=k+1 zi)/2c push. Thus, the access time for retrieving the data item d i through the roadcast channel can e otained as: W (i) = ³ PM j=k+1 zj 2c push + zi. (4) The average access time of the roadcast channel can e derived y considering the expected value of W (i) : W = E[W (i) ]= X M ³ zj PM j=k+1 = 2c push i=k+1 + X M piw (i) i=k+1 pi z i, (5) where p i represents the proaility that the data item d i is accessed in the roadcast channel. i.e., p i = λ i/λ push. Special case of MGBC : A special case occurs when z k+1 = z k+2 =... = z M = z. We have W = (M k)z 2c push + z,whichisthesameastheresult in Eq. (1), the access time of the roadcast channels in the conventional environment. For ease of exposition, the flat roadcast program is considered in this paper. Nevertheless, our improved analytical models are also applicale to the design of other roadcast programs. In the conventional model in which each data item is of the same size, the flat roadcast program can e generated y a round-roin scheduling. However, when the sizes of data items are different, the round-roin approach will cause the difference of the aggregate size among each roadcast channel. The practical access time of round-roin scheduling cannot meet the result derived in Eq. (5). Although there are several techniques such as DRR [10] proposedtoimprovetheround-roininotherresearchfields, those techniques are not suitale to e used here since the roadcasting environment is considered. Therefore, a scheduling approach called MinError Scheduling is proposed with MGBC analysis so that the characteristics of the roadcast channels can e modeled y the aove derivations in Eq. (5). Let U push represent the data items disseminated in the roadcast approach, and the set {U 1,U 2,..., U cpush e the scheduling result. Each element U i in the set corresponds to the data items in an individual roadcast channel c i. The algorithmic form of the MGBC scheduling is outlined as follows: Procedure MinErrorScheduling : Input: U push, c push Output: {U i, 1 i c push 1. egin 2. DescendSort(U push) yitem_size 3. cur_item 1 4. while cur_item sizeof(u push) 5. for i =1;i c push ;i ++

4 Tale 1: Symols and corresponding descriptions Symol Description D The disseminated dataase c Numer of channels M Numer of items in the disseminated dataase d i The i-th item in the dataase λ i The request arrival rate for d i z i The size of d i D push The items disseminated in roadcast channels D pull The items disseminated in on-demand channels c push Numer of roadcast channels c pull Numer of on-demand channels λ push The aggregate access frequency of the pushed items The aggregate access frequency of pulled items λ pull 6. U i U i +{d cur_ item 7. aggr_itemsize of c i+=sizeof(d cur_ item) 8. cur_item ++; 9. end for 10. AscendSort({U i 1 i c push ) y aggr_itemsize 11. end while 12. return {U i 13.end Example 1: The intuition of the MinError Scheduling is to make the aggregate size of data items in each channel e the same as others. Consider the data items d i, 1 i 9 with corresponding sizes {1, 2, 3, 2, 2, 4, 1, 3, 4 and three roadcast channels. The scheduling result of the conventional round-roin approach is shown in Figure 2(a). The major drawack of the approach is the unfair distriution of the aggregate sizes among the roadcast channels. Since the aggregate size of each channel is different from one another, the practical access time of the roadcast channels cannot meet the result in Eq. (4). In contrast, in the proposed MinError Scheduling approach is depicted in Figure 2(). In the eginning, the data items {d i are sorted according to the item size in a descending order. After that, in each iteration, the data items are assigned to each channel according to their sorted order. During the scheduling, the aggregate item size of each channel is also considered. i.e., the channel order in which a data item is assigned will e sorted dynamically according to the aggregate item size of the channel. When the MinError Scheduling procedure terminates, the aggregate size of each channel will tend to e close to each other. Therefore, the error etween the practical access time of the roadcast channels and the result in Eq. (4) can e reduced to the minimum. 3.2 MGOD for On-demand Channels: In the general dissemination environment, since the item size varies from data oject to another, it is clear that the service time will no longer e exponential distriuted. Therefore, the conventional M/M/c queueing model can not suitale to e adopted here. In order to characterize the service ehavior, we use an M/G/c queue to model the general ondemand dissemination. According to Eq. (3) in Section 2, the queueing length of an M/G/c queue can e approximated as Q M/G/c Q M/M/c E[S 2 ]/(2E 2 [S]), wheres is a service time. Q M/G/c is the expected numer of customers in the queue. In the righ-hand side of Eq. (3), Q M/M/c is the corresponding queueing length for an M/M/c queue withthesamearrivalandservicerate. Sincethesizeofdata item d i is z i units, if we let the andwidth of each channel e, the average service time per unit of each channel will e 1/. S i,theservicetimeforthedataitemd i, will have the mean value z i /. Assume the data items d i, 1 i k with arrival rate λ i. Each random variale S i is exponentially distriuted, and E[S i ], the average service time of the data item d i, can also e viewed as a random variale. Therefore, the average service time E[S] can e derived as: E[S] = E[E[S i]] = X Z k pi( i=1 = X k i=1 0 s i z i exp( z i s i)ds i) pizi/, (6) where p i = λ i /λ pull. Moreover, the value of E[S 2 ] can also e derived similarly: E[S 2 ] = E[E[Si 2 ]] = X Z k pi( s 2 i exp( s i=1 i)ds i) 0 z i z i = X k i=1 2piz2 i / 2. (7) Also, the term Q M/M/c can e derived from Eq. (2). Since the value of Q M/M/c, E[S 2 ] and E[S] can e derived exactly. According to the Little s Formula [8], the average waiting time in an M/G/c queue can e directly derived as: W qm/g/c = Q M/G/c /λ pull Q M/M/cE[S 2 ] 2E 2 [S]λ pull.

5 C 1 C 2 C 3 d 1 d 4 d 7 d 2 d 5 d 8 d 3 d 6 d 9 item freq size item freq size d d d d d d d d d d (a) Round-Roin Scheduling Figure 3: The profile of example 2 C 1 C 2 C 3 d 3 d 6 d 9 d 8 d 4 d 5 d 7 d 1 d 2 () MinError Scheduling Figure 2: Illustration of the scheduling approaches of the roadcast channels Then the average waiting time in the on-demand channels can e derived as: W om/g/c = E[S]+W qm/g/c E[S]+ Q M/M/cE[S 2 ]. (8) 2E 2 [S]λ pull Special case of MGOD: A special case occurs when the data items are of the same size. Assume that z 1 = z 2 =... = z k = z. The average servicetimeofeachdataitemisz/. SinceS is exponentially distriuted, we have E[S] =z/ and E[S 2 ]=2(z/) 2. The Eq. (8) can e rewritten as follows. W om/g/c = z/ + Q M/M/c 2(z/) 2 2(z/) 2 1 λ pull = z/ + W qm/m/c = W om/m/c. The aove derivation is also the same as the formula in the on-demand channel modeled y an M/M/c queue as in Eq. (2). Example 2: Consider the data items {d i, 1 i 10 with corresponding access frequency and item size shown in Figure 3. The channel numer and the andwidth of each channel are assumed to e 6 and 100, respectively. The parameter λ pull can e derived from the summation of the access frequency of each data item. i.e., λ pull =26.First,we adopt the conventional M/M/c queue to analyze the channel characteristics. The parameter µ in Eq. (2) is viewed as the reciprocal of the average service time E[S], which can e derived from Eq. (6). Since E[S] =0.109, wehave µ = According to Eq. (2), we have the average access time of on-demand channels W om/m/c = On the other hand, if the channel is modeled y MGOD, the average access time can e derived from Eq. (8). The parameter Q M/M/c can e otained y Little s Formula and Eq. (2). i.e., Q M/M/c = Also,wehaveE[S 2 ]=0.036 according to Eq. (7). Finally, we derive W om/g/c = Comparing W om/g/c with W om/m/c,wefind that there is significant error for M/M/c queueing analysis. The misestimate of M/M/c modelwillaffect the later channel estimation and item classification. 3.3 GDS for Dhannel Allocation and Data Classification: This susection discusses the proposed scheme GDS, which is used to disseminate data items according to the models MGBC and MGOD. Scheme GDS comprises three parts: channel allocation, data classification, and item scheduling. For channel allocation, the scheme determines the numer of roadcast and on-demand channels. For data classification, the data are grouped into two clusters, the roadcast cluster and the on-demand cluster, so that the two clusters are disseminated via roadcast and on-demand channels, respectively. As for item scheduling, the items in roadcast cluster are rescheduled to each roadcast channel according to MinError Scheduling in Section 3. Unlike the conventional schemes, the scheme GDS considers oth the access frequency and the data item size. Therefore, the alance point with the near-optimal access time is achieved. The algorithmic for of scheme GDS is given as follows: Scheme GDS: Input: D, c Output: U push,u pull,c push,c pull, {U i 1 i c push 1. egin 2. GloalMinT ime 3. U push {,U pull { 4. c push 0,c push 0 5. c push c, c pull 0 6. while c push 0 7. if c push== c 8. Upush D, Upull { 9. LocalMinT ime EvaluatePushTime(U push,c push ) 10. else if c pull == c 11. Upush {, Upull D

6 12. LocalMinT ime EvaluatePullTime(Upull,c pull) 13. else 14. (U push,u pull ) DataClassification(D, c push,c pull ) 15. t push EvaluatePushTime(Upush,c push ) 16. t pull EvaluatePullTime(Upull,c pull ) 17. t EvaluateTime(t push,t pull,u push,u pull ) 18. LocalMinT ime t 19. end if 20. if LocalMinT ime < GloalMinTime 21. U push U push, U pull U pull 22. c push c push, c pull c pull 23. end if 24. c push c push 1, c pull c pull continue 26. end while 27. {U i MinErrorScheduling(U push,c push ) 28. return U push,u pull,c push,c pull, {U i 29.end Given the channel numer c and the dataase D, scheme GDS will return c push and c pull, which represent the numers of roadcast and on-demand channels, respectively, and U push and U pull, which stand for the roadcast and ondemand clusters. According to the derived U pull and c push, GDS also returns the scheduling results of the roadcast program. There are three conditions in channel allocation. The first condition occurs when all channels are allocated for roadcast dissemination, where U push = D and U pull = {, The procedure EvaluatePushTime is used to derive the average access time from Eq. (5). The second condition occurs when all channels are for on-demand dissemination, in which all data items will e delivered via on-demand channel. Similarly, we use the procedure EvaluatePullTime to derive the average access time according to Eq. (8). In the final condition, since c push > 0 and c pull > 0, U push and U pull are non-empty disjoint susets of D. The data classification procedure in Line 14, named as Procedure DataClassification(D, c push,c pull ), is outlined elow. Procedure DataClassification(D, c push,c pull ) Input: D, c push,c pull Output: U push,u pull 1. egin 2. U push D, U pull { 3. Upush D, Upull { 4. DescendSort(U push ) y access_freq MinTime EvaluatePushTime(Upush,c push ) while push 0 7. d LastElement(Upush) 8. Upush U push d, Upull U pull+d 9. λ pull EvaluateAvrgFreq(Upull) 10. E(S) EvaluateServiceTime(Upull) 11. if λ pull E (S) <c pull 12. t push EvaluatePushTime(Upush,c push ) 13. t pull EvaluatePullTime(Upull,c pull ) 14. t EvaluateTime(t push,t pull,u push,u pull ) 15. if t<mintime 16. U push U push 17. U pull U pull 18. end if 19. else 20. reak 21. end if 22. end while 23. d LastElement(U push ) 24. alance_ound Freq(d) Size(d) 25. U c Filter(U push, alance_ound) 26. {d i SusetSelect(U c, alance_ound, MinT ime) 27. U push U push {d i 28. U pull U pull +{d i 29. return U push,u pull 30.end In the proposed data classification procedure, oth the access frequency and the data item size are considered. The innovation of the proposed data classification procedure is the adoption of two-phase classification: coarse phase and fine phase. In the coarse phase, which is from Step 6 to Step 22, the data items are classified according to the access frequency. In the fine phase, as shown from Step 23 to Step 28, the items are re-classified y the product value of the access frequency and the item size. Each phase is illustrated as follows. The coarse phase starts as all data items are initially assigned to U push, and U pull = {. In each iteration of the phase, the data item with minimum access frequency in U push is moved to U pull. The corresponding push access time, pull access time and average access time are derived. The minimum access time in the current iteration is otained y comparing the average access time in the current iteration and the minimum access time in the previous iteration. Moreover, the condition in Step 11 is used to detect the load of the on-demand channel in order to reduce the computation complexity. The coarse alance point, which represents the lowest average access time in the coarse phase, is then reached at Step 22. As for the fine phase, the fine alance point of fine classification is achieved. In this phase, the product value of the access frequency and the item size is viewed as the classification metric. In the previous phase, since the coarse alance point is reached, the access frequencies of the data items near the coarse alance point are similar. Therefore, the size of the data item will dominate the classification in the fine phase. In this phase, several items in U push will further e moved to U pull in order to achieve fine alance point. Considering U push as the solution space, the goal of this phase is to find the most suitale suset {d i d i U push. Initially, the product of the access frequency and the item size of the last data item in U push is viewed as the alance ound. The filtering process in Step 25 is then executed to reduce the complexity: If the frequency-size product of a data item in U push exceeds the alance ound, thisitemisfiltered out. The filtering result of U push is U c, which is viewed as the candidate space, where U c U push. The SusetSelect(U c )

7 procedure is responsile for finding the most suitale suset {d i. Finally, the fine alance point is achieved y moving the data set {d i from U push to U pull. The procedure SusetSelect(U c ), which is outlined elow, plays an important role in the fine classification of the second phase. Given the candidate space U c, the procedure will search and find out the most suitale suset {d i from U c. Since each element in U c contriutes to achieving the fine alance point, the est solution with the closest position to the alance point will e the suset of U c. TheinsightoftheprocedureisviewingthesusetofU c as an integer i in inary representation, where 0 i<2 Uc. The inary representation is stored in a one dimensional array of oolean variales with array size U c. Each element of the array corresponds to a data item. If the the value of an element is 1, the suset contains the corresponding data item, and vice versa. For example, suppose that the U c contains the items {d 0,d 1,..., d 9, which are sorted according to the product of frequency and size in ascending order. Each suset will e mapped to a distinct integer etween 0 and That is, the integer 1023 corresponds to {d 0,d 1,...,d 9 while the integer 0 corresponds to {. In each suset, AggrFreqSize, which represents the aggregate of the frequency-size product of each data item in the suset, is compared with alance_ound. If the value AggrF reqsize of a suset exceeds alance_ound, itwill not e considered as a suitale solution. Generally speaking, the suset with larger AggrFreqSize corresponds to a larger integer. Therefore, from Step 17 to Step 30, the possile susets are searched according to the corresponding integer i. In order to reduce the computation complexity, from Step 7 to Step 15, an approach is applied to reduce the length of the inary array from U c to MinLength. Usually, the length of the inary array is set etween U c and MinLength, as shown in Step 16, in order to compromise etween complexity and accuracy. Procedure SusetSelect(U c ) Input: U c,alance_ound,mintime Gloal Parameter: U push,u pull,c push,c pull Output: {d i d i U c 1. egin 2. {d i { 3. MaxLength U c 4. MinLength 0 5. AggrF reqsize 0 6. AscSortFreqSize(U c ) 7. while true 8. d GetElement(U c,minlength) 9. AggrFreqSize AggrF reqsize+freq(d) Size(d) 10. if AggrF reqsize < alance_ound 11. MinLength MinLength else 13. reak 14. end if 15. end while t t 1 t 2 t 4 t 3 P 1 P 2 P 3 P 4 Figure 4: Coarse classifcation of GDS 16. Length SetLength(M inlength, MaxLength) 17. for i =0;i<2 Length ; i {d i BinaryMapping(U c,i) 19. AggrF reqsize GetAggrFreqSize({d i ) 20. if AggrFreqSize <alance_ound 21. Upush U push {d i 22. Upull U pull +{d i 23. t push EvaluatePushTime(Upush,c push ) 24. t pull EvaluatePullTime(Upull,c pull ) 25. t EvaluateTime(t push,t pull,u push,u pull ) 26. if t<mintime 27. {d i {d i 28. end if 29. end if 30. end for 31.end Example 3: The illustration of the data classification procedure can e shown in Figure 4 and Figure 5. Figure 4 depicts the coarse phase of the classification. In this phase, the data items are classified according to the access frequency. Each point P i, representing a classification result, corresponds to the access time t i. The first phasereturnsthecoarsealancepointp 3,whichleadsto the minimum access time t 3. However, this phase cannot achieve the alance point with optimal access time ecause only access frequency is considered. Therefore, in the fine phase, the data items are reclassifiedasedonthecoarse alance point, as depicted in Figure 5. Unlike the previous phase, the fine classification is performed according to the value of AggrF reqsize. In the second phase, each {d i, the suset of U c, U c U pull, is filtered according to the value of alance_ound, which corresponds to the point P 4. If the value AggrF reqsize of a suset is smaller than alance_ound, it is called a suitale solution. Every suitale solution is sanned y the procedure SusetSelect(U c). The solution which leads to the minimum access time is called the most suitale solution. Therefore, the most suitale solution will e the closest point to P opt among all suitale solutions. 4. EXPERIMENTAL RESULTS λ pull

8 t t 4 t 3 t opt P 3 P opt P 4 alance_ ound AggrFreqSize Figure 5: Fine classification of GDS access time GDS ABS CS Φ 4.1 Simulation Model Tale 2 lists the simulation parameters. The access pattern of the data is generated y Zipf distriution expressed p i = ( 1 i )θ Á P M j=1 ( 1 j )θ, where θ is a skew coefficient and 1 i M. The size of each data item is represented y 10 φ Kytes, where φ is uniformly distriuted over the interval [0, Φ]. Φ can e regarded as the diversity of the item size. More specifically, Φ varies from 0 to 2.5 during the experiment. The case of Φ =0implies that all data items are of the same size (i.e., 1 KBytes). On the other hand, when Φ =2.5, the size of each data item is located in the interval 10 0, KBytes. Therefore, the diversity of the data item size increases as Φ increases. 4.2 GDS Analysis In the experiment, we discuss the performance of the proposed disseminating scheme GDS. Since there is no prior work, we adopt a direct perceived scheme CS (standing for Contrast Scheme) for comparison purposes. Both the access frequency and the item size are still taken into consideration for scheme CS. Compared to GDS, CS has the following properties. First, like GDS, CS adopts the proposed MGBC/MGOD. Moreover, scheme CS is a one-phase classification technique, which classifies the data items directly according to the product value of the access frequency and the item size of data items. The second property differentiates CS from GDS. In order to investigate how the analytical models affect the schemes, the scheme ABS [4], which adopts the conventional analytical models instead of MGBC/MGOD, is also used for comparison. Figure 6 shows the performance of different schemes as the value Φ varies. Compared to the scheme ABS and CS, the scheme GDS achieves the smallest average access time. The reason that the curve of ABS goes up drastically can e explained as follows. First, as the experimental results shown in the previous susections, the increasing Φ causes larger errors of oth roadcast and on-demand channels in the con- Figure 6: The access time v.s. the diversity of the data items for different schemes with fixed θ =1 ventional analytical models. Moreover, scheme ABS classifies the data only according to the access frequency. When thevarianceofthesizeofthedataitemsecomeslarger, this classification approach will tend to work inefficiently. Therefore, the error etween ABS and GDS increases as Φ increases. On the other hand, scheme CS, which adopts the proposed MGBC/MGOD models, is a one-phase classification scheme ased only on the frequency-size product of each data item. Like ABS, the inefficiency is indistinct when Φ is small. However, when Φ ecomes larger, the increasing variance of the size among data items will make the one-phase classification inefficient. Therefore the error etween CS and GDS increases as Φ increases. This experiment shows that the GDS is more suitale for disseminating the highly diverse data items. Figure 7 depicts the access time of different schemes as the value θ varies. Like the previous experiment in this susection, the GDS achieves the est performance, while the ABS results in the worst performance. There are several oservations as follows. First, the access time of each scheme decreases as θ increases. It is ecause that highly skewed access frequency enhances the degree of request locality. This phenomenon will reduce the traffic load of oth the roadcast and the on-demand channels. Second, the difference of the access time etween scheme ABS and GDS changes slightly. The value θ caneviewedanminorfactorofinefficiency for scheme ABS. As for scheme CS, the difference from GDS changes irregularly. It can e ascried to the inaccuracy of the one-phase classification of CS. Therefore, the classification approach dominates the inefficiency with the fixed diversity of the item size. 5. CONCLUSION In this paper, we focus on the general model of hyrid dissemination. Unlike the previous works in which each data

9 Tale 2: Description of simulation parameters notation meaning value z data item size 10 φ Kytes c numer of channels 6 channel andwidth 100 Kytes/sec M numer of data items in dataase 300 θ skew coefficient, access pattern y Zipf Φ exponent range of the data item size access time GDS ABS CS θ Figure 7: The access time v.s. the skewness of access pattern for different schemes with fixed Φ =1.5 item is assumed to e of the same size, this paper discarded the assumption ecause of the variety and the diversity of the modern wireless communication environment. Specifically, we first proposed two analytical models, MGBC and MGOD, to descrie the roadcast and on-demand channels, respectively. After that, ased on the proposed analytical models, the dissemination scheme GDS was adopted to achieve the minimum access time. The experimental result has shown that what we proposed achieved etter performances than the conventional works. Therefore, the proposed models and scheme will e more suitale to disseminate data in the modern information system. 6. ACKNOWLEDGEMENTS The work was supported in part y the National Science Council of Taiwan, R.O.C., under Contracts NSC E PAE. 7. REFERENCES [1] S.Acharya,R.Alonso,M.J.Franklin,andS.B. Zdonik. Broadcast disks: Data management for asymmetric communications environments. In Proceedings of the 1995 ACM International Conference on Management of Data, pages , May [2] S.Acharya,M.J.Franklin,andS.B.Zdonik. Balancing push and pull for data roadcast. In Proceedings of the 1997 ACM International Conference on Management of Data, pages , May [3] O.J.B.J.W.CohenandN.Huffels. Approximations of the mean waiting time in an m/g/s queueing system. Oper. Res., 27: , [4] C.-L. Hu and M.-S. Chen. Adaptive alanced hyrid data delivery for multi-channel data roadcast. In Proceedings of the IEEE 2002 International Conference on Communications, April [5] J.-L. Huang and M.-S. Chen. Dependent data roadcasting for unordered queries in a multiple channel moile environment. IEEE Trans. on Knowledge and Data Engineering, 16(6), Jun [6] J.-L. Huang, W.-C. Peng, and M.-S. Chen. Binary interpolation search for solution mapping on roadcast and on-demand channels in a moile computing environment. In Proceedings of the 10th ACM International Conference on Information and Knowledge Management, Novemer [7] W.-C. Lee, Q. Hu, and D. L. Lee. A study on channel allocation for data dissemination in moile computing environments. ACM/Baltzer Moile Networks and Applications, Special Issue on Resource Management in Wireless Systems, 4(2): , [8] J.D.C.Little.Aproofforthequeueingformula: L=λw. Oper. Res., 9: , [9] W.-C. Peng and M.-S. Chen. Efficient channel allocation tree generation for data roadcasting in a moile computing environment. Wireless Networks, 9(2): , [10] M. Shreedhar and G. Varghese. Efficient fair queueing using deficit round roin. In SIGCOMM, [11] C.-L. Wang and R. W. Wolff. The m/g/c queue in light traffic. Queueing Systems: Theory and Applications, 29(1):17 34, [12] R. W. Wolff. Stochastic modeling and the theory of queues. Prentice-Hall, Englewood Cliffs, NJ, [13] D. D. Yao. Refining the diffusion approximation for the m/g/c queue. Oper. Res., 33: , 1985.

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