Exploring aggregate effect with weighted transcoding graphs for efficient cache replacement in transcoding proxies

Exploring Aggregate Effect with Weighted Transcoding Graphs for Efficient

Cache Replacement in Transcoding Proxies

Cheng-Yue Chang and Ming-Syan Chen

Department of Electrical Engineering

National Taiwan University

Taipei, Taiwan, ROC

E-mail: {cychang@arbor.ee.ntu.edu.tw

; mschen@cc.ee.ntu.edu.tw}

Abstract

This paper explores the aggregate effect when caching multiple versions of the same Web object in the transcoding proxy. Explicitly, the aggregate profit from caching multi-ple versions of an object is not simply the sum of the profits from caching individual versions, but rather, depends on the transcoding relationships among them. Hence, to evaluate the profit from caching each version of an object efficiently, we devise the notion of a weighted transcoding graph and formulate a generalized profit function which explicitly con-siders the aggregate effect and several new emerging factors in the transcoding proxy. Based on the weighted transcod-ing graph and the generalized profit function, an innova-tive cache replacement algorithm for transcoding proxies is proposed in this paper. Experimental results show that the algorithm proposed consistently outperforms compan-ion schemes in terms of the delay saving ratios and cache hit ratios.

1. Introduction

Recent technology advances in mobile communication have ushered in a new era of personal communication. Users can ubiquitously access the Internet via many mobile appliances, such as handheld PCs, personal digital assis-tants (PDAs), and WAP-enabled cellular phones. As these devices are divergent in size, weight, input/output capabil-ities, network connectivity and computing power, differen-tiated services should be tailored and delivered in a certain way to meet their diverse needs. In addition, users may have different content presentation preferences. Both lead to the demand of transcoding technologies to adapt the same Web object to various mobile appliances [6].

Transcoding is defined as a transformation that is used to convert a multimedia object from one form to another,

fre-quently trading off object fidelity for size. For the mobile appliances featured with lower-bandwidth network connec-tivity, transcoding can be used to reduce the object size by lowering the image resolution or downscaling the image size. For the mobile appliances which only accept a text, transcoding can be used to covert the image or speech into a text. As pointed in [5], from the aspect of the place where transcoding is performed, the transcoding technologies can be classified into three categories, i.e., server-based, client-based, and proxy-based approaches. In the server-based ap-proaches [12], Web objects are off-line transcoded to mul-tiple versions and stored in the server disks. The advantage of this approach is that no additional delay will be incurred by transcoding during the time between the client issues a request and the server responses to it. The drawback is, however, keeping several versions of the same object in the server may cost too much storage space. Further, this ap-proach is notflexible in dealing with the future change of clients’ needs. Conversely, in the client-based approaches, transcoding is left for mobile clients for considerations. The advantage of this approach is that it can preserve the origi-nal semantic of system architecture and transport protocols. However, transcoding at the client side is extremely costly due to the limited connection bandwidth and computing power of a mobile device. For these reasons, it will be bet-ter to transcode the Web objects at the inbet-termediate proxies. Many studies have recently been conducted to explore the advantages of proxy-based approaches [6][7][9][10] where an intermediate proxy is able to on-the-fly transcode the re-quested object to a proper version according to the client’s specification before it sends this object to the client. Such an intermediate proxy which possesses the transcoding ca-pability is referred to as a transcoding proxy in this paper.

While the transcoding proxy is attracting more and more attention, it is noted that a transcoding proxy, just as a

tra-ditional Web proxy, plays an important role in the func-tionality of caching. To enable the cache replacement algo-rithms devised for traditional Web proxies [1][2][3][14][17] to handle the situations in transcoding proxies, extensions to these traditional algorithms are needed. In [5], two extended strategies, which are called coverage-based and

demand-based strategies, are proposed. Note that when transcoding

the requested object to the specified version before sending this object to the client, the transcoding proxy has the oppor-tunity of caching either the original version, the transcoded version, or both versions of the object in the local memory. In view of this, the coverage-based strategy in [5] is de-signed to choose the original version of the object to cache, whereas the demand-based strategy is designed to choose the transcoded one to cache. In addition, most cache re-placement algorithms employ an eviction function to de-termine the priorities of objects to be replaced. The LRU (Least Recently Used) algorithm, for example, evicts the page with the largest elapsed time since the last access of that page. The LFU (Least Frequently Used) algorithm, on the other hand, evicts the page with the smallest refer-ence rate in the cache. Collaborating the eviction function of the traditional cache replacement algorithm with these two strategies, we can decide the priority of the cached object to be evicted and the version of the new coming object to be cached. For example, the demand-based extension to LRU algorithm in [5] replaces the object of the largest elapsed time since its last access with the transcoded version of the new coming object. The coverage-based extension works similarly except the evicted object is replaced with the orig-inal version of the new coming object.

Note that both the coverage-based and demand-based extensions in [5] are designed for efficient Web caching. As a research work along this direction, we shall design in this paper a cache replacement algorithm for transcod-ing proxies that maximizes the performance of Web object caching. The motivation for our study is mainly due to the new emerging factors in the environment of transcod-ing proxies. First, transcodtranscod-ing incurs additional delays, and this factor should be included in the eviction functions for consideration whenever the eviction priority is determined. For example, suppose there are two cached objects, say, A and B, having the same eviction priority according to some eviction function which does not consider the transcoding delay. Clearly, if the transcoding delay of A is longer than that of B, A should be given a higher eviction priority than B so as to shorten the response time. Without considering the transcoding delays, the traditional cache replacement al-gorithms could make a wrong replacement decision. Sec-ond, different versions of the same object are of different

(a)

A1 A2

A3

Reference sequence 1: A2, A1, A3, A2, A3, A1, A1, A3, A2 Delay of fetching A from Web server: 10 ms

5

5

3

Reference sequence 2: A2, A1, A2, A2, A3, A1, A2, A3, A2

A1 A3 A2 5 5 3 0 0 0 (b) Server 10

Figure 1. Illustrative examples.

sizes. Specifically, the transcoded version is usually of a smaller size than the original one. Thus, the strategy which chooses the transcoded version to cache may admit more objects with the same cache capacity. An eviction function is thus required to take this object size factor into consider-ation. Third, the reference rates to different versions of an object should be considered separately because the distribu-tion of them could affect the caching decision. This can be seen by two contrary examples illustrated in Figure 1. As-sume that Lena image has three versions (i.e., A1, A2 and A3). A1 is the original version with full resolution and size. A1 can be transcoded to A2 which is of the same size but a lower resolution. A1 can be also transcoded to A3 which is of a lower resolution and a smaller size. In addition, A2 can be transcoded to A3, whereas A3 is the least detailed ver-sion which cannot be transcoded any more. The transcoding delay for A1 to A2 and that for A1 to A3 are both assumed to be 5 ms, and the transcoding delay for A2 to A3 is 3 ms. The transmission delay of fetching object A from Web server is assumed to be 10 ms. Consider the reference se-quence 1 in Figure 1, where the numbers of references to A1, A2 and A3 are all equal to 3. By caching A1, we can get the delay saving equal to 90 ms, which is obtained by 3 ∗ 10 + 3 ∗ (10 + 5 − 5) + 3 ∗ (10 + 5 − 5). By caching

A2, we can, however, get the delay saving only equal to 81 ms, which is obtained by3 ∗ (10 + 5) + 3 ∗ (10 + 5 − 3). As such, caching a more detailed version can get more benefit in this case. However, in reference sequence 2 in Figure 1, the numbers of references to A1, A2 and A3 are 2, 5 and 2 respectively. By caching A1, we can get the delay saving equal to 90, which is obtained by 2 ∗ 10 + 5 ∗ (10 + 5 − 5) + 2 ∗ (10 + 5 − 5). In con-trast, by caching A2, we can get the delay saving equal to 99, which is obtained by5 ∗ (10 + 5) + 2 ∗ (10 + 5 − 3). In this case, caching a less detailed version will get more benefit than caching a more detailed one.

Also, consider the example in reference sequence 1 in Figure 1, the individual delay savings from caching A1 and A2 are 90 and 81 ms respectively. However, the aggre-gate delay saving from caching A1 and A2 at the same time is not simply the sum of the individual delay savings (i.e., 90 + 80 = 171 ms), but rather, depends on the transcoding relationship among them. Note that the transcoding rela-tionship in Figure 1(a) can be depicted by a directed graph with weights on edges as shown in Figure 1(b). Such a di-rected graph is called a weighted transcoding graph, which will be formally defined in Section 2. In the example of Figure 1, when caching A1 and A2 at the same time, the subsequent references to A2 and A3 are not necessary to be supported by the transcoding from A1 any more. Instead, they can be supported by the transcoding from A2 because there is no transcoding between A2’s, and the transcoding cost from A2 to A3 is smaller than that from A1 to A3. Hence, the delay saving from caching A1 is over-evaluated, and should be revised as30 = 3 ∗ 10. The aggregate delay saving from caching A1 and A2 together is thus 111 rather than 171. Such an over-evaluation will probably result in a wrong cache replacement decision as will be explained below.

Object Delay saving Size

W 90 8 A1 90 10 X 85 8 Y 75 8 Replace Y with A2

Object Delay saving Size

W 90 8

A1, A2 111 18

X 85 8

Figure 2. Wrong cache replacement decision.

In the case of Figure 2, assume that the cache capacity is 34 Kbytes, and the cache is filled with W , A1, X and Y . The computed delay saving and size of each object is listed in the left table of Figure 2. Suppose here comes a new object A2. In the example of Figure 1, we have calculated

the delay saving from caching it is 122. Without consider-ing the aggregate effect of cachconsider-ing A1 and A2 together, the cache replacement algorithm will choose to replace Y with A2 because the delay saving from caching Y is smaller than that from caching A2. The resulting cache will, thus, be-come the one as listed in the right table of Figure 2. Recall that the aggregate delay saving from caching both A1 and A2 is 111 rather than 171. Thus, the overall delay saving of the entire cache decreases from 340 to 286 as Y is replaced with A2. Obviously, replacing Y with A2 is a wrong cache replacement decision.

Consequently, we propose an efficient cache replace-ment algorithm for transcoding proxies in this paper. Specifically, we formulate a generalized profit function to evaluate the profit from caching a version of an object. This

generalized profit function explicitly considers several new

emerging factors in the transcoding proxy and the aggre-gate effect of caching multiple versions of the same object. As explained, the aggregate effect is not simply the sum of the delay savings from caching individual versions of an object, but rather, depends on the transcoding relation-ship among these versions. Thus, the notion of a weighted

transcoding graph is defined to represents the

transcod-ing relationship among the different versions of an object. In addition, to evaluate the aggregate effect properly, we devise Procedure MATC (standing for Minimal Aggregate

Transcoding Cost) to find the subgraph of the weighted

transcoding graph. This subgraph shows the transcoding re-lationship which minimizes the aggregate transcoding cost when caching a certain set of versions of the object. Uti-lizing the generalized profit function as the eviction func-tion, we propose in this paper an innovative cache replace-ment algorithm for transcoding proxies. This algorithm is referred to as algorithm AE (standing for Aggregate

Ef-fect) for the reason that it explores the aggregate effect of

caching multiple versions of an object in the transcoding proxy. Using an event-driven simulation, we evaluate the performance of our algorithm under several circumstances. By varying the simulation parameters, we observe the per-formance impacts of two important parameters, including the cache capacity and γ_{F D} ratio. In the simulation, al-gorithm AE is compared with several companion schemes (i.e., based LRU, demand-based LRU,

coverage-based LRUMIN and demand-coverage-based LRUMIN). The

exper-imental results shows that algorithm AE proposed consis-tently outperforms the companion schemes in terms of the primary performance metric, delay saving ratio, and also the secondary performance metrics, hit ratios.

The rest of this paper is organized as follows. In Section 2, we introduce the system environment where the caching

issues in the transcoding proxy are considered. We propose our cache replacement algorithm for transcoding proxies in Section 3. In Section 4, we empirically evaluate the perfor-mance of several algorithms. We conclude this paper with Section 5.

2. System Environment

To facilitate our later discussion in this paper, we de-scribe the system environment in this section. The system architecture is presented in Section 2.1. The Web object transcoding is described in Section 2.2.

2.1. System Architecture

This paper studies cache replacement algorithms for the transcoding proxy which typically interconnects two het-erogeneous networks. A well-known example of such a transcoding proxy is the WAP proxy or a gateway, which in-terconnects the wireless network and the Internet. The mo-bile clients connect to the WAP proxy via the wireless net-work, whereas the WAP proxy connects to the Web servers via IP network. The mobile clients are capable of request-ing and renderrequest-ing WAP content. They vary widely in their features such as the screen size/color, the processing power, storage, and user interface. The clients can describe their capabilities via User Agent Profiles (UAProfs) [16] sup-ported by WAP. UAProfs are initially conveyed when WSP sessions are established with the WAP proxy. Keeping these UAProfs in the proxy, the WAP proxy knows the preference of each client, and will on-the-fly transcode the requested object to a proper version before delivering to it. Moreover, as pointed out in [15], to speed up wireless data access, the WAP proxy also acts as the role of an caching proxy. In such a circumstance, the WAP proxy has the opportunity of caching either the original version, the transcoded version, or both versions of the object for future references. This emerging caching model justifies the problem we deal with in this paper. Note that while being applicable to the WAP architecture, the cache replacement algorithm we shall de-vise is for general transcoding proxies and by no means re-stricted to the WAP applications.

2.2. Web Object Transcoding

Transcoding is defined as a transformation that is used to convert a multimedia object from one form to another, frequently trading off object fidelity for size. The original object contains the full information of the content, and usu-ally corresponds to the original version or the most detailed version of the object. The original version of object can

be transcoded to a less detailed one, and such a transcoded object is called the transcoded version or the less detailed version of the object. Without loss of generality, we as-sume that each object can be represented in n versions. The original version of object is denoted as V1 (i.e.,

ver-sion1), whereas the least detailed version which cannot be transcoded any more is denoted as Vn(i.e., version n). The

intermediate versions are, in turn, denoted as V2, ..., Vn−1,

in which Viis a more detailed version than Vjfor each i, j

if i < j.

It is noted that not every Vican be transcoded to Vj for

i < j. It is possible that Vidoes not contain enough

con-tent information for the transcoding to Vj. The transcoding

proxy thus must a priori know the transcodable relationship of an object whenever it performs the transcoding. To this end, we devise the notion of weighted transcoding graph as defined in Definition 1.

Definition 1 The weighted transcoding graph, Gi, is a

directed graph with weight funtion wi. Gi depicts the

transcoding relationship among transcodable versions of object i. For each vertex v ∈ V [Gi], v represents a

transcodable version of object i. Version u of object i is transcodable to version v iff there exists a directed edge

(u, v) ∈ E[Gi]. The transcoding cost from version u to

v is given by wi(u, v) which is the weight of the edge from

u to v. 1 2 3 4 10 10 10 8 8 0 0 0 0

Figure 3. Weighted transcoding graph. The weighted transcoding graph is in essence an ex-tension to the transcoding relation graph proposed in [5]. For each object, we maintain a corresponding weighted transcoding graph in the transcoding proxy. Note that since each version is surely transcodable to itself with null transcoding, there is a directed edge pointed from each ver-sion to itself with the corresponding weight being 0. An example of weighted transcoding graph is given in Figure 3 where the original version V1can be transcoded to each

less detailed versions V2, V3 and V4. However, V3 cannot be transcoded to V4due to insufficient content information.

3.

Cache

Replacement

Algorithms

for

Transcoding Proxies

In Section 3.1, we define the generalized profit function which will be used to determine the eviction priorities of the cached objects in our algorithm. To evaluate the aggre-gate effect properly, we devise Procedure MATC (standing

for Minimal Aggregate Transcoding Cost) to find the

sub-graph of the weighted transcoding sub-graph. This subsub-graph shows the transcoding relationship which minimizes the ag-gregate transcoding cost when caching a certain set of ver-sions of the object. Utilizing this generalized profit function as the eviction function, we design an innovative cache re-placement algorithm for transcoding proxies in Section 3.2. This algorithm is referred to as algorithm AE (standing for

Aggregate Effect) for the reason that it explores the

aggre-gate effect of caching multiple versions of an object in the transcoding proxy.

3.1. Generalized Profit Function

As mentioned in Section 1, most cache replacement al-gorithms employ an eviction function to determine the pri-orities of objects to be replaced. In our algorithm, we de-termine the eviction priorities according to the generalized

profit function, which will be defined later. Initially, we will

derive the individual profit function for the calculation of the profit from caching a single version of an object. Then, we derive the aggregate profit function for the calculation of the profit from caching multiple versions of an object at the same time. Based on the aggregate profit function, we for-mulate the marginal profit function for the calculation of the profit from caching a version of an object, given other ver-sions of the object are already cached. Finally, we conclude these profit functions as the generalized profit function.

Let oi,jdenote version j of object i. The reference rates

to different versions of objects are assumed to be statis-tically independent and denoted by ri,j, where ri,j is the

mean reference rate to version j of object i. The mean de-lay to fetch object i from the original server to the transcod-ing proxy is denoted by di, where we define the delay to

fetch an object as the time interval between sending the re-quest and receiving the last byte of response. The size of version j of object i is denoted by si,j. The corresponding

weighted transcoding graph of object i is denoted by Gi,

and the transcoding delay of object i from version x to ver-sion y is given by the weight on the edge (x, y) in E[Gi]

which is denoted by wi(x, y). A list of the symbols used is

give in Table 1. With these given parameters, we can deter-mine the profit from caching version j of object i.

Symbol Description

oi,j versionjof objecti

ri,j the mean reference rate to versionjof objecti di the delay of fetching objectifrom the original

server to the proxy

si,j the size of versionjof objecti

Gi the corresponding transcoding relation graph of objecti

wi(u, v) the weight on each edge(u, v)inGi Table 1. A list of the symbols used.

First, we consider the profit from caching a single ver-sion of an object in the transcoding proxy (i.e., no other versions are cached). From the standpoint of client users, an optimal cache replacement algorithm is expected to min-imize the response time perceived. In other words, an opti-mal cache replacement algorithm will be designed to max-imize the delay saving by caching a certain set of Web ob-jects. Thus, the individual profit from caching version j of object i is defined in terms of the delay saving obtained as below.

Definition 2 P F (oi,j) is a function for the individual profit

from caching oi,j while no other version of object i is cached. P F (oi,j) =
(j,x)∈E[Gi] ri,x∗ (di+ wi(1, x) − wi(j, x)) (1) Note in Eq. (1), each(j, x) ∈ E[G_i] represents each transcodable version from version j to version x. Expres-sion di+ wi(1, x) represents the delay if oi,jis not cached

(i.e., the delay of fetching object i from the original server plus the delay of transcoding from the original version to version x). On the other hand, wi(j, x) represents the

de-lay if oi,j is cached. Thus, di+ wi(1, x) − wi(j, x) is the

delay saving from caching oi,j in terms of the subsequent

references to oi,x. Consider the scenario in Figure 4 for

ex-ample. Suppose we would like to calculate the individual profit from caching version 2 of object i. The transcodable versions from version 2 is pointed by bold arrows in Fig-ure 4. Without caching version 2, the transcoding proxy has to fetch object i from the original server and to perform transcoding to satisfy the subsequent references to versions 2, 3 or 4. The delays for them in this circumstance are all equal to 30. On the other hand, with caching version 2,

the transcoding proxy only has to perform transcoding from version 2 to versions 2, 3 or 4 to satisfy the subsequent refer-ences to them with the corresponding delays being 0, 8 and 8 respectively. Thus, the delay saving from caching version 2 is30 = 30−0 for each reference to version 2, 22 = 30−8 for each reference to version 3, and also22 = 30 − 8 for each reference to version 4. Multiplying the delay savings by the reference rates, we can get the individual profit from caching version 2. In this example, suppose the reference rates to versions 1, 2, 3 and 4 are all equal to 10. The delay saving from caching version 2 of object i is thus equal to 740 (i.e.,10 ∗ 30 + 10 ∗ 22 + 10 ∗ 22 = 740). 1 2 3 4 10 10 10 8 8 Server G_i(V,E) 20 0 0 0 0

Figure 4. An example illustrating Definition 1.

As a matter of fact, however, the transcoding proxy may cache two or more versions of an object at the same time if the aggregate profit from caching them together is valu-able. However, as pointed out in Section 1, the aggregate profit depends on the transcoding relationship among dif-ferent versions of an object. Hence, we shall devise a pro-cedure, Procedure MATC (standing for Minimal Aggregate

Transcoding Cost), to find a subgraph Gi ⊆ Githat shows

the transcoding relationship which maximizes the aggregate profit when caching two or more versions of an object at the same time.

Procedure MATC(G, w, C) 1 A ←− ∅

2 for each vertex u ∈ C 3 do color[u] ←− GRAY 4 for each vertex v ∈ V [G]

5 do for each(x, v) ∈ E[G] and color[x] = GRAY 6 do find the minimum w(x, v)

7 A ←− A ∪ (x, v)

8 return G(V [G], A)

The input of Procedure MATC consists of 3 arguments in which G is the corresponding weighted transcoding graph, w is the weights on the edges in G, and C is the set of

the versions that the transcoding proxy tries to cache. The operations of Procedure MATC can be best understood by the example of Figure 5. Suppose G and w are given by the graph in Figure 3, and the input of C is {1, 2}. As shown in Figure 5(a), lines 1-3 initialize the set A to the empty set and color vertex 1 and 2 gray, where A is the resulting edge set of the subgraph to be returned. The for loop in lines 4-7 finds, for each vertex v ∈ V [G], the edge (x, v) with minimum weight w(x, v), and adds (x, v) into A. In our example, since (1,1) and (2,2) are the only edges pointed from gray vertexes to themselves, the iterations for vertex 1 and vertex 2 add (1,1) and (2,2) in A as shown in Figure 5(b) and 5(c) respectively. As shown in Figure 5(d), the iteration for vertex 3 adds (2,3) in A because w(2, 3) < w(1, 3). The iteration for vertex 4 works similarly and adds (2,4) in A as shown in Figure 5(e), leading to the resulting subgraph of Procedure MATC in Figure 5(f). It can be verified that the time complexity of Procedure MATC is O(V E), mainly contributed by the for loop in lines 4-7.

(a) (b) 1 2 3 4 10 10 10 8 8 0 0 0 0 1 2 3 4 10 10 10 8 8 0 0 0 0 (c) 1 2 3 4 10 10 10 8 8 0 0 0 0 (d) 1 2 3 4 10 10 10 8 8 0 0 0 0 (e) 1 2 3 4 10 10 10 8 8 0 0 0 0 (f) 1 2 3 4 10 8 8 0 0

Figure 5. The execution of Procedure MATC. With Procedure MATC, we can thereafter define the ag-gregate profit from caching multiple versions of an object. Definition 3 P F (oi,j1, oi,j2, ..., oi,jk) is a function for

the calculation of the aggregate profit from caching

oi,j1, oi,j2, ..., oi,jkat the same time.

P F (oi,j1, oi,j2, ..., oi,jk)

=
v∈V [G
_]
(v,x)∈E[G
_] ri,x∗ (di+ wi(1, x) − wi(v, x)) (2)

where Gis the resulting subgraph of Procedure MATC(G, w, {j1, j2,..., jk}).

Following the example in Figure 5 and with the resulting subgraph in Figure 5(f), the aggregate profit from caching versions 1 and 2, P F (oi,1, oi,2), can be determined as 940

(i.e.,10∗(20−0)+10∗(20+10−0)+10∗(20+10−8)+ 10 ∗ (20 + 10 −8) = 940). After the aggregate profit is cal-culated, we can determine the marginal profit from caching oi,j, given oi,j1, oi,j2, ..., oi,jkare already cached.

Definition 4 P F (oi,j|oi,j1, oi,j2, ..., oi,jk) is a function for

the calculation of the marginal profit from caching oi,j,

given oi,j1, oi,j2, ..., oi,jk are already cached where j =

j1, j2, ..., jk.

P F (oi,j|oi,j1, ..., oi,jk) = PF(oi,j, oi,j1, ..., oi,jk)

−PF (oi,j1, oi,j2, ..., oi,jk)

From the above example, we can obtain that the marginal profit from caching version 1, given version 2 is already cached, is only 200 (i.e., 940 − 740 = 200). Finally, we can integrate all the profit functions into a generalized profit

function as P FG(oi,j) =                      P F (oi,j) si,j

if no other version of object i cached;

P F (oi,j|oi,j1,...,oi,jk)

si,j

if there are other versions oi,j1, ..., oi,jkcached.

                     (3)

It is noted that the generalized profit function is further nor-malized by the size of version j of object i to reflect the object size factor. The generalized profit function defined in Eq. (3) explicitly considers the new emerging factors in the transcoding proxy and the aggregate effect of caching multiple versions of an object. As such, we can evaluate the profit from caching a certain version of an object, and then, in view of the cost model, develop the optimal cache replacement algorithm.

3.2. Design of Algorithm AE

In Section 3.1, we have formulated the generalized profit

function, P FG. Based on this generalized profit function, we design algorithm AE in this subsection. The main idea behind algorithm AE is to first order the objects according to their values of the generalized profit function, and then se-lect the object with the highest value, one by one, until there is not enough space to hold any object more. The objects se-lected are thus the objects to be cached in the transcoding proxy, and the rest objects are to be evicted.

Note, however, that some assumptions made in Section 3.1 to facilitate our presentation may have to be relaxed

when an algorithm is designed. First, the mean reference rates of Web objects are not static but may change as time advances. This phenomenon is quite common, especially in a news Web site where the reference rates to the news docu-ments decrease as time goes by. Second, the mean reference rates to different Web objects are not independent from one to another. For instance, if there is a hyperlink from one Web object to another, the mean reference rates to them are somewhat dependent. Third, the delays of fetching Web objects from the original servers are not known a priori. To address these issues, we shall employ the concept of sliding

average functions [14][18] to practically estimate the

run-ning parameters, diand ri,j, in the transcoding proxy in this

paper. With the estimated running parameters, we can de-vise the pseudo-code of algorithm AE. Algorithm AE takes 4 arguments as the inputs. C is a priority queue which is used to hold the objects cached in the transcoding proxy. The cached objects in C are maintained by a heap data structure in nondecreasing order according to their values of the generalized profits. Snowis the accumulated size of the objects currently cached in the priority queue. oi,1and oi,jare the original and transcoded versions to be cached. Algorithm AE(C, S_now, o_i,1, o_i,j)

1 insert oi,1and oi,jinto C

2 for each version of object i in C

3 do calculate or revise its generalized profit 4 BuildHeap(C)

5 while Snow> cacheCapacity

6 do exclude the first object om,nfrom C 7 Snow←− Snow− sm,n

8 for each version of object m 9 do revise its generalized profits 10 BuildHeap(C)

4. Performance Analysis

In this section, we will evaluate the performance of our cache replacement algorithm by conducting an event-driven simulation. The primary goal of performing an event-driven simulation is to assess the effect on the performance of our cache replacement algorithm by varying different system parameters. We will describe the simulation model in Sec-tion 4.1. The experimental results will be shown in SecSec-tion 4.2.

4.1. Simulation Model

The simulation model is designed to reflect the system environment of the transcoding proxy as described in Sec-tion 2. In the client model, as in [5], we assume that the

mobile appliances can be classified into five classes. That is, all Web objects could be transcoded to five different ver-sions by the transcoding proxy to satisfy the users’ need. Without loss of generality, the sizes of the five versions are assumed to be 100%, 80%, 60%, 40% and 20% of the orig-inal object size. Also, we assume a more detailed version can be transcoded to a less detailed one, and the transcod-ing delay is determined as the quotient of the object size to the transcoding rate. The distribution of these five classes of mobile clients is modeled as a device vector of <15%, 20%, 30%, 20%, 15%>.

Next, we consider the workload of the mobile clients’ requests. We assume that the number of total Web ob-jects is 1000. The sizes of the obob-jects are lognormally dis-tributed with the mean value of 15 Kbytes. The popularity of the Web objects follows a Zipf-like distribution, which is widely adopted to be a model for real Web traces [3][4][13] [14]. The default value of parameter α in Zipf-like distribu-tion is set to be 0.75 with a reference to the analyses of real Web traces in [4]. As shown in [8][11], small files are much more frequently referenced than large files. Hence, we as-sume that there is negative correlation between the object size and its popularity. Finally, the interarrival time of two consecutive clients’ requests is modeled by exponential dis-tribution with the mean value of 0.1 second. As to the model of the transcoding proxy, we choose the default cache ca-pacity to be0.01 ∗ ( object size). The delays of fetching objects from Web servers are given by an exponential distri-bution. Table 2 provides a summary of the parameters used in the simulation model.

Parameter Dist./Default value

Number of Web objects 1000

Object size Lognormal (µ = 20)

Object popularity Zipf-like (α = 0.75)

Interarrival time of clients’ requests Exp. (µ = 0.1)

Cache capacity 0.01 ∗ ( object size)

Object fetching delay Exp. (µ = 2)

Transcoding rate 20 KB/sec

Table 2. Parameters of the simulation model.

4.2. Experimental Results

Based on the simulation model devised, we implement a simulator using Microsoft Visual C++ package on IBM compatible PC with Pentium III 450 CPU and 128 Mbytes RAM. Each set of the experimental results is obtained by ten simulation runs with different seeds. The primary per-formance metric employed in our experiments is the delay

saving ratio which is defined as the ratio of total delay saved

from cache hits to the total delay incurred under the circum-stance without caching. In addition, we also use the exact

hit ratio, the useful hit ratio, and the overall hit ratio as the

secondary performance metrics in our experiments. The ex-act hit ratio corresponds to the frex-action of requests which are satisfied by the exact versions of the objects cached (i.e., no transcoding is needed for such requests), whereas the use-ful hit ratio is the fraction of requests which are satisfied by the more detailed versions of the objects cached (i.e., ad-ditional transcoding is required for them). Our yardsticks are the coverage-based and demand-based extensions to the traditional LRU and LRUMIN cache replacement algo-rithms. We denote the coverage-based LRU as LRUCVand

the demand-based LRU as LRUDM. The coverage-based

and demand-based extensions to LRUMIN are denoted as LRUMIN

CV and LRUMINDM respectively. In addition, to

under-stand the benefit of algorithm AE achieved by considering the aggregate effect, we also compare algorithm AE with algorithm AE0where algorithm AE0 is a modified version of algorithm AE, in which the eviction priorities are deter-mined by the profit function in Definition 1 instead of the

generalized profit function. Recall that the profit function in

Definition 1 did not consider the aggregate effect in the en-vironment of transcoding proxy. By comparing algorithm

AE with AE₀, the advantage of considering the aggregate effect can be observed. A list of the symbols used is given in Table 3.

Symbol Description LRUCV coverage-based LRU

LRUDM demand-based LRU

LRUMIN_CV coverage-based LRUMIN LRUMIN_DM demand-based LRUMIN

AE Algorithm AE

AE₀ Modified algorithm AE without consideration of the aggregate effect

γ_{F D} The ratio of the mean object fetching delay to the mean transcoding delay

Table 3. A list of the symbols used.

4.2.1. Impact of Cache Capacity

In the first experiment set, we investigate the perfor-mance of our cache replacement algorithm by varying the cache capacity. The simulation results are shown in Figure 6 and Figure 7. As presented in Figure 6, our algorithm out-performs the other ones in terms of the primary performance metric, i.e., the delay saving ratio (DSR). Specifically, the

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 3 5 7 9 ˖˴˶˻˸ʳ˖˴̃˴˶˼̇̌ʳʻʸʼ DS R ˔˘ ˣ˙ ˄˅ˆˇˈˉˊˋ ˅ ˆ ˇ AE₀ LRUMINCV LRUMINDM LRUCV LRUDM

Figure 6. DSR under various cache capacity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6

Exact hit ratio Us eful hit ratio Overall hit ratio

AE₀ AE

LRUMINCV

LRUMINDM

LRUCV

LRUDM

Figure 7. Hit ratios of various algorithms.

mean improvements of the delay saving ratios over LRUCV

and LRUDM algorithms are 162.4% and 221.3%

respec-tively, whereas the mean improvements over the LRUMIN CV

and LRUMIN

DM are 22.4% and 36.4%. The main reason that

the extensions to LRU algorithm did not perform well is that the LRU algorithm considers neither the sizes of objects nor the delays of fetching objects from original servers. This drawback is more severe when it comes to the circumstance of the transcoding proxy. On the other hand, the advantage of algorithm AE over LRUMIN

CV and LRUMINDM is contributed

by the devised generalized profit function which explicitly considers the new emerging factors in the environment of transcoding proxies, especially the aggregate effect when observing the difference between AE and AE₀.

Figure 7 provides more insights of the relationship be-tween algorithms and performance. Observe that the exact hit ratios of the demand-based algorithms are higher than those of the coverage-based algorithms. Conversely, the useful hit ratios of the coverage-based algorithms are higher

than those of the demand-based algorithms. This can be ex-plained by the reason that demand-based algorithms always cache the transcoded versions to avoid repeating transcod-ing operations, thus leadtranscod-ing to higher exact hit ratios. How-ever, the coverage-based algorithms always cache the orig-inal versions to provide maximum coverage on subsequent requests, accounting for the reason that they perform bet-ter in bet-terms of useful hit ratios. Although algorithm AE is designed to maximize the DSRrather than the hit ratios, al-gorithm AE still outperforms others in terms of the overall hit ratio.

4.2.2. Impact of γ_{F D}Ratio

We define the ratio of the mean object fetching delay to the mean transcoding delay as the γ_{F D}ratio. Formally, the value of γ_{F D}can be determined by

γ_{F D}= mean f etching delay

mean object size/transcoding rate. The second experiment set examines the performance of our algorithm by varying the value of γ_{F D}. The simulation re-sults are given in Figure 8 and Figure 9. As shown in Figure 8, the overall hit ratio of each algorithm almost remains the same for various γ_{F D} ratios. This is because all simula-tion parameters except the mean fetching delay are fixed, and the mean fetching delay will not affect the overall hit ratio. However, the delay saving ratio increases as the γ_{F D} ratio increases in Figure 9. In addition, the performance difference between the coverage-based algorithm and the

demand-based algorithms (e.g. LRUCV and LRUDM)

in-creases at the same time. The former phenomenon is more intuitive because the delay saving ratio is, of course, af-fected by the mean fetching delay. We explain the latter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.25 0.5 1 2 4 Ratio O v er al l Hi t Ra tio _AE 12345678 1 2 3 γFD LRUMINCV LRUMINDM LRUCV LRUDM AE

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.25 0.5 1 2 4 Ratio DS R AE 12345678 1 2 3 γFD LRUMIN CV LRUMINDM LRUCV LRUDM AE

Figure 9. DSR under various γ_{F D}ratios.

phenomenon as follows. The delay saving ratio is con-tributed by cache hits which can be classified into exact cache hits and useful cache hits. Although the ratios of exact and useful cache hits do not change by γ_{F D} ratio, the weights of their contribution to DSRdo change. When the γ_{F D} ratio is small, the delay incurred by the transcod-ing is more significant than the delay incurred by fetchtranscod-ing from the original server. In other words, the contribution of the exact hit ratio is more weighted than that of the useful hit ratio. Thus, the demand-based algorithm which avoids repeating transcoding operations performs relatively well. The situation is reversed when the value of the γ_{F D}ratio is large. Unlike either the demand-based algorithms which maximize the exact hit ratio or the coverage-based ones which maximize the useful hit ratio, algorithm AE deals with the caching problem of transcoding proxies in a com-prehensive way. Consequently, the performance of algo-rithm AE remains robust against various values of γ_{F D}.

5. Conclusions

In this paper, we developed an efficient cache replace-ment algorithm for transcoding proxies. We formulated a generalized profit function to evaluate the profit from caching each version of object. Based on the weighted transcoding graph and the generalized profit function, we proposed algorithm AE as the new cache replacement algo-rithm for transcoding proxies. Using an event-driven sim-ulation, we showed that algorithm AE consistently outper-forms companion schemes in terms of the delay saving ra-tios and cache hit rara-tios.

Acknowledgment

The authors are supported in part by the National Science Council, Project No. NSC 89-2219-E-002-007 and NSC

89-2213-E-002-032, Taiwan, Republic of China.

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