access time minimization for distributed multimedia applications

Multimedia Tools and Applications, 12, 235–256, 2000c© 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Access Time Minimization for DistributedMultimedia Applications

BHARADWAJ VEERAVALLI [email protected] of Electrical Engineering, The National University of Singapore, 10, Kent Ridge Crescent, Singapore119260

GERASSIMOS BARLAS [email protected] of Informatics, University of Cyprus, P.O.B. 537, 1678 Nicosia, Cyprus

Abstract. The problem of minimizing the access time of a requested multimedia (MM) document on a networkbased environment is addressed. A generalized version of this problem is formulated and retrieval strategiesthat minimize the access time of the user-requested MM document from a pool of MM servers are proposed. Tothis end, we designsingle-installmentandmulti-installmentMM document retrieval strategies, through which theminimization of access time can be carried out. The main idea is to utilize more than one MM server in downloadingthe requested document. Each server assumes the responsibility of uploading a predetermined portion of the entiredocument in a particular order. Single- and multi-installment strategies differ in the number of disjoint documentpieces each server sends to the client. We first introduce a directed flow graph (DFG) model to represent theretrieval process and generate a set of recursive equations using this DFG. Then, we derive closed-form solutionsfor the portions of the MM document downloaded from the various servers and the corresponding access time.We present rigorous analysis for these two strategies and show their performance under MPEG-I and MPEG-IIvideo streams playback rates. Their behavior under different network bandwidths is also examined, revealingin-depth information about their expected performance. We also show that in the case of a multi-installmentstrategy, the access time can be completely controlled byfine tuningthe number of installments. Since thenumber of installments is software tunable, the adaptive nature of the strategies to different channel bandwidthsis also demonstrated. Important trade-off studies with respect to the number of servers involved in the retrievalprocess and the number of installments are presented. In the case of a heterogeneous network employing asingle-installment strategy, we prove that the access time is independent of the server sequence used. Illustrativeexamples are provided for ease of understanding.

Keywords: video distribution, bandwidth, access time, sequencing, multi-installments, retrieval schedule

1. Introduction

With the increasing popularity of multimedia (MM) services on network based environ-ments, there is a continuous thrust in achieving an optimized design for MM servers andnetwork service providers [13]. The main attraction of these services is that viewing andpresentation control is handed over to the user, in contrast with conventional video broad-cast services as cable TV. A particular movie can not only be made available to a user athis/her convenience (as with video cassettes), but the user can also have complete controlon all aspects of the different media involved (audio, video), as for example the physicallayout of the screen. Also, with an increase in demand for a particular movie, depending onthe popularity profile, viewing cost per user can also be reduced considerably when clever

236 VEERAVALLI AND BARLAS

placement of movies at strategic locations on the network is carried out [1, 2]. Most of theservices are also attractive with respect to the monetary cost involved, as these services aremade available on low-cost PC platforms. The effort for the development of such systemsand services would be futile without the availability of high performance computers andhigh speed fiber optic networks that offer the capability of supporting such demands.

The above mentioned application and other futuristic MM applications like Movie on-Demand [8], collaborative video editing and synthesis of MM objects [4], and other networkbased distributed applications [3, 6, 7], will be attractive only when the available networkbandwidth and other necessary resources are cleverly utilized. In [2], a multimedia distribu-tion network has been presented. Here, the authors introduce a three tier network architec-ture to the video distribution problem on networks. While using state-of-the-art technologyto realize such applications is always a solution, the established—network mostly—andslowly changing infrastructure leaves no alternative but to carefully plan and utilize what iscurrently available. Owing to the continuous thrust in developing MM services on networkbased environments, the service providers situated at geographically large distances canco-operate and share their MM documents in order to serve their local subscribers. Oncea document is available locally, in turn, each service provider can choose the appropriateadmission control and scheduling algorithms [10, 12, 14, 15] to maximize their servicingcapabilities. We believe that this two-tier service architecture, provides an elegant solutionfor developing such MM services, since the data sizes that are involved are very large. An-other motivation underlying this support is from monetary aspects. It will be prohibitivelycostly if one tries to replicate all the MM documents across all the sites, to provide efficientservice. Of course, replication of the documents may take care of the performance to certainextent, although monetarily such a scheme may not be desirable.

Also, applications like virtual MM conferencing or virtual group discussions, will beattractive and viable, only when the on-line discussions are captured and presented withoutany temporal delays. If however, the subject of discussion is currently not available at thelocal service provider, then the corresponding document would have to be downloaded froma remote site in which it is available. Since the sites are typically geographically apart anddownloading would involve the transmission of large volumes of data, it would take quitean amount of time before the entire document is downloaded and the discussion at the localsite initiated. This waiting time may be annoying to end users, especially if the discussiondepends on making time critical decisions. An illustrative example presented in the nextsection clearly describes the motivation for this research. Our research contribution inthis paper precisely addresses the minimization of this waiting time, or the access time forsimilar and allied network based MM services. Also, our approach, to a large extent presentsa unified theoretical framework that suits most of the existing network-based distributedMM services and also opens a new avenue for the researchers in this field. The theoreticalframework is accompanied by an in-depth analysis of the behavior of the proposed strategiesthat can lead to the design of more refined approaches.

1.1. Related work

There has been an extensive study on many allied problems related to our current research.A Video on-Demand (VoD) or Movie on-Demand (MoD) service which is usually offered

ACCESS TIME MINIMIZATION 237

on hi-speed network based environments is one of the natural candidates [12, 14] for theproblem presented in this paper. We refer to these applications as VoD, hereafter. Therehas been a continuous effort to optimize the performance of MM servers designed for VoDapplications. These essentially focus in optimizing the storage and retrieval of video blockson the disks [10] and also in the design of servers to maximize the number of customers.Depending on the popularity of the movies in a networking domain, physical placement ofmovies on particular server sites is important in optimizing the monetary cost incurred forviewing that movie [1].

Very recently a novel technique, referred to as pyramid broadcasting, is proposed in [16],as a means of serving a large pool of customers in metropolitan area networks (MANs).This technique typically supports services like VoD or MoD on MANs. Another closelyrelated theme can be found in a study by [5] in which retrieval schedules are generated basedon the availability of the network and other necessary resources with flexible presentationrequirements. The influence of the network service providers and on the buffer resources atthe the client site are also addressed in this study. A slight variant of this service architecturewas proposed in the study on MM presentation planing by [9]. As mentioned earlier, ourstudy is a generalization of the basic MM document retrieval process which typically suitsapplications in which very large volumes of MM data needs to be transported from one siteto another on very large scale networks involving considerable communication delays.

1.2. Network architecture

We envision a network consisting of a set of service providers serving each locality (seefigure 1). Each service provider has a directory facility, which registers the available doc-uments at various sites. Whenever a user requests a movie, this directory service willproduce a list of servers that can supply the requested MM document. Thus, if the requested

Figure 1. Networked multimedia servers servicing customers.


MM document is not locally available, the service provider will request the other serviceproviders to upload that document to its local site. Thereafter, whenever a request for thisdocument arrives, the local server can use the stored document. This service providercould be the MM server itself, if it has adequate local memory to support this service. Insuch a large network, the user requests may originate anywhere, and servicing this requestmust incur minimum possible time. If not, such a MM service becomes less attractive. Arecent survey [11] on monetary issues provides a good hope of realizing such MM applica-tions on network based environments and opens avenues to pursue further research in thisdomain.

In the case of networks that span a large area wherein the server sites are geographicallydistributed, communication from one site to the other will incur a finite amount of non-zerodelay. One of the attractive features of such MM services on networks lies in keepingup the promise of a smooth presentation without any audio-visual discontinuities. Unlessclever strategies are adopted in retrieving the video blocks amidst the presence of thesecommunication delays, this objective may not be met. We model the communication delayas a quantity that is directly proportional to the length of the video data that is carried overthat established communication path (virtual or circuit switched) to the service provider.This is unlike the model proposed by [16], in which the video blocks in the successiveretrievals are of increasing sizes. In our model, the available communication bandwidthand the display/playback rate of the video clip are the two major system parameters thatare considered, and the contribution in this paper is in the design and analysis of retrievalstrategies that minimize the access or the wait time of the users.

The paper is organized as follows. Section 2 presents the problem setting and intro-duces the necessary notations and terminology. Section 3 presents the design and analysisof a single installment servicing policy to minimize the access time. Section 4 extendsthe above study and presents the design and analysis of a multi-installment servicing pol-icy to minimize the access time, which provides a fine tuning control of the access timeas a function of the number of installments. Section 5 discusses all significant contri-butions of this paper and presents several interesting observations. Section 6 concludesthe paper.

2. Problem formulation and preliminary remarks

In this section, we present the problem more formally, describe the network architecturethat is considered, and introduce the necessary definitions, notations and terminology.

The network model consists of a pool ofN MM servers each serving their respectivecustomers (see figure 1). The requests for viewing a movie of a long duration (typically of100 to 120 minutes) arrive at these servers from its local customers. These servers are byand large, powerful workstations with sufficient amount of bandwidth capacity and memoryspace to serve a maximum number of users concurrently by employing efficient admissioncontrol algorithms [14]. Upon an arrival of a request, the server seeks the requested MMdocument. We interchangeably use the terms service providers and servers as per the con-text. If the document is available locally, then usual retrieval and presentation techniquesas described in the so far literature can be employed to serve the request. However, if the


requested document is not available, then the server with its directory service facility, akind of look-up table procedure, determines the server sites at which the requested MMdocument is present. It then obtains a set of server indices from which the document may beretrieved. The requested MM document is then retrieved by employing our strategy demon-strated through the following motivating example. We introduce the necessary notations andterminology in the example for the ease of understanding.

2.1. Motivating example

Consider a scenario in which a requested MM document is not available locally at a serverdenoted as,S. Let the requested MM document be present at the sitesS0, S1, andS2. Letthe total size of the movie requested beL = 1GB. Further, let the channel bandwidthsin terms of the time delay encountered per unit load transfer, measured insecs per unitload, between each of these servers toS be denoted asbwi i = 0, 1, 2. It may be notedthat we are referring to the inverse of the channel bandwidth, however, we will continueto use the terminologybandwidthwhile referring to this quantity. Let these quantities(expressed in secs per Mbits or in secs per Mbytes) bebw0 = 1, bw1 = 2, andbw2 = 3,respectively. Thus, with our definition, in this example,bw0 is the fastest channel,bw1 isthe next fastest and so on. Hence, sending a unit load onbw0 takes less time to reachSthan from others. We assume that when the serverS receives the document from anotherserver, it starts the playback simultaneously at the user terminal. A discussion on this aspectis presented in Section 5. After locating the respective servers having the requested MMdocument (in this case servers 0 to 2), serverS adopts the following strategy. From eachserver a portion of the entire document is retrieved and are collected byS in a particularorder. Upon receiving a portion fromS0, the playback is started at the user terminal. Letthe inverse of the playback rate (expressed in the same units as thebwi ’s), denoted asRp be 5.3333 units (MPEG I stream). Now, the retrieval strategy is such that before theplayback of this portion comes to an end, next portion of the requested MM document iscollected fromS1. This process is repeated for all the servers participating in the retrievalprocess.

This example describes our first strategy namely, thesingle installmentretrieval strategy.In Section 4, we will see an example for the case of multi-installment strategy. This strategyhas some inherent advantages. Firstly, it retrieves disjoint portions from different servers andthus, minimizes the retrieval time. Secondly, the strategy inherently takes care of continuityrequirements, which are crucial when implementing such a strategy on network basedenvironments. Thus, the continuity in the presentation is one of the aspects that our strategyguarantees in the retrieval process apart from access time minimization. A fundamentalassumption of all the analysis that follows is that the playback of a portion of the documentis (or can be) initiated after this portion is completely received from the correspondingserver. Theaccess timeor thewait time is defined as the time between the start of thedownloading and the start of the playback [16]. The wait time or the access time is directlyproportional to the size of the portion retrieved from serverS0, i.e., starting from the timeat which the downloading starts to the time at which the playback starts. An immediatenaive choice would be to make this size as small as possible to minimize the access time of


Figure 2. Directed flow graph representation for the case of three servers.

the entire MM document. However, in that case we will later show in our analysis that thepresentation continuitycannot be guaranteed by choosing the first retrieved portion as smallas we desire. Hence, using this strategy, the problem now is to decide on the optimal sizesof the portions of the MM document to be retrieved from each of the servers, satisfying thepresentation continuity, using the bandwidth constraints, and the playback rate constraintsto minimize the access time.

In the above example, we see that the following size distributionm0 = 85.997 MBytes,m1 = 272.3235 MBytes,m2 = 665.679 MBytes satisfies the constraints, wheremi is thesize of the data retrieved from serverSi , i = 0, 1, 2, respectively. For this distribution, theaccess time (following the definition) is given bym0bw0 = 85.997 secs.

An elegant representation of this retrieval strategy is by means ofdirected flow graphs(DFGs). Figure 2 shows the directed flow graph for this example. The communicationnodes at the first level are assigned a weight equal to the total communication time of theportions of the MM document they are transferring toS. The dots on these communicationnodes indicate that all these servers start their downloading simultaneously at timet units.Without loss of generality, we assumet = 0. If m0 is the portion of the MM documentcommunicated byS0, then the total communication delay (which is the weight of the node 0(see figure 2) is given bym0bw0. The second level of nodes are referred to asplaybacknodes. The weight of the playback node 0 is proportional to the total time of playback of thatportion of video, given bym0Rp, whereRp is the inverse of the rate of playback expressedin seconds per unit load of display. The directed arrows depict thecausal precedencerelationships between the node events. Thus, atS, the display of the portion fromS2 startsonly after the display of the portion fromS1 is completed and also the portion fromS2 mustbe completely available.


2.2. Some definitions

Throughout the paper we will use the following definitions.

1. Retrieval schedule distribution: This is defined as anN ordered tuplem given by

m= (m0,m1, . . . ,mN−1) (1)

where,mi is the portion of the MM document downloaded from serverSi , i = 0, 1, 2, . . . ,N − 1. Further,

N−1∑k=0

mk = L (2)

and

0≤ mi ≤ L , i = 0, 1, . . . , N − 1, (3)

The set of all such retrieval schedule distributions is denoted as0.2. Access Timeor thewait time is defined as the time between the instant at which the

servers start uploading their portions to the time at which the presentation starts. Thisis denoted as, AT(m).

Typically, this is the time to access the first portion of the downloaded data, given bym0bw0, wherebw0 is the bandwidth of the established communication path fromS0 toS. This is similar to the definition used by Viswanathan et al. [16]. Hereafter, we shalluse the term “access time” throughout the paper.

3. Minimum access timeis defined as,

AT∗ = minm∈0AT(m) (4)

Thus, from the above set of definitions and the strategy illustrated in the above example,our objective is to minimize the access time by determining the optimal sizes of the portionsof the video to be retrieved from different servers involved in the retrieval process.

3. Single installment retrieval policy

In this section, following the retrieval strategy mentioned in the previous section, we shallanalytically determine the sizes of the various portions retrieved from all theN servers. Wethen derive a closed-form solution for the minimum access time. It will be shown that thecontinuity and the bandwidth constraints are implicitly considered, stemming naturally asa property of the optimal solution in the analysis.

Figure 3 shows a generalized version of the example illustrated in the previous section,with N servers. From figure 3, we can derive a relationship between the communication


Figure 3. Directed flow graph representation using single-installment strategy for MM document retrieval fromN servers.

nodesi andi +1 and the playback time of the portionmi with the use of causal precedencerelations and continuity constraint as,

mi+1bwi+1 ≤ mi bwi +mi Rp, i = 0, 1, . . . , N − 2. (5)

Let us denote(bwi + Rp)/bwi+1 = ρi . Using in (5), we have,

mi+1 ≤ miρi , i = 0, . . . , N − 2. (6)

One can solve these set of recursive relations with equality conditions. Using equalityrelations in (5) and (6) produces themaximumsize of all document portions other thanm0.So under the constraint of (2), this yields theminimum m0 or equivalently the minimumaccess time. Thus, we have a recursive set of (N − 1) equations with equality relationsfrom (6). Eachmj in (6) can be expressed in termsm0 as,

mj = m0

j−1∏k=0

ρk, j = 1, . . . , N − 1. (7)

In general, the continuity relationship [14] states thatthe playback duration of a portion ofa video must be greater than or equal to the total retrieval time of the immediate successiveportion of the video. Otherwise, the continuity of the presentation will be lost and leadsto a considerable performance degradation. In the above set of recursive equations, itmay be noted that we have taken care of this continuity constraint by using the equalityrelationships rather than the inequality relationships. That is, the next playback will startimmediately after the current playback comes to an end. Thus, when these set of linearrecursive equations are solved with equality relationships, we obtain the minimum accesstime, as the solution set (m0 to mN−1 through this procedure) gives the minimum value ofm0. Equation (7) generatesN − 1 equations, but we haveN media portions. However,


along with (2), we haveN equations which can be solved to obtain the individual disjointportions of the requested MM document. Substituting eachmi from (7) into (2), we obtain,

m0 = L(1+∑N−1

p=1

∏p−1k=0 ρk

) (8)

Substituting (8) in (7), we obtain the individual sizes of the portions as,

mj = L∏ j−1

k=0 ρk(1+∑N−1

p=1

∏p−1k=0 ρk

) (9)

for all j = 1, . . . , N − 1. Note that the access time (see figure 3) is given by,

AT(m) = m0bw0 = L bw0(1+∑N−1

p=1

∏p−1k=0 ρk

) (10)

where we have used (8) form0. The retrieval schedule distribution demonstrated in themotivating example in the previous section is derived through these set of formulae.

3.1. Homogeneous channels

We consider a network with identical channel bandwidths or a channel that is shared byseveral servers. In other words, we have,bwi = bw, for all i = 0, . . . , N − 1. Theindividual sizes of the portions retrieved from the serversS0 till SN−1 are given by,

m0 = L(ρ − 1)

ρN − 1(11)

mj = L(ρ − 1)ρ j

ρN − 1, j = 1, . . . , N − 1, (12)

where we have usedbwi = bw for all i = 0, 1, . . . , N − 1 in (8) and (9) to obtain theseexpressions. Hence, the access time is given by,

AT(m) = m0bw0 = L bw(ρ − 1)

ρN − 1(13)

We shall present a detailed discussion in Section 5 on the behavior of this system.

3.2. Effect of sequencing on the access time

It is worth noting at this juncture that throughout the above analysis we have assumed thatretrieval follows a particular order, referred to asfixed sequence, S0 to SN−1. Given a set ofN servers, we haveN! retrieval sequences possible. However, one may wonder if it could


be possible to gain performance by varying the sequence in which the MM document isretrieved from the servers. The following lemma and theorem prove that such a behavioris not possible when a single installment strategy is employed.

Lemma 1. Let the access time of a requested MM document by the server S be denotedas AT(m, σ (k, k + 1)), whereσ(k, k + 1) = (S0, . . . , Sk−1, Sk, Sk+1, Sk+2, . . . , SN−1),

denotes the sequence in which the requested MM document is retrieved from the servers.Then, for a sequenceσ ′(k, k+ 1) = (S0, . . . , Sk−1, Sk+1, Sk, Sk+2, . . . , SN−1), the accesstime AT(m′, σ ′(k, k + 1)) is equal to AT(m, σ (k, k + 1)) where, σ ′(k, k + 1) denotes aretrieval sequence in which the adjacent channels k and k+ 1 are swapped, i.e., portionfrom server Sk+1 is retrieved first and then from server Sk.

Proof: The denominator of (10) can be written as:

denom(m) = 1+ ρ0+ ρ0ρ1+ ρ0ρ1ρ2+ · · · = 1+ ρ0(1+ ρ1(1+ ρ2(· · ·))) (14)

We can distinguish two cases depending on whether the first serverS0 is involved or not.If S0 is not involved, when a switch is made between two successive servers, the newdenominator is different from the original one in threeρ terms. This difference can bewritten as

denom(m′)− denom(m) = Rp+ bwi−1

bwi+1

(1+ Rp+ bwi+1

bwi

(1+ Rp+ bwi

bwi+2

))− Rp+ bwi−1

bwi

(1+ Rp+ bwi

bwi+1

(1+ Rp+ bwi+1

bwi+2

))(15)

With little algebraic manipulation, the above equation returns zero.In the second case, where the first two serversS0 andS1 are switched, the difference in

access time is,

AT(m)− AT(m′) = L bw0

denom(m)− L bw1

denom(m′)

= L(bw0 denom(m′)− bw1denom(m))

denom(m) denom(m′)(16)

By using (14) (denom(m) and denom(m′) differ in twoρ-terms), the numerator of the abovefraction becomes equal to

bw0

(1+ Rp+ bw1

bw0

(1+ Rp+ bw0

bw2

))− bw1

(1+ Rp+ bw0

bw1

(1+ Rp+ bw1

bw2

))(17)

which in turn can be immediately proven to be equal to zero. 2

The significance of the lemma is that the order in which the portions are downloadedaffects only the respective size distribution, but not the access time when adjacent serversare swapped. We prove this claim in general for the case ofN servers, as follows.


Theorem 1. Given a pool of N MM servers capable of rendering the requested MM doc-ument, using the single installment strategy, the access time is independent of the retrievalsequence used.

Proof: Direct application of Lemma 1 proves the theorem. Any valid sequence of serverscan be derived from a single sequence by iteratively switching the positions between adjacentservers. Lemma 1 guarantees that these operations do not affect access time. 2

4. Multi-installment servicing policy

In this section, we present a generalized servicing policy, which provides a tuning con-trol mechanism for the access time. This tuning provides a better control on the retrievalprocess and allows the system to adapt to the variations in the bandwidth of the network.This policy constitutes of retrieving the MM data from each of the serversS0 to SN−1

in more than one installment. This means that every server participates more than oncein the process of uploading disjoint portions of the requested document one after other, ina particular order. Since this facilitates the accessing of the data byS at a much earliertime, the access time decreases. As in the case of the single installment policy, the con-tinuity of the presentation must be guaranteed. The necessary and sufficient conditionsfor the multi-installment policy, can be derived by extending the directed graph represen-tation of the previous section. By assuming that each server uploads a disjoint portion ofthe MM data inn installments, we can use the extended graph to derive a set of recursiveequations.

4.1. Recursive equations and solution methodology

Figure 4 shows the directed graph for this policy. Note that we have extended the commu-nication and playback processes from single installment to multi-installment using causalprecedence relations (time orderliness). Letmi, j represents the part of theL in totalMM data, that is downloaded from theSi server during thej -th installment, wherej =0, . . . ,n− 1. Thus, there is a total ofNn portions of the MM document that are retrievedfrom serversS0 to SN−1 in n installments. These arem0,0,m1,0, . . . ,mN−1,0,m0,1, . . . ,

mN−1,1, . . . ,m0,n−1, . . . ,mN−1,n−1. It can be easily deducted from (see figure 4) thatthe causal precedence relations and the continuity relationships impose the followinginequalities:

mk,0bwk ≤ mk−1,0bwk−1+mk−1,0Rp, (18)

for all k = 1, 2, . . . , N − 1. Fori = 1, 2, . . . ,n− 1, we have,

mk,i bwk ≤(

N−1∑p=k

mp,i−1+k−1∑p=1

mp,i

)Rp (19)

for all k = 0, 1, . . . , N − 1.


Figure 4. Directed flow graph representation using multi-installment strategy for MM document retrieval fromN servers.

Themi, j parts are also connected by the normalizing equation:

N−1∑i=0

n−1∑j=0

mi, j = L (20)

Theminimumsize ofm0,0, which determines the minimum access time, can be obtainedby seeking themaximizationof all othermi, j . This goal can be achieved by using theequality relations in Eqs. (19) and (20). Deriving a closed-form solution from the above setof recursive equations is too tedious. However, since

m1,0 = m0,0 · bw0+ Rp

bw1(21)

mk,0 = m0,0 ·k∏

j=1

bw j−1+ Rp

bw j(22)

mk,i =(

N−1∑j=k

mj,i−1+k−1∑j=0

mj,i

)Rp

bwk(23)

we can reach a solution by using the following procedure. If we assume thatm0,0 = 1 wecan obtain from (21)m1,0, and from Eqs. (22) and (23) successively allmk,i . The originalassumption is the equivalent of multiplyingm0,0 by a constantK so that it equals unity.Because of the waymi, j are related through Eqs. (21), (22) and (23), this process returnsall mk,i also multiplied byK. K can in turn be estimated from the normalizing Eq. (20),which becomes:

N−1∑j=0

n−1∑l=0

mj,l = K · L (24)


and this completes the solution. We denote the access time, using multi-installment strategy,as AT(N, n), as we have now two parametersN ann to control the access time. The accesstime is given by,

AT(N, n) = m0,0bw0, (25)

wherem0,0 is obtained by solving the recursive equations a mentioned above. Followingexample demonstrates this strategy.

Example 2. Suppose that the client atS requests a 2GB MM document from serversS0, S1, S2 andS3, which are linked withS with 128 KByte/sec connections. If we assumea playback rate of 4 Mbits/sec, typical of a MPEG II document, then a single installment,i.e. n = 1, results in an access time of 2841.67 seconds. However, doubling the numberof installments results in an access time of 1362.0 seconds. Thus, we gain a significantdecrease (of 52%) in the access time.

4.2. Homogeneous channels

Here too, we shall consider analysis for homogeneous channels. Thus, the above set ofrecursive relations ((18) and (19)) reduces to,

mk,0bw≤ mk−1,0bw+mk−1,0Rp, (26)

for all k = 1, 2, . . . , N − 1. Then, fori = 1, 2, . . . ,n− 1, we have,

mk,i bw≤(

N∑p=k

mp,i−1+k−1∑p=1

mp,i

)Rp (27)

for all k = 0, 1, . . . , N − 1. Denoting,Rp/bw asσ , we have,

mk,0 ≤ mk−1,0(1+ σ) (28)


mk,i ≤(

N∑p=k

mp,i−1+k−1∑p=1

mp,i

)σ (29)

for all k = 0, 1, . . . , N − 1. Using the equality relationship, we write,

mk,0 = mk−1,0(1+ σ) (30)


mk,i =(

N∑p=k

mp,i−1+k−1∑p=1

mp,i

)σ (31)


for all k = 0, 1, . . . , N−1. Each of themk,0, k = 1, . . . , N−1 from (30) can be expressedas a function ofm0,0 as,

mk,0 = m0,0P(σ, k), (32)

where, P(σ, k) = (1 + σ)k−1. It is worth noting that the polynomialP(σ, k) containsbinomial coefficients when expanded. Also, it may be observed from (31) that to ob-tain mk,i for any i > 0, we need to simply addN of its preceding terms and multiplyby aσ .

We define a transformationk = i (n − 1) + j N , and denote the portions of the MMdocument retrieved fromS0, . . . , SN−1 in n installments as,Qk, k = 0, 1, . . . , Nn− 1,wherek is as defined above. For instance, whenN = 4 andn = 2, m1,1 which is actuallythe second installment for serverS1 is denoted byQ5, ask = 1.(1) + 1.(4) = 5. Thus,with this notation and transformation, and with (32) and (31), we can easily generate thefollowing table. We have shown the table forN = 4 andn = 2 case. The entries in eachrow of the table are the coefficients of the respective powers ofσ . Thus, the maximumnumber of columns will beNn− 1. As an example,m1,1 corresponds to the rowQ5, givenby (31) as 3σ + 10σ 2 + 10σ 3 + 5σ 4 + σ 5, and the entries are precisely these coefficientsof the various powers ofσ .

j

mi, j k 0 1 2 3 4 5 6 7

0,0 0 1 0 0 0 0 0 0 0

1,0 1 1 1 0 0 0 0 0 0

2,0 2 1 2 1 0 0 0 0 0

3,0 3 1 3 3 1 0 0 0 0

0,1 4 0 4 6 4 1 0 0 0

1,1 5 0 3 10 10 5 1 0 0

2,1 6 0 2 12 20 15 6 1 0

3,1 7 0 1 12 31 35 21 7 1

Thus, generalizing this idea, we have the following boundary conditions and a recursivedefinition to generate a particular entryE(i, j ) in the table for arbitraryN andn.

The boundary conditions, which generate entries for the first installmentn = 0 aregiven by,

E(k, 0) = 1, ∀k = 0, 1, . . . , N − 1, (33)

E(k, 0) = 0, ∀k = N, . . . , Nn− 1, (34)

E(k, j ) = 0, ∀ j > k, and k, j = 0, 1, . . . , Nn− 1, (35)

E(k, j ) = E(k− 1, j − 1)+ E(k− 1, j ), ∀k = 1, 2, . . . , N − 1 (36)


Note that the first entryE(0, 0) is assumed to be equal to 1. This is just for the purpose ofgenerating the table entries as a function ofm0,0 which is assumed to be unity. However,when actually solving the recursive equations, we can express each of themi, j as a functionof m0,0. Now, for the remaining rows,QN, . . . , QNn−1, we have,

E(k, j ) =k−1∑

p=k−N

E(p, j−1), ∀k = N, . . . , Nn−1, j = 1, 2, . . . , Nn−1, (37)

Thus, we have forQ5 = m1,1 = E(5, 0) + E(5, 1)σ + · · · + E(5, 4)σ 4 + E(5, 5)σ 5.This is nothing but the polynomial shown above. Following this notion, we can writemi, j as,

mi, j = Qk =i∑

j=0

E(i, j )σ j , ∀i = 0, 1, . . . , Nn− 1, (38)

We have a total ofNn unknowns withNn− 1 equations. As in the previous section, weuse the normalizing equation,

Nn−1∑i=0

i∑j=0

E(i, j )σ j = L (39)

to have a total ofNn equations to solve for all the unknowns. Note that each of themi, j

can be expressed in terms ofm0,0, by using the recursive definition of (38) and using (39),we obtain,

m0,0 = L∑Nn−1i=0

∑ij=0 E(i, j )σ j

, (40)

where,E(i, j ) is generated by using Eqs. (34) to (37).Thus, given a set ofN MM servers having identical channel bandwidth connections, it

easier to obtain the optimal sizes of the portions of the MM document to be retrieved fromeach server by immediately using Eqs. (34) to (40).

5. Discussion of the results

The problem tackled in this paper presents a generalized approach to the theory of minimiz-ing the access time for network based MM document retrieval/distribution. Researchersin this field have addressed the problem from different perspective, however, on differentapplication specific requirements. For instance, the results equally apply to a single servermultiple client MM video-On demand system by replacing the bandwidths specified inthis paper by the disk bandwidths [10, 14]. On a network based environment, one of thecrucial bottlenecks being the available bandwidth, the treatment presented in this paper


Figure 5. Access time vs number of MM servers using MPEG-I and MPEG-II streams.

suggests an elegant solution to minimize the access time. Future network based MM ap-plications demanding a trade-off between the available network bandwidth and the servicefrom geographically well separated servers become the natural candidates of the problemaddressed in this paper. We shall now present some interesting observations made during ouranalysis.

Using single-installment strategy, we have derived the closed-form solution. By using therecursive procedure, obtaining the optimal sizes of the media portions (solution set) retrievedfrom the servers, can be achieved withO(N) time complexity. An attractive and interestingfeature of this strategy is that the optimal solution evolves naturally as we solve the set ofrecursive Eq. (7) with equality constraints (capturing the continuity relationships) togetherwith (2). As it is evident from the closed-form solution, the access time monotonicallydecreases as we tend to utilize more and more number of servers. Figure 5 shows thebehavior of the access time with respect to the number of servers utilized for the case ofhomogeneous channels, i.e., whenbwi = bw, for all the channels. As expected, as therequested MM document is available on more servers, the access time decreases, as we tendto utilize all the probable servers. In the figure, we have shown the plots for typical MPEG-Iand MPEG-II video streams with 1.5 Mbytes per sec and 4 Mbytes per sec playback rates,respectively. Another important contribution in this research is the proof ofaccess timeinvariance property on sequencing. The non-triviality lies in identifying such a behaviorof the access time. The result of Lemma 1, though not astonishing, clears the fact that theaccess time is independent of the order in which the MM document is retrieved from theserver pool. Alternative to this strategy and as a future extension to the problem consideredin this paper is the following. It may be possible with the current day technology that a


client may be able to download multiple streams at the same time depending on its availablebuffer and the bandwidth. This scheme is attractive as it is adaptable to varying systemloads. However, the implementation of this scheme involves the design of efficient resourcemanagement and scheduling policies at the client site.

In the case of multi-installment strategy, obtaining a closed-form as in single-installmentstrategy, is tedious as equations are complex to solve. However, the recursive procedureproposed in Section 4.1 generates the optimal sizes of the media portions from the serversin O(Nn) time complexity. Also, the recursive expressions presented in Section 4.2 for thecase of homogeneous channels are easier to implement as the entries of the table (shownfor N = 4 andn = 2 case) can be automatically generated and can be reused when thenumber of installments or the servers varies. Another attractive feature of this methodologyis that the speed parameters (channel bandwidth and the playback rates) do not influence oraffect the entries of the table, and hence these entries (E(i, j ), ∀i, j can be generated mucha priori and can be stored using the best possible data structure for computation purposes.

As far as the performance of this strategy is concerned, a set of questions that might ariseare the following:

• What is the relation between the number of servers used and the number of installments?• Can we expect that ever-increasing the number of installments will always result in gains

with respect to access time?

Answers to the above questions are attempted by means of an experiment. An iterativeprocedure was employed in order to compute the access time under ever-increasing numberof installments. The document size was set to 2 GBytes and the playback rateRp to 2sec/MByte, typical of a MPEG II stream. The number of servers varied from 2 to 30 andtheir bandwidth from 1 sec/MByte to 32 sec/MByte. All connections were consideredidentical, i.e. bwi ≡ bw. The iteration stopped, either when increasing the number ofinstallments by 1 resulted in less than 5% gain in access time, or when the access timefell below 1 second. The number of installments at which the process ended is plotted infigure 6(a). Figure 6(b) shows the corresponding access times. They-axis is labeled asL bw in order to give a measure of the time necessary to download the MM documentfrom a single server. The usage of theL bw product is not a way for normalizing things.Actually, using different bandwidths with loads that have identicalL bw products resultsin different (quantitative) behaviors.

A striking observation (see figure 6) is that there is a barrier beyond which increasingthe number of installments and/or the number of servers cannot benefit the access time.This barrier is network capacity related and it can result in very poor performance. Justbefore this barrier is reached, the number of installments that are suitable for near optimumperformance grows rapidly and falls the same way after the barrier is exceeded. Before thebarrier is reached, the network connections are fast enough to guarantee that even with asmallm0,0 the portions of the document will reach the client prior to their turn for playback.Increasing the number of installments has just that effect, i.e. keepingm0,0 small. Afterthe barrier is exceeded,m0,0 must be a large portion of the document in order to guaranteethat the continuity constraints hold. As a result, the access time escalates while increasingthe number of installments does not serve the purpose.


Figure 6. (a) Number of installments where the access time falls below 1 sec, or where an additional installmentwould improve the access time less than 5%, against the number of servers and connection speed. (b) Thecorresponding access times for the installments shown in (a).

The relationship between the number of servers and number of installments is moreclearly seen in figure 7(a) and (b) which depict an experiment identical to the one describedabove, with the exceptions of

1. Adding more servers in the previous case, actually scales the client’s connection band-width, which is not the usual case—for Internet at least—connections. So in this case,there is a shared connection whose bandwidth is divided among all servers.

2. The shared bandwidth ranged from 0.5 sec/MByte to 4 sec/MByte.

The number of servers and installments are perpendicular ways for maximizing serverutilization under the multi-installment strategy. This is clearly seen in figure 7(a), where asthe number of servers increases, the near optimum number of installments decreases. Thiscan work the other way around, i.e., instead of increasing the number of servers, the numberof installments can be modified to achieve a target access time. Of course, there are limitsto this approach especially if each connection is independent of the others (see figure 6).As a final remark, the multi-installment strategy has an inherent advantage in managingthe available buffer at the client’s site. If the buffer size available is limited, then the bestpossible way to optimize the performance is by the use of multi-installment strategy, as thesizes of the individual chunks retrieved are smaller than the individual sizes recommendedby the single installment strategy. This in a way makes the scheme attractive, as the existinglimited resources (buffer capacities) are cleverly utilized without any additional resourceinvestment. One may suggest that our multi-installment strategy fails to address the need


Figure 7. (a) Number of installments where the access time falls below 1 sec, or where an additional installmentwould improve the access time less than 5%, against the number of servers and connection speed. The differ-ence from figure 6(a) is that the servers share the same connection to the document requesting party. (b) Thecorresponding access times for the installments shown in (a).

to use the available MM servers for servicing multiple client requests. This, however,falls outside the scope of this paper. Serving multiple clients, calls for employing otheradministration policies for allocating the bandwidth of each server. The strategies presentedhere are suited for analytical or otherwise systematical evaluation of such policies, whichcould very well be the subject of further research. For example, as figures 6 and 7 indicate,our strategies can be used to determine the minimum bandwidth that each server shouldallocate for a single request while keeping the access time to a minimum.

6. Conclusions

In this paper, we have presented a generalized approach to the theory of minimizing theaccess time of retrieving a MM document requested by a client. On a network basedenvironment, we have shown the impact of non-zero communication delays on the perfor-mance. We have designed and analyzed two different retrieval strategies that minimize theaccess time of the MM document. The single and multi-installment strategies proposed,can provide a basis for improved quality services in the emerging markets of VoD or MoD.Given the limiting technological aspects of the current network—mainly—infrastructure,these strategies form not just the basis for improved services, but can be the only way torealize them. In general, these strategies are designed to suit most of the network based MM


applications in which a large volume of data needs to be transferred between sites incurringa considerable amount of communication delays. This is the first time that such a unifiedapproach has been attempted. Approximations to the strategies and the model presented inthis paper precisely addresses the problems in the domain of MM applications in which theimpact of network delays and bandwidth are significant.

In the case of single-installment strategy, we have derived closed-form solution for theindividual sizes of the portions retrieved from various servers and the corresponding accesstime. We have extended the analysis to the case of homogeneous or shared channels. In thecase of multi-installment strategy, we have presented a recursive procedure to obtain the indi-vidual sizes of the document portions from various servers. It has been shown that the multi-installment strategy has the added advantage of fine tuning the access time, which is cru-cial to time and delay sensitive applications. Rigorous performance tests were conducted tostudy the trade-off relationships between the use of several servers in the network as opposedto the use of multiple installments. It was observed that the gain achieved by increasingthe number of installments is in effect similar to employing more servers for the job. Ofcourse, the former approach is far more attractive as a means towards minimization of theaccess time. This also means that the available servers can be appropriately partitioned toattend to different service requests, while at the same time keeping the access time as lowas possible. This observation is not only non-trivial, but also allows to increase the overallservicing capability of the network and makes such network based MM applications viableand more attractive. In the case of single installment strategy, we have proved an importanttheorem which eliminates the need for seeking a particular server sequence. Although, theaccess time is independent of the sequence chosen, a deployed system may elaborate onthe sequence in order to ‘release’ some servers before others. Of course, as the number ofinstallments increases such actions become pointless.

The analysis presented in the discussion section can be the stimulus for developingmore fine-tuned approaches, that take into account the ever-changing nature of the networkenvironment parameters. Constant bandwidths is rarely the case for the heavily loadedInternet connections that are shared by thousands of users. The volatility of the networkconnections is the key factor for future research. Although, the multi-installment strategy isthe way to go, insuring QOS requires that a deployed system assigns and possibly modifiesduring a session, the document downloading responsibilities, subject to server availabilityand bandwidth irregularities.

References

1. C.C. Bisdikian and B.V. Patel, “Issues on movie allocation in distributed video-on-demand systems,” inProceeding on IEEE International Conference on Communications, IEEE Communications Society, NewYork, 1995, pp. 250–255.

2. C.C. Bisdikian and B.V. Patel, “Cost-based program allocation for distributed multimedia-on-demand sys-tems,” IEEE Multimedia, pp. 62–72, Fall issue, 1996.

3. M.M. Buddhikot and G. Parulkar, “Distributed data layout, scheduling, and playout control in a large scalemultimedia storage server,” Technical Report WUCS-94-33, January 1995.

4. K.S. Candan, B. Prabhakaran, and V.S. Subrahmanian, “Collaborative multimedia documents: Authoring andpresentation,” Technical Report: CS-TR-3596, UMIACS-TR-96-9, University of Maryland, College Park,Computer Science Technical Series Report, January 1996.


5. K.S. Candan, B. Prabhakaran, and V.S. Subrahmanian, “Retrieval schedules based on resource availabilityand flexible presentation specifications,” ACM/Springer-Verlag Multimedia Systems Journal, Vol. 6, No. 4,pp. 232–250, 1996.

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demand servers,” in Proceedings of th International Conference of Communications 1991, Denver-Colorado,IEEE Press, 1991, pp. 842–846.

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Bharadwaj Veeravalli received his B.Sc. in Physics, from Madurai-Kamaraj Uiversity, India in 1987, Master’sin Electrical Communication Engineering from Indian Institute of Science, Bangalore, India in 1991 and PhDfrom Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India in 1994. He did hispost-doctoral research in the Department of Computer Science, Concordia University, Montreal, Canada, in 1996.He is currently with the Department of Electrical Engineering in the Computer Engineering Division atTheNational University of Singapore, Singapore, as an Assistant Professor. He is a member of IEEE and IEEEComputer Society. His research interests includeHigh speed heterogeneous computing, Scheduling in paralleland distributed systems, andMultimedia computing. He is one of the earliest researchers in the field of divisibleload theory and is the principal author of the book entitledScheduling Divisible Loads in Parallel and DistributedSystems, IEEE Computer Society, Los Alamitos, CA, USA, August 1996.


Gerassimos Barlaswas born in 1967 in Athens, Greece. In 1989 he received the B.Sc. degree in physics from theNational University of Athens and earned a scholarship with the Institute of Informatics and Telecommunicationsof the NCSR “DEMOCRITOS”. In 1990 he joined as a graduate student the National Technical University ofAthens, from which he received a Ph.D. in computer science in 1996. He is currently a visiting assistant professorwith the Informatics Department of the University of Cyprus. His research interests include parallel algorithms,compression of biomedical signals and medical informatics. G. Barlas is a member of the IEEE ComputerSociety.

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