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Peer-to-Peer Networking andApplications ISSN 1936-6442 Peer-to-Peer Netw. Appl.DOI 10.1007/s12083-012-0188-9

On the expected file download time of therandom time-based switching algorithm inP2P networks

Keqin Li

1 23

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Peer-to-Peer Netw. Appl.DOI 10.1007/s12083-012-0188-9

On the expected file download time of the randomtime-based switching algorithm in P2P networks

Keqin Li

Received: 26 July 2011 / Accepted: 21 November 2012© Springer Science+Business Media New York 2012

Abstract The expected file download time of the ran-dom time-based switching algorithm for peer selection andfile downloading in a peer-to-peer (P2P) network is stillunknown. The main contribution of this paper is to analyzethe expected file download time of the time-based switch-ing algorithm for file sharing in P2P networks when theservice capacity of a source peer is totally correlated overtime, namely, the service capacities of a source peer in dif-ferent time slots are a fixed value. A recurrence relation isdeveloped to characterize the expected file download timeof the time-based switching algorithm. It is proved that fortwo or more heterogeneous source peers and sufficientlylarge file size, the expected file download time of the time-based switching algorithm is less than and can be arbitrarilyless than the expected download time of the chunk-basedswitching algorithm and the expected download time ofthe permanent connection algorithm. It is shown that theexpected file download time of the time-based switchingalgorithm is in the range of the file size divided by the har-monic mean of service capacities and the file size divided bythe arithmetic mean of service capacities. Numerical exam-ples and data are presented to demonstrate our analyticalresults.

Keywords Chunk-based switching · Download time ·File sharing · Peer-to-peer network · Peer selection ·Time-based switching

K. Li (�)Department of Computer Science, State University of New York,New Paltz, NY 12561, USAe-mail: lik@newpaltz.edu

1 Introduction

A peer-to-peer (P2P) network employs diverse connectivityamong participating peers and the combined resources ofparticipants to provide various services [3]. A P2P networkprovides services in a way different from that of central-ized resources where a small number of servers provide allservices. A pure P2P network does not have the notion ofclients and servers, but only equal peers that simultaneouslyfunction as both clients and servers to other peers. Thismodel of network architecture differs from the traditionalclient-server model where communication is among manyclients and a single central server.

A unique feature of P2P networks is that all peerscontribute resources, including storage space, computingpower, and communication bandwidth. Therefore, as par-ticipating peers in a network increases, the total servicecapacity of the network also increases. This is not the case ina client-server system with a small number of servers, whereadding more clients reduces the speed of data transfer forall clients and degrades the overall system performance. Inaddition to the above advantage of scalability, the distributednature of a P2P network also increases the robustness of thenetwork and the capability of fault tolerance in case of peerfailures by replicating data over multiple peers. In a pureP2P network, peers find locations of data without relyingon a centralized index server, which means that there is nosingle point of failure in the network.

File sharing using application layer protocols such as Bit-Torrent is the most popular application of P2P networks, inaddition to many other applications such as telephony, mul-timedia (audio, video, radio) streaming, discussion forums,instant messaging and online chat, and software publica-tion and distribution. File sharing means distributing andaccessing digitally stored information such as computer

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programs, multimedia, documents, and electronic books.It can be implemented in various storage, transmission,and distribution models. Common file sharing methodsare manual sharing using removable memory, centralizedfile servers, WWW-based hyperlinked documents, and dis-tributed P2P networking. The increasing popularity of theMP3 music format in the late 1990s led to Napster andother software designed to aid in the sharing of electronicfiles. Current popular P2P networks/protocols include AresGalaxy, eDonkey, Gnutella, and Kazaa [2].

Performance measurement, modeling, analysis, and opti-mization of file sharing in P2P networks has been a veryactive research area in the last few years. Research hasbeen conducted at three different levels, i.e., system level,peer group level, and individual peer level. At the systemlevel, research is focused on establishing models of P2P net-works such as queueing models [12, 26] and fluid models[11], so that overall system characterizations such as systemthroughput and average file download time can be obtained.At the peer group level, research is focused on distributinga file from a set of source peers to a set of user peers so thatthe overall distribution time is minimized [14, 16, 20, 21,25, 27]. At the individual peer level, research is focused onanalyzing and minimizing the file download time of a singlepeer [9, 17, 18].

It is clear that the vast majority of file downloads are per-formed by individual users. Fine system level modeling andefficient group file distribution methods do not help individ-ual users to minimize their file download times. Therefore,P2P network performance optimization from a single peer’spoint of view becomes an interesting and important issue.File download includes two parts, namely, file searching andfile transfer. Since file searching takes a very small portionof file download time, by file download time, we mean filetransfer time. In a P2P network with heterogeneous sourcepeers, after searching and determining the source peers ofa file of interest, the major problem for an individual userpeer is the peer selection problem, namely, switching amongsource peers and finally settling on one, while keeping thetotal time of probing and downloading to a minimum [6].The problem is called the server selection problem in WWWclient-server applications [8, 10]. The peer selection prob-lem is also studied in the context of free-market economywith additional consideration of cost of download [4, 5].

A number of randomized peer selection and file down-loading algorithms were proposed in [9], including thepermanent connection algorithm, the chunk-based switch-ing algorithm, and the time-based switching (also calledperiodic switching) algorithm. The expected file downloadtime of the first two algorithms is easy to obtain, namely, thefile size divided by the harmonic mean of service capacities.However, the expected file download time of the time-basedswitching algorithm is still unknown. In [9], it is mentioned

that if the service capacities of a source peer in different timeperiods are lightly correlated or almost independent, thenthe expected file download time of the time-based switch-ing algorithm is approximately the file size divided by thearithmetic mean of expected service capacities. However,the assumption of almost independent service capacities indifferent time periods is unrealistic. Therefore, finding theexpected file download time of the time-based switchingalgorithm for highly or totally correlated service capacitiesbecomes an interesting and important problem.

The motivation and significance of our study is twofold.First, there has been no analysis for the time-based switch-ing algorithm when service capacities are highly or totallycorrelated. We contribute new findings in this direction.Second, such analysis is necessary when the time-basedswitching algorithm is to be compared with other algo-rithms, such as the permanent connection algorithm and thechunk-based switching algorithm. We reveal the fact that thetime-based switching algorithm does perform better than theother two methods.

The main contribution of this paper is to analyze theexpected file download time of the time-based switchingalgorithm for file sharing in P2P networks when the ser-vice capacity of a source peer is totally correlated over time,namely, the service capacities of a source peer in differenttime slots are a fixed value. We give a recurrence rela-tion to characterize the expected file download time of thetime-based switching algorithm. We show that for a fixedgroup of source peers and a given length of time period,the expected file download time of the time-based switchingalgorithm is a piecewise linear function of file size.

Furthermore, we compare the performance of the time-based switching algorithm with the permanent connectionalgorithm and the chunk-based switching algorithm. Weprove that for two or more heterogeneous source peersand sufficiently large file size, the expected file downloadtime of the time-based switching algorithm is less than theexpected download time of the chunk-based switching algo-rithm and the expected download time of the permanentconnection algorithm. More specifically, for a fixed file size,the expected file download time of the time-based switchingalgorithm is a nondecreasing function of the length of timeperiod. When the length of time period is sufficiently large,the time-based switching algorithm is identical to the chunk-based switching algorithm and the permanent connectionalgorithm, namely, the expected file download time of thetime-based switching algorithm is the file size divided bythe harmonic mean of service capacities. When the lengthof time period approaches zero, the expected file downloadtime of the time-based switching algorithm approaches thefile size divided by the arithmetic mean of service capaci-ties. The performance of the random permanent connectionalgorithm and the random chunk-based switching algorithm

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can be arbitrarily worse than the performance of the randomtime-based switching algorithm.

Our analytical results are very clean and have intu-itive explanations. For very long length of time period, thetime-based switching algorithm behaves no differently fromthe permanent connection algorithm and the chunk-basedswitching algorithm, because all source peers are chosenwith the same chance (or, equivalent, roughly the same per-centage of a file is downloaded from all source peers). Asthe length of time period decreases, the source peers haveincreased chance of being used for the same amount of time.In the ideal case, when all source peers are chosen with thesame amount of time t , the download time T must be thefile size divided by the arithmetic mean of service capaci-ties, since the time t is the download time T divided by thenumber of source peers, and the file size is the product of t

and the total service capacity.In addition to analytical results, we also present numeri-

cal examples and data to demonstrate our analytical results.Furthermore, we derive approximate closed form expres-sions for the expected file download time of the time-basedswitching algorithm and demonstrate that they are veryaccurate.

We notice that it is very effective to employ chunk-basedswitching and peer selection by using the method of prob-ing high-capacity peers [18]. Furthermore, the method ofparallel downloading has been used in reducing file down-load times [7, 9, 13, 15, 19, 22, 23]. However, these topicsare beyond the scope of this paper. Due to space limita-tion, analysis of and comparison with these methods deserveseparate papers. We will propose and analyze and com-pare the methods of probing high-capacity peers and parallelfile download algorithms in P2P networks with random ser-vice capacities in separate papers. The focus of this paperis to analyze the random time-based switching algorithmand compare its performance with the random chunk-basedswitching algorithm.

2 Algorithms and analysis

Assume that n peers 1, 2, ..., n have been identified assource peers of a file of interest, such that any part of thefile can be downloaded from any of these n source peers.We assume that the service capacity of source peer i is Ci ,which is unknown when a file is to be downloaded. The n

source peers are homogeneous if C1 = C2 = · · · = Cn.It is assumed that all source peers are stable and remain

in a P2P network for significant amount of time. There is noeffect of peer churn [24] for downloading the file of interest,i.e., all source peers are available during downloading of thefile. Moreover, all the n source peers are seed peers, i.e.,they all hold a complete copy of a file, such that any chunk

of the file can be obtained from any source peer. Furtheranalysis of the effect of peer churn and non-seed peers canbe a direction for future investigation.

We use S to represent the size as well as the name ofa file. Let Ti(S) be the download time of a file of size S

from source peer i. It is clear that Ti(S) = S/Ci . Let TA(S)

denote the expected download time for a file of size S byusing a randomized algorithm A from n source peers withservice capacities C1, C2, ..., Cn.

2.1 Random permanent connection

The random permanent connection (PC) algorithm worksas follows. To download a file, a source peer i is chosenrandomly from n source peers with a uniform distribution,that is, each source peer is selected with equal probability1/n. Proposition 1 states that the expected file downloadtime of the random permanent connection algorithm is thefile size divided by the harmonic mean of service capacitiesC1, C2, ..., Cn.

Proposition 1 The expected file download time TPC(S) ofthe random permanent connection algorithm is

TPC(S) = 1

n

n∑

i=1

S

Ci

= S

(1

n

n∑

i=1

1

Ci

).

Proof If source peer i is chosen, the file download time isTPC(S) = Ti(S) = S/Ci , which happens with probability1/n. Thus, the expected file download time of the randompermanent connection algorithm is

TPC(S) = 1

n

n∑

i=1

Ti(S) = 1

n

n∑

i=1

S

Ci

= S

(1

n

n∑

i=1

1

Ci

).

This proves the result.

2.2 Random chunk-based switching

In a chunk-based switching algorithm, a file to be down-loaded is divided into chunks of size S∗, where S∗ is anetwork-wide parameter agreed by and acceptable to all ser-vice and user peers. Without loss of generality, it is assumedthat S can be divided by S∗ and m = S/S∗ is the number ofchunks, such that the chunks are numbered by 1, 2, ..., m.Given a file of size S and n source peers, a downloadschedule specifies a source peer for each chunk.

In the random chunk-based switching (CBS) algorithm,a source peer is randomly and uniformly chosen fromthe n source peers for each chunk. Proposition 2 statesthat the random chunk-based switching algorithm has thesame expected file download time as that of the randompermanent connection algorithm.

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Proposition 2 The expected download time TCBS(S) of therandom chunk-based switching algorithm is

TCBS(S) = 1

n

n∑

i=1

S

Ci

= S

(1

n

n∑

i=1

1

Ci

).

Proof It is clear that the expected time to download onechunk is TPC(S∗). Since there are m chunks, we get

TCBS(S) = mTPC(S∗) = mS∗(

1

n

n∑

i=1

1

Ci

)

= S

(1

n

n∑

i=1

1

Ci

).

This proves the result.

2.3 Random time-based switching

The random time-based switching (TBS) algorithm dividestime into periods of equal length τ . During each time period,a fragment of a file is downloaded from a source peerwhich is randomly chosen from the n source peers withequal probability. Due to different service capacities of thesource peers, the sizes of the fragments downloaded duringdifferent time slots may be different.

It seems that the uniform distribution in selecting asource peer in each time period does not make the TBSalgorithm to be different from the PC and CBS algorithms.However, it turns out that the TBS algorithm performs moreefficiently than the PC and CBS algorithms.

Before making such a comparison, we need to knowthe expected file download time of the time-based switch-ing algorithm. The following theorem gives a recurrencerelation for TTBS(S).

Theorem 1 The expected file download time TTBS(S) of therandom time-based switching algorithm is characterized bythe following recurrence relation:

TTBS(S) = 1

n

n∑

i=1

(S > τCi ? τ + TTBS(S − τCi) : S

Ci

).

(Note: The value of an expression c ? a : b is a if conditionc is satisfied and b otherwise.)

Proof Consider the first period τ . Assume that source peeri is chosen. This happens with probability 1/n. If S > τCi ,a fragment of a file of size τCi is downloaded from source

peer i, and the expected download time for the rest of thefile of size S−τCi is TTBS(S−τCi), and the total downloadtime is τ +TTBS(S − τCi). If S ≤ τCi , a file is downloadedin one period of time with length S/Ci . Summarizing theabove discussion, the expected file download time TTBS(S)

of the random time-based switching algorithm is

TTBS(S) = 1

n

n∑

i=1

(S > τCi ? τ + TTBS(S − τCi) : S

Ci

).

This proves the theorem.

It is always interesting to solve a recurrence relation.Unfortunately, it seems unlikely that there exists a closedform expression for TTBS(S). To show this, consider n = 2source peers with C1 = C and C2 = 2C. It is easy to verifythat

TTBS(S) = 1

2

(S

C1+ S

C2

)

= 1

2

(S

C+ S

2C

)

= 0.75S

C, 0 < S ≤ τC;

TTBS(S) = 1

2

(τ + TTBS(S − τC1) + S

C2

)

= 1

2

(τ + TTBS(S − τC) + S

2C

)

= 1

2

(τ + 0.75

S − τC

C+ 0.5

S

C

)

= 1

2

(1.25

S

C+ 0.25τ

)

= 0.625S

C+ 0.125τ, τC < S ≤ 2τC;

TTBS(S) = 1

2(τ + TTBS(S − τC1) + τ + TTBS(S − τC2))

= 1

2(τ + TTBS(S − τC) + τ + TTBS(S − 2τC))

= 1

2

(τ + 0.625

S − τC

C+ 0.125τ

+ τ + 0.75S − 2τC

C

)

= 0.6875S

C, 2τC < S ≤ 3τC;

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TTBS(S) = 1

2(τ + TTBS(S − τC1) + τ + TTBS(S − τC2))

= 1

2(τ + TTBS(S − τC) + τ + TTBS(S − 2τC))

= 1

2

(τ + 0.6875

S − τC

C

+ τ + 0.625S − 2τC

C+ 0.125τ

)

= 1

2

(1.3125

S

C+ 0.1875τ

)

= 0.65625S

C+ 0.09375τ, 3τC < S ≤ 4τC;

...

The above calculation shows that TTBS(S) has a differentexpression in each subinterval ((j − 1)τC, jτC], wherej ≥ 1. If we represent TTBS(S) as αj (S/C) + βj τ insubinterval ((j − 1)τC, jτC], then as j → ∞, we haveα∞ = 0.6666666....

Although there is no closed form expression of TTBS(S),we at least know that the expected file download timeTTBS(S) is a piecewise linear function of S. Let (0, ∞)

be cut into subintervals I1 = (S0, S1], I2 = (S1, S2],I3 = (S2, S3], ..., Ij = (Sj−1, Sj ], ..., where S0 < S1 <

S2 < S3 < · · · < Sj < · · · , and Sj = γj τ for allj ≥ 0, with γj determined by C1, C2, ..., Cn. Without lossof generality, we assume that C1 ≤ C2 ≤ · · · ≤ Cn.The values of S0, S1, S2, S3, ..., Sj , ... are defined induc-tively as follows. First, we have S0 = 0 and S1 = τC1.Based on S0, S1, S2, S3, ..., Sj−1, we define Sj to be largestvalue which satisfies the following condition, namely, for allSj−1 < S ≤ Sj , the size S − τCi falls into Iji

with ji < j ,where 1 ≤ i ≤ kj , and kj is the largest integer such that forall Sj−1 < S ≤ Sj , we have S > τCi for all 1 ≤ i ≤ kj ,and S ≤ τCi for all kj + 1 ≤ i ≤ n.

Theorem 2 The expected file download time TTBS(S) is apiecewise linear function of S in the subintervals of (0, ∞)

cut by S0, S1, S2, S3, ..., Sj , ....

Proof We prove by induction on j ≥ 1, that in thesubinterval Ij = (Sj−1, Sj ], we have

TTBS(S) = αj (C1, C2, ..., Cn)S + βj (C1, C2, ..., Cn)τ,

where αj (C1, C2, ..., Cn) is a positive function ofC1, C2, ..., Cn and βj (C1, C2, ..., Cn) is a nonnegative

function of C1, C2, ..., Cn. First, it is clear that when j = 1(i.e., for subinterval I1), we have S1 = τC1 and

TTBS(S) =(

1

n

n∑

i=1

1

Ci

)S,

that is,

α1(C1, C2, ..., Cn) = 1

n

n∑

i=1

1

Ci

,

and

β1(C1, C2, ..., Cn) = 0.

Next, we consider subinterval Ij with j > 1, based on theinduction hypothesis that the expected file download timeTTBS(S) is

TTBS(S) = αj ′(C1, C2, ..., Cn)S + βj ′(C1, C2, ..., Cn)τ,

in subinterval Ij ′ , for all 1 ≤ j ′ ≤ j − 1. Recall that kj isdefined in such a way that S > τCi for all 1 ≤ i ≤ kj ,and S ≤ τCi for all kj + 1 ≤ i ≤ n, where 1 ≤ kj ≤ n.The recurrence relation for TTBS(S) in Theorem 1 can berewritten as

TTBS(S) = 1

n

⎛

⎝kj∑

i=1

(τ +TTBS(S−τCi))+⎛

⎝n∑

i=kj +1

1

Ci

⎞

⎠S

⎞

⎠.

We notice that for all S ∈ Ij , the size S − τCi falls into Iji

with ji < j , where 1 ≤ i ≤ kj . By the induction hypothesis,we get

TTBS(S − τCi) = αji(C1, C2, ..., Cn)(S − τCi)

+ βji(C1, C2, ..., Cn)τ,

for all 1 ≤ i ≤ kj . Then, in subinterval Ij , we have

TTBS(S) = 1

n

⎛

⎝kj∑

i=1

(τ + αji(C1, C2, ..., Cn)(S − τCi)

+ βji(C1, C2, ..., Cn)τ )

+⎛

⎝n∑

i=kj +1

1

Ci

⎞

⎠ S

⎞

⎠ ,

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which is actually

TTBS(S) = 1

n

⎛

⎝kj∑

i=1

αji(C1, C2, ..., Cn) +

n∑

i=kj +1

1

Ci

⎞

⎠ S

+ 1

n

⎛

⎝kj∑

i=1

(1 − αji(C1, C2, ..., Cn)Ci

+ βji(C1, C2, ..., Cn))

⎞

⎠ τ

= αj (C1, C2, ..., Cn)S + βj (C1, C2, ..., Cn)τ,

where

αj (C1, C2, ..., Cn)

= 1

n

⎛

⎝kj∑

i=1

αji(C1, C2, ..., Cn) +

n∑

i=kj +1

1

Ci

⎞

⎠ ,

and

βj (C1, C2, ..., Cn) = 1

n

⎛

⎝kj∑

i=1

(1 − αji(C1, C2, ..., Cn)Ci

+ βji(C1, C2, ..., Cn))

⎞

⎠ .

This proves the theorem.

3 Performance comparison

Now, we are ready to compare the performance of algorithmTBS with algorithms PC and CBS. The following theoremstates that the expected file download time of the time-basedswitching algorithm is no longer than that of the perma-nent connection algorithm and the chunk-based switchingalgorithm, where the equality holds only for homogeneoussource peers or files of very small sizes (or equivalently,very long length of time period τ ).

Theorem 3 For any file S, we have TTBS(S) ≤ TPC(S) =TCBS(S), where the equality holds only for homogeneoussource peers or S ≤ τ min{C1, C2, ..., Cn}.

Proof We prove the theorem by induction on S. First, it iseasy to observe that

TTBS(S) = TPC(S) = TCBS(S),

for all 0 ≤ S ≤ τ min{C1, C2, ..., Cn}. Next, for arbi-trary S > τ min{C1, C2, ..., Cn}, we are going to showthat TTBS(S) ≤ TPC(S) = TCBS(S) based on the induc-tion hypothesis that TTBS(S′) ≤ TPC(S′) = TCBS(S′) for allS′ < S. Without loss of generality, we assume that S > τCi

for all 1 ≤ i ≤ k, and S ≤ τCi for all k + 1 ≤ i ≤ n, where1 ≤ k ≤ n. The recurrence relation for TTBS(S) in Theorem1 can be rewritten as

TTBS(S) = 1

n

⎛

⎝k∑

i=1

(τ + TTBS(S − τCi)) +n∑

i=k+1

S

Ci

⎞

⎠ .

By the induction hypothesis, we have

TTBS(S − τCi) ≤ TPC(S − τCi) = TCBS(S − τCi)

= (S − τCi)

(1

n

n∑

i=1

1

Ci

),

for all 1 ≤ i ≤ k. Hence, we get

TTBS(S)

≤ 1

n

⎛

⎝k∑

i=1

(τ + (S − τCi)

(1

n

n∑

i=1

1

Ci

))+

n∑

i=k+1

S

Ci

⎞

⎠

= 1

n

⎛

⎝kτ +(

k∑

i=1

(S − τCi)

)(1

n

n∑

i=1

1

Ci

)+

n∑

i=k+1

S

Ci

⎞

⎠

= 1

n

⎛

⎝kτ +(

kS − τ

k∑

i=1

Ci

)(1

n

n∑

i=1

1

Ci

)+

n∑

i=k+1

S

Ci

⎞

⎠.

Since

1

n

n∑

i=1

1

Ci

≤ 1

k

k∑

i=1

1

Ci

,

where the equality holds only when k = n, we obtain

TTBS(S)

≤ 1

n

⎛

⎝kτ +(

kS − τ

k∑

i=1

Ci

)(1

k

k∑

i=1

1

Ci

)+

n∑

i=k+1

S

Ci

⎞

⎠

= 1

n

⎛

⎝kτ −(τ

k∑

i=1

Ci

)(1

k

k∑

i=1

1

Ci

)+

k∑

i=1

S

Ci

+n∑

i=k+1

S

Ci

⎞

⎠

= 1

n

(kτ

(1 −

(1

k

k∑

i=1

Ci

)(1

k

k∑

i=1

1

Ci

))+

n∑

i=1

S

Ci

).

Since(

1

k

k∑

i=1

Ci

)≥ 1

(1

k

k∑

i=1

1

Ci

) ,

that is, the harmonic mean of the Ci’s is no greater than thearithmetic mean of the Ci’s, where the equality holds onlywhen C1 = C2 = · · · = Ck , we have(

1

k

k∑

i=1

Ci

)(1

k

k∑

i=1

1

Ci

)≥ 1,

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Table 1 Comparison of Analytical and Simulation Data of TTBS(S)

(τ = 15)

S Analytical Simulation Relative

data data error (%)

100 24.63 24.66 0.12150 36.56 36.57 0.03200 48.29 48.22 −0.14250 60.06 60.00 −0.10300 71.83 71.83 0.00350 83.59 83.57 −0.02400 95.35 95.30 −0.05450 107.12 107.01 −0.10500 118.88 118.78 −0.08550 130.65 130.58 −0.05600 142.41 142.48 0.05650 154.18 154.15 −0.02700 165.94 165.89 −0.03750 177.71 177.65 −0.03800 189.47 189.54 0.04850 201.24 201.16 −0.04900 213.00 212.94 −0.03950 224.77 224.77 0.001000 236.53 236.43 −0.041050 248.30 248.19 −0.041100 260.06 259.93 −0.051150 271.83 271.76 −0.031200 283.59 283.67 0.031250 295.35 295.38 0.011300 307.12 307.19 0.021350 318.88 318.89 0.001400 330.65 330.55 −0.031450 342.41 342.29 −0.041500 354.18 354.00 −0.05

which yields

TTBS(S) ≤ 1

n

n∑

i=1

S

Ci

= TPC(S) = TCBS(S).

Notice that TTBS(S) = TPC(S) = TCBS(S) only when C1 =C2 = · · · = Ck with k = n, that is, the n source peers arehomogeneous. The theorem is proven.

4 Numerical and simulation data

In this section, we present a numerical example to comparethe performance of algorithms PC, CBS, and TBS. As inmost P2P file sharing and exchange systems, the file sizesare in the range 10 ∼ 1500 MB [1]. The service capacity ofa source peer is in the range 50 ∼ 1, 000 kbps, i.e., 0.375 ∼7.5 MB/min.

Let us consider a P2P file sharing system with n = 12source peers. The service capacities of the n source peers are1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0 MB/min.

In Table 1, we show simulation results and their accuracycompared with the analytical results for the expected filedownload time of the time-based switching algorithm. Thetime period is τ = 15 min. The analytical data are calculatedby using Theorem 1. Each simulation datum is obtainedby repeating a simulation (i.e., execution of the time-basedswitching algorithm) for sufficient number of times, suchthat the 99 % confidence interval is sufficiently small, whichis ±1.03598 % in our case. The relative error of eachsimulation datum is also given, which is no greater than±0.14 %. These simulation data validate the correctness ofour analytical data.

In Fig. 1, we demonstrate the expected file downloadtime of the time-based switching algorithm as a function

Fig. 1 The expected filedownload time TTBS(S) vs. filesize S.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

S (MB)

0

50

100

150

200

250

300

350

400

450

The

exp

ecte

d do

wnl

oad

time

TT

BS(

S) (

min

utes

)

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..... = 62

= 1000

= 500

= 125 = 250

= 31 = 1

Author's personal copy

Peer-to-Peer Netw. Appl.

of file size S, where 0 ≤ S ≤ 1500, for τ = 1000, 500,250, 125, 62, 31, 1 min. Notice that in Fig. 1, the curvefor τ = 1000 is precisely the performance of the perma-nent connection algorithm and the chunk-based switchingalgorithm.

In Fig. 2, we demonstrate the expected file downloadtime of the time-based switching algorithm as a functionof the length of time period τ , where 0 ≤ τ ≤ 1000, forS = 300, 600, 900, 1200, 1500 MB.

It is observed from Figs. 1 and 2 that shorter timeperiod τ yields better performance, i.e., shorter expected filedownload time. When τ = 1000, i.e.,

S ≤ τ min{C1, C2, ..., Cn},the random time-based switching algorithm has the sameexpected file download time as the random permanent con-nection algorithm and the random chunk-based switchingalgorithm. When τ = 1, i.e., almost zero, the randomtime-based switching algorithm approaches its minimumexpected file download time.

The following facts formally describe our observations,which provide further details to Theorem 3.

Fact 1 If the time period τ is

τ ≥ max

{S

C1,

S

C2, ...,

S

Cn

}= S

min{C1, C2, ..., Cn} ,

the expected file download time TTBS(S) of the randomtime-based switching algorithm is

TTBS(S) = S(

1

n

n∑

i=1

1

Ci

)−1,

namely, the file size divided by the harmonic mean of servicecapacities, which is the same as that of the random per-manent connection algorithm and the random chunk-basedswitching algorithm.

Proof Fact 1 is a direct consequence of Theorem 1, fromwhich we know that if

τ ≥ max

{S

C1,

S

C2, ...,

S

Cn

},

we have

TTBS(S) = 1

n

n∑

i=1

S

Ci

,

since S ≤ τCi , for all 1 ≤ i ≤ n.

Fact 2 For a fixed S and n ≥ 2 heterogeneous source peers,if

τ < max

{S

C1,

S

C2, ...,

S

Cn

}= S

min{C1, C2, ..., Cn} ,the expected file download time TTBS(S) is an increasingfunction of τ .

Proof From Theorem 2, we know that the expected filedownload time TTBS(S) is a piecewise linear function of τ ,i.e.,

TTBS(S) = αj (C1, C2, ..., Cn)S + βj (C1, C2, ..., Cn)τ,

in the subinterval S/γj ≤ τ < S/γj−1, and hence, is anondecreasing function of τ . Without loss of generality, weassume that C1 ≤ C2 ≤ · · · ≤ Cn, i.e., S/C1 ≥ S/C2 ≥· · · ≥ S/Cn. For n ≥ 2 heterogeneous source peers, if

τ < max

{S

C1,

S

C2, ...,

S

Cn

},

Fig. 2 The expected filedownload time TTBS(S) vs.period τ .

0 100 200 300 400 500 600 700 800 900 1000

(minutes)

0

50

100

150

200

250

300

350

400

450

The

exp

ecte

d do

wnl

oad

time

TT

BS(

S) (

min

utes

)

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S = 1500

S = 1200

S = 900

S = 600

S = 300

Author's personal copy

Peer-to-Peer Netw. Appl.

there must be k, where 1 ≤ k ≤ n, such that S > τCi for all1 ≤ i ≤ k, and S ≤ τCi for all k + 1 ≤ i ≤ n. By Theorem1, we get

TTBS(S)= 1

n

⎛

⎝k∑

i=1

(τ +TTBS(S−τCi))+⎛

⎝n∑

i=k+1

1

Ci

⎞

⎠S

⎞

⎠,

where, we can easily see that TTBS(S) is an increasing func-tion of τ , because all the TTBS(S − τCi))’s are at leastnondecreasing functions of τ .

Fact 3 As the time period τ approaches zero, the expectedfile download time TTBS(S) approaches its minimum valuewhich is

limτ→0

TTBS(S) = S

1

n(C1 + C2 + · · · + Cn)

,

namely, the file size divided by the arithmetic mean ofservice capacities.

Proof As the length of the time period τ approaches zero,there will be increasing chance that all the source peersare used for the same amount of time. When all the sourcepeers are used for the same amount of time t , we will haveTTBS(S) = nt , i.e., the expected file download time TTBS(S)

is the number of source peers n times the time t . Further-more, we have S = tC1 + tC2 + · · · + tCn = t (C1 + C2 +· · · + Cn), which implies that

t = S

C1 + C2 + · · · + Cn

,

i.e., the time t is the file size divided by the total servicecapacity. Consequently, we get

TTBS(S) = S

1

n(C1 + C2 + · · · + Cn)

,

i.e., the expected file download time TTBS(S) is the file sizedivided by the arithmetic mean of service capacities.

Fact 4 The ratios TTBS(S)/TPC(S) and TTBS(S)/TCBS(S)

can be arbitrarily small, that is, the performance of therandom permanent connection algorithm and the randomchunk-based switching algorithm can be arbitrarily worsethan the performance of the random time-based switchingalgorithm.

Proof Fact 4 is based on the fact that the ratio of the arith-metic mean of service capacities to the harmonic mean of

service capacities can be arbitrarily large. For instance, letus consider n = 2 source peers with C1 = 1 and C2 = C.The ratio of the arithmetic mean (C + 1)/2 to the harmonicmean 2C/(C + 1) is

(C + 1)2

4C,

which can be arbitrarily large as C → 0 or C → ∞.Consequently, since by Fact 3,

TTBS(S) ≈ 2S

C + 1for very small τ,

and by Propositions 1 and 2,

TPC(S) = TCBS(S) = S(C + 1)

2C,

we get

TTBS(S)/TPC(S) = TTBS(S)/TCBS(S) ≈ 4C

(C + 1)2,

which can be arbitrarily small when C → 0 or C → ∞.

5 Closed form approximations

An approximate closed form expression for TTBS(S) canbe obtained and its quality can be examined numerically.Assuming without loss of generality that C1 ≤ C2 ≤· · · ≤ Cn. A good approximation of TTBS(S) is to con-sider the subinterval (S1, S2], where S1 = τC1 and S2 =min{2τC1, τC2}, in which, we have

TTBS(S) = 1

n

(τ + TTBS(S − τC1) + S

C2+ · · · + S

Cn

)

= 1

n

(τ + (S − τC1)

1

n

(1

C1+ 1

C2+ · · · + 1

Cn

)

+ S

(1

C2+ · · · + 1

Cn

))

= 1

n

((1

n

(1

C1+ 1

C2+ · · · + 1

Cn

)

+(

1

C2+ · · · + 1

Cn

))S

+(

1− C1

n

(1

C1+ 1

C2+ · · · + 1

Cn

))τ

).

Author's personal copy

Peer-to-Peer Netw. Appl.

For our example in the last section, the above approximationgives an over-estimation of TTBS(S) by no more than 10.2 %for τ = 50 and 100 ≤ S ≤ 1500. It is very accurate for S

close to τC1 or for τ close to S/C1.A more accurate approximation of TTBS(S) can be

obtained by considering the subinterval (S2, S3], where S2

and S3 depend on the relationship among C1, C2, and C3.For our example in the last section, we have S2 = C2 andS3 = C3. Therefore, we get

TTBS(S) = 1

n

(τ + TTBS(S − τC1) + τ + TTBS(S − τC2)

+ S

C3+ · · · + S

Cn

)

= 1

n

(τ + (S − τC1)

1

n

(1

C1+ 1

C2+ · · · + 1

Cn

)

+ τ + (S −τC2)1

n

(1

C1+ 1

C2+· · ·+ 1

Cn

)

+ S

(1

C3+ · · · + 1

Cn

))

= 1

n

((2

n

(1

C1+ 1

C2+ · · · + 1

Cn

)

+(

1

C3+ · · · + 1

Cn

))S

+(

2− C1+C2

n

(1

C1+ 1

C2+· · ·+ 1

Cn

))τ

).

The above approximation gives an over-estimation ofTTBS(S) by no more than 3.8 % for τ = 50 and 100 ≤ S ≤1500.

If we further consider the subinterval (S3, S4], whereS3 = C3 and S4 = C4, we have

TTBS(S) = 1

n

((3

n

(1

C1+ 1

C2+ · · · + 1

Cn

)

+(

1

C4+ · · · + 1

Cn

))S

+(

3 − C1 + C2 + C3

n

×(

1

C1+ 1

C2+ · · · + 1

Cn

))τ

).

The relative error of the above approximation is in the range[−0.09 %, 1.69 %] for τ = 50 and 100 ≤ S ≤ 1500. InTable 2, we give the numerical values of the above approx-imation and the actual values of the expected file downloadtime TTBS(S) when τ = 50 for the same P2P file sharingsystem of Section 4.

Table 2 Accurate approximation of TTBS(S) (τ = 50)

S Actual Approximate Relative

value value error (%)

100 28.41 28.64 0.79150 40.35 40.35 0.00200 51.68 52.07 0.76250 62.87 63.79 1.47300 74.26 75.51 1.69350 86.12 87.23 1.28400 98.38 98.94 0.58450 110.21 110.66 0.41500 121.86 122.38 0.43550 133.51 134.10 0.44600 145.22 145.81 0.41650 157.01 157.53 0.33700 168.84 169.25 0.24750 180.63 180.97 0.18800 192.37 192.68 0.16850 204.12 204.40 0.14900 215.87 216.12 0.11950 227.64 227.84 0.081000 239.42 239.56 0.061050 251.19 251.27 0.031100 262.95 262.99 0.021150 274.71 274.71 −0.001200 286.47 286.43 −0.021250 298.24 298.14 −0.031300 310.01 309.86 −0.051350 321.77 321.58 −0.061400 333.54 333.30 −0.071450 345.30 345.01 −0.081500 357.06 356.73 −0.09

6 Conclusions

We have analyzed the expected file download time of thetime-based switching algorithm for file sharing in P2P net-works when the service capacity of a source peer is totallycorrelated over time, namely, the service capacities of asource peer in different time slots are a fixed value. We havedeveloped a recurrence relation to characterize the expectedfile download time of the time-based switching algorithm.We have shown that the expected file download time of thetime-based switching algorithm is in the range of the filesize divided by the harmonic mean of service capacitiesand the file size divided by the arithmetic mean of servicecapacities, i.e.,

S

1

n(C1 + C2 + · · · + Cn)

< TTBS(S) ≤ S(

1

n

n∑

i=1

1

Ci

)−1.

Unless for very long time period, the time-based switchingalgorithm performs better than the permanent connectionalgorithm and the chunk-based switching algorithm.

Author's personal copy

Peer-to-Peer Netw. Appl.

The performance of the time-based switching algorithmis now well understood when the service capacities of asource peer in different time periods are lightly correlatedor almost independent [9], or when the service capacities ofa source peer in different time periods are highly or totallycorrelated, i.e., the service capacity of a source peer remainsstable at least for a reasonable amount of time (e.g., withinthe time of downloading a file). However, the performanceof the time-based switching algorithm is still not clear whenthe service capacities of a source peer in different time peri-ods are partially correlated. It is therefore an interestingand important and challenging open problem to analyze theexpected file download time of the time-based switchingalgorithm for file sharing in P2P networks when the ser-vice capacities of a source peer in different time periods arepartially correlated, i.e., to include temporal fluctuation ofsource peer capacities into consideration. New mathemat-ical models are required to describe temporal correlationin the service capacity of a source peer. Such analysis iseven more difficult if the effects of peer churn and non-seedpeers are also included. It is conceivable that such investiga-tion needs significantly new insights which are well beyondthe scope of this paper, and deserves separate papers. Inthis sense, our effort in this paper is only an initial attempttowards this direction and should inspire subsequent studies.

Acknowledgments The criticism and comments from three review-ers are greatly appreciated.

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Author's personal copy

Peer-to-Peer Netw. Appl.

Keqin Li is a SUNY Dis-tinguished Professor. Profes-sor Li’s research interests aremainly in the areas of designand analysis of algorithms,parallel and distributed com-puting, and computer net-working. He has contributedextensively to processor allo-cation and resource manage-ment; design and analysis ofsequential/parallel, determin-istic/probabilistic, and approx-imation algorithms; parallel

and distributed computing systems performance analysis, prediction,and evaluation; job scheduling, task dispatching, and load balancingin heterogeneous distributed systems; dynamic tree embedding andrandomized load distribution in static networks; parallel computingusing optical interconnections; dynamic location management in wire-less communication networks; routing and wavelength assignment inWDM optical networks; and energy-efficient power management andperformance optimization. His current research interests include life-time maximization in sensor networks, file sharing in peer-to-peersystems, and cloud computing. He has published over 245 journalarticles, book chapters, and research papers in refereed internationalconference proceedings. He has received several Best Paper Awardsfor his highest quality work. He is currently on the editorial board ofIEEE Transactions on Parallel and Distributed Systems, Journal ofParallel and Distributed Computing, International Journal of Paral-lel, Emergent and Distributed Systems, International Journal of HighPerformance Computing and Networking, and Optimization Letters.

Author's personal copy