Analyzing the Resilience-Complexity Tradeoff of Network Coding in Dynamic P2P Networks

Abstract

Most current-generation P2P content distribution protocols use fine-granularity blocks to distribute content to all the peers in a decentralized fashion. Such protocols often suffer from a significant degree of imbalance in block distributions, especially when the users are highly dynamic. As certain blocks become rare or even unavailable, content availability and download efficiency are adversely affected. Randomized network coding may improve block diversity and availability in P2P networks, as coded blocks are equally innovative and useful to peers. However, the computational complexity of network coding mandates that, in reality, network coding needs to be performed within segments, each containing a subset of blocks. In this paper, we quantitatively evaluate how network coding may improve content availability, block diversity, and download performance in the presence of churn, as the number of blocks in each segment for coding varies. Based on stochastic models and a differential equation approach, we explore the fundamental tradeoff between the resilience gain of network coding to peer dynamics and its inherent coding complexity. We conclude that a small number of blocks in each segment is sufficient to realize the major benefits of network coding, with acceptable coding cost.

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On the Resilience-Complexity Tradeoff of
Network Coding in Dynamic P2P Networks
Di Niu, Baochun Li
Department of Electrical and Computer Engineering
University of Toronto
{dniu, bli}@eecg.toronto.edu
Abstract—Most current-generation P2P content distribution
protocols use fine-granularity blocks to distribute content to all
the peers in a decentralized fashion. Such protocols often suffer
from a significant degree of imbalance in block distributions, such
that certain blocks become rare or even unavailable, adversely
affecting content availability. It has been pointed out that
randomized network coding may improve block availability in
P2P networks, as coded blocks are equally innovative and useful
to peers. However, the computational complexity of network
coding mandates that, in reality, network coding needs to be
performed within segments, each containing a subset of blocks.
In this paper, using both theoretical analysis and simulations,
we quantitatively evaluate how segment-based network coding
may improve resilience to peer dynamics and content availability.
The objective of this paper is to explore the fundamental
tradeoff between the resilience gain of network coding and its
inherent coding complexity. We introduce a differential equations
approach to quantify the resilience gain of network coding as a
function of the number of blocks in a segment, as well as various
other tunable parameters. We conclude that a small number of
blocks in each segment is sufficient to realize the major benefits
of network coding, with acceptable coding complexity.
I. INTRODUCTION
Peer-to-peer (P2P) content distribution has become the
de facto standard in current-generation content distribution
protocols. The basic idea in P2P content distribution protocols
is to segment large volumes of data (usually hundreds of
megabytes or even gigabytes) into fine-granularity blocks, and
then distribute these blocks in an efficient manner by letting
peers exchange them with one another.
In reality, however, P2P protocols often suffer from severe
peer dynamics (i.e., arrivals and departures), as peers are
inherently unreliable. As the fundamental philosophy of peer-
to-peer protocols is to use resources on the peers to store
blocks of content, content distribution sessions may continue
to proceed even when the original peers (sometimes referred
to as seeds) are no longer available. However, in these cases,
blocks may become rare or even unavailable when peers arrive
and depart frequently with short lifetimes. Such significant
degrees of imbalance with respect to block availability —
henceforth referred to as block variation — will adversely af-
fect the availability of content, leading to longer downloading
times at each peer.
As end hosts at the edge of the Internet possess abundant
computational resources with modern processors, it is natural
to take advantage of the power of network coding in P2P
applications, by allowing end hosts to not only forward and
replicate, but to code as well. Network coding has been
The completion of the research was made possible thanks to Bell Canada’s
support through its Bell University Laboratories R&D program.
originally proposed in information theory [1], [2], [3], and
has been more recently used to improve the resilience to peer
dynamics in P2P content distribution protocols [4], leading
to shorter downloading times. Intuitively, one may observe
that, since network coding distributes coded blocks rather than
original blocks, and all coded blocks are equally innovative
and useful to any peer, the challenge of locating rare original
blocks may indeed be addressed.
However, network coding may not realize its benefits
without introducing significant computational complexities at
peers. It has been shown in recent work [5] that, network
coding may not be computationally feasible if one is to
code more than a few hundred blocks, even with modern
processors! In order to reduce coding complexity, Chou et
al. [6] has proposed the concept of group network coding,
which performs coding on the blocks within the same group
or segment, while each segment contains a prescribed (and
arguably small) number of blocks. Though group network
coding helps to reduce coding complexity, its negative effects
on the resilience to peer dynamics — the main advantage of
using network coding in the first place in P2P networks — are
not fully understood.
Let us consider two extremes of P2P protocol design. The
first one does not use network coding at all, and the second
uses network coding across all existing blocks. If we consider
the number of blocks in each segment (referred to as the
segment size) in group network coding, we may observe that
the first extreme corresponds to a segment size of one, with as
many segments as blocks, while the other extreme corresponds
to the case of grouping all blocks into the same segment.
Intuitively, the degree of resilience to peer dynamics that
network coding has to offer improves as we increase the
segment size; but the coding complexity increases as well.
If we vary the segment size when network coding is used, we
are fundamentally moving from one extreme to another, and
making our choice in the challenge of resilience-complexity
tradeoff of network coding in P2P networks. It would be best
if we may operate with an appropriate segment size to enjoy
most of the resilience advantage of using network coding, but
with acceptable coding complexity.
Motivated by our curiosity on choosing the “sweet spot” in
the resilience-complexity tradeoff, we quantitatively evaluate
the content availability with different segment sizes, using both
theoretical analysis and simulations. In our theoretical analy-
sis, we consider large-scale dynamic P2P systems, where each
peer is to have a random lifetime following an arbitrary yet
general distribution. We operate a linear system of differential
equations, and derived closed-form results with respect to the

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variations of block availability. We use simulations to not only
substantiate our theoretical conclusions, but also shed further
insights into the problem in a wide range of scenarios. To the
best of our knowledge, such a resilience-complexity tradeoff
introduced by network coding in P2P content distribution has
never been studied with theoretical rigor in previous work.
The remainder of this paper is organized as follows. In
Sec. II, we present related work. In Sec. III, we present our
system model, formulate the problem and outline the main
theoretical results. In Sec. IV, we derive the differential equa-
tions on which we obtain results regarding content availability
and block variations. In Sec. V, we present theoretical results
regarding steady state block variation when data sources
are available and content availability in the absence of data
sources. In Sec. VI, we carry out simulations to corroborate
our theoretical results. We conclude the paper in Sec. VII.
II. RELATED WORK
The landmark papers on randomized network coding by Ho
et al. [7] and Chou et al. [6] have claimed that randomized
network coding can be designed to be robust to random
packet loss, delay, as well as any changes in network topology
and capacity. Avalanche [4], [8] has further proposed that
randomized network coding can be used for elastic content
distribution. However, it has also been shown [5] that the
coding complexity escalates when an increasing number of
blocks are used in network coding, and that even with modern
processors, coding more than a few hundred blocks may not
be computationally feasible.
In Wu [9], it has been argued that a key advantage of
network coding is its inherent ability to adapt to network
dynamics, both to ergodic changes such as random packet
loss and to non-ergodic changes such as link failures. Koetter
et al. [3] also discussed robust networking and analyzed the
resilience of network coding to non-ergodic link failures. Lun
et al. [10] theoretically analyzed the benefit of network coding
on a directed acyclic hypergraph with lossy links. Acedanski
et al. [11] showed through analysis that with network coding,
a peer downloading a file may randomly connect to fewer
other peers to retrieve the entire file. The relationship between
block selection policies and the imbalance among different
blocks has been discussed for P2P content distribution without
network coding in Fan et al. [12]. In this paper, we focus on
the resilience-complexity tradeoff of network coding, when the
number of blocks in a segment varies from one extreme to an-
other. To our knowledge, this is a first attempt to quantitatively
evaluate the resilience of network coding to block variations
due to peer dynamics.
III. BACKGROUND, MODEL AND MAIN RESULTS
Modern P2P bulk content distribution systems (e.g., Bit-
Torrent [13]) are organized as an application-layer overlay of
peers, each with a number of neighbors. A peer is referred
to as a server if it owns a complete copy of the content
of interest, otherwise it remains as a user. Data exchanges
may only occur between a peer and its neighbors. Normally,
a peer only uploads to a small number (e.g., less than 5) of its
neighbors at a time which are called its downstream peers,
since excessive concurrent uploading may negatively affect
throughput [14]. However, a peer may still frequently change
its downstream peers, either because it intentionally does so
to take advantage of idle capacities, or due to peer dynamics.
We now present a framework that allows us to analyze
P2P content distribution with variable coding complexities. A
large file of size F bytes (usually on the order of hundreds of
Megabytes or several Gigabytes) is to be broadcast to every
online peer. The content is segmented into G segments, each
of which are further broken into m blocks, referred to as the
segment size. Thus, there are M = G · m different original
blocks in the content, each of size k = F/M bytes. In group
network coding [6], random linear coding (RLC) is applied
to each segment of m blocks. Assume segment i has original
blocks B(i)= [Bi
segment i is a linear combination of [Bi
Galois field GF(28). Coding operation is not limited to the
source: if a peer (including the source) has l (l ≤ m) coded
blocks of segment i [bi
p, it independently and randomly chooses a set of coding
coefficients [cp
encodes all its blocks from segment i, and produces one coded
block x of k bytes: x =?l
A coded block x is self-contained, in that the coding coeffi-
cients used to encode original blocks to x are embedded in the
header of the coded block. As soon as a peer has received a
total of m coded blocks from segment i x = [xi
which are linearly independent, it will be able to decode
segment i with coding coefficients embedded in each of the
m coded blocks. To decode segment i, we first need to
compute the inverse of the m×m coefficient matrix Aiusing
Gaussian elimination, which requires O(m3) operations (or
O(m2) operations per input block). To obtain the original m
blocks B(i), it then needs to multiply Ai−1and x, which takes
m2· k multiplications of two bytes in GF(28), which runs in
time O(m2) (or O(m) operations per input block). It turns out
the latter cost dominates the overall decoding time, although
the first phase has a higher computational complexity in terms
of m. This is because the cost of the latter phase also depends
on the block size k, which is usually on the order of Kilobytes.
Naturally, the overall decoding complexity increases as the
segment size m increases.
Assume there are N user peers and Nsservers online, each
with an average upload capacity µ and a separate downlink of
sufficiently large capacity, then a peer will upload at a rate of
λ = µ/k blocks on average. With respect to reducing block
variation in P2P networks, network coding and block balanc-
ing schemes (e.g., the local rarest first policy in BitTorrent
[13]) interact in complicated ways. It is therefore impossible
to assess the resilience of network coding without reference
to block balancing schemes. However, in order to focus on
the effect of network coding, we will use a gossiping protocol
as the reference. Specifically, whenever a peer is to upload a
block, it randomly chooses a downstream peer from all other
peers. (This assumption can be relaxed to randomly choosing
a peer from its neighborhood, as long as the neighbors are
uniform samples of the entire network.) After that, block
uploading is performed in three steps: segment reconciliation,
1,Bi
2,...,Bi
m], then a coded block b from
1,Bi
2,...,Bi
m] in the
1,bi
2,...,bi
l], when serving another peer
1,cp
2,...,cp
l] in the Galois field GF(28), and then
j=1cp
j· bi
j.
1,xi
2,...,xi
m],

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encoding, and transmission. First, the peer will compare its
buffer with that of the downstream peer p to find out which
segments that it has are needed by p. As a result, a difference
set of segments is formed. It then randomly chooses a segment
in the difference set and encodes all the blocks from that
segment. Finally, it transmits the encoded block to p.
To model peer dynamics, we fix the online peer number
while allowing peers to join and leave the network frequently.
Specifically, we assume that there are always N users and Ns
servers simultaneously online, even though user departures and
joins occur constantly. Servers are always online. Each user
has a random lifetime L. A user will leave the network when
its lifetime has expired. In the meantime, a new user with a
random lifetime L as well will join the network to replace the
departed user. No peer will return after it leaves. A similar
model for churn can be found in [15]. As will be shown, by
modeling peer dynamics this way, the total number of blocks
in the network will remain constant while the distribution of
blocks from each segment could be changing in time. This
enables us to focus on studying the block variation. In fact, the
assumption on a fixed number of online peers can be relaxed
to accommodate slow changes of N and Nsin time. As long
as such changes in N(t) and Ns(t) take place at a much
slower rate than the average upload rate of a peer, the analysis
will hold the same except that N and Ns are considered as
functions of t in the results. At a given observation time to,
we also define a peer’s age A as the time till tosince it joined.
In our analysis, the peer age will be an important parameter
(note its difference from the lifetime L).
i
3
2
1
0
ri
r1
r0= G
n0
r2
r3
n3
n2
n1
a b c d e f g h i j k l ...Segments:
(Block number
of each
segment)
()
(a)
i
3
2
1
0
p2
p3
p1
p0
s3
s2
s1
s0= 1 si
a b c d e f g h i j k l ...Segments:
(Block number
of each
segment)
()
(b)
Fig. 1.An illustration for the notations pi(t), si(t), ni(t) and ri(t).
Our main goal is to derive the steady-state distribution
of the number of blocks in each segment when N and M
approaches infinity. If we denote by a random variable I the
number of blocks each segment has in users with mean µI
and variance σ2
for I. The derivation of this block distribution will allow us to
evaluate important metrics such as block variation, download
time, content loss rate and content lifetime. To characterize
the distribution of I, we therefore introduce a system of
important notations that represent the system’s state. Since
servers always have complete copies of the content, we are
only interested in knowing the block statistics in users:
⊲ ni(t): the number of segments which have i blocks in all
the users.?∞
⊲ pi(t): the fraction of segments which have i blocks in all
I, we are interested in deriving the distribution
i=0ni(t) = G for any t.
the users. pi(t) = ni(t)/G.
⊲ ri(t): the number of segments which have at least i blocks
in all the users. ni(t) = ri(t) − ri+1(t).
⊲ si(t): the fraction of segments which have at least i
blocks in all the users. We have si(t) = ri(t)/G and
pi(t) = si(t) − si+1(t).
These notations, illustrated in Fig. 1, will be the main
variables of interest in our analysis with differential equations.
Another helpful way to view Fig. 1 is that the vertical axis i
represents the number of blocks each segment has in all the
users, and the horizontal axis represents segments arranged
in the decreasing order of i. It is not hard to see that as the
segment number G → ∞, pi(t) = ni(t)/G will approach the
PMF of the variable I. These notations cover both non-coding
and coding cases as the segment size m varies. For simplicity,
we may omit t in the following context. From Fig. 1, we
also get the following simple facts: 1) the total number of
blocks in users at time t is Y (t) =?∞
average number of blocks each segment has in users at time t
is?∞
line in Fig. 1(b).
The state of the system could be represented by the vector
S(t) = (s0(t),s1(t),...) or P(t) = (p0(t),p1(t),...). If we
use S(t) to denote the system’s states for instance, at any given
time t, the future state of the system is solely decided by S(t),
in that both block increase and block loss of each segment
depend merely on the current values of s0(t),s1(t),..., given
the memoryless property of the gossiping protocol and of
the exponential block upload time. Therefore, vector S(t)
represents a density dependent family of jump Markov pro-
cesses [16], with the number of segments G being the total
population size and (s0,s1,...) being densities. Apparently,
once we obtain the expressions for S(t), we can obtain all
the block statistics at any time t. However, since such Markov
processes are too complicated to handle technically, we will
introduce in the next section a system of differential equations
which asymptotically approximate the corresponding Markov
processes described above with an arbitrarily small error.
To evaluate degrees of imbalance with respect to block
availability of different segments, we define an important
parameter called block variation as:
i=1ri(t), and 2) the
i=1si(t) = Y (t)/G, which equals to the area under the
γ2
I=var(I)
E2{I}=
?∞
i=0i2pi− (?∞
(?∞
i=0i · pi)2
i=0i · pi)2
,
(1)
which is essentially the variance of the number of blocks each
segment has in users divided by the square of its mean. var(I)
alone does not serve as a good indicator of block variation,
in that its effectiveness in assessing block imbalance may
be biased by the value of E{I}. Thus, we adopt the above
definition of block variation, which is inspired by the typical
fairness measure in resource allocation [17]. Block variation
γ2
0 and ∞, and that it is 0 for the balanced block distribution
when each segment has the same number of blocks in the
network. As we will show through simulations, download time
is directly related to block variation, which is a good indicator
of the availability of different segments. As a major theoretical
contribution of this paper, we have derived a surprisingly
Ihas the nice property that its value always lies between

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concise yet powerful formula which sheds many insights into
network coding’s behaviour in face of churn:
Theorem: For large N, M, in steady state network, the block
variation γ2
blocks within a segment for coding:
Iis inversely proportional to m, the number of
γ2
I=
F
NsmµA,
(2)
where A is the average age of a user in a steady state network
that will be determined by Lemma 1.
As the segment size increases, the block variation is sub-
dued. An intuition behind the benefit of network coding is
that without network coding (m = 1), departing peers may
take away important blocks that have already become rare in
the network, while with pure network coding (m = M), all
the blocks are equally useful and thus there is no variation of
block availability at all. As segment size m grows, the total
number of distinct segments is reduced. Hence, intuitively, one
may observe that different segments are distributed in a more
balanced way in face of churn.
IV. CHARACTERIZING SYSTEM STATES WITH
DIFFERENTIAL EQUATIONS
In this section, we formulate a set of differential equations
to characterize the network’s states by asymptotically approx-
imating the underlying Markov processes. The correctness of
this approach was proved by Kurtz [18]. As an application, it
was then successfully applied into the solution to a Markov
queuing model in load balancing problems by Mitzenmacher
[16], [19]. With proper simplifications at several points, we
are able to use this method to model the seemingly complex
system of P2P content distribution with node churn. The
main results regarding content availability drawn from the
differential equations are presented in the next section.
First, we will show the proposed network model has an
equilibrium, in which the total number of blocks in the
network stays constant. But before showing this in Lemma
2, let us first cite a useful lemma proposed by [15] and
demonstrated again in [20].
Lemma 1. Let L denote a peer’s lifetime with mean L and
variance σ2. Given an observation time to, let random variable
A denote a peer’s age at to, with expectation E[A] = A. If
N is fixed, then as to→ ∞, the probability density function
of A is given by fA(x) = [1−FL(x)]/L. Moreover, E[A] =
A = (L
2+ σ2)/2L.
It is easy to check that if the peer lifetime L follows an
exponential distribution, the distribution of peer age A will
be exactly the same as that of L. In the following lemma, we
show that after the network has evolved for a sufficiently long
time, the total number of blocks in users will almost always
remain constant.
Lemma 2. Let Y (t) denote the total number of blocks in
users from all the segments at a given time t for t sufficiently
large, then
Y (t)
(N+Ns)→ λA in probability, i.e., for any ǫ > 0,
P{|
Y (t)
(N + Ns)−λA| > ǫ} → 0,M → ∞,N = αM,Ns= αsM.
(3)
Proof. Please refer to our technical report [21].
From Lemma 2, we know that, for a large segment number
G, the average number of blocks each segment has in users is
?∞
We now give the system of ODEs that characterize the block
distributions of different segments. Under the condition that
M → ∞, N → ∞, Ns → ∞, N/M → α, Ns/M → αs,
and m remains finite, though αsis often far less than α and
can also be 0, the block distributions are characterized by
the following system of ODEs:
⊓ ⊔
i=1si(t) = Y (t)/G = (N + Ns)λA/G = (α + αs)mλA.
A ·dsi
dt
s0= 1
= αsmλA · pi−1+
α
α + αs(i − 1)pi−1− i · pi
∀i ≥ 1
(4)
Let us explain the reasoning behind these ODEs. We con-
sider a small time interval ∆t and decide the expected change
of riin ∆t, denoted by ∆ri. We denote by I(ri) the expected
increase in ri due to the upload from users, by Is(ri) the
expected increase in ridue to the upload from servers, and by
D(ri) the expected decrease in ridue to block losses caused
by peer departures. We have ∆ri= Is(ri) + I(ri) − D(ri).
We first determine Is(ri). Since the time to upload a block
is exponentially distributed, the expected number of blocks the
servers have uploaded is Nsλ∆t. Now we wish to know how
many of these Nsλ∆t blocks contribute to the increase in ri.
From Fig. 1, we see that ri increases by one if and only if
a segment with i − 1 blocks increases a block. We choose
Ns,M,∆t such that when Ns → ∞,M → ∞,Ns/M →
αs and ∆t → 0, the total number of blocks servers have
uploaded Nsλ∆t will also approach to infinity, but is less than
the segment number G =M
achieve this). With this requirement, each segment can either
increase 1 block or not increase at all in ∆t. Since all the
segments are perfectly balanced in servers, each segment has
an equal chance of being chosen, as all the blocks are largely
needed by most peers, which are highly dynamic. Thus, we
obtain
Is(ri) = Nsλ∆t · pi−1
Note that to simplify the analysis, we have implicitly assumed
that no linear dependency will occur when a peer is updating
another with network coding. This assumption is reasonable
because according to Lemma 2.1 in [22], a random linear
combination of all the blocks from the same segment at a peer
p is useful to another randomly chosen peer in the network
with probability at least 1 − 1/q if network coding is done
in Fq. And this argument is true regardless of whether peer p
is a server or a user. In this paper, the field size q has been
assumed to be 28.
We then consider the value of I(ri). Similarly, the total
block increase of all the segments due to users’ upload is
Nλ∆t. However, the increase in riis no longer Nλ∆t·pi−1,
in that each segment does not enjoy an equal chance of being
m(e.g., letting ∆t = Θ(
1
√Ns) can
(5)

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chosen by the users. On the contrary, however, the more blocks
a segment has in all the users, the more frequently it will be
encoded and uploaded. Therefore,
I(ri) = Nλ∆t·(i − 1)ni−1
Y
→
α
α + αs
∆t(i − 1)ni−1
A
, (6)
where Y is the total number of blocks in users given by
Lemma 2. In fact, the experimental results in Sec. VI have
substantiated this observation.
Finally, we determine D(ri). First, we know that the total
loss of blocks in ∆t is (Ns+ N)λ∆t, because the total
number of blocks remains unchanged by Lemma 2. Similarly,
ri decreases by one if and only if a segment with i blocks
loses a block. And the more blocks a segment has in all the
users, the more blocks it loses due to peer departures. Hence,
we have
D(ri) = (Ns+ N)λ∆t ·i · ni
Y
→∆ti · ni
A
(7)
Substituting Is(ri),I(ri),D(ri) into ∆ri= Is(ri) + I(ri) −
D(ri), we get
∆ri= Nsλ∆t·pi−1+
Dividing by G·∆t on both sides, we get the system of linear
differential equations (4).
We have omitted the rigorous proof of the concentration of
(4) around their underlying Markov processes, since such a
proof deviates from the thesis of this paper and can be done
in a standardized way [18], [16]. According to Mitzenmacher
[16], [19], one necessary condition for these differential equa-
tions to hold in theory is that the average number of blocks
each segment has in users?∞
times, which implies cases with severe churn. This condition
is satisfied in our model, since?∞
in probability, which we require to remain finite.
α
α + αs
∆t(i − 1)ni−1
A
−i · ni
A
·∆t (8)
i=1si(t) remains finite at all
i=1si(t) → (α + αs)mλA
V. QUANTIFYING THE RESILIENCE OF NETWORK CODING
In this section, we operate on the system of ODEs (4) to
analyze the resilience of network coding in both the steady
state when servers are online, and the transient stage after
servers’ departure.
A. Steady State Block Variation and Block Distribution
Assuming the presence of servers, let us now determine
the block variation and block distribution in steady state
networks. Our main findings are that the steady-state block
variation is inversely proportional to the segment size for
network coding, and that the steady-state block distribution
follows a form of negative binomial distribution, which can
be well approximated by a Gaussian distribution. The block
variation is a good indicator of the download time, which we
will show in the simulations. And the block distribution in
steady state not only offers an intuitive concept with respect
to the imbalance of block availability, but also serves as a
starting point for analyzing content availability in the absence
of servers.
Let us denote the steady-state solutions to (4) by
p0,p1,...,pG. By settingdsi
dt= 0 in the differential equations
(4), we get pi= [B +β(i−1)]·pi−1/i, where B = αsmλA,
β =
by,
i?
j=1
Under some relatively mild conditions on B and β, we could
derive piin a simple closed form in the following theorem.
Theorem 1. (Steady-State Block Distribution) Let B =
αsmλA and β =
and
the fraction of segments with a total number of i blocks in
users is given by,
α
α+αs. Hence, the steady-state block distribution is given
pi= p0·
(β +B − β
j
),
for i ≥ 1.
(9)
α
α+αs. If β is a positive rational number,
β∈ Z++(Z++= {1,2,...}), then in the steady state,
B
pi=
?i +B
β− 1
i
?
βi(1 − β)
B
β,i = 0,1,2,....
(10)
In particular, the fraction of empty segments (segments which
have no block among users) is given by p0= (1 − β)
Remarks. In practice, the requirement that β is a rational
number is naturally satisfied, since both the server number
and the user number in the network are integers. Moreover, B
is usually much larger than β and thus substituting B/β with
the integer nearest to it will hardly affect the outcome of pi.
Therefore, the value of pifor nearly all valid β and B can
be approximated by using the nearest β and B satisfying the
conditions set in Theorem 1. Recall that we use I to denote
the total number of blocks a segment has in users. Since the
total number of segments G is large, the PMF of I will be
asymptotically approximated by the values of pi, i.e., P{I =
i} = pi.
Proof. Let C = B/β−1. We first prove pi= βi?i+C
i ≥ 1, and then determine p0from the fact that?∞
Let β =
integer, we have
B
β.
⊓ ⊔
i
i=0pi= 1.
?·p0, for
l
n< 1, where l,n ∈ N. Because C is a non-negative
pi= p0
i?
j=1
(l
n+B − β
j
)
By straightforward induction, we can finally get,
pi= p0(l
n)i·(i + C)!
i!C!
= βi
?i + C
i
?
· p0
We now determine p0. Let ai = pi/p0= βi
0,1,2,.... Then we have 1/p0=?∞
need to do is to determine?∞
?i + C
i
i
we can get,
?i+C
i
?, i =
i=0ai. Hence, what we
i=0ai. Because
?
i − 1
?
=
?C + i − 1
+
?C + i − 1
?
,i ≥ 1
∞
?
i=0
If we view ai as a function of C and let f(C)
?∞
iteration for f(C),
?0 = (β − 1)f(C) + f(C − 1) + β,
f(0) =?∞
ai=
∞
?
i=1
βi
?(C − 1) + i
i
?
+β
∞
?
i−1=0
βi−1
?C + i − 1
i − 1
?
+a0
=
i=1ai(C) =
?∞
i=1βi?i+C
i
?, we can get the following
for C ≥ 1
i=1βi=
β
1−β