Stochastic Fluid Theory for
P2P Streaming Systems
Rakesh Kumar
Department of Electrical and
Computer Engineering,
Polytechnic University,
Brooklyn, NY, USA 11201
Email: rkumar04@utopia.poly.edu
Yong Liu
Department of Electrical and
Computer Engineering,
Polytechnic University,
Brooklyn, NY, USA 11201
Email: yongliu@poly.edu
Keith Ross
Department of Computer and
Information Science,
Polytechnic University,
Brooklyn, NY, USA 11201
Email: ross@poly.edu
Abstract—We develop a simple stochastic fluid model that seeks
to expose the fundamental characteristics and limitations of P2P
streaming systems. This model accounts for many of the essential
features of a P2P streaming system, including the peers’ real-
time demand for content, peer churn (peers joining and leaving),
peers with heterogeneous upload capacity, limited infrastructure
capacity, and peer buffering and playback delay. The model is
tractable, providing closed-form expressions which can be used
to shed insight on the fundamental behavior of P2P streaming
systems.
The model shows that performance is largely determined by
a critical value. When the system is of moderate-to-large size,
if a certain ratio of traffic loads exceeds the critical value, the
system performs well; otherwise, the system performs poorly.
Furthermore, large systems have better performance than small
systems since they are more resilient to bandwidth fluctuations
caused by peer churn. Finally, buffering can dramatically im-
prove performance in the critical region, for both small and large
systems. In particular, buffering can bring more improvement
than can additional infrastructure bandwidth.
I. INTRODUCTION
With the widespread adoption of broadband residential
access, live video streaming may be the next disruptive IP
communication technology. As an indication of the potential
of live video streaming, recently a commercial P2P streaming
system broadcasted the 2006 Chinese New Year’s celebration
to over 200,000 users over the Internet at bit rate in the
400-800 kbps range [9], generating an aggregate bit rate in
the vicinity of 100 gigabits/sec. In the future, we expect to
see thousands of live video streaming channels available on
the Internet, each with a bit rate of 500 kbps or more, each
supporting tens of users to hundreds of thousands of users.
There are several classes of delivery architecture for live
video streaming, including native IP multicast [6], application-
level infrastructure overlays such as those provided by CDN
companies [1], [7], and P2P architectures. Requiring minimal
infrastructure, the P2P architectures offer the possibility of
rapid deployment at lowest cost. P2P streaming architectures
roughly fall into two categories: (i) multicast trees such as in
end-system multicast [3]; (ii) and pull-driven P2P streaming
such as CoolStreaming [18], PPLive [13] and PPStream [14].
Bearing strong similarities to BitTorrent [5], pull-driven P2P
streaming architectures have the following characteristics:
• A video (live or stored) is divided into media chunks of
about one second in duration, and the chunks are made
available at an origin server.
• A peer, interested in viewing the video stream, obtains
from the system a list of peers currently watching the
video. The peer then establishes partner relationships
(TCP connections) with the peers on the list.
• The peer requests media chunks from its partners and
(possibly) from the server. But because the chunks have
playback deadlines, a peer only requests chunks that can
likely be received before their playback deadlines.
• Once a peer has obtained a chunk, it makes the chunk
available for downloading by other peers.
An important characteristic of pull-driven P2P streaming is the
lack of an (application-level) multicast tree - a characteristic
particularly desirable for the highly dynamic, high-churn P2P
environment [18]. Although pull-driven P2P streaming has
similarities with BitTorrent, BitTorrent in itself is not a feasible
delivery architecture, since it does not account for the real-time
needs of streaming.
Several pull-driven P2P streaming systems have been suc-
cessfully deployed to date, accommodating thousands of si-
multaneous users. Most of the these deployments have origi-
nated from China (including Hong Kong). The pioneer in the
field, CoolStreaming, reported more than 4, 000 simultaneous
users in 2003. More recently, a number of second-generation
pull-driven P2P streaming systems have reported phenomenal
success on their Web sites, advertising tens of thousands of
simultaneous users who watch channels at rates between 300
kbps to 1 Mbps. These systems include PPLive [13], PPStream
[14], VVSky [17], TVAnts [16] and FeiDian [8].
In P2P streaming systems, participating nodes are very
heterogeneous, particularly in terms of the amount of upload
bandwidth they contribute [9]. Today there are roughly two
classes of peers participating in P2P streaming systems: broad-
band residential peers with DSL and cable access; and institu-
tional peers with high-bandwidth Ethernet access. In addition
to being heterogeneous, nodes churn, with peers randomly
joining the system, watching the video for a random period
of time, and then leaving the system. As the peers churn,
both the system’s demand for video as well as the system’s
overall ability to supply video changes. Another important
characteristic of P2P live streaming systems is that they can
allow for small buffering delays, which potentially mitigate
against the short-term bandwidth variations due to peer churn.
Broadly speaking, a P2P streaming system performs well if
all participating peers can continuously playback the video
(without freezing or skipping) with a small playback delay.
In this paper we develop a simple stochastic fluid model
that seeks to expose of the fundamental characteristics and
limitations of P2P streaming systems. This model accounts
for many of the essential features of a P2P streaming system,
including the peers’ real-time demand, peer churn (peers join-
ing and leaving), peers with heterogeneous upload capacity,
limited server upload capacity, and buffering and playback
delay. Additionally, the model is tractable, providing closed-
form expressions which can be used to shed insight on the
fundamental behavior of P2P streaming systems. We use
the stochastic fluid model to seek answers to the following
questions.
• What are the key parameters that determine the perfor-
mance of the P2P streaming system? Is there a threshold
effect for which the performance switches from poor to
excellent as the threshold is crossed?
• It has been observed that large P2P streaming systems
generally perform better than small systems [9]. Why do
large systems perform better?
• What happens to performance as the system scales? In
particular, for a dynamic system with churn, what is the
asymptotic performance of the system as the average
number of participating peers become very large?
• Can buffering and playback lag significantly improve
performance? If so, by how much and in what circum-
stances?
• Can we quantify the benefit of additional infrastructure
resources? Will increasing the server upload rate signifi-
cantly improve performance?
• Finally, how can admission control be applied to provide
adequate service to all peers while minimizing the num-
ber of rejected peers?
This paper is organized as follows. In Section 2 we intro-
duce the basic model for P2P streaming with peer churn. In
Section 3, we take a brief interlude and derive necessary and
sufficient conditions for a churnless system. These conditions
are not only central to our stochastic model with peer churn
but are also of independent interest. In Section 4 we return to
systems with churn. We first determine an explicit expression
for the probability of degraded service. We then employ an
asymptotic model to study large P2P streaming systems. In
Section 5 we explore the potential for improvement with
playback buffering and lag at the peers. In Section 6 we use
the results from Section 4 to develop an effective admission
control scheme for P2P streaming systems. We summarize the
contributions of this paper and conclude in Section 7.
A. Related Work
To our knowledge, this is the first paper that presents
an analytical model for P2P streaming systems (fluid or
otherwise). Here, we briefly describe other papers that propose
fluid models for P2P download systems. Qui and Srikant
[15] developed and solved a fluid model for BitTorrent-like
systems. The model accounts for churn, and views the number
of seeds and leechers as fluid quantities. They develop simple
differential equations for the fluids and solve the equations in
steady state. Clevenot et al [4] develop a multiclass fluid model
for BitTorrent systems. The multiclass fluid model leads to a
non-linear system of differential equations with special struc-
ture. They prove the system of differential equations admits a
unique stable equilibrium, which is computed in closed-form.
The fluid models in [15] and [4] are not applicable to streaming
systems with heterogeneous upload rates, since there is no
notion of leechers transitioning to seeds.
There is also recent work in modeling the time it takes to
distribute a file from seeds to leechers in churnless download
systems. Mundinger et al have studied this problem for hetero-
geneous peers with infinite download capacity, both for chunk-
based and fluid-based systems [12]. Kumar and Ross derived
an explicit expression for the minimum download time in a
general heterogenous fluid system with finite download rates.
They also extended this result to multi-class systems with first
and second-class leechers. Biersack et al used a chunk-based
model to derive expressions for the distribution time for a
several practical overlay topologies [2].
II. MODELING P2P STREAMING
In this section we provide our basic model and notation
of P2P streaming. The video originates from a server node;
denote by u
s
(in bps) for the upload rate of the server. Let
r denote the rate (in bps) of the video. The video is to be
streamed to all participating peers.
We classify each peer as either a super peer or an ordinary
peer. Super peers provide high-speed access rates in excess of
a few Mbps; ordinary peers have residential broadband access,
typically with upload rates of 500 kbps or less. In our model,
all super peers have the same upload capacity u
1
; and all
ordinary peers have the same upload capacity u
2
with u
2
< u
1
.
Unless explicitly stated, we assume that u
2
< r < u
1
. In other
words, a super peer can upload at a rate higher than the video
rate, and an ordinary peer’s uploading capacity is smaller than
the video rate. We often refer to super peers and ordinary
peers as class-1 and class-2 peers, respectively. Although we
are assuming only two classes of the peers, the theory and
results presented here can easily be extended to any number of
classes, with each class having its own upload rate. However,
we shall see that a two-class model suffices to expose many
of the key issues underlying P2P streaming.
Peers join and leave the P2P streaming system at random
times. As in existing P2P streaming systems, whenever a peer
joins the system and receives chunks of video, it is obligated
to redistribute the chunks it receives [18] [13] [14]. Denote by
λ
i
for the rate at which class-i peers join the P2P streaming
system. Denote by 1/µ
i
for the average amount of time a class-
i peer views the video (and hence sojourns in the system). We
make no assumptions on the distribution of the peer sojourn
(viewing) times. Peers from the two classes join the system as
two independent Poison processes. Let P
i
(t) be the number of
class-i peers in the system at time t. Clearly, P
1
(t) and P
2
(t)
are two independent M/G/ ∞ processes [10].
Having described the model for peer churn, we now turn
to streaming. We adopt a fluid flow model and focus on the
instantaneous rate at which peers receive and transfer bits.
Initially we assume a bufferless system, that is, bits cannot
be buffered before playback or before copying to other peers.
(In Section 5, we extend the model to allow for buffers and
playback lag.) In this bufferless model, a peer can playback the
video whenever it receives fresh content bits at rate r. When
all participating peers receive the video at rate r, we say that
the system provides universal streaming. When the system is
not providing universal streaming, we say that system operates
in degraded service mode.
At any given instant of time, whether universal streaming
can be accomplished or not depends of number of super peers
and ordinary peers in the system at time t, that is, it is a
function of P
1
(t) and P
2
(t). The more super peers in the
system, the greater the average upload capacity per peer and
the easier it is to accomplish universal streaming. Denote by
Φ(P
1
(t), P
2
(t)) for the maximal rate at which the system can
deliver fresh content to each of the peers when the system
is in state (P
1
(t), P
2
(t). The function Φ(·, ·) depends on the
efficiency or the distribution scheme in the pull-driven P2P
streaming protocol. Universal streaming occurs at time t if
and only if Φ(P
1
(t), P
2
(t)) ≥ r. To complete the stochastic
fluid model, we need to specify Φ(·, ·), which we refer to as
the fluid function. In the next section we provide an explicit
expression for Φ(·, ·) for an optimized system.
III. UNIVERSAL STREAMING FOR CHURNLESS SYSTEMS
In this section we seek to derive the maximum streaming
rate r for which universal streaming is possible for a churnless
system, that is, for a system in with a fixed set of peers. We do
this for a system that is more general than that described in the
previous section - namely, we consider general heterogeneous
systems, with each peer having its own upload rate.
The results derived in this section will play a central role
in the analysis of the stochastic system with churn. However,
they are also of independent interest, as they provide simple
and explicit expressions for the maximum rate of a churnless
P2P streaming system.
Denote by n for the number of peers in the system and let
u
i
denote of the upload capacity of peer i for i = 1, . . . , n.
Viewing bits as fluid, bits arrive to the server at rate r. As the
bits arrive to the server, they can be copied to one or more
peers. As bits arrive to a peer, they can also be copied to one
or more of the remaining peers. The aggregate bit rate out of
the server cannot exceed u
s
; the aggregate bit rate out of a
peer cannot exceed u
i
, i = 1, . . . , n. The system can perform
universal streaming if it is possible to copy and route the bits
so that all peers receive fresh bits at rate r. We define the
maximum achievable rate to be the maximum value of r such
that the system can perform universal streaming.
Theorem 1: The maximum achievable streaming rate, r
max
,
is given by
r
max
= min{u
s
,
u
s
+
P
n
1=1
u
i
n
}.
Proof: Denote
u(P) =
n
X
1=1
u
i
Clearly, the maximum streaming rate cannot exceed the aggre-
gate upload rate of the server; thus, r
max
≤ u
s
. Furthermore,
the maximum aggregate rate that bits can flow out the server
and out of the n peers is bounded by u
s
+u(P). This maximum
aggregate rate, if it could be achieved, needs to be distributed
to the n peers. Thus the maximum streaming rate to an
individual peer is also bounded by u
s
+ u(P)/n . Combining
these two bounds gives
r
max
≤ min{u
s
,
u
s
+ u(P)
n
}. (1)
It remains to show that if (i) u
s
≤ (u
s
+ u(P))/n, then the
streaming rate r = u
s
can be supported; and if (i i) u
s
>
(u
s
+ u(P))/n, then the streaming rate r = (u
s
+ u(P))/n
can be supported.
Suppose u
s
≤ (u
s
+ u(P))/n. Consider a video stream of
rate r = u
s
. Divide this video stream into n substreams, with
the ith substream having rate
s
i
=
u
i
u(P)
u
s
for all i ∈ P
Note that the aggregate rate of the n substreams is equal to
the rate of the stream, that is,
n
X
i=1
s
i
= u
s
= r.
We have the server copy the ith substream to the ith peer.
Furthermore, because (n − 1)s
i
≤ u
i
, we can have the ith
peer copy its substream to each of the other n −1 peers. Thus
each peer receives a substream directly from the server and
also receives n −1 additional substreams from the other n −1
peers. The total rate at which peer i receives is
r
i
= s
i
+
X
k6=i
s
k
= u
s
.
Hence the rate r = u
s
can be supported.
Now suppose u
s
> (u
s
+u(P))/n. Consider a video stream
of rate r = (u
s
+u(P))/n. Divide this video stream into n+1
substreams, with the ith substream, i = 1, . . . , n, having rate
s
i
= u
i
/(n − 1) and the (n + 1)st substream having rate
s
n+1
= (u
s
−
u(P)
n − 1
)/n
Clearly s
i
≥ 0 for all i = 1, . . . , n + 1. Now have the server
copy two substreams to each peer i: the ith substream and the
(n + 1)st substream. The server can do this because
n
X
i=1
(s
i
+ s
n+1
) = u
s
Furthermore, have each peer i stream a copy of the ith
substream to each of the n − 1 other peers. Each peer i can
do this because (n −1)s
i
= u
i
for i = 1, . . . , n. The total rate
at which peer i receives is
r
i
= s
i
+ s
n+1
+
X
k6=i
s
k
= (u
s
+ u(P))/n.
Hence the rate r = (u
s
+ u(P))/n can be supported.
We remark that this result can be extended to include finite
download rates at each of the n peers. In particular, suppose
peer i has download rate d
i
. Let d
min
= min{d
i
: i =
1, . . . , n}. Then
r
max
= min{u
s
,
u
s
+ u(P)
n
, d
min
} (2)
We omit the proof of (2) for brevity; see [11] for a related
result concerning the minimum time to distribute a file to all
peers in a P2P system.
A. Two-Class Model
Let us now return to the original model in which every peer
is either a super peer or an ordinary peer. Denote by n
1
and
n
2
the number of super peers and ordinary peers in the P2P
streaming system. The following result is a consequence of
Theorem 1 and will be used repeatedly in this paper:
Corollary 1: For any rate r such that u
2
< r < u
1
, universal
streaming is achievable by some fluid distribution scheme if
and only if
r ≤ φ(n
1
, n
2
) (3)
where
φ(n
1
, n
2
) = min{u
s
,
u
s
+ n
1
u
1
+ n
2
u
2
n
1
+ n
2
}.
IV. P2P STREAMING WITH PEER CHURN
We now return to original model as given in Section 2 with
peers joining and departing at random times. Denote ρ
i
=
λ
i
/µ
i
. Recall that with P
i
(t) denoting the number of active
type-i peers at time t, the two stochastic processes (P
1
(t), t ≥
0) and (P
2
(t), t ≥ 0) are independent M/G/∞ processes with
arrival rates λ
1
and λ
2
and departure rates µ
1
and µ
2
. Recall
that to complete the stochastic fluid model, we need to specify
a fluid function Φ(·, ·). Henceforth, we use Φ(·, ·) = φ(·, ·),
where φ(·, ·) is given in Corollary 1. Thus, we assume an
optimized P2P distribution scheme. We remark, however, that
the theory developed in this paper can be easily extended to
any relevant fluid function Φ(·, ·) ≤ φ(·, ·). For example, we
could use Φ(·, ·) = .8 · φ(·, ·), modeling a P2P distribution
scheme that is only 80% efficient.
A natural question is, in steady-state, for what fraction of
time do we have universal streaming? We refer to this fraction
of time as the universal streaming probability. Let P
i
be the
random variable denoting the number of active type-i peers in
steady state. It is well-known that P
i
has a Poisson distribution
with mean E[P
i
] = ρ
i
. From Corollary 1, we have
P (universal streaming) = P (P
1
≥ cP
2
− u
′
s
) (4)
where
c =
r − u
2
u
1
− r
and u
′
s
=
u
s
u
1
− r
Since P
1
and P
2
are independent Poison random variables, we
can explicitly calculate the universal streaming probability as
follows. Let M = ⌊
u
′
s
c
⌋. We have
P (P
1
≥ cP
2
− u
′
s
) =
∞
X
l=0
P (P
1
≥ cl −u
′
s
|P
2
= l)P (P
2
= l)
= P (P
2
≤ M ) +
∞
X
l=M+1
P (P
1
≥ cl −u
′
s
)P (P
2
= l)
= P (P
2
≤ M ) +
∞
X
l=M+1
P (P
1
≥ ⌈cl − u
′
s
⌉)P (P
2
= l)
= F
2
(M)+
∞
X
l=M+1
(1−F
1
(⌈cl−u
′
s
⌉)+f
1
(⌈cl−u
′
s
⌉))f
2
(l) (5)
where
f
i
(n) =
e
−ρ
i
ρ
n
i
n!
and F
i
(n) =
n
X
l=0
f
i
(n)
We will return to this result when we present numerical results
at the end of this section.
A. Large System Analysis
We now scale the system by letting ρ
1
→ ∞ and ρ
2
→ ∞.
It is natural to consider scaling regimes in which ρ
1
/ρ
2
= K
for some constant K. We consider here a more generalized
regime in which
ρ
1
= Kρ
2
+ β
√
ρ
2
(6)
for some K > 0 and some β (positive or negative). We will
see that this more generalized scaling will enable us to glean
additional insight into the fundamental characteristics of P2P
streaming systems.
Theorem 2: In an asymptotic regime with ρ
1
= Kρ
2
+β
√
ρ
2
,
the asymptotic probability of universal streaming is given by
lim
ρ
2
→∞
P (P
1
≥ cP
2
− u
′
s
) =
1 K > c
F (
−β
√
c+c
2
) K = c
0 K < c
(7)
where 1 − F (·) is the distribution function of the standard
normal random variable.
Proof: Define the normalized random variables
X
1
=
P
1
− ρ
1
√
ρ
1
, X
2
=
P
2
− ρ
2
√
ρ
2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.2 0.28 0.36 0.44 0.52 0.6 0.68 0.76
ȡ
1
/ȡ
2
Degraded-Service Probability
u_s = 7
u_s = 14
approximation
(a) Small System λ
2
= 100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55
ȡ
1
/ȡ
2
Degraded-Service Probability
u_s = 7
u_s = 70
u_s = 140
approximation
(b) Large System λ
2
= 10000
Fig. 1. Degraded-Service probability for small and large systems
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0 10 20 30 40 50 60 70 80 90 100
Scaling Factor
Degraded-Service Probability
u_s = 7
u_s = 14
approximation
Fig. 2. Degraded-Service probability with ρ
1
/ρ
2
= 0.54 fixed. The system
size is scaled in multiples of 10.
Note that
P (P
1
≥ cP
2
−u
′
s
) = P (
√
ρ
1
X
1
+ ρ
1
≥ c(
√
ρ
2
X
2
+ ρ
2
) −u
′
s
)
= P (
s
K +
β
√
ρ
2
X
1
+ (K − c)
√
ρ
2
+ β ≥ cX
2
−
u
′
s
√
ρ
2
)
Now, as ρ
2
→ ∞, clearly this probability goes to 1 if K > c
and goes to 0 if K < c. Now consider the case K = c. As
ρ
2
→ ∞, this probability goes to
P (
√
KZ
1
− cZ
2
> −β)
where Z
1
and Z
2
are two independent standard normal random
variables. But since a linear combination of independent
normal random variables is also a normal random variable,
we have
P (
√
KZ
1
− cZ
2
> −β) = F (
−β
√
c + c
2
)
Theorem 2 indicates that P2P streaming systems exhibit a
critical threshold. For a large system, if ρ
1
/ρ
2
+ǫ < c, then the
performs poorly, rarely providing universal streaming. On the
other hand, if ρ
1
/ρ
2
> c + ǫ, then the system almost always
provides universal streaming. This critical threshold c plays an
important role in the design and operation of P2P streaming
systems. We say that when ρ
1
/ρ
2
≈ c, we say that system
operates in the critical region.
Theorem 2 also leads to useful and simple approximation
for medium and large systems when operating in the critical
region Given ρ
1
, ρ
2
and c = (r −u
2
)/(u
1
−r), we set K = c
and solve for β in (6):
β =
ρ
1
− cρ
2
√
ρ
2
.
We then plug this expression for β into F (−β/
√
c + c
2
) from
Theorem 2 and obtain the explicit approximation:
P(universal streaming) ≈ F (
ρ
1
− cρ
2
/
√
ρ
2
√
c + c
2
). (8)
B. Numerical Results and Insights
We now explore how the equation (5) and the approximation
(8) can be used to study the performance of P2P streaming
systems. We do this for two systems: a “small system” with a
number of concurrent peers in the vicinity of 75; and a large
system with the number of concurrent peers in the vicinity of
7,500.
For both the small and large systems, we use the rates r = 3,
u
2
= 1 and u
2
= 7. These rates could be, for example, in units
of 100 kbps. The chosen rates reflect current streaming rates,
residential upload rates, and enterprise/university rates. For
this choice of rates, the video rate is three times the upload rate
of the ordinary peers; and the upload rate of the superpowers is
7 times that of the ordinary peers. (For example, most PPLive
channels are currently in the vicinity of 400 kbps, which is
about 2-4 times the upload rate of many residential broadband
peers. University access rates vary, depending on traffic and
university-to-ISP bandwidths; for most cases, we expect u
2
/u
1
to be in the 5 to 20 range.) These values give c = 0.5 for the
critical factor.
In these numerical examples, we use different server rates
u
s
, all multiples of the u
1
. By using different multiples for the
server rate, we explore how additional infrastructure resources
can potentially improve performance.
