A bandwidth adaptive method for estimating end-to-end available bandwidth

Abstract

The probe gap model (PGM) was proposed as a lightweight and fast available bandwidth estimation method. Compared to the probe rate model (PRM) which requires multiple iterations with different probing rates, PGM uses a single probing rate and infers the available bandwidth from the relationship between the input and output rates of probing packet pairs. In this paper, we proved that PGM is accurate for multi-hop path under the case of path persistent cross traffic, and even for the one-hop persistent case, we show that PGM can be accurate as long as the input probing rate is set properly. According to our analysis, a bandwidth adaptive method is introduced by adjusting the input probing rate. The measurement results show that the improved Spruce algorithm is more accurate than the original one and can estimate the end-to-end available bandwidth accurately.

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A Bandwidth Adaptive Method for Estimating
End-to-End Available Bandwidth
Dawei XU1
1Dept. of Computer Science and Technology
Xi’an Jiaotong University
Xi’an, 710049, China
Email: xudawei@263.net
Depei QIAN1,2
2Sino-German Joint Software Institute
Beihang University
Beijing, 100191, China
Email: depeiq@263.net
Abstract—The Probe Gap Model (PGM) was proposed as
a lightweight and fast available bandwidth estimation method.
Compared to the Probe Rate Model (PRM) which requires
multiple iterations with different probing rates, PGM uses a
single probing rate and infers the available bandwidth from the
relationship between the input and output rates of probing packet
pairs. In this paper, we proved that PGM is accurate for multi-
hop path under the case of path persistent cross traffic, and
even for the one-hop persistent case, we show that PGM can
be accurate as long as the input probing rate is set properly.
According to our analysis, a bandwidth adaptive method is
introduced by adjusting the input probing rate. The measurement
results show that the improved Spruce algorithm is more accurate
than the original one and can estimate the end-to-end available
bandwidth accurately.
Keywords—Network capacity; available bandwidth; packet
pair dispersion; Probe Gap Model; Probe Rate Model; cross
traffic
I. INTRODUCTION
The techniques for estimating end-to-end available band-
width have received significant research interest recently [1]–
[7]. The available bandwidth of a link is defined as the unused
or spare capacity of the link during a certain time period [8].
Consider a store-and-forward network link i with capacity Ci.
Let ui(t) be the instantaneous utilization of the link at time
t, i.e., 0 if the link is idle or 1 if the link transmits a packet
at t. The average utilization ui(t,t + τ) of link i during time
interval (t,t + τ) is,
ui(t,t + τ) =1
τ
?t+τ
t
ui(x)dx
(1)
The available bandwidth Ai(t,t + τ) of link i during the
time interval (t,t + τ) is defined as the average unutilized
capacity in that duration,
Ai(t,t + τ) = Ci[1 − ui(t,t + τ)]
For a network path with n hops, the end-to-end available
bandwidth A(t,t + τ) is defined as the minimum of all the
link available bandwidths in the same time interval,
(2)
A(t,t + τ) = min
i=1...n{Ai(t,t + τ)} = C[1 − u(t,t + τ)] (3)
This work is supported by National Science Foundation of China under
Grant No. 60673180, and the National High-Tech Research and Development
(863 Project) under Grant No. 2006AA01A118.
where C and u(t,t+τ) are the capacity and the utilization of
the bottleneck link respectively.
The link with the minimum available bandwidth along a
network path is called tight link of the path, whereas the link
with the minimum capacity is called narrow link. The end-
to-end available bandwidth of a network path is equal to the
available bandwidth of the tight link. The bottleneck link is
referred to the link which is both the tight link and the narrow
link.
Many available bandwidth estimation tools have been in-
troduced such as Pathload [1], [2], ToPP [3], IGI/PTR [4],
Delphi [5], pathChirp [6] and Spruce [7]. All the existing
estimation techniques and tools can be grouped into two
classes. The first is Probe Rate Model (PRM), also known
as Iterative Probing. In PRM, packet pairs or packet trains are
sent at different rates, while the receiver compares the input
and output rates. When the output rate is lower than the input
rate, the probing rate is higher than the available bandwidth.
PRM is based on the concept of self-induced congestion.
The second is Probe Gap Model (PGM), also known as
Direct Probing. In PGM, packet pairs or packet trains are
sent at a single rate. The probing rate needs to be set as
larger than the available bandwidth of the path, so that the
dispersion of packet pair increases at the tight link while the
receiver calculates the available bandwidth by the input and
output rates of a probing packet pair. PGM is based on the
relationship between the dispersion of the probing packets at
the input and output of the measured path.
Lao et al. studied the accuracy of PGM method, and of
Spruce in particular, in multi-hop paths in [9]. They proved
that Spruce can underestimate the available bandwidth of
multi-hop paths. But they confused PGM with Spruce. Spruce
is the special case of PGM that the input probing rate is
equal to the capacity of the tight link. In this circumstance,
Spruce can underestimate the available bandwidth in multi-
hop paths when the utilization of non-bottleneck link is high.
We found that PGM is accurate in most circumstances as long
as the input probing rate is set properly. That means that the
underestimation error occurs in Spruce does not always occur
in PGM.
The rest of this paper is organized as follows. In section 2,
we provide some relevant background. Then, in section 3, we
1-4244-2424-5/08/$20.00 ©2008 IEEEICCS 2008
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Figure 1.The Probe Gap Model (PGM)
prove PGM is accurate when cross traffic is path persistent,
and show that PGM can estimate the available bandwidth
accurately as long as we set the input probing rate properly
when cross traffic is one-hop persistent. A bandwidth adaptive
method is introduced in section 4. Simulation analysis is
presented in section 5. We conclude in section 6.
II. BACKGROUND
Both PGM and PRM approaches assume:
• FIFO queuing at all routers along the path;
• Cross traffic follows a fluid model;
• The average rates of cross traffic change slowly and are
constant in the duration of a single measurement.
A. The Probe Gap Model
In PGM, it is assumed that there is a single bottleneck link
along the path. Further, PGM assumes that the capacity of the
bottleneck is known in advance. We can use BProbe [10] to
get the capacity of the tight link. PGM exploits the information
in the time gaps between the arrivals of probing packets at the
receiver. The probing packets are sent at rate Ri= L/ΔTi,
and reach the receiver at rate Ro= L/ΔTo, as shown in Fig.
1. When Ri> A, the available bandwidth can be estimated
in (4):
A = C − Ri(C
Ro
− 1) = C(1 −ΔTo− λΔTi
ΔTi
)
(4)
where λ = Ri/C. Specially, Spruce sets input rate Ri= C,
and the available bandwidth can be estimated as follows:
ASpruce= C(1 −ΔTo− ΔTi
ΔTi
)
(5)
B. The Probe Rate Model
PRM assumes a single tight link too, but it does not need
the capacity of the tight link in advance and does not assume
that the tight link is the same with the narrow link. If the
sender sends probing packets at a rate lower than the available
bandwidth along the path, the arrival rate of probe traffic at
receiver will be the same as their rate at the sender. When the
probing rate is higher than the available bandwidth, the tight
link becomes saturated and a queue builds up. The receiver can
detect that the tight link is saturated because the time gaps of
successive probing packets increase, or equivalently, because
the probing rate at the receiver is less than the probing rate
at the sender. Thus, the receiver can estimate the available
bandwidth by searching for the turning point at which the
input and output of probing rates start matching.
Figure 2. Path persistent (a) and one-hop persistent (b) cross traffic
III. ANALYSIS OF PGM
According to the standard assumptions to the Probe Gap
Model, we assume that each link is modeled as a FIFO queue
and the cross traffic in the measured path follows the fluid
model. This means that if the cross traffic at a link has rate Rc,
then the amount of cross traffic that arrives in that link during
any interval of length ΔT is RcΔT. Further, the capacity C
of the single bottleneck link is known.
A. Cross traffic types
In the case of multi-hop paths, the routing of cross traffic
relative to the measured path is important, as shown in Fig. 2.
Path persistent traffic follows the same path with the probing
traffic. One-hop persistent traffic shares only a single queue
with the probing traffic. In the path persistent case, cross traffic
only intervenes between the packet pair at the first queue,
and the amount of traffic between the packet pair remains
constant in the rest of the path. In the one-hop persistent case,
the amount of cross traffic intervenes between the packet pair
at hop i is equal to (Ci− Ai)ΔTi−1 = uiCiΔTi−1. In the
following, we prove that PGM is accurate for multi-hop path in
the path persistent case. In the one-hop persistent case, PGM,
not Spruce, does not absolutely underestimate the available
bandwidth of multi-hop path.
B. One-hop path
First consider a single queue. The two probing packets arrive
at the link back-to-back with dispersion ΔTi= L/Ri. We set
λ = Ri/C and 0 < λ ≤ 1, then ΔTi= L/λC. The amount
of cross traffic that arrives at the queue between the packet
pair is X = (C − A)ΔTi= uL/λ, and so the dispersion of
the packet pair at the receiver is increased to:
ΔTo=L + X
C
= (λ + u)ΔTi
(6)
Then the available bandwidth can be estimated by (4):
APGM= C[1 −(λ + u)ΔTi− λΔTi
ΔTi
] = A
(7)
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So PGM can accurately estimate the available bandwidth of
a single hop path.
C. Two-hop path
The cross traffic that intervenes between the packet pair is
the traffic that arrives just after the arrival of the first probing
packet and before the arrival of the second. So, the cross traffic
between the packet pair at the first hop is X = u1C1ΔTi. Due
to the nature of the path persistent cross traffic, the arrival rate
of the latter is the same at both the first and second links, i.e.,
u1C1= u2C2= Rc.
If the bottleneck is the first link (C = C1), the dispersion
at the output of the first link is:
ΔT1=L + X
C1
= (λ + u1)ΔTi
(8)
Because C1 < C2, the dispersion stays constant at the
second link. So, ΔTo= (λ + u1)ΔTi, which is same as (6).
Thus, APGM= C1(1 − u1) = A.
If the second link is the bottleneck (C = C2), we need to
consider the following two cases. First, if (L+X)/C1< ΔTi,
which is equivalent to u1 < 1 − λC2/C1, the first link can
transmit the first probing packet and the intervening cross
traffic before the second probing packet arrives. In this case,
the dispersion of the packet pair remains same as the input
gap, i.e., ΔT1 = ΔTi. At the second link the dispersion is
increased to:
ΔTo=L + X
C2
= (λ + u2)ΔTi
(9)
Thus, APGM= C2(1 − u2) = A.
Second, if (L + X)/C1 ≥ ΔTi, which is equivalent to,
the second probing packet queues at the first link. Then the
dispersion of the packet pair at the output of the first link is
increased to:
ΔT1=L + X
C1
(10)
The dispersion at the output of the second link is further
increased to:
ΔTo=L + X
C2
C2
=λC2ΔTi+ u2C2ΔTi
= (λ + u2)ΔTi
(11)
Thus, APGM= C2(1 − u2) = A.
D. Multi-hop path
In the path with n hops case, we have proved PGM is
accurate for estimating the available bandwidth when n = 1,2.
Assuming PGM is accurate when n = k, i.e., PGM can
estimate the available bandwidth of the path with k hops,
let us prove PGM can also accurately estimate the available
bandwidth of the path with k + 1 hops.
If the bottleneck is located in the former k links, link
m is the bottleneck link, i.e., C = Cm< Ck+1(m ≤ k). The
dispersion of the probing packet pair remains constant after
the link m, so the dispersion of the packet pair at the output
of the link k + 1 is:
ΔTo= ΔTk= ΔTm= (λ + um)ΔTi
(12)
Thus, APGM= Cm(1 − um) = A.
If the link k + 1 is the bottleneck link along the path,
i.e., C = Ck+1< Cm< Ci(i ?= m,i ∈ {1,...,k}), we need
to consider the following two cases. First, if Ri< Am, then
the dispersion of the packet pair does not increase at the first
k links, i.e., ΔTk= ΔTi. At the link k + 1, the dispersion is
increased to:
ΔTk+1=L + X
Ck+1
= (λ + uk+1)ΔTi
(13)
Thus, APGM= A.
Otherwise, if Ri ≥ Am, as in the previous case, the
dispersion of the packet pair at link k is:
ΔTk= ΔTm= (λ + um)ΔTi
(14)
As we mentioned in 3.1, in the path persistent case, cross
traffic only intervenes between the packet pair at the first
queue, and the amount of cross traffic remains constant in the
rest of the path. Assuming it queues on link l, so the amount
of the intervening cross traffic is:
X = ulClΔTi= uk+1Ck+1ΔTi
(15)
So the dispersion of the packet pair at the link k + 1 is:
ΔTk+1=L + X
Ck+1
= (λ + uk+1)ΔTi
(16)
Thus, APGM= A.
To sum up, if PGM is accurate when n = k, we proved that
PGM is absolutely accurate too when n = k + 1. According
to mathematical induction, PGM is absolutely accurate to
estimate the available bandwidth of a path with n hops.
So, no matter where the bottleneck is located at which link,
and for any utilization at the non-bottleneck link, PGM can
accurately estimate the available bandwidth of a multi-hop
path with path persistent cross traffic.
E. One-hop persistent cross traffic
[9] analyzes a two-hop model for simplicity because the
underestimation error for Spruce takes place even in this case.
We analyze two-hop model also in order to compare the two
situations for PGM and Spruce.
With one-hop persistent cross traffic, the amount of cross
traffic that arrives between the packet pair at hop i is Xi=
(Ci−Ai)ΔTi−1. It takes (L+Xi)/Citime units to transmit
the first probing packet and the cross traffic. If this time units
is not smaller than the dispersion of the packet pair at link
i − 1, the dispersion will be increased. On the other hand,
if Ri−1 < Ai, link i can complete the transmission of the
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first packet and Xibefore the second packet arrives. So, the
dispersion of the packet pair is:
⎧
⎩
We distinguish two cases depending on whether the bottle-
neck is the first or the second link. In the following, we will
compare the two situations for PGM and Spruce.
The bottleneck is the first link. In this case, we compare
the output probing rates R1and R?
the two cases Ri= L/ΔTi= λC1which we discuss in this
paper, and R?
R?
ΔTi=
⎨
ΔTi−1
ΔTi−1,
?
1 +Ri−1− Ai
Ci
?
,
if Ri−1≥ Ai;
if Ri−1< Ai.
(17)
1after the first queue under
i= L/ΔT?
i= C1which [9] uses. If both R1and
1are larger than A1, then based on Eq. 17,
ΔT1= ΔTi(1 +Ri− A1
C1
) = (λ + u1)ΔTi
(18)
ΔT?
1= ΔTi(1 +R?
i− A1
C1
) = (1 + u1)ΔT?
i
(19)
So, the probing rates after the first queue are:
R1=
L
ΔT1
=
L
(λ + u1)ΔTi
(20)
R?
1=
L
ΔT?
1
=
L
(1 + u1)ΔT?
i
(21)
Because λΔTi = ΔT?
It shows that the possibility of the output probing rate at the
first link larger than the available bandwidth of second link is
decreased by setting the lower input probing rate in this case.
When R1< A2, PGM can estimate the available bandwidth
accurately.
Even if R1≥ A2, when the capacity of the bottleneck falls
in the following range:
iand ΔTi ≥ ΔT?
i, so R1 ≤ R?
1.
(1 + u1/λ)A2≤ C1< min{C2,A2/(1 − u1)}
(22)
The dispersion of the packet pair at the second queue
increases to:
ΔT2= ΔT1(1 +R1− A2
C2
)
(23)
The PGM estimation is:
APGM= A −C1
C2[λC1− (λ + u1)A2]
(24)
And the Spruce estimation is:
ASpruce= A −C1
C2[C1− (1 + u1)A2]
(25)
The bias in (24) is still smaller than the bias in (25).
The bottleneck is the second link. In this case Ri =
λC2 ≤ C2 = R?
input probing rate is larger than the available bandwidth of
i, it still decreases the possibility that the
the first link. When Ri < A1, the dispersion of the packet
pair does not increase at the first queue, and so the PGM is
accurate.
Even if Ri≥ A1, when the capacity of the bottleneck falls
in the following range:
A1/λ ≤ C2< min{C1,A1/(1 − u2)}
The dispersion of the packet pair at the first queue increases
to:
ΔT1= ΔTi(1 +Ri− A1
(26)
C1
)
(27)
The probing rate after the first queue is:
R1=
λC1C2
C1+ λC2− A1
(28)
The dispersion of the packet pair at the second queue
increases further to:
ΔT2= ΔT1(1+R1− A2
C2
) = ΔTi(1+Ri− A1
C1
)(1+R1− A2
C2
)
(29)
After some algebra, the available bandwidth is:
APGM= A −C2
C1(λA2− u2A1)
(30)
And the Spruce estimation is:
ASpruce= A −C2
C1(A2− u2A1)
(31)
Comparing the bias in (30) and (31), and noticed that
0 < λ ≤ 1, it can be shown that PGM’s bias is smaller than
the Spruce’s bias.
To summarize the analysis of the two-hop path, the original
PGM method decreases the underestimation error greatly if we
set the input probing rate properly. The reason that error occurs
is the second probing packet will encounter more queues at
the non-bottleneck link if the utilization of non-bottleneck
link and the input probing rate are both high. If we can set
the input probing rate just a little larger than the available
bandwidth, and smaller than available bandwidth of all other
non-bottleneck links, the underestimation error will be gone.
IV. BANDWIDTH ADAPTIVE METHOD
In order to let the input probing rate larger than the available
bandwidth of the tight link, Spruce set it as the capacity C of
the bottleneck link. But it tremendously increases the possibil-
ity that the second probing packet queues at the non-bottleneck
links, and the underestimation error can be significant.
According to the analysis above, if the input probing rate
is only more than the available bandwidth, and less than
the available bandwidth of all other non-bottleneck links, the
estimation is accurate. By borrowing a concept from PRM, we
introduce a bandwidth adaptive method by modifying Spruce
here.
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Given ΔTini, γ, where 0 < γ < 1, Rini= L/ΔTini< C,
we can get a sequence of ΔTi = γiΔTini, i = 0,1,...,n.
In the meantime, we can get a sequence of the input probing
rates Ri = L/γiΔTini, i = 0,1,...,n. To increase the i
from zero to n, the input probing rate increases too. The
receiver observers the dispersion of the probing packet pair at
the output link. When ΔTo= ΔTi, the input probing rate of
packet pair is lower than the available bandwidth of the path.
When ΔTo≥ ΔTi, the input probing rate is larger than the
available bandwidth, i.e., Ri≥ A. So, we can set the input
probing rate to start measurement. It both makes sure that
the input probing rate is more than the available bandwidth,
and keeps the input probing rate low which prevents the extra
queues on the non-bottleneck links.
We modified Spruce into two phases. The first phase is
pre-probing phase, which detects the proper input probing
rate by comparing the dispersions of packet pair at the
input and output of the path. After proper input rate is set,
the second phase starts measuring procedure to estimate the
available bandwidth. Because the input dispersion increases
exponentially from ΔTinito γkΔTini, it only needs to send k
more packet pairs, and it won’t affect the efficiency of Spruce.
The semantic description of the pre-probing phase is:
1) set ΔTinput= ΔTini, γ = 0.9 ( γ can be any value
as long as 0 < γ < 1);
2) send probing packet pair and get ΔToutput;
3) if ΔToutput= ΔTinput,
4)
ΔTinput= γΔTinputand jump to line 2;
5) else
6) set the input probing rate Rinput= L/ΔTinput;
7) end if;
8) start the second measuring phase to estimate the
available bandwidth at input rate Rinput.
V. SIMULATION ANALYSIS
In this section, we present simulation results of PGM
probing in two paths with two-hops each, which is same
as in [9]. The link capacities are {6, 10}Mbps and {10,
6}Mbps, respectively. The cross traffic is constant-bit-rate
(CBR) traffic with a smaller packet size of 40 bytes, similar
to the fluid model. The probing packet size is 1500 bytes,
and L/C = 2ms. In both cases, we set ΔTini= 40ms which
means the initial input probing rate begins one twentieth of
the bottleneck link capacity. This parameter can be adjusted
for other paths. We focus on the case of one-hop persistent
traffic only.
In the case of one-hop persistent traffic, we vary the
utilization of the bottleneck and of the non-bottleneck link,
and plot the real and the estimated available bandwidth
according to both Spruce and to improved bandwidth adaptive
method. Fig. 3 shows the results of the first path and Fig. 4
for the second path. In both cases, Spruce underestimates the
end-to-end available bandwidth when the available bandwidth
of the non-bottleneck link is less than a certain threshold.
This threshold is determined by the conditions derived in Sec.
Figure 3. Estimated available bandwidth for a two-hop path with C1=6Mbps
and C2=10Mbps. The available bandwidth A1 of the bottleneck link is: (a)
3Mbps, (b) 1.8Mbps and (c) 0.6Mbps. The cross traffic follows the one-hop
persistent model.
3.5 under λ = 1.
In the case of Fig. 3, the second probing packet takes
longer time in queue on the link 2 for Spruce when
A2 < C1/(1 + u1). In the case of Fig. 4, underestimation
occurs for Spruce when A1 < C2. The bandwidth adaptive
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