MR-BART: Multi-Rate Available Bandwidth Estimation in Real-Time

Journal of Network and Computer Applications 35 (2012) 731–742

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MR-BART: Multi-Rate Available Bandwidth Estimation in Real-Time Mahboobeh Sedighizad a,1, Babak Seyfe a,n, Keivan Navaie b,2 a b

Department of Electrical Engineering, Shahed University, Tehran, Iran School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK

a r t i c l e i n f o

abstract

Article history: Received 19 May 2011 Received in revised form 14 October 2011 Accepted 1 November 2011 Available online 11 November 2011

In this paper, we propose Multi-Rate Bandwidth Available in Real Time (MR-BART) to estimate the endto-end Available Bandwidth (AB) of a network path. The proposed scheme is an extension of the Bandwidth Available in Real Time (BART) which employs multi-rate (MR) probe packet sequences with Kalman filtering. Comparing to BART, we show that the proposed method is more robust and converges faster than that of BART and achieves a more AB accurate estimation. Furthermore, we analyze the estimation error in MR-BART and obtain analytical formula and empirical expression for the AB estimation error based on the system parameters. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Available bandwidth Kalman filter Network path Probing sequence

1. Introduction Accurate estimation of the Available Bandwidth (AB) over a network path with a temporally variable cross-traffic is a challenging problem (Prasad et al., 2003; Welsh et al., 2001; Kang et al., 2004). The capability of measuring available bandwidth is useful in several contexts; including service level agreement verification, network monitoring and server selection. Measurement of available bandwidth in real-time, with per-sample update of the bandwidth estimate, opens up for adaptation based on available bandwidth directly (rather than ‘‘first-order’’ measures such as loss or delay) in congestion control and streaming of audio and video (Ekelin et al., 2006). In such cases an accurate estimation of the available bandwidth has a crucial role in providing Quality-ofService (QoS) guarantees to the users (Hussain, 2006; Hu and Steenkiste, 2003; Dovrolis and Jain, 2003). In practice, several users share the network bandwidth and under some circumstances the bandwidth demand is higher than that of the link capacity which causes network congestion. Network congestion results in degradation of some QoS parameters such as transmission delay and packet loss. Therefore, an accurate estimation of the available bandwidth at each time instant has an important role in designing more efficient network resource management schemes (Blefari Melazzi and Femminella, 2008; Bergfeldt et al., 2009; Liebeherr et al., 2010).

n

Corresponding author. Tel.: þ98 21 51212051. E-mail addresses: [email protected] (M. Sedighizad), [email protected], [email protected] (B. Seyfe), [email protected] (K. Navaie). 1 Tel.: þ98 21 55277400. 2 Tel.: þ44 113 3431422. 1084-8045/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jnca.2011.11.006

In practice, a perfect estimation of the available bandwidth between two arbitrary nodes in the network requires access to the temporal traffic information of each node along that network path, which is not always possible for the end-users. To tackle this issue, probing schemes are utilized. The basic idea of probing is to inject a sequence of packets namely probe packets with predefined inter-packet time interval into the network path. Hereafter, we refer to the sequence of probe packets as the probing sequence. In the receiver side, the inter-packet times of the probe packets are affected by the actual available bandwidth. The available bandwidth is then estimated utilizing the relationship between the inter-packet time dispersions at the two ends of the network path. If the probe packet transmission rate exceeds the available bit rate of the network path, the probe packets are backlogged at some intermediate nodes, resulting in an increased transmission delay; otherwise, the probe packets is received with no delay. The available bit rate is then estimated as the probing rate at the beginning of congestion (Melander et al., 2000). The main components of the above mentioned methods consist of the pattern of probing sequences and the available bandwidth estimation method. Various probing schemes are proposed in the literature to estimate the available bandwidth through the information available at the network edges. TOPP (Melander et al., 2000) employs trains of Packet Pairs and varies the rate of probing traffic to induce congestion in the network. After the probing session, TOPP deploys linear regression in order to get an estimate of the available bandwidth and bottleneck link capacity. Pathload (Jain and Dovrolis, 2002) uses a sequence of the constant rate packet trains, where the transmission rate of consecutive trains is iteratively varied until it converges to the available bandwidth.

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Pathchirp (Ribeiro et al., 2003) is another tool that measures the end-to-end available bandwidth and exploits the iterative method. In pathchirp, the rate is varied within a single packet train using geometrically decreasing inter-packet gaps. Both above methods interpret increasing delays as an indication of overload, i.e., to detect if the probing rate exceeds the available bandwidth. Jain and Dovrolis (2005) presented a study of how to measure the available bandwidth variation range. The available bandwidth is a mean value over some time and the variation range describes how much the cross traffic fluctuates during this interval. The variation range is defined using second order statistics such as the variance. That method was implemented in a tool called Pathvar. PathMon (Kiwior et al., 2004), Bandwidth Available in Real-Time (BART) (Ekelin et al., 2006), Machiraju et al. (2007) are other instances of probing schemes. Most of the existing available bandwidth measurement techniques impose a large amount of extra traffic load because of injecting the probing sequences. Furthermore, they usually require a long observation interval to estimate the AB with an acceptable level of accuracy. In this paper, we propose a method which is an extension of the BART and is based on injecting multi-rate (MR) probing sequence which we call MR-BART. Our proposed method then utilizes Kalman filtering (KF) for AB estimation. In MR-BART, in addition to the changing the probing rate from one probing sequence to another, we also alter the probing rate within each probing sequence. This technique enables us to obtain a rich set of observations by injecting each probing sequence. The observed set of data is then utilized by a KF to adaptively estimate the available bit rate. We show that MR-BART obtains a more accurate AB estimation and less sensitive to the initial state of the KF than that of BART. The increment of the accuracy of estimation comparing to BART is mainly due to the increasing the dimension of the KF input data caused by injecting multi-rate probe packets. Selecting multi-rate for any probing sequence causes a smoother AB estimation and prevents from sharp increments which sometimes occur in BART. The proposed method also introduces a set of adjustable parameters which increases its applicability into different scenarios. We compare our proposed method with BART approach which has been presented by Ekelin et al. (2006) that outperforms other methods in the literature. Simulation results show that using our proposed approach a higher level of accuracy is achieved rather than BART, whereby the extra traffic load because of probing sequence is equal to the BART. In the simple and first version of the proposed method in this paper (Sedighizad et al., 2008), we introduced MR-BART in details. In the present paper, we performed a complete performance analysis, and derived an analytical model for estimation error (mean square error MSE) based on the system parameters M and P in Section 4. Where M and P are the total number of the packets and the number of portions, in each probing sequences, respectively. Based on this analysis, the impact of the probing sequence parameters on the accuracy of estimation is investigated. In other words, we offered a theoretical base for our proposed method, in which by analyzing the interaction of cross traffic and probe packets, we drive a formula for MSE based on system parameters and cross traffic characteristics. Also, we completed the comparison of our method with BART in various scenarios and we added the related simulation results in the last section of the paper (Section 5.4). In this subsection, BART in a special situation (MPCR-BART as an extension of BART) was compared with our method, and the preference of our method was shown. Moreover, the impact of adjusting the Q (the covariance matrixes of the process noise), was thoroughly analyzed and related simulation

results were added in Section 5.4. Finally, in Section 5.5, the last subsection of the Section 5, we explain the impact of MR-BART probing traffic on TCP performance in some details. The rest of the paper is organized as follows; in Section 2 we elaborate on the probing procedure and present the system model. Then in Section 3, our proposed method for available bandwidth estimation is presented. The performance of the proposed method is studied in Section 4. We present the simulation results in Section 5. The paper is then concluded in Section 6.

2. Available bandwidth estimation in a network path A network path from the transmitter to the receiver consists of a number of nodes. A node itself consists of a queue connected to an input link and an output link. The queues in the nodes have infinite buffer length and first in first out (FIFO) is the serving discipline (we explain briefly about what happen to our estimation in the case of using traffic prioritization and finite buffer length, at the end of Section 3). A network path consists of L links, each with capacity of Cl(bits/s), (l¼1, 2, y, L). The link capacity, Cl, is determined based on the physical layer interfaces of the transmitter and the receiver. The temporal variations of Cl is slow so that it can be assumed constant in time-scale of interest. For link l, a cross-traffic with time varying rate of yl ¼yl(t,dT)(bits/s) is considered. Here dT is the observation time at which we are interested in describing the traffic fluctuations. Therefore, the available (residual) bandwidth of link l, Al, is Al ¼ C l yl :

ð1Þ

For a network path consisting of L links, A, is defined as A ¼ min Al :

ð2Þ

l

As it is seen in (2), for a network path, the Available Bandwidth (AB), is mainly determined by the link which has the minimum residual bandwidth. The link with the minimum residual bandwidth is called the bottleneck link. Consider a fluid flow crosstraffic at rate y that is transported through a single hop network with the link capacity C. In the probing scenario, the transmitter injects probe packets with instantaneous rate u into the output link. Hereafter, we simply refer to the instantaneous rate as rate, unless otherwise stated. Because of the impact of the cross-traffic in the network path, in the receiver, the receiving time interval between the consequent probe packets may be changed. By measuring the time interval between the received probe packets, the receiver is able to obtain the probe rate, r. For cases where urC y no congestion is experienced, thus r ¼u. However, in cases where u 4C  y, network is in an overload status. Therefore (Melander et al., 2000), (  u þ y 1 u r Cy u ¼ max 1, ¼ 1 : ð3Þ y u þ u 4 Cy r C C C The inter packet strain parameter, e, is then defined as (Ekelin et al., 2006) u r

e ¼ 1:

ð4Þ

Therefore, for u 4C  y 1 yC uþ : C C

e¼

ð5Þ

By setting

a¼

1 ; C

b¼

yC , C

ð6Þ

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

the inter packet strain parameter in (4) is obtained as

e ¼ au þ b:

ð7Þ

Based on the AB definition, the value of u at which (u/r) starts to deviate from unity is an estimate of AB. This definition of AB can be interpreted as follows: if sending rate is smaller than the AB, the packets do not cause congestion on the network path and the transmitting and receiving rates are equal. Otherwise, the packets are backlogged, thus the congestion is experienced according to (7). Note that the fluid flow model is an asymptotic model for an actual packet transmission scenario; therefore, (7) expresses the asymptotic relation between the transmission probe rate, u and the inter-packet strain, e (Liu et al., 2007). Following the same argument for several concatenated links, it was shown by Melander et al., (2000) that the AB of the entire path from the transmitter to the receiver can be estimated by investigating (u/r) and determining the point at which (u/r) starts deviating from unity. The corresponding obtained value of u at this point is considered as an estimate of the path AB.

733

probe pair i(i¼ 1, 2, y, M  1), by measuring the inter-arrival time of the consequent packets at the receiver

ei ¼

ðg O Þi 1, ðg I Þi

ð8Þ

where, (gI)i and (gO)i are the initial inter-packet time for probing pair i at the transmitter, and the inter-arrival time between the packets of probing pair i received at the receiver side, respectively. After receiving the last packet of the probe sequence, and obtaining the strain of all probing pairs, we obtain an AB estimation using Kalman filtering. In order to use KF, similar to Ekelin et al., (2006) we model the system by the following linear equation: xk ¼ f ðxk1 Þ þ wk1 ,

ð9Þ

where xk is the state vector at the kth step of the estimation which is defined as xk ¼ ½ak , bk T ,

ð10Þ

f(U)is a known function of xk  1, and wk  1 is the process noise at the kth step. We also define the measurement equation

3. AB estimation using multi-rate probing

zk ¼ hðxk Þ þ vk ,

By utilizing the sequence of single rate for sampling the network path, in some cases (u rC  y) we exert the load of probe packet to the path without obtaining any information about AB (Ekelin et al., 2006). In this situation, another probing sequence should be sent and therefore the time and bandwidth source is consumed. While, if the probing rate can be varied in each sequence, we have further chance for obtaining a sample of AB using each sequence. Although, variation of probing rate in each sequence increases the variance of probing sequences strain and degrades the statistical precision. In the consequent section we explain that how we can handle this problem by efficient and simple method.

where h(U) is a known function of xk, and vk is the measurement noise. In this equation, zk is a P  1 vector of measured strains

ð11Þ

zk ¼ ½ðz1 Þk ðz2 Þk . . .ðzP Þk T ,

where (zp)k is the average of the measured strains of probe packets in portion p, at kth step of estimation. Assuming an unchanged network setting in the observation interval f(U) can be defined as f ðxk þ 1 Þ ¼ Axk ,

ð13Þ

where, transition matrix A is equal to the unit matrix I. In addition, we consider h(U) as hðxk Þ ¼ Hk xk :

3.1. Probing procedure In this paper, we propose a new method which employs multirate probing sequence as it is shown in Fig. 1. Employing multirate probe packet transmission enables us to probe the network path over several rates in each probing sequence. Let M be the total number of the packets in each probing sequence, and S be the packet size (in bytes). The probing sequence consists of P portions, each containing (M 1)/P probe packet pairs so that the second packet in one pair also constitutes the first packet in the next pair. In each portion, the inter-packet times are equal and considered so that the actual probe packet transmission rate is up(p ¼1, 2, y, P). We refer to each two consequent probe packets as a probe pair; therefore, probe pair i refers to the ith two consequent probe packets in a sequence. 3.2. Estimation by Kalman filtering To estimate the available bit-rate of a path, a sequence of the abovementioned probe packets is injected into the path of interest. Then, the inter-packet strain ei, will be obtained for each

M M-1

(M-1)/P

uP

u2

2

u1

Fig. 1. MR-BART probing sequence scheme.

1

ð12Þ

ð14Þ

In this equation, Hk is 2 3 ðu1 Þk 1 6 7 6 ðu2 Þk 1 7 7, Hk ¼ 6 6 ^ ^7 4 5 ðuP Þk 1

ð15Þ

where (up)k is the rate of probe packets in portion p, at the kth step of estimation. Considering the above definitions and assumptions, we get xk ¼ xk1 þwk1 ,

ð16Þ

zk ¼ Hk xk þ vk :

ð17Þ

In our model we assume that vk  N ð0,RÞ;

wk  N ð0,Q Þ,

and N ð0, hÞ denotes a zero mean Gaussian random variable with covariance matrix h, Q and R are the covariance matrixes of the process and measurement noise, respectively. As mentioned in the above section, varying the probing rate in each probing sequence results in increasing the variance of the strain of the sequence. For handling this problem, we compute the strain of each portion separately, and define the covariance matrix of probing sequence strain, R (P  P matrix), as 2 3 ðR1 Þk 0 ... 0 6 7 0 7 ðR2 Þk 6 0 7, R¼6 ð18Þ 6 ^ & ^ 7 4 5 0 0    ðRP Þk

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M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

where (Rp)k is the variance of the measured strain of probe packets in portion p, at the kth step of estimation. Note that, in practice, the cross-traffic is not fluid flow with constant rate because of its bursty nature. Deviation from fluid flow model due to the burstiness of the cross-traffic is taken care of by considering the noise process wk in (16). To apply Kalman filtering method in (16), we need to find an appropriate value for Q which is a 2  2 symmetric matrix. This matrix describes the intrinsic fluctuations in the system, and is related to the cross-traffic temporal fluctuation. For a and b which are independent random variables, matrix Q can take the following simple form:   l 0 Q¼ , ð19Þ 0 l where l is an adjustable parameter and related to the cross-traffic statistics. In our proposed approach, the KF takes Hk, zk and x^ k as inputs, and then obtains a new state vector x^ k þ 1 . Using the new state vector x^ k þ 1 an estimation of the AB is then obtained as

b^ A^ ¼  , a^

ð20Þ

and then immediately we can estimate the link capacity as follows: 1 C^ ¼ : a^

the mean square error (MSE) of our estimation N 1X ðAk A^ k Þ2 , N-1 N k¼1

^ 2 ¼ lim x ¼ EðAAÞ

where Ak refers to the AB of the bottleneck link obtained by the kth probing sequence. Here, N is the number of probing sequences which are injected into the network path. Equality in the second line of (22) holds due to the weak law of large numbers (Billingsley, 1999). Utilizing (1), we can write, A ¼ CydT ðtÞ,

dp ¼

ðM1ÞS : Pup

4.1. AB estimation error In this section, we derive an analytical model for estimation error based on the system parameters M and P. Here, we consider

ð24Þ

Total observation time of probing sequence with respect to the observation time of its portions is as follows:

dT ¼

P X

dp :

ð25Þ

p¼1

Utilizing (23), we can write ^ y^ ðtÞ, A^ ¼ C dT

ð26Þ

^ C^ and y^ ðtÞ are the estimation of A, C and y ðtÞ, where A, dT dT respectively. Using (23) and (26) in (22), we get, ^ ^ dT ðtÞÞÞ2 : x ¼ EððCCÞðy dT ðtÞy

ð27Þ

Here, we assume that the real bottleneck link capacity is C, and analyze the behavior of MSE based on probing sequence parameter and behavior of cross-traffic. Under such assumption, the estimation problem is reduced to finding an interpretation of the following equation with respect to the probing sequence parameters:

x ¼ EðydT ðtÞy^ dT ðtÞÞ2 :

ð28Þ

By applying the above assumption to the measurement Eq. (17), we get 2 3 2 3 2 3 ðz1 Þk  C1 ðu1 Þk ðv1 Þk 1 6 7 6 7 6 ðz2 Þk  C1 ðu2 Þk 7 6 6 ðv2 Þk 7 17 7 6 7¼6 7, ð29Þ 6 7b k þ 6 6 7 6 7 ^ 4 5 4^5 4 ^ 5 ðvP Þk ðzP Þk  C1 ðuP Þk 1 or equivalently, z~ k ¼ hk x~ k þ vk :

ð30Þ

In the above equation, pth element of z~ k (P  1 vector) is as follows: 1 ðz~ p Þk ¼ ðzp Þk  ðup Þk , C

4. Performance analysis

ð23Þ

where C and A are the capacity of bottleneck link, and AB of the path, respectively. In this equation, ydT ðtÞ is the average crosstraffic rate in the time interval [t,tþ dT]. In this time interval, dT is the inter-packet time between the first and the last packet of the probing sequence (which we referred to as observation time of probing sequence), and t is the arrival time of the first packet in a probing sequence to the bottleneck link. If dp is considered as observation time of pth portion of probing sequence, then it can be defined as

ð21Þ

If the bottleneck link used with its capacity then, bandwidth is not available and available bandwidth is zero. If we consider the assumption of finite buffer, some packets of a probing sequence may be lost due to the limited buffer length. When a probe packet is dropped, the measurement for this packet is ignored and the adjacent inter packet strain (ei  1) can be used instead of its real strain (ei) (if we know that the dropped packet is belonged to which portion of the probing sequence). Since each portion of a sequence in our method has (M 1)/P probe packet pairs, so if we miss all the packets of one portion totally, we only lose one rate out of P rates (one element of measurement vector). Therefore, this event can not affect our method’s performance significantly. In the case of dropping all the probe packets of a sequence we consider the previous AB (available bandwidth) estimation as the current one, then another sequence will be sent. As mentioned in Section 2, we assume that first in first out (FIFO) is the serving discipline of the queues. This means that we assume that there is no prioritization in the router, so that all traffic is served on a first-come first-served (FCFS) basis. In the case of using traffic prioritization, the probe packets may be queued for such a long time that they dropped finally due to the time expiry. In this situation, we may lose some elements of the measured strains vector zk in Eq. (12), and we cannot rely on this vector. So, we use previous AB estimation and send another probing sequence. If this situation occurs successively, then it can affect our estimation results negatively, otherwise we can use the valid estimated AB of the previous step until we obtain a new estimation of AB.

ð22Þ

and hk (P  1 vector) is 2 3 1 617 6 7 hk ¼ 6 7: 4^5 1

ð31Þ

ð32Þ

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

By considering the above assumption about the real bottleneck link capacity, the state vector of our problem is reduced to a scalar as, x~ k ¼ bk . Therefore, (16) changes as follows: x~ k ¼ x~ k1 þ wk1 ,

ð33Þ

where wk  1 is a Gaussian variable with zero mean and variance l. In our model, the noise covariance is a diagonal matrix, and thus the components of v are uncorrelated. For such condition, it is advantageous to consider the components of z as independent scalar measurements, rather than as a vector measurement in KF (Grewal and Andrews, 2001). Based on Grewal and Andrews (2001), if we consider the components of z as independent scalar measurements, then the filter implementation requires P iterations. The updating process can be implemented iteratively using the rows of Hk and the diagonal elements of R as the corresponding (scalar) measurement noise covariance as the following equations:

735

Ddp ðt p Þ Ddp ðt p Þ S þðg I Þp rðg O Þp r þ ðg I Þp þ , ðM1Þ=P ðM1Þ=P C

ð46Þ

where (gI)p and (gO)p are the average inter-packet time of pth portion at the transmitter, and the receiver, respectively. For defining the Ddp ðt p Þ in (46), we first define the hop-workload process W(t), as sum of service time of all packets in the queue and the remaining service time of the packet in service (Liu et al., 2007). By this definition, Ddp ðt p Þ is defined as the difference between hop-workload at time tp and (tp þ dp), i.e., Ddp ðt p Þ ¼ Wðt p þ dp ÞWðt p Þ:

ð47Þ

Rearranging the (45) and (46), we get the strain of pth portion as 8 yd ðtp Þ ðg O Þp > ðg I Þp r CS < ðg I Þp 1 ¼ pC þ ðgISÞp C 1 , ð48Þ y ðtp Þ y ðt p Þ ðg Þ > : dpC þ ðg SÞ C 1 r ðgOÞ p 1 r dpC ðg I Þp 4 CS I p

I p

ð34Þ

Ddp ðt p Þ Ddp ðt p Þ ðg O Þp S : r 1 r þ ððM1Þ=PÞðg I Þp ððM1Þ=PÞðg I Þp ðg I Þp C ðg I Þp

½p ½p1 ½p1 w½p kk ðHp Þk wk , k ¼ wk

ð35Þ

In order to find the bounds of strain, we analyze the behavior of Ddp ðt p Þ. We can write W(tp þ dp) as follows:

½p ½p1 ½p1 ½p þ kk ½z½p ðHp Þk x^ k , x^ k ¼ x^ k k

ð36Þ

Wðt p þ dp Þ ¼ Wðt p Þ þ

1

½p

kk ¼

½p1 ðHp Þk wk ðHp ÞTk

þ ðRp Þk

w½p1 ðHp ÞTk , k

½0 ½0 ½P for p¼1, 2,y, P, using the initial values wk and x^ k . Where wk is ½p state estimate error covariance matrix, x^ k is the estimated state vector for pth portion of kth probing sequence, and k is the Kalman gain. The intermediate variable (Rp)k is the pth diagonal element of the P  P diagonal matrix Rk and (Hp)k is the pth row of ½p ½P the P  2 matrix Hk. The final value x^ k , i.e., x^ k is an estimate of the state vector obtained from sending one probing sequence consists of P portions. Combining Eqs. (31–33) with Eqs. (34–36) results in

1

½p

kk ¼

c½p1 þ ðRp Þk k

c½p1 , k

ð37Þ

½p ½p1 c½p , k ¼ ð1kk Þck ½p

½p1

x^~ k ¼ x^~ k

½p

½p

½p1

ck ¼ Eðx~ k x^~ k Þ2 ,

From (6) and (41) we have,  2 ðy ðtÞy^ dT ðtÞÞk ck ¼ E dT : C

ð40Þ

N C2 X c : N k¼1 k

ð51Þ

ð52Þ

Combining (47), (50) and (51) we get,

dp C

ydp ðt p Þdp þ Idp ðt p Þ:

ð53Þ

C

ydp ðt p Þdp rDdp ðt p Þ r

dp C

ydp ðt p Þ:

ð54Þ

Combining (49) and (54), we get ð41Þ

ydp ðt p Þ C

ð42Þ

1 r

yd ðt p Þ ðg O Þp S þ : 1 r p ðg I Þp C C ðg I Þp

Collecting (48) and (55) leads to 8 yd ðtp Þ ðg O Þp > ðg I Þp r CS < ðg I Þp 1 ¼ pC þ ðgISÞp C 1 y ðtp Þ y ðtp Þ ðg Þ > : dpC 1r ðgOÞ p 1r dpC þ ðg SÞ I p

ð43Þ

Combining (22), (28) and (43) we get

N-1

:

Applying the bounds of Idp ðt p Þ on (53) results in

Note that,

ck ¼ c½P k :

dp

0 r Idp ðt p Þ r dp :

dp

thanks to x~ k ¼ bk ,

ck ¼ Eðbk b^ k Þ2 :

bðt p þ dp Þbðt p Þ

In addition, Idp ðt p Þ is the total amount of idle time of the link in the time interval [tp,tp þ dp] (Liu et al., 2007). If the link is busy for transmitting the cross-traffic packets in the all time of this time interval, then Idp ðt p Þ ¼ 0. If in this time interval link is idle, then Idp ðt p Þ ¼ dp , thus,

ð39Þ

Utilizing the KF equations yields

x ¼ lim

ydp ðt p Þ ¼

Ddp ðt p Þ ¼ :

ð50Þ

where b(tp) is the total volume of cross-traffic (measured in bits) receive by the bottleneck link up to time instant t. Using this definition we can write the ydp ðt p Þ as

ð38Þ

þ kk ½z~ k x^~ k

bðt p þ dp Þbðt p Þ dp þIdp ðt p Þ, C

ð49Þ

ð44Þ

If we can find the value of (Rp)k analytically, then using initial ½0 value for ck the relationship between x and system parameters will be obtained. From Liu et al. (2007) we have 8 ðg Þ y ðt Þ < ðg O Þp ¼ I p Cdp p þ CS ðg I Þp r CS ð45Þ , ðg Þ y ðt Þ ðg Þ y ðt Þ p p I p d I p d p p : þ CS r ðg O Þp r þ ðg I Þp ðg I Þp 4 CS C C

I pC

ðg I Þp 4 CS

ð55Þ

:

ð56Þ

In each portion of probing sequence, C, S and (gI)p are constant; so the variance of the above equation is ! ðg O Þp 1 Var 1 ¼ 2 Varðydp ðt p ÞÞ, ð57Þ ðg I Þp C or equivalently, Rp ¼

1 C2

Varðydp ðt p ÞÞ,

ð58Þ

where Rp is the variance of the measured strain of probe packets in portion p. Eq. (57) is the final equation of the interaction between cross-traffic and probe packet traffic for obtaining the Rp.

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M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

In this equation Rp is valid for up ZA, because, if (gI)p Z(S/C) then up ZA can be concluded. Therefore we can use this value of Rp in our MSE analysis. Note that in our model the cross-traffic is self-similar which is generated from fBm distribution, so we can write (Erramilli et al., 2002; Billingsley, 1999; Prakasa Rao, 2004), bðtÞ ¼ mt þ soðtÞ,

C

MSE

3.5 3

1.5 ð61Þ

By substituting the above equation in to the (58) we have 2H2

4

2

Varðydp ðt p ÞÞ ¼ s

ðs2 dp 2

4.5

ð60Þ

2 2H2 dp :

1

P = 1 (simulation) P = 2 (simulation) P = 3 (simulation) P = 4 (simulation) P = 1 (analytic) P = 2 (analytic) P = 3 (analytic) P = 4 (analytic)

2.5

1 2H 2H 2H ð9t þ t9 þ 9t9 9t9 Þ, 2

where H denotes the self-similarity index. From (51) and (60) we get

Rp ¼

x 10-3

5

ð59Þ

where m is the average rate, s is a controlling factor of fluctuation and o(t) is the fractional Brownian motion (fBm) process describing the cross-traffic. fBm process o(t) is a Gaussian process with zero mean, which is stationary increment and its covariance function is (Mikosch et al., 2002) Eðoðt þ tÞoðtÞÞ ¼

5.5

Þ:

ð62Þ

40

50

60

70 M

80

90

100

Fig. 3. Impact of varying M on the accuracy of MR-BART for fixed values of P (C ¼10 Mbist/s, H¼ 0.7).

Utilizing (62), (37), (38) and (44), and normalizing by C2 we will have ½P1 N ðck Þ2 1X : c½P1  k N-1 N c½P1 þ 12 ðc½P1 Þ k¼1

x ¼ lim

k

C

ð63Þ

k

Therefore, we obtained x with respect to the probing sequence parameters and features of cross-traffic. In order to analyze the behavior of MSE with respect to the probing sequence parameter, we plot the MSE versus P and M in Figs. 2 and 3, respectively. In addition, in these figures comparison between simulation and analytical results is made simultaneously. The simulated network has one bottleneck link as shown in Fig. 4. The bottleneck of the network path is the link between the two considered routers. In our model, the bottleneck link has the minimum capacity along the network path of interest and its capacity is C ¼10(Mbits/s) (MSE is normalized by C2). For the cross-traffic, we consider a fractional Brownian motion (fBm) model with H¼0.7 (Duncan et al., 2000). In addition, the number

5.5

x 10-3 M = 34 (analytic) M = 51 (analytic) M = 85 (analytic) M = 102 (analytic) M = 34 (simulation) M = 51 (simulation) M = 85 (simulation) M = 102 (simulation)

5 4.5

MSE

4 3.5 3 2.5 2 1.5 1

2

3

4

P Fig. 2. Impact of varying P on the accuracy of MR-BART for fixed values of M (C¼ 10 Mbits/s, H¼ 0.7).

Fig. 4. A schematic view of the considered network path in our experiments

of probing sequences which are injected into the network path considered as N ¼1000. Figure 2 shows the impact of the increasing P on the accuracy of estimation. As it is seen in this figure, increasing the P results in decreasing MSE both in the simulation and theoretical results. By increasing P the dimension of the measurement vector of KF will increase and therefore we can obtain a more accurate estimation. Although, the slop of the decreasing the MSE in theoretical results is larger than that of simulation. The reason behind this phenomenon will be discussed later. Figure 3 shows the impact of the increasing M on the behavior of MSE. For all P, MSE decreases as the number of probe packets increase. This is because of the increasing the observation interval (due to the increasing the M) and therefore obtaining an appropriate estimate for the average cross-traffic arrival rate. The experimental results we obtained agree with our analytical findings. Although, the slop of decreasing the MSE of analytical results are restively larger than that of simulation. The interpretation of this phenomenon is as follows. As mentioned in Section 2, the fluid flow model is an asymptotic model for an actual packet transmission scenario; therefore (7) expresses the asymptotic relation between the transmission probe rate, u and the inter-packet strain, e. In the actual and therefore in the simulated network, we have bursty arrivals of discrete cross-traffic packets. Therefore, the relationship between e and u deviate from (7). The amount of this deviation is influenced by the probing sequence parameters (Liu et al., 2007). As the P increases, the variation of probe packet rate in each sequence will increase (increasing the burstiness). On the other hand since the deviation of fluid flow model is originated from the burstiness of the traffic, therefore the deviation from

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

4.2. Impact of P on the complexity of computation As mentioned above, in our model the noise covariance is a diagonal matrix, which makes the components of v uncorrelated. For such condition, we consider the components of z as independent scalar measurements. According to Grewal and Andrews (2001), the number of operations which is required for processing z (a vector by dimension P  1) as P successive scalar measurements, is significantly less than that of the corresponding number of operations required for vector measurement processing. It is shown that the complexity of computations for the vector implementation of KF equations grows as P3 (Grewal and Andrews, 2001), whereas that of the scalar implementation these equations grows only as P. Furthermore, if we consider the components of z as independent scalar measurements, we can avoid matrix inversion in the implementation of the KF equations and improve the robustness of the computation against error (Grewal and Andrews, 2001). Therefore, by processing z as P successive scalar measurements we can reduce the computation time and improve numerical accuracy. By utilizing above implementation, we can reduce the complexity of computation, so that, in our method the complexity of computation is only P times more than that of BART.

5. Simulation results We consider a general model for a network path with one bottleneck link which is shown in Fig. 4. The bottleneck of the network path is the link between the two considered routers. In our model, the bottleneck link has the minimum capacity along the network path of interest. A similar model is also considered by Ekelin et al. (2006 ) for a network path with one bottleneck link. For the cross-traffic, in this paper, we consider self-similar traffic which is explored from a fractional Brownian motion (fBm) model with H¼0.7 (Duncan et al., 2000). This model is shown to be a good fit to the aggregated traffic in a data network with bursty traffic sources (Mikosch et al., 2002). Unless otherwise specified, in all experiments in this paper, we consider l ¼10  4 for matrix Q. All links in the simulated network have a nominal capacity of 100 (Mbits/s), except the bottleneck link between the two routers, which has the capacity of 10 (Mbits/s). Probe packet transmission rate for each portion of probing sequence was randomly chosen from a uniform distribution, over the interval from 1 Mbit/s to 20 Mbit/s. MR-BART was configured to produce an estimate every second, i.e., the inter-departure time between two consecutive probing sequences is one second, so this method injects probe traffic with an average density of M  S  8 bit/sto the network path (where M is the total number of the packets in each probing sequence, and S is the packet size). For example in the case of M ¼17 and S ¼1500 bytes, MR-BART injects probe traffic with an average density of

0.204 Mbit/s, which is 2.1% of the bottleneck link capacity. In our simulations, we set N¼ 1000 and MSE is normalized to the square of bottleneck link capacity (Ekelin et al., 2006). Furthermore, in our simulations we find that the accuracy of estimation will not change significantly for S 41500 (bytes), hence we set S¼1500 (bytes) in all experiments unless otherwise stated. We used MATLAB for our simulation. We wrote a program to generate cross traffic with fBm model. As mentioned above we generated probe packets traffic with transmission rate for each portion of probing sequence which was randomly chosen from a uniformly distribution, over the interval from 1 Mbit/s to 20 Mbit/s. We also wrote a program in order to simulate the network path, probe packet and cross traffic sender and receiver, and routers. Figure 5 shows the AB estimation utilizing the MR-BART method. In this figure, in order to estimate AB, we use probing sequence with length M¼33 and P¼2. As it is seen in this figure, we obtain reasonable level of accuracy of AB estimation by using two portions in each probing sequence. This figure shows that our method is able to track both slow and fast variation of crosstraffic rate simultaneously. 5.1. Impact of M and S on the estimation accuracy Figure 6 depicts the AB estimation by using the proposed method in different values of M, and S. In this figure, we set P¼3. As seen in this figure, increasing M and S leads to the better estimation of AB. This is because of the increasing the observation interval (due to the increasing the M and S) and therefore obtaining an appropriate estimate for the average cross-traffic arrival rate. In Table 1, the MSE of estimation has been reported. In our experiments we investigate the behavior of MSE versus M and S, and find that larger M and S, results in smaller MSE. In addition we observe that the accuracy of estimation for S4 1500 bytes and M434 will not change significantly. Therefore, hereafter we use S¼1500 bytes in our simulations. 5.2. Impact of M and P on the estimation accuracy Here we study the impact of parameters P and M, on the accuracy of the AB estimation. As it is shown in Fig. 7, the simulation result confirms that, increasing P in small values of M leads to decreasing the accuracy of estimation. But if we choose M large enough, then selecting the large value of P, results in accurate

10 Real A MR-BART (P = 2, M = 33)

9 Available Bandwidth [Mbits/sec]

fluid flow model will increase. This effect decreases the accuracy of the network modeling. However, it should be noted that, by increasing the P accuracy of the KF estimation will increase. Based on our observation, increasing P, at first increases the accuracy of estimation (by increasing the accuracy of KF) but further increasing the P cause to degrading the accuracy of modeling and influence the accuracy of estimation. Since in our theoretical analysis deviation from fluid flow model due to the burstiness of the cross-traffic is only taken care of by considering the process noise, therefore the effect of increasing the KF accuracy by increasing P is more obvious than degrading the deviation of modeling. Based on this reason, by increasing P the MSE of the analytical curves will decrease faster than the simulation curves.

737

8 7 6 5 4 3 2 1 0 0

0.5

1

1.5 2 Time [msec]

2.5

Fig. 5. MR-BART estimation of AB (P¼ 2, M ¼33).

3

3.5 x 105

738

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

10

Table 2 MSE of MR-BART for PZ 5 (S¼ 1500 bytes).

9 Available bandwidth [Mbit/sec]

M

P

8 5

6

7

8

0.009 0.008 0.006

0.009 0.009 0.006

0.010 0.009 0.006

0.011 0.010 0.005

7 34 45 100

6 5 4

0.06

3 MR-BART (M = 13, s = 500 bytes) MR-BART (M = 22, s = 900 bytes) MR-BART (M = 34, s = 1500 bytes) Real A

2 1

P = 1 (simulation) P = 2 (simulation) P = 3 (simulation) P = 4 (simulation) P = 5 (simulation) P = 1 (modeling) P = 2 (modeling) P = 3 (modeling P = 4 (modeling) P = 5 (modeling)

0.05

0.04

0 1

2

3

4 5 Time [msec]

6

7

8

9 x 104

Fig. 6. Impact of varying M and S on accuracy of MR-BART estimate (P ¼3).

MSE

0

0.02

Table 1 Impact of variation M and S (P¼ 3) on accuracy of MR-BART estimation.

MSE

0.03

S ¼ 500 bytes M ¼ 13

S¼ 900 bytes M ¼22

S ¼1500 bytes M ¼ 34

0.190

0.013

0.009

0.01

0

Available bandwidth [Mbits/s]

20

40

50

60 M

70

80

90

100

10 Fig. 8. Impact of varying M on the accuracy of MR-BART for fixed values of P (C ¼10 Mbits/s).

5

0.16

Real A MR-BART (P = 4, M = 17) MR-BART (P = 3, M = 17)

1

2

3 4 Time [msec]

5

6

P = 1 (simulation) P = 2 (simulation) P = 3 (simulation) P = 4 (simulation) P = 5 (simulation) P = 1 (modeling) P = 2 (modeling) P = 3 (modeling) P = 4 (modeling) P = 5 (modeling)

0.14

0 0

7 x 104

0.12

10

0.1

8

MSE

Available bandwidth [Mbits/s]

30

6 4

0.06

Real A MR-BART (P = 4, M = 34) MR-BART (P = 3, M = 34)

2

0.08

0

0.04 0

1

2

3 4 Time [msec]

5

6

7 x 104

Fig. 7. Impact of varying P and M on accuracy of MR- BART estimate (S¼ 1500 bytes).

0.02 0 20

estimation of AB. From Fig. 7 we see that for M¼17, P¼3 shows better performance rather than P¼4. But if we select M large enough (M¼34), then P¼4 results in better estimation of AB. As mentioned above, if we choose M large enough then increasing P results in more accurate estimation of AB. However, our simulation results show that, for PZ5, error of estimation will not change significantly by increasing P. The results of simulation have been reported in Table 2. 5.3. Modeling of error based on the system parameters In this subsection, we derive an empirical model for estimation error based on the system parameters M and P. We consider a

30

40

50

60 M

70

80

90

100

Fig. 9. Impact of varying M on the accuracy of MR-BART for fixed values of P (C ¼70 Mbits/s).

general model for a network path with one bottleneck link which is shown in Fig. 4. The bottleneck of the network path is the link between the two considered routers. In our model, the bottleneck link has the minimum capacity along the network path of interest. In the following we use the above scenario in order to obtaining x from (22) using 16rM r100 probe packets, and 1rPr5 portions. Based on our observations from simulation results (see Figs. 8–11), we obtain the following expression by

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

0.06

0.04 MSE

Table 3 Values of a and b in different P and C.

M = 16 (simulation) M = 34 (simulation) M = 45 (simulation) M = 61(simulation) M = 100 (simulation) M = 16 (modeling) M = 34 (modeling) M = 45 (modeling) M = 61 (modeling) M = 100 (modeling)

0.05

739

C (Mbits/s)

0.03

P 1

2

3

4

5

10

a ¼0.04 b ¼0.04

a ¼0.06 b ¼0.33

a¼ 0.01 b ¼0.33

a¼ 0.26 b¼ 1.26

a¼ 0.15 b¼ 1.26

30

a ¼0.07 b ¼0.21

a ¼0.10 b ¼0.16

a¼ 0.02 b ¼0.25

a¼ 0.08 b¼ 0.63

a¼ 0.05 b¼ 0.63

50

a ¼0.10 b ¼0.08

a ¼0.16 b ¼0.33

a¼ 0.02 b ¼0.51

a¼ 0.41 b¼ 0.94

a¼ 0.41 b¼ 0.94

70

a ¼0.32 b ¼0.45

a ¼0.29 b ¼0.53

a¼ 0.27 b ¼0.73

a¼ 0.44 b¼ 1

a¼ 0.36 b¼ 1.14

0.02

0.01

0 1

1.5

2

2.5

3 P

3.5

4

4.5

5

Fig. 10. Impact of varying P on the accuracy of MR-BART for fixed values of M (C¼ 10 Mbits/s).

0.16

M = 16 (similation) M = 43 (simulation) M = 45 (simulation) M = 61 (simulation) M = 100 (simulation) M = 16 (modeling) M = 34 (modeling) M = 45 (modeling) M = 61 (modeling) M = 100 (modeling)

0.14 0.12

Eq. (65) denotes that x is a descending function of M. We also note that the minimum of x for a constant M, depends on not only P but also a and b. Therefore, to obtain the minimum x for a given M, we can find the suitable values of a and b for P ¼1,2, y, 5, from Table 3, and then utilizing (64) to find the value of P for which the minimum of x occurs. If we rearrange the (64), then M is M¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ae1:1P b

xðP 2 þPÞ

:

ð66Þ

From (66), we can obtain the appropriate value of M for a given

x in a bottleneck capacity of interest, through the following steps:

MSE

0.1 1st step: Considering a value for P, by regarding the constrain of computational complexity. 2nd step: Finding suitable values of a and b for selected P, using Table 3. 3rd step: Computing M from (66), which results in to the given x.

0.08 0.06 0.04 0.02 0 1

1.5

2

2.5

3 P

3.5

4

4.5

5

Fig. 11. Impact of varying P on the accuracy of MR-BART for fixed values of M (C¼ 70 Mbits/s).

curve fitting on the simulation results of x based on M and P as follows:

x¼a

e1:1P b

M ðP 2 þ PÞ

,

ð64Þ

where a and b are two positive parameters. The values of a and b depend on P and the bottleneck capacity, as shown in Table 3. In Table 3, we report suitable values of a and b for P¼1, 2, y, 5, and bottleneck capacity 10 rCr 70(Mbits/s). We showed in Table 2 that for values of P greater than 5, we do not achieve significant improvement on the estimation performance. Here after, we analyze the impact of parameters M and P on the accuracy of the proposed AB estimation method. Based on (64), x is a function of two parameters, M and P. The partial derivative of x(P,M) with respect to M is @x e1:1P M b þ 1 ¼ ab : @M ðP 2 þPÞ

ð65Þ

In order to gain more insight on the behavior of x based on M and P, we plot x(M) for the fixed values of P in Figs. 8 and 9, and x(P) for the fixed values of M in Figs. 10 and 11 utilizing the (64) and simulation in the above scenario. In Figs. 8 and 10 we consider bottleneck link capacity C¼ 10 (Mbits/s) and in Figs. 9 and 11, C ¼70 (Mbits/s). As it is seen in Figs. 8 and 9, by increasing M for a constant P, the estimation error x is decreased. Comparing to the BART (where P¼1), in our proposed method the input vector of KF has a larger dimension (P41). Therefore, we expect higher estimation accuracy than that of obtained by the conventional BART method, which is confirmed by the simulation results in Figs. 8 and 9. In Figs. 10 and 11, the behavior of x for different values of M has been plotted. As it is seen, x for a constant M has a minimum versus P. That is, if M remains unchanged and P increases, then x decreases to its minimum value but thereafter, with increasing the P, x will increase. It should be noted that, this problem is more obvious for small values of M. If we choose a large enough M (i.e., MZ34), then the increase in x beyond its minimum point can be ignored. The interpretation of this phenomenon is as the following. As mentioned in previous section, by increasing P the dimension of the measurement vector of KF is increased therefore, we can obtain a more accurate estimation. But, if the increase in the estimation accuracy is obtained for a constant M, then the observation time interval for each portion of probe packets is decreased. Thus, we are not able to obtain an appropriate estimate of the average cross-traffic arrival rate in each

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

5.4. Comparison of MR-BART and BART Here, we compare the performance of our method with BART method in different situations. In Fig. 12 we show the impact of increasing the M on the accuracy of estimation. As seen in this figure, the accuracy of estimation for M434 will not change significantly. But, since in the BART method M¼17 has been selected as the number of packet in which the accuracy of estimation is reasonable, therefore we also select this number for comparing our method with BART method. Figs. 13 and 14 depict the AB versus time, and compare the behavior of MR-BART (P¼2 and P¼3, respectively) and BART when synthetic cross-traffic has been used. From these figures, it is quite clear that MR-BART, estimates the available bandwidth more accurately than BART. In addition, these figures show that our proposed method is able to track slow and fast variation of cross-traffic rate simultaneously. Whereas in BART method, the parameter Q (covariance matrixes of the noise process) should be tune for each type of fluctuation in cross-traffic. Now, we show that the changing the value of Q does not significantly affect the accuracy of our results. In our method, due to the portioning idea we will be able to track fluctuation of crosstraffic and therefore we can estimate AB more accurately and the dependency to Q matrix is reduced. For explaining this issue we compare our method with BART in the encountering cross-traffic. BART: In this method, probing sequence is sent to the network path at a constant rate. If the probing sequence rate is less than AB, no new AB estimation will be produced. In this method, the Q matrix causes improvement in the accuracy of the AB estimation. But in order to tune the Q matrix we should take time and regularly impose extra traffic load to the network path.

10 Real A MR-BART (P = 2,M = 17) BART (M = 17)

9 Available bandwidth [Mbits/sec]

observation time interval (Liu et al., 2007). Consequently, the accuracy of estimation may be degraded. Simulation results confirm that for small number of probe packets (Mr 45) the minimum of x occurs in P ¼3. However, for M445, the minimum value of x occurs at P¼4 or P¼5. Note that, the error estimation at P¼4 or P ¼5 is marginally smaller than that of achieved for P¼3. Therefore, in practice, we have a tradeoff between the accuracy of estimation, the cost of overload of probe packets and computational complexity due to the increasing P.

8 7 6 5 4 3 2 1 0

0.5

1

1.5 Time [msec]

2

2.5

3 x 105

Fig. 13. Comparison of MR-BART and BART(P ¼2, M¼ 17, S¼ 1500 bytes).

10 Real A MR-BART (P = 3, M = 17) BART (M = 17)

9 Available bandwidth [Mbits/sec]

740

8 7 6 5 4 3 2 1 0

0.5

1

1.5 Time [msec]

2

2.5

3 x 105

Fig. 14. Comparison of MR-BART and BART(P ¼3, M¼ 17, S¼ 1500 bytes).

10 MR-BART (M = 16) MR-BART (M = 34) MR-BART (M = 46) Real A

Available bandwidth [Mbits/s]

9 8 7 6 5 4 3 2 0

1

2

3 4 Time [msec]

5

6

7 x 104

Fig. 12. Impact of variation of M on accuracy of MR-BART estimate (P ¼3, S¼ 1500 bytes).

MR-BART: In this method we divide the probing sequence into P portions so that each portion has a random rate which is independent from other portions’ rates. In this situation the probability that the rate of all portions is less than AB will be reduced. Therefore we expect that sending each probing sequence leads to a new AB estimation. In other words, due to the portioning, the chance of accurate observation of the cross-traffic increases and the probability that the traffic do not be observed decreases (see Figs. 13 and 14). Equivalently, by portioning we reduce the dependency of the estimation process on the Q in inferring the traffic fluctuation. Figure 15 confirms this issue. This figure shows the effect of adjusting the matrix Q matrix on the accuracy of our estimation. As it is seen from this figure, since MR-BART uses KF like BART method, then adjusting the Q matrix can improve the performance of our method too. But you should regard that, our method is not dependent mainly on Q matrix, and can estimate AB more accurate than BART even without adjusting the matrix Q matrix accurately (Fig. 14 shows that the increasing the P leads to the accurate estimation without need for accurately adjusting the Q matrix). This claim is shown in Fig. 16. In this figure, MSE of (unadjusted Q) MR-BART is equal with 0.01. MSE of

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

10

8

^ ðMbits=sÞ ðAÞ 0

MR-BART

BART

7

2.5 8.5 5

0.033 0.008 0.006

0.057 0.022 0.009

6 5 4

10

3

9

2 1 0 0

0.5

1

1.5 2 Time [msec]

2.5

3 x 105

Fig. 15. The impact of adjusting Q matrix on accuracy of estimation in MR-BART (P¼ 3, M ¼ 34).

10

Available bandwidth [Mbits/sec]

8

Real A MP-CR-BART (P=2, M = 17) MR-BART (P = 2, M = 17)

8 7 6 5 4 3 2

Real A BART (adjusted Q) MR-BART (adjusted Q) MR-BART (unadjusted Q)

9

Available bandwidth [Mbits/sec]

Available bandwidth [Mbits/sec]

Table 4 MSE of MR-BART (P ¼2) and BART estimation when using three arbitrary initial states for Kalman filter S¼ 1500 bytes, M ¼ 17, (A)0 ¼6.2 Mbits/s.

Real A MR-BART (adjusted Q) MR-BART (unadjusted Q)

9

741

1 0

7

1

2

3 4 Time [msec]

5

6

7 x 104

Fig. 17. Comparison of MR-BART and MP-CR-BART (P¼ 2, M ¼ 17).

6 5 4 3 2 1 0 0

0.5

1

1.5 2 Time [msec]

2.5

3 x 105

Fig. 16. Impact of adjusting Q matrix on MR-BART method in comparison with BART method (P ¼3, M ¼34).

(adjusted Q) MR-BART is equal with 0.008. MSE of (adjusted Q) BART is equal with 0.02. As it is seen from Fig. 16, BART method can track only one type of the variation of cross-traffic accurately depend on the value of Q (adjusted for low or fast variation of cross-traffic), and can not accurately track both types of variation simultaneously. In utilizing KF as an estimator, we need to consider an initial value for state vector. In our estimation problem, we need to ^ as an initial estimate of initialize available bandwidth, i.e., ðAÞ 0 initial real value, i.e., (A)0. Therefore in our simulations, we select ^ different values for ðAÞ 0 and analyze the behavior of x for ^ . In Table 4 we report the results inappropriate initial value of ðAÞ 0 of these experiments and compare the robustness of MR-BART and BART against the deflection from the real values of initial available bandwidth. ^ which is near to the (A)0 From this Table, we see that each ðAÞ 0 leads to the smaller MSE than that of is further. In addition, it can

be concluded that MR-BART shows better performance in case of inappropriate initial state of KF compared to BART. Here, we show the preference of our method in compare with BART method in a new situation. Suppose that in BART method M packets divide into P portions with constant rate and are sent back-to-back, then BART method will be applied to the observations. We name this as MP-CR-BART (multi-portion constant-rate BART). In Figs. 17 and 18, we compare our method with MP-CRBART for P¼2 and P¼3, respectively. As we expect, the results show that MR-BART estimates AB more accurately than BART method (in particular for P ¼3). In our method, we use probing sequences with P independent rates that means we probe the path with P different independent rates by each probing sequence. But, in MP-CR-BART for each probing sequence only one rate is chosen and therefore produced observations are originated from one constant rate. As it is seen, the preference of our method is more obvious in Fig. 18, because in this figure the multi-rate characteristic of MR-BART reveals itself more clearly. In Fig. 17, MSE (MR-BART) is equal with 0.007 and MSE (MP-CR-BART) is equal with 0.01. In Fig. 18, MSE (MR-BART) is equal with 0.01 and MSE (MP-CR-BART) is equal with 0.05. 5.5. Impact of MR-BART probing traffic on TCP performance However measuring the impact of MR-BART probing traffic on TCP is not within the scope of this paper, we use a related work as a reference and describe the expected impact of our method on TCP performance. Since TCP is a dynamic protocol that varies the send rate depending on network latency and loss rate, the injection of probe packets will affect its performance. The reduction of TCP performance is due to the fact that the probe packets cause TCP packet loss, by long probe packet trains (sequences), and increased round-trip time, mainly caused by injecting many

742

M. Sedighizad et al. / Journal of Network and Computer Applications 35 (2012) 731–742

10 MR-BART (P = 3, M = 34) MP-CR-BART (P = 3, M = 34) Real A

Available bandwidth [Mbits/sec]

9 8

measurement noise, the number of computations grows only as the dimension of the measurement vector. Therefore, we can still obtain a real-time more accurate estimate of the AB than the conventional BART estimate, with marginal addition to the complexity of computation.

7 6

References

5 4 3 2 1 0 0

2

4

6 8 Time [msec]

10

12 x 104

Fig. 18. Comparison of MR-BART and MP-CR-BART (P ¼3, M ¼ 34).

¨ probe packet pairs (Johnsson and Bjorkman, 2006). As it is ¨ investigated by Johnsson and Bjorkman (2006), the medium length probe packet sequences are the pattern to be used in order to minimize the reduction in TCP performance. Johnsson and ¨ Bjorkman (2006) analyzed the impact of the probing sequence pattern (length) on TCP performance. The scenarios are the following: injection of 16 probe packet pairs, 2 probing sequence of length 9 and one probing sequence of length 17, once a second, respectively (using BART method). Each probing sequence pattern results in 16 samples per second. It is shown that the reduction in TCP performance is less when 2 probing sequence with length 9 is used compared to the case of probe packet pairs or probing sequence of length 17 is used. Finally they concluded that to minimize the impact of probe packet on TCP one might choose to use medium length probing sequence. In the view of affecting the TCP performance our method (MR-BART) and BART method are alike. Because we use the probing sequence with the same length of BART method. However, if we use the medium length probing sequence in order to reduction in TCP performance, then it seems our method will estimate AB more accurately than BART. Because, BART method calculates the average of all measured strain as the measurement strain and it causes the increment in variance. But our method calculates the average strain of each portion separately.

6. Conclusion In this paper we proposed a new method called MR-BART. This method is an efficient method for real-time estimation of the available bit-rate in a network path with concurrent cross-traffic using Kalman filtering. In the proposed method a probing sequence consists of multi-rate probe packets is utilized. Indeed, in this method by using the new parameter, i.e., P, we have more degree of freedom to design an estimator of Available Bandwidth. The proposed method is highly accurate and converges quickly in compared with conventional BART method. In addition, this method is robust against inappropriate initial value of KF. Due to the special feature of the covariance matrixes of the

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