Bandwidth Provisioning in ADSL Access Networks

Transcription

1 Bandwidth Provisioning in ADSL Access Networks Kaiqi Xiong, Harry Perros, and Steven Blake 2 Department of Computer Science, NC State University, Raleigh, NC , USA {xiong,hp}@csc.ncsu.edu 2 Extreme Networks Raleigh, NC, USA sblake@extremenetworks.com Abstract. We consider an ADSL access network consisting of subscribers, s, metro Ethernet switches and a broadband remote access aggregation server (BRAS). We obtain expressions for dimensioning the access network in the upstream direction. Specifically, we show that the bandwidth required at each, metro Ethernet switch, and BRAS can be expressed as an exponential function of the subscribers in a log scale when the number of subscribers does not surpass a certain value. After this value, it grows linearly as a function of the subscribers. Keywords: Network provisioning, ADSL networks, equivalent bandwidth INTRODUCTION Triple play services have dramatically increased the bandwidth required in an ADSL access network. In addition, it is anticipated that the number of subscribers that can be supported by a single ADSL access network will easily reach to a,. In this paper, we obtain efficient expressions for dimensioning the upstream links of the s, metro Ethernet switches, and a broadband remote access aggregation server (BRAS) in an access network. Network provisioning has been widely studied in the literature. For example, [3], [4], [9], [], and [2]. Cao and Zegura [3] considered the bandwidth allocation scheme for an available bit rate service. In Liao and Campbell [4], a mechanism was developed with the capability of delivering capacity provisioning in an efficient manner providing quantitative delay-bounds with differentiated loss across per-aggregate service classes. Nowak, et al. [9] developed a C++ event driven simulator to measure these values in optical networks. In paper [], the author proposed a dimensioning model of an access network using the bufferless Elrang loss model. Resource allocation has been studied for differentiated services in Xiong and Perros [2]. One of the challenging problems is the dimensioning of an output port buffer into which multiple bursty and correlated arrival streams flow. This buffer is modeled by a queueing system, known as the statistical multiplexer, which has

2 2 Xiong et al. been widely studied particularly within the context of ATM networks. For instance, Anick et al. [] analyzed an ATM multiplexer using fluid flow models. Both statistical multiplexing and statistical bandwidth allocation are quantitatively evaluated for ATM traffic in Saito et al. []. Jelenkovic and Lazar [8] analyzed the network traffic of multiplexing sub-exponential on-off sources. Hasslinger [6] provided a performance evaluation of statistical multiplexing, where traffic flows were modeled by autoregressive processes producing autocorrelated and Gaussian distributed workloads. The equivalent bandwidth provides a simple way for the calculation of the bandwidth required by a variable bit connection in a statistical multiplexer, so that it experiences a given packet loss. The calculation of equivalent bandwidth was originally proposed by Guérin and Ahmadi [5]. For a discussion of its behavior, see Perros [7]. Zukerman [3] used a Poisson Pareto burst process (PPBP) to fit Internet traffic traffic stream for the calculation of the parameters of the PPBP, and predict the queueing performance of a sample trace of aggregated Internet traffic. Recently, Cho et al. [2] reported aggregated traffic measurements collected over 2 months from seven ISPs covering 42% of the Japanese backbone traffic. Characterizing the Internet traffic of an entire country in [2] will certainly be very useful in bandwidth provisioning. In this paper, we obtain expressions for dimensioning the access network in the upstream direction, i.e., from the subscribers to the network. We decompose the problem of dimensioning an ADSL access network into three levels of hierarchically interconnected statistical multiplexers. The first level consists of statistical multiplexers each representing a number of subscribers served by a. Each subscriber is represented by an interrupted Poisson process (IPP), and there may be a large number of these multiplexers since we assume a large number of subscribers. The second level consists of a smaller number of multiplexers, each representing a number of s served by a metro Ethernet switch. Finally, the third level consists of a single multiplexer that represents the metro Ethernet switches served by the BRAS. We use the equivalent bandwidth expressions to dimension each multiplexer. A key requirement for this analysis is the characterization of the departure process from one multiplexer which becomes part of the arrival process to a higher level multiplexer. We note that despite the extensive literature on statistical multiplexers, the problem of characterizing the departure process from a multiplexer has not been adequately addressed. In this paper, we show that when the number of IPP arriving streams into a multiplexer is large, the departure process can also be approximated by an IPP as well, where the mean on and off periods can be easily computed approximately. This simplified the analysis of these multiplexers and permitted us to use the equivalent bandwidth for dimensioning the access network. Based on these dimensioning results, we show that the bandwidth requirement at a multiplexer can be predicted by the following formula: BW = { rca log U, for < U U ˆrCU + b for U > U ()

3 Bandwidth Provisioning in ADSL Access Networks 3 where U is the number of subscribers, C is the subscriber peak bit rate, and the base of log is. a, r, ˆr, and b are constants with a, r, ˆr>. All of them are determined for the multiplexer under study. Formula () states that the bandwidth requirement at a multiplexer can be expressed as an exponential function of the subscribers U in a log scale when U is less than a certain value U. For values of U > U, it grows linearly in U. Note that a log U = U log a is not equivalent to U when a is not equal to. To the best of our knowledge, no one has studied and verified () before. The rest of the paper is organized as follows. In Section 2, we describe the dimensioning problem for the upstream case of an ADSL access network. In Section 3, we characterize the departure process from a statistical multiplexer, and in Section 4 we give the equivalent bandwidth expressions for each of the three level multiplexers. Numerical results are given in Section 5. Finally, our conclusion and further work are given in Section 6. 2 The Bandwidth Provisioning Problem An ADSL access network consists of subscribers which are served by s. Groups of s are connected to a metro Ethernet switch, and all the switches are connected to a BRAS, as shown in Figure. We provide formulae for dimensioning such an access network. Central Office Metro Metro Ethernet Ethernet BRAS Internet Application Provider Central Office Fig.. An ADSL access network The upstream ADSL access network shown in Figure can be modeled by the queueing network depicted in Figure 2. The finite queues represent the upstream output ports of the s, metro Ethernet switches and BRAS. Each queue is served by a single server which represents an upstream link. The service rate, which is the capacity of the link, is the same for all queues, and it is the same for all the metro Ethernet switch queues. Let µ j, j =, 2, 3, be the service rate of a, metro Ethernet switch and BRAS. The queueing network consists of three levels of statistical multiplexers. The first level represents the s, where groups of subscribers are connected to a. Each subscriber generates traffic which is modelled as an interrupted Poisson process (IPP), i.e., it is an on-off process where the on and off periods are exponentially

4 4 Xiong et al. distributed. Packets arrive only during the on period in a Poisson fashion. All subscribers are assumed to have an identical IPP. Let us assume that the total number of s is N. Groups of s are connected to a metro Ethernet switch, and let N 2 be the total number of switches. These switches are all connected to a BRAS which is connected to the Internet. µ Metro switch µ 2 µ BRAS µ 3 µ Metro switch µ 2 µ Fig. 2. The Upstream Queueing Network Model To analyze this queueing network, we decompose it into the three levels of multiplexers. Since all the multiplexers are similar, it suffices to consider a single one. We use the equivalent bandwidth to dimension the queue for a fixed number of subscribers and IPP parameters. Likewise, it suffices to consider a single metro Ethernet switch multiplexer. The input to this multiplexer is the output of all the multiplexers connected to it. In view of this, we need to characterize the output process of the multiplexer. In the following section, we characterize the output process from a multiplexer as an IPP process. This characterization is correct for a large number of input sources, i.e. subscribers. This approach now permits us to treat the metro Ethernet switch multiplexer in the same way we treat the multiplexer. Consequently, we can use the equivalent bandwidth to dimension a switch. The same process applies to the BRAS multiplexer. 3 The Departure Process From a Multiplexer In this section, we characterize the departure process from a and a metro Ethernet switch. On-off sources have been widely studied in the literature as they capture the bursty nature of network traffic. Hence, in this paper, we assume that the traffic generated by each subscriber is an IPP. We consider the multiplexer shown in Figure 3. Let N be the number of IPP identical independent sources flowing into the multiplexer, and let µ be the service rate expressed in bits/sec. We assume exponentially distributed service times. Obviously, the departure process is an on-off process. The on and off periods correspond to the busy and idle periods of the multiplexer. Of interest is to identify the distribution of those two periods. In view of this, we simulated the multiplexer for different traffic

5 Bandwidth Provisioning in ADSL Access Networks 5 intensities and different values of N. In order to construct an efficient simulator, we did not simulate individual packets. Rather, since an on period of an IPP arrival stream represents a burst of packets, we treat each burst as a single customer. (Customers do not arrive instantaneously. It takes the equivalent of an on period for a customer to arrive.) This simplification may have an impact on the waiting time of an individual packet, but it does not alter the duration of the busy and idle periods of the multiplexer. N Sources µ Fig. 3. The Multiplexer Let us assume that the on and off periods of an IPP source have a mean of α and β respectively, expressed in seconds. Then, each source transmits one burst for a period of time which is equal to α on the average followed by an off period with a mean of β. As a result, the rate at which it transmits bursts is ( α )+( ). β Therefore, the total rate of transmission from all N sources in bursts/sec is N λ = ( α )+( β ). Let R be the peak bit rate of a source. Then, the total arrival rate to the multiplexer expressed in bits/sec is ˆλ N = ( α )+( )R α = λr α. Furthermore, β let τ be the mean idle period of the multiplexer expressed in seconds and ρ its utilization. We approximate τ by the burst inter-arrival time, i.e., τ = λ = ( ) N α β α+β = α+β N α β. Now, let s be the mean of the busy period of the multiplexer expressed in seconds. Then, is the percent of time that the multiplexer is s s+τ busy, which is equal to ρ, where ρ = ˆλ. (We recall that µ is also expressed in bits/sec.) That is, s s+τ = ρ, which derives that s = τ µ /ρ = ˆλ µ ˆλ λ = R µ N R β α. α+β From a large number of simulation experiments, we have concluded that the distribution of the on and off periods of the departure process can be approximated by an exponential distribution with a mean on and mean off as specified by τ and s above when N is not too small. In Figures 4, 5 and 6, we plot the cumulative distribution function of the busy and idle periods for N =5, 5, respectively. In addition, we have fitted an exponential distribution to the simulation data which is also shown in Figures 4, 5 and 6 as well. We note that the exponential distribution fits the simulation data very well when N = 5 and N = 5, but not when N =. A large number of our experiments showed that the larger the value of N, the better the fit. 4 Network Bandwidth Provisioning We calculate the required bandwidth at each multiplexer level, using the notion of the equivalent bandwidth. Let us consider that a source produces packets that

6 6 Xiong et al. Cumulative Distribution Function F(t) On Period(N=5).8.6 CDF.4 exponential fit Time t Cummulative Distribution Function F(t) Off Period (N=5)..8.6 data exponential fit Time t (a) On period (b) Off period Fig. 4. The departure process when N = 5 Cumulative Distribution Function F(t) On Period (N=5).8.6 CDF.4 exponential fit Time t Cummulative Distribution Function F(t) Off Period (N=5)..8 data.6 exponential fit Time t (a) On period (b) Off period Fig. 5. The departure process when N = 5 Cumulative Distribution Function F(t) On Period (N=).8.6 CDF.4 exponential fit Time t Cummulative Distribution Function F(t) Off Period (N=)..8.6 data exponential fit Time t (a) On period (b) Off period Fig. 6. The departure process when N =

7 Bandwidth Provisioning in ADSL Access Networks 7 are queued in a finite buffer which is served at a given rate. Then, the equivalent bandwidth of the source is defined as the service rate at which the queue is served that corresponds to a packet loss rate of ε. The equivalent bandwidth of a source falls somewhere between its average arrival rate and its peak arrival rate. If the source is very bursty, then it is closer to its peak service rate. Otherwise, it is closer to its average arrival rate. Various approximations have been suggested to compute the equivalent bandwidth of a source. One of the most commonly used approximations adopts the assumption that each source is an interrupted fluid process (IFP) characterized by the triplet (R j, ρ j, α j ) for j =, 2,, N, where R j is the peak bit rate of source j, ρ j the fraction of time that source j is active, defined as the ratio of the mean length of the on period divided by the sum of the mean on and off periods, and α j the mean duration of the on period. We further assume that the N sources flow into a finite-capacity queue with a constant service time, and let K be the size of the queue expressed in bits. Then, the equivalent bandwidth e j of source j is calculated by (refer to [7]): e j = [η j K + (η j K) 2 + 4Kη j ρ j ] R j 2η j (2) for j =, 2,, N, where η j = α j ( ρ j )R j ln ε. The equivalent bandwidth of N sources can be given by the expression (refer to [7]): E = min{γ + σ 2 ln(ε) ln(2π), N e j } (3) where γ is the average bit rate of all the sources, and σ the sum of the standard deviation of the bit rate of all the sources. σ is obtained below. Denote by ξ j the random variable representing the bit rate generated by source j. Then, the probability distribution of ξ j can be expressed by p(ξ j = R j ) = ρ j and p(ξ j = ) = ρ j. This implies that the expectation of ξ j is given by E(ξ j ) = ρ j R j + ( ρ j ) = ρ j R j. That is, the average bit rate of source j is ρ j R j denoted by δ j. Moreover, let σ j be the standard deviation of the bit rate of the j-th source. Then, σj 2 = E(ξ j E(ξ j )) 2 = ρ j (R j δ j ) 2 +( ρ j )( δ j ) 2 = ρ j R j (R j ρ j R j ) = δ j (R j δ j ), which means that σ j = δ j (R j δ j ). Subsequently, the sum of the standard deviation of the bit rate of N identical sources σ is computed by σ 2 = N j= σ2 j = N j= δ j(r j δ j ). N That is, σ = j= δ j(r j δ j ). Bandwidth Provisioning in the : In order that the has a packet loss rate of less than ε its bandwidth B should be more than the equivalent bandwidth E, i.e., B µ = min{γ + σ 2 ln(ε) ln(2π), N j= e j}.we have assumed that all N sources are assumed to be identical IPPs, with a mean on and off period given by α and β respectively. Therefore, for the j- th source we have R j = c j, where c j is the link speed between the j-th source and the multiplexer. ρ j = /α /α+/β = β α+β, η j = α ( ρ j)r j ln ε = R j α+β ln ε for j=

8 8 Xiong et al. j =, 2,, N. Let K d be the size of the finite-capacity queue of the expressed in bits. Thus, we have B min{γ + σ 2 ln(ε) ln(2π), Ne } (4) where σ = R Nρ ( ρ ) and e is determined by { [ e = R ln ] { [ ε K d (α + β) + R ln ε K d (α + β) +4K d R β ln ε This is because we have that } 2 } ( 2 ln ε ) e = η K d + (η K d ) 2 + 4K d η ρ R 2η = R α+β ln ε K d + ( R α+β ln ε K d) 2 + 4K d R α+β ln ε 2R α+β ln ε ] 2 β α+β from which the above expression of e can be easily derived. Bandwidth Provisioning in the Metro Ethernet Switch: As discussed in Section 3, the departure from each is approximated by an IPP process. This implies that the arrival process from each to a metro Ethernet switch is an IPP as well. According to Section 3, the off period of the process has a mean of α+β a mean of ρ = ˆλ Nαβ ρ α+β ρ Nαβ N R /α+/β α µ = α d = µ NR β α+β denoted by β d denoted by α d µ = NRβ µ (α+β) R expressed in seconds, and its on period has expressed in seconds, where ρ is given by, which derives that the on period has a mean of R α. Similarly, the equivalent bandwidth e m of a source at the metro Ethernet switch is given by e m = [η m K m + (η m K m ) 2 + 4K m η m ρ m ] R m 2η m (5) where η m = α d ( ρ m )R m ln ε, K m is the size of the finite-capacity queue of the metro Ethernet switch expressed in bits, R m is the peak bit rate of a source in the metro Ethernet switch, ρ m is the fraction of time that the source is active, defined as the ratio of the mean length of the on period divided by the sum of the mean on and off periods, that is, ρ m is given by ρ m = β d α d +β d. Notice that the traffic of N s flows into a metro Ethernet switch and we assume that all these flows are identical. Thus, in order that the has a packet loss rate of less than ε its bandwidth B 2 should be more than the equivalent bandwidth of all N sources, i.e., B 2 min{γ m + σ m 2 ln(ε) ln(2π), N e m } (6)

9 Bandwidth Provisioning in ADSL Access Networks 9 where γ m is the average bit rate of all the N sources flowing into a metro Ethernet switch, and σ m is the sum of the standard deviation of the bit rate of all the sources given by σ m = N δ m (R m δ m ) = R m N ρ m ( ρ m ) due to δ m = ρ m R m. Similarly, we can express e m as a function of inputs α and β like e. We omit it here. Bandwidth Provisioning in the BRAS: Assuming that more than 5 s are connected to a metro Ethernet switch, the departure process from a switch can be approximated by an IPP. The mean off period is expressed in burst/sec, and the mean on pe- α d +β d α N α d β d, or β m = N d β d α d +β d riod α m = ρ d α d +β d N ρ d, or α m = N ρ d α d β d NRmβ d µ 2 (α d +β d ). β m = ρ d α d β d α d +β d expressed in burst/sec, where ρ d is given by ρ d = Notice that the traffic of N 2 metro Ethernet switches flows into a BRAS router and assume that all these flows are identical. Then, in order for the BRAS to have a packet loss rate of less than ε its bandwidth B 3 should be more than the equivalent bandwidth of all N 2 sources, i.e., B 3 min{γ b + σ b 2 ln(ε) ln(2π), N2 e b } (7) where γ b is the average bit rate of all the N 2 metro switch s sources flowing into a BRAS router, and σ b is the sum of the standard deviation of the bit rate of all the sources given by σ b = R b N2 ρ b ( ρ b ). R b is the peak bit rate of a source at the BRAS, and e b is the equivalent bandwidth of a metro Ethernet switch source expressed by the equivalent bandwidth e m of a source at the metro Ethernet switch is calculated by [η b K b + ] (η b K b ) 2 + 4K b η b ρ b R b e b = (8) 2η b where η b = α m ( ρ b )R b ln ε, K b is the size of the finite-capacity queue of the metro Ethernet switch expressed in bits, and ρ b the fraction of time that the source is active, defined as the ratio of the mean length of the on period divided by the sum of the mean on and off periods, that is, ρ b is computed by ρ b = βm α m +β m. 5 Numerical Results In this section, we provide numerical results that illustrate the bandwidth required in the upstream link of a, a metro Ethernet switch and a BRAS based on the model assumptions given in Section 2. The bandwidth required was calculated using the equivalent bandwidth expressions (4), (6) and (7). Denote by K d, K m, and K b the buffer in the, the metro switch and the BRAS respectively. Let us first study the bandwidth required in the upstream link of a. Suppose that K d = 5 Mbits. That is, the buffer size is such that the maximum delay in the uplink output port is no more than 5 msec when the uplink capacity is Gbps. We also choose R = 25 Mbps, α =.667 sec, β =.6 sec, and ε = 6. The average size of a burst is equal to R /α, or

10 Xiong et al EB Expon. (EB) y = 6.24e.89x R 2 = EB Linear (EB) y = 2.677x R 2 = #Subscribers in a log scale #Subscribers (a) < U 272 (b) U > 272 Fig. 7. Bandwidth required in the upstream link Mbits, on the average. Figures 7(a) and 7(b) show the bandwidth required in the upstream link based on the expression (4) for the equivalent bandwidth as a function of the number of subscribers U when < U U = 272 and U > 272 respectively. We fitted various curves into the results obtained of which the best-fit curve is y = 6.24e.89x, where x = log U, with R 2 =.997, when < U 272. The value of U is estimated through a brute force search in the curve fitting. Note that Figure 7(a) is in log scale but Figure 7(b) is not. We use the R 2 value to indicate how close the curve fits the experimental data. (R 2 = means a perfect fit, R 2 = implies that the curve does not fit the experimental data at all.) That is, when < U 272 the bandwidth required in the upstream link of a as a function of the number of subscribers U is given as follows. B = 6.24e.89 log U = log U (9) Furthermore, we fitted various curves into the results obtained of which the bestfit curve is y = 2.677x , where x = U, with R 2 =.9999 when U > U = 272. That is, the bandwidth required in the upstream link of a as a function of the number of subscribers U is B = 2.677U , when U > 272. Actually, this linear relation between B and U can be mathematically confirmed below. Since the average load (γ) dominates the square root of N in the standard deviation (σ) for large values of N, (3) is O(N) in the case of a large number (N) of subscribers. This means that (4) is O(N) in the case of a large number (N) of subscribers as well. That is, B is a linear function of N or U n the case of a large number (U) of subscribers. However, it is not straightforward to mathematically prove that the rest of results given in the form () hold for, metro Ethernet switch and BRAS. Below we continue to focus on the numerical validations of the results given in the form (). As can be seen, the above expressions have the same form as () if a = 6.657, r =.6486, b = and ˆr =.43, since C = R = 25 Mbps. Thus, it follows from (9) that we can get U = 89 when B = Gbps. That is, a with GE uplink to a metro Ethernet switch can serve 89 subscribers. Of course, the number of subscribers will change if we change the subscriber s traffic characteristics /α and /β. (The arrival rate of packet during the on period is not of interest since we assume that the source transmits continuously during the on period.) Subsequently, we analyze a metro Ethernet switch. Specifically, we will calculate how many upstream links can be supported by a single GE

11 Bandwidth Provisioning in ADSL Access Networks EB Expon. (EB) x y = 49.7e R 2 = EB 4 Linear (EB) 3 2 y = 483.4x R 2 = #s in a log scale #s (a) < U 999 (b) U > 999 Fig. 8. Bandwidth required in the metro Ethernet switch uplink from a metro Ethernet switch to BRAS. We also assume that K m = Mbits. That is, the buffer size is such that the maximum delay in the uplink output port is no more than msec when the metro uplink capacity is Gbps. Using the formulas derived in Section 3, we have that the on and off means of the departure process from a are given by α d =.36 sec and β d =.35 = 3.6 Mbits. Figures 8(a) and 8(b) show the bandwidth requirement in the metro switch as a function of the number of s uplink D based on the implementation of (6). As shown in Figures 8(a) and 8(b), the best-fit curves to the calculated required bandwidth are y = 49.7e 2.326x, where x = log D, with R 2 =, when < x 5, and y = 483.4x + 295, where x = D, with R 2 =, when < x 5. That is, the bandwidth required in the metro Ethernet switch is given by sec. The size of a burst is equal to α d B 2 = { 49.7e log D = log D, for < D D 483.4D + 295, for D > D () where D is the total number of uplinks to the metro Ethernet switch served by the GbE uplink to BRAS, and D = 5 is estimated in the same way as U. Moreover, notice that a permits 89 subscribers. Hence, given a number of subscribers U, we need D = U 89 uplinks. Therefore, log D = log( U ) = log U and the bandwidth required in the metro Ethernet switch can be rewritten as { log B 2 = U, for < U 9639 () U + 295, for U > 9639 where U is the number of subscribers. This expression has the form of () if a =.2, r =.38, ˆr =.23, and b = 295, since C = R = 25 Mbps. It follows from () that D = 2 when B 2 = Gbps. Thus, one GE link from a metro Ethernet switch to BRAS can serve 2 s. Finally, we analyze the BRAS. As above, we will calculate how many metro Ethernet switch GE uplinks can be supported by a single GE uplink from BRAS to WAN. Let us also assume that K d = 2 Mbits. That is, the buffer size is such that the maximum delay in the uplink output GE port is no more than 2 msec. The departure traffic of the metro Ethernet switch can be determined

12 2 Xiong et al. 2 5 EB Expon. (EB) y = 9452e 2.326x 5 R 2 = EB y = 9452x + E-8 Linear (EB) R 2 = #Metro sw itches in a log scale #Metro switches (a) < U 684 (b) U > 684 Fig. 9. Bandwidth required in the BRAS by the use of the formulas presented in Section 3. They are respectively given by α m = / sec and β m = / burst/sec. The size of a burst is Mbits. Figures 9(a) and 9(b) show the bandwidth requirement in the BRAS upstream link as a function of the number of the metro Ethernet switch GE uplinks to BRAS, based on the implementation of (7). As we see in these figures, the best-fit curves are y = 9452e 2.326x, where x = log S, with R 2 = when < x 8, and y = 9452x + 8, where x = S, with R 2 = when x > 8. That is, the bandwidth required in the BRAS is given by B 3 = { 9452e log S = log S, for < S S 9452S + 8, for S > S (2) where S is the total number of GE uplinks from the metro Ethernet switches to the BRAS, and S = 8 is estimated in the same way as U and D. Again, notice that one GE uplink from a permits 89 subscribers and one GE uplink from a metro Ethernet switch permits 2 s. Hence, given U, we need U 89 D s, i.e., GE uplinks, and D 2 S s, i.e., GE metrio switch uplinks. Therefore, log S = log( D 2 ) = log( U ) log 2 = log U This means that the bandwidth required in the BRAS can be rewritten as { log B 3 = U, for < U U + 8 (3), for U > 684 where U is the number of subscribers. This expression has the same form as () when a =.2, r =.6, ˆr =.2, and b = 8, since C = R = 25 Mbps. When choosing B 3 = GE in (2), we get that S =. That is, one GE uplink from BRAS to WAN can support incoming GEs. It is interesting that in equation (3) the first expression is almost the same as the second one. This means that the traffic smoothes out after two levels of multiplexing and as a result there is no statistical gain when U is small. Of course, the result is dependent on how bursty the subscribers are. Thus, suppose that there are N s connected to a metro switch each via a GE link. Then, the number M of the required GE links from the switch to the BRAS is M = N 2. For instance, if N = 2, then M = 6.

13 Bandwidth Provisioning in ADSL Access Networks X 4X 6X #Subscribers in a log scale (a) at a metro Ethernet switch, one GE uplink X 4X 6X #s in a log scale (b) at a metro Ethernet switch, one GE uplink Fig.. Bandwidth required with the buffer sizes of X, 4X, and 6X Similarly, suppose that there are M GE uplinks from metro switches to a BRAS. Then, the number of the required GE uplinks from the BRAS to the WAN is BR = M. Hence, if M = 2, then BR = 2. Notice that the above conclusions hold based on our assumption about the subscriber traffic and buffer on the uplinks of the, the metro Ethernet switch and the BRAS. Formulas (9), (), and (3) have the same form as (). They also give the relationship between bandwidth requirements and the number of subscribers at each aggregation layer. For a given number of subscribers, the bandwidth required at each aggregation level can be estimated using these three formulas. Alternatively, given the capacity of each aggregation layer we can determine how many subscribers can be supported. We also studied the dimensioning problem with varied buffer sizes that are not reported in this paper due to space limit. Sensitivity Analysis The bandwidth required at each aggregation level was re-calculated assuming that the buffers at a, metro Ethernet switch, and BRAS were 4X and 6X, where X was the original buffer size, as shown in Table. Table. Buffer Sizes in the, the metro Ethernet switch and the BRAS Router Buffer Size (Mbits) X 4X 6X metro Ethernet switch BRAS router Figure (a) shows the bandwidth required at the with the buffer sizes of X, 4X and 6X. As indicated in the figure, the bandwidth requirement is almost the same for the buffer sizes of 5.2 Mbits and 2.48 Mbits when the number of subscribers U is more than (or log U = 2), but it requires about 3% less bandwidth for the buffer size of 8.92 Mbit (6X) compared to the buffer size of 5.2 Mbits (X).

14 4 Xiong et al. Fig.. Bandwidth Required at BRAS, one GbE uplink with the buffer sizes of X, 4X, and 6X The bandwidth requirements at a metro Ethernet switch and BRAS router are shown in Figures (b) and respectively. In the case of the metro Ethernet switch, it requires about 6.66% less bandwidth for a buffer size of 4X and about 8.25% less bandwidth for a buffer size of 6X compared to the metro Ethernet switch with a buffer size of X. However, the bandwidth required in BRAS is almost the same for all buffer sizes, i.e, X, 4X and 6X, as shown in Figures, which suggest that the X buffer size is sufficient. 6 Conclusions and Further Work In this paper, we considered the dimensioning of an ADSL access network in the upstream direction. The problem was decomposed into a number of statistical multiplexers, each depicting the s, metro Ethernet switches and BRAS that make up an ADSL access network. We used the equivalent bandwidth for dimensioning each multiplexer. We showed that the departure process from each multiplexer can be approximated by an IPP. We also demonstrated that the bandwidth required at an uplink of a, metro Ethernet switch, or BRAS can be expressed as an exponential function of the number of subscribers in a log scale when the number of subscribers does not surpass a certain value. After this value, it grows linearly as a function of the subscribers. Acknowledgments. The authors would like to thank the reviewers for their helpful comments that help improve the paper. This research was funded by a grant from the Center of Advanced Computing and Communication (CACC), NC State University.

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