Characterization of the Burst Aggregation Process in Optical Burst Switching.

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1 See discussions, stats, and author profiles for this pulication at: Characterization of the Burst Aggregation Process in Optical Burst Switching. CONFERENCE PAPER in LECTURE NOTES IN COMPUTER SCIENCE JANUARY 2006 Impact Factor: 0.51 DOI: / _63 Source: DBLP CITATIONS 11 READS 18 2 AUTHORS, INCLUDING: Xenia Mountrouidou Jacksonville University 14 PUBLICATIONS 61 CITATIONS SEE PROFILE Availale from: Xenia Mountrouidou Retrieved on: 02 January 2016

2 Characterization of the Burst Aggregation Process in Optical Burst Switching Xenia Mountrouidou and Harry G. Perros Computer Science Department, North Carolina State University, Raleigh, NC 27695, USA {pmountr, Astract. We descrie an analytic approach for the calculation of the departure process from a urst ggregation algorithm that uses oth a timer and maximum/minimum urst size. The arrival process of packets is assumed to e Poisson or ursty modelled y an Interrupted Poisson Process (IPP). The analytic results are approximate and validation against simulation data showed that they have good accuracy. 1 Introduction An important design aspect of an OBS network is the urst aggregation process performed at the edge nodes. This process concentrates upper layer packets which are then transmitted optically over the OBS network. In view of this, the urst aggregation strategy defines the urst arrival process to the OBS network. This process depends on the parameters of the aggregation process, and so far it has not een adequately studied. However, it is important that the urst arrival process to an OBS network is well characterized if we are to understand etter the performance of OBS networks. The main parameters of a urst assemly algorithm are a timer and the maximum and minimum urst size. When the timer expires, the edge node assemles a urst that consists of packets in the edge s packet queue that have the same destination. This procedure takes place in the electrical domain, and the resulting ursts are transmitted in the optical domain. If the arrival rate at an edge node is very high, then each time the timer expires, there may e a large numer of packets waiting in the packet queue. This would lead to large data ursts if all packets are assemled into a single urst. Large ursts decrease the performance in the OBS network, since they occupy the resources for long intervals. As a result, they lock other ursts thus leading to a high urst loss proaility [1]. In order to avoid this prolem, a maximum urst size is used to ound the size of a urst. Another drawack of the urst assemly algorithm, if it is driven entirely y a timer, is that ursts can e very small if the arrival rate to the edge node is very low. This results to high overheads, since the OBS network has to set up a path for each urst. The solution to this prolem is to use a lower ound for the urst size. If this lower limit is not reached, when the timer expires, then the urst is not transmitted.

3 2 Various algorithms have een proposed to aggregate packets into ursts. Most of these assemly algorithms use either an assemly timer or a maximum and minimum urst length or oth as a way of creating ursts. Let T e the length of the timer, B max the maximum urst length and B min the minimum urst length. We can classify the assemly algorithms into the following three categories: Time-ased aggregation algorithms: In this case a fixed-threshold T is used to create a urst. In some implementations, a minimum length B min is required [2]. If the urst is shorter than B min then padding is used to increase the length to B min. The shortcoming of the time-ased assemly algorithm is that, under heavy traffic load, the numer of packets that are gathered until the timer expires may e high, thus resulting to large ursts. Burst-length ased aggregation algorithms: In this case, the urst is sent out as soon as the urst length exceeds a given maximum urst length B max. Thus, the packets are uffered until the total size reaches the maximum threshold. The main disadvantage of this algorithm is that it does not constraint the waiting time of the packets in the packet queue. Therefore, when the traffic is low, waiting time may e large. Time and urst-length ased urst aggregation algorithms: The disadvantages of the aggregation algorithms ased on a timer or a maximum urst length can e overcome using a comination of a timer and maximum and minimum urst lengths. In this case, the packets are uffered until the timer expires. Then, we compare the total size of the packets in the queue with the upper and lower limits, B max and B min. If the size is less than B min, then we keep the packets in the packet queue until the next aggregation period, i.e. until the next time when the timer expires. If the size is greater than B min ut less than B max we aggregate all the packets in one urst. If the size is greater than B max, we make one urst of maximum size and then we repeat this process with the remaining its. We note that an adaptation scheme has to e created which will assemle packets into ursts at the transmitter s side, and correctly recover these packets at the receiver s side. There are several examples of adaptation schemes, such as the schemes used for the formation of AAL 5 PDU and AAL 2 CPS packets in ATM networks. In OBS depending upon the adaptation scheme, a packet may straddle over two successive ursts. Alternatively, a urst may e allowed to exceed a maximum urst size, so that the last packet is included in its entirety. In this paper we assume that the maximum urst size is strictly enforced, and therefore a packet may straddle over two successive ursts. The urst aggregation process has een studied in [3], [4], [5], [6] and [2]. Papers [4], [5], [6] and [2] study the effect of urst aggregation algorithms on the self-similarity characteristics of the input traffic. [3] gives an analytical method to calculate the aggregated urst size for various algorithms, and assuming Poisson arrivals of packets to the edge node. The authors did not consider the aggregation algorithm that uses oth a timer and a maximum/minimum urst size analyzed in this paper. In this paper we otain analytically the distriution of the numer

4 3 of ursts created y the aggregation algorithm which uses oth a timer and a maximum/minimum urst size. We first assume that the arrival of packets to the edge node is Poisson, and then we extend the analysis to the case where packets arrive according to an Interrupted Poisson Process (IPP). This process models ursty traffic such as video, voice or data. The rest of this paper is organized as follows. In Section 2 we otain analytically the distriution of the numer of ursts created at each aggregation period assuming Poisson arrivals. In Section 3 we extend these results to IPP arrivals. The results otained in this paper are approximate and in Section 4 we compare our analytical results with simulation data. Finally, Section 5 gives the conclusions. 2 The case of Poisson Arrivals We consider an edge node which receives packets in the electronic domain and transmits them to destination edge nodes optically over an OBS network. The arriving packets are queued to different packet queues, each associated with a different destination edge node. We only consider a single packet queue in which packets are queued for a specific destination. We assume that the arrival process of packets to the queue is Poisson with a rate of λ. Packet size is exponentially distriuted with a mean of 1/ ytes. We recall that the length of the aggregation period, i.e. the time after which the timer expires, is T. Since packets arrive in a Poisson fashion, the proaility that n packets arrive within T is: P [X = n] = e λt (λt ) n n!. Therefore, the pdf of the numer of ytes B that arrive during the i th aggregation period ((i 1)T, it ] is: f B (x) = P [X = n]f Sn (x), (1) n=1 where f Sn (x) is the proaility that the total numer of ytes associated with n packets is x. This is otained y convoluting n i.i.d. exponentially distriuted variales, which in fact is the pdf of an n-stage Erlang distriution [7], given y: f Sn (x) = (x)n 1 e x (n 1)!. Thus, the pdf of the numer of ytes that arrive during the i th aggregation period is: f B (x) = e λt (λt ) n (x) n 1 e x n=1 n!(n 1)! The cumulative distriution function (cdf) of the numer of ytes in the packet queue at the end of the period T is: F B (x) = 0 f B (x) where f B (x) is given y (2). Using F B (x) we can calculate the proaility of the numer of ursts that are formed at the end of each period T. For instance, the (2)

5 4 numer of ytes x has to e within the interval [0, B min 1] in order to have zero ursts, it has to e within [B min, B min +B max 1] in order to have one urst, and in general it has to e within the interval [B min +(k 1)B max, B min +kb max 1] in order to have k ursts. That is: P [k = 0 ursts] = Bmin 1 0 Bmin+kB max 1 f B (x) (3) P [k ursts] = f B (x), k >= 1 (4) B min+(k 1)B max Expression 2 is quite difficult to work with, and whenever possile it is approximated y a simple mixture of exponential distriution as descried elow. As will e seen in order to do this, we need the first three moments of the numer of ytes in the packet queue at the end of each aggregation period. We note that the numer of ytes that arrive during period T is a random sum of exponentially distriuted variales. Therefore, the moment generating function (MGF) of the numer of ytes is [8]: M B (t) = M N (ln(m S (t))) (5) where M B (t) is the MGF of the numer of ytes during interval ((i 1)T, it ], M N (t) is the MGF of the numer of packets N, and M S (t) the MGF of the packet size S. Thus, we have: M N (t) = e λt (eln(m S (t)) 1), M S (t) = and therefore: M B (t) = e λt ( t t 1) (6) At the end of each aggregation period there may e a residual numer of ytes r which are not transmitted in a urst ecause of the condition that a urst has to e at least greater than B min. We have found empirically that if B min <<< B max, then the residual numer of ytes is typically zero. On the other hand, if B min is close to B max, then we have oserved that the residual numer of ytes is uniformly distriuted within [0, B min ). For instance, if B min = 16 Kytes and B max = 112 Kytes, then there is a high proaility that the last urst will e larger than 16 Kytes, which means that the residual numer of ytes will e zero. However, if we set B min = 85Kytes then there is a high proaility that the last urst will not e greater than B min, which means that it will not e transmitted out. This remainder can e safely assumed that it is uniformly distriuted within [0, B min ). In view of the aove empirical oservations, we distinguished two cases. If B min <<< B max, then we assume that there is zero left over ytes from the previous aggregation period, in which case the pdf of f B (x) and its MGF M B (t) are given y expressions (4) and (8). If B min is close to B max, then the pdf f X (x) of the numer of ytes at the end of an aggregation period is given y the convolution of f B (x) and f r (x). We have: f X (x) = f B (x) f r (x). Therefore, the MGF of f X (x), M X (t) is given y ([9]): M X (t) = M B (t)m r (t) or M X (t) = { e λt ( t 1) etb min 1 e λt ( tb min if t > 0 t 1) if t = 0 (7)

6 5 We can now calculate the moments of the distriution of the numer of ytes availale in an aggregation period for oth models. In the first model, where we do not include the residual numer of ytes, we have: m 3 = M (3) B m 1 = M B(0) = λt, (8) m 2 = M B(0) = λt (2 + λt ) (9) 2 λt (0) = [(2 + λt )(3 + λt ) + λt ] (10) 3 In the second model, where we include the residual of the numer of ytes, we have: m 1 = M X(0) = λt + B min, (11) 2 m 2 = M X(0) = λt 2 (2 + λt ) + B2 min + λt B min (12) 3 m 3 = M (3) λt X (0) = 3 [(2+λT )(3+λT )+λt ]+B3 min + 3λT B [ min 1 ] 4 2 (2+λT )+B min 3 (13) From the first three moments of these different models, we see that the numer of ytes that arrive within a period T and the residual from the previous period are independent. This is ecause of the way we calculated the total numer of ytes availale at the end of period T. Using the three moments, we can now approximate the pdf f B (x) or f X (x) of the total numer of ursts in the packet queue at the end of a period T y a twostage Coxian, C 2 [10]. For this we set the first three moments, m 1, m 2, m 3 equal to the first three moments of C 2 with parameters (µ 1, µ 2, α). The three moment fit, can e used if 3m 2 2 > 2m 1 m 3 and c 2 > 1, where c 2 is the squared coefficient of variation. Alternatively a two moment fit can e used if the condition 3m 2 2 > 2m 1 m 3 does not hold or 0.5 < c 2 < 1. The pdf of a C 2 is given y the expression: ( f Y (y) = (1 α)µ 1 e µ1y µ1 µ 2 + α e µ1y + µ 1µ 2 e µ2y) (14) µ 2 µ 1 µ 1 µ 2 where in the case of the three-moment fit: µ 1 = L+(L2 4K) 1/2 2, µ 2 = L µ 1 and α = ((µ 1 m 1 ) 1), K = 6m1 3(m2/m1) ((6m, L = 1/m 2 2 )/(4m1) m3 1 + m2k 2m 1 and in the case of the two-moment fit: µ 1 = 2 µ 1, µ 2 = 1 µ 1c and α = 1 2 2c. Using the C 2 2 pdf we can easily calculate the cumulative distriution F X (x) and from there the proaility of creating k ursts at the end of each period T (see equations 3, 4). When c 2 < 0.5, we can fit an Erlang distriution or a generalized Erlang distriution (see [10]). However we oserved empirically, that the numer of ytes in the packet queue at the end of an aggregation period has a small variaility.

7 6 In view of this, we have found that it is sufficient to evaluate f B (x), given y equation 2, for only a small range of values of n. That is, we limit the sum to: f B (x) = avgnump acks+10σ n=avgnump acks 10σ e λt (λt ) n (x) n 1 e x n!(n 1)! (15) where avgnump acks = λt, is the average numer of packets that arrive during a period T, and σ = λt is the variance of the numer of packets that arrive during T. From 15 we can then numerically compute the cumulative distriution of the pdf f B (x) or f X (x) and susequently the proaility of having k ursts, where k 0 (see equations 3, 4). Due to the limited variaility of the numer of ytes in T, we have also found that the following approximation gives good results: P [k ursts] = where k = m 1 /B max. 3 The case of IPP Arrivals m 1/B max, P [(k 1) ursts] = 1 P [k ursts] (16) m 1 /B max The IPP is a modified Poisson process. It is similar to a Markov Modulated Poisson Process with two states (MMPP2). The main difference etween IPP and MMPP2 is that the arrival rate in the second state of the MMPP2 is zero, which means there are no arrivals in this state [11]. An IPP is an ON/OFF process, where the ON and OFF periods are exponentially distriuted with rates σ 1 and σ 2 respectively. We also define the vector π: π = (π 1, π 2 ) = 1 σ 1+σ 2 (σ 2, σ 1 ) which gives the average duration of the ON and OFF periods. During the ON period there are Poisson arrivals with rate λ, and during the OFF period there are no arrivals. This is a very useful model for data/voice and video transfers over the Internet, where ursty arrivals of packets occur for a period of time followed y an idle interval. We assume that the packet sizes are exponentially distriuted with an average size 1/ ytes. The IPP process is also characterized y the squared coefficient of variation, c 2 IP P, of the packet interarrival time, that measures the urstiness of the arrival process, given y the expression: c 2 IP P = 1 + 2λσ1 (σ 1+σ 2). From this equation we can oserve that the 2 value of c 2 IP P is affected y the duration of the ON and OFF periods. We also define the quantity: average transmission rate = transmissionspeed σ2 σ 1+σ 2 The average transmission rate depends on the transmission speed, it is in fact the transmission speed within the ON period. In our implementation, we set average packet size 1/λ = transmissionspeed = 1/ 10Gps, where 1/ = 500 ytes. Thus, 1/λ is the time needed to transmit one packet during the ON period. Given the c 2 IP P and the 1 average transmission rate we calculate the ON and OFF periods : σ 1 and 1 σ 2 respectively. In this section we calculate the pdf of the numer of ursts during an aggregation period assuming that packets arrive in IPP fashion. We follow the same

8 7 approach as in the case of Poisson arrivals. We first calculate the MGF of the numer of ytes that arrive during a period T. Similarly to the Poisson arrival case, we have a random sum of packets whose size is exponentially distriuted, and therefore we use equation 5: M B (t) = M N (ln(m S (t))) where M B (t) is the MGF of the numer of ytes during interval ((i 1)T, it ], M N (t) is the MGF of the numer of packets N, and M S (t) the MGF of the packet size. The MGF of the numer of packets that arrive during a period T is otained as follows. Let: P ij = P ro{n t = n, J t = j N 0 = 0, J 0 = i} e the proaility that N t arrivals occur during (0, t] given that at time 0 there were 0 arrivals and the IPP was in state J 0 = i and at time t the IPP was in state J t = j. The z-transform of P ij [11] is: P (z, t) = e (Q (1 z)λ)t where Q is the infenitesimal generator of the IPP and Λ the matrix of arrival rates, i.e. Q = ( ) σ1 σ 1, Λ = σ 2 σ 2 ( ) λ Now we can use this z-transform to form the generating function of the numer of packets. We know that the MGF of a discrete random variale x, is its z- transform when z = e s. Therefore, M X (s, t) = P (e s, t), where P (z, t) the z-transform of function P (n, t). Thus, the MGF of the numer of packets that arrive during (0, t] is: M N (s, t) = e (Q (1 es)λ)t. Sustituting s with ln(m S ( s)), where M S (s) = +s, is the Laplace transform of the exponentially distriuted packet size, we otain: M B (s, t) = e (Q (1 s )Λ)t (17) We can now calculate the first two moments m 1,IP P and m 2,IP P of the numer of ytes that arrive during a period T, y setting t = T. We have: m 1,IP P = e T µ(s) = e T M (T )e (18) [ ] where: M (T ) = s M B(s, T ) and e is a column vector of 1 s and s=0 et a row vector of 1 s. In order to differentiate the MGF we use the eigenvalue decomposition of the matrix exponential given in 17. The eigenvalue decomposition always exists for this MGF. We have: e At = P e Dt P 1 where A = (Q (1 s )Λ)t D is the diagonal matrix of the eigenvalues of A, P is the matrix composed of eigenvectors and P 1 the inverse matrix of P. After differentiating and using the chain rule we get: M (T ) = eat s = P s edt P 1 1 DT P DT D + P e + T P e s s P 1 (19) Sustituting the aove in equation 18 we have: m 1,IP P = π 1 λt (20) The aove expression is intuitively ovious, as it is the mean duration of the ON period π 1 multiplied y the mean numer of ytes that arrive during this period

9 8 λt. The second moment is given y: m 2,IP P = e T µ 2 (s) = e T M (2) (T )e (21) [ ] where the second derivative M (2) (T ) = 2 s M B(s, T ) is otained y applying s=0 the chain rule to equation 19. We have: M (2) (T ) = 2 P s edt P 1 + 2T P D edt s s P P 1 P edt s s ) 2P 1 DT D P 1 +2T P e + P e DT 2 P 1 s s s + T 2 P e DT ( D s +T P e DT 2 D s P 1 Now we can use the two moments to approximate the pdf f B (x) of the numer of ytes that arrive during a period T, with a C 2 distriution as in the previous section. If c 2 < 0.5 then we used the generalized Erlang k approximation, where: 1 k c2 1 k 1 In the generalized Erlang k with proaility a, after the first exponential phase the service continues for the rest of the k 1 stages or it ends with proaility 1 a. Proaility a is given y [10]: 1 a = 2kc2 + k 2 k kc 2 2(c 2 + 1)(k 1) (22) and service rate: µ = 1+(k 1)a m 1 The proaility of having k ursts at the end of an aggregation period T is: P [k ursts] = kbmax 1 (k 1)B max f Y (y), k >= 1 where f Y (y) is the pdf of the C 2 or the generalized Erlang pdf as aove. Notice that B min = 0 in this model, since there are no residual ytes included. The case of B min > 0 can e modelled as in the previous section. Finally, we note that the numer of ytes that arrive during each interval T is approximated y the numer of ytes that arrive in (0, t]. In reality, the IPP state at the eginning of each interval T is the state at the end of the previous period T. In our model, we approximate this y assuming that the IPP is at state i at the eginning of the interval T and then uncondition on this event. 4 Numerical Results In this section we compare our approximate analytic results to simulation data for oth Poisson and IPP arrivals. A simulation program was writen in C to simulate the ehavior of the urst aggregation algorithm under study, with Poisson and IPP arrivals. 95% confidence intervals were computed using the method of atch means. The numer of atches was fixed to 30 and each atch consisted of 100,000 aggregation periods T. The confidence intervals were extremely small and they are not discernile in the figures.

10 9 Fig. 1. Proaility distriution of the numer of ursts. Poisson Arrivals, no residual, B min = Poisson Arrivals Figures 1, 2 and 3 give approximation and simulation results of the proaility distriution of the numer of ursts for the case where c 2 > 0.5. In this case the proaility distriution is computed using the fitted C 2 distriution. Figures 1 (a) and 1 () give results for T = and T = respectively. B min = 0, which means that the proaility of having zero ursts is 0, B max = 112 Kytes. The arrival rate λ was 2.5 packets/µsec and 0.5 packets/µsec for 1 (a) and 1 () respectively. The average packet size 1/ is otained from the expression: 8(1/) (its) transmission speed (Gps) 1/λ =, where the transmission speed is 10 Gps. We oserve that the results are quite accurate. In the case where B min = 16 Kytes, we have oserved that our approximation is slightly affected y the residual ytes. This is ecause in the case where B min = 16 Kytes we may have residual ytes ut this model does not include them since they are very few and usually 0. This is why we have a variation from the simulation results. Figures 2 (a) and 2 () give the analytical and simulation results of the proaility distriution of the numer of ursts when B min is close to B max. In this case, we include the residual ytes from the previous aggregation period in our calculation. B max = 200 Kytes, B min = 150 Kytes, the average packet size 1/ = 125 Kytes and transmissionspeed = 1 T ps. These parameters could e meaningful in very high speed networks with dedicated connections where large file transfers may occur, such as in a Grid environment. In Figure 2 (a) and in Figure 2 () T = µsec and T = µsec. Our approximation is very accurate in this case, only a slight variation is oserved for a low numer of ursts that could e justified since the assumption that the residual ytes are uniformly distriuted in [0, B min ) is not always accurate. This assumption is more accurate when B min is higher (180 Kytes) and therefore the difference B max B min is smaller, as can e viewed in Figures 3 (a) and 3 (). Figure 4 gives results for a case where c 2 < 0.5. As mentioned aove, we analyze this case, either y computing equation 2 for a limited range of values, or using the approximation given in (18). In this case the approximate results match the simulation data. The latter approach is very fast ut it does not give accurate results in all cases. The former method is much slower, ut gives very accurate results.

11 10 Fig. 2. Proaility distriution of the numer of ursts. Poisson Arrivals, with residual, B min = 150 Kytes Fig. 3. Proaility distriution of the numer of ursts. Poisson Arrivals, with residual, B min = 180 Kytes Figures 4 (a), 4 (), 4 (c) and 4 (d) give the analytic results otained using oth methods and the simulation results for T = 128, 256, 512 and 1024 µsec respectively. No residual is included and B min = 16 Kytes. The transmission speed was 10 Gps, the average packet size, ased on IP packets, was 500 ytes. In the case where T = 128, 256, 512 µsec oth analytic methods give accurate results. When T = 1024 µsec the aggregation period is high and the variaility in the numer of ursts increases. Thus in this case the approximation method is not accurate. If the residual ytes are included then we have oserved that oth our analytic models give almost the same results as the simulation model for a variale aggregation period T. We note that when c 2 < 0.5 the distriution of the availale numer of ytes at the end of each aggregation period is almost constant. This explains why the proaility distriution of the numer of ursts is almost constant. The result can prove useful in traffic engineering as it may simplify the architecture. 4.2 IPP arrivals The results that are given in Figure 5 were otained under the following assumptions: transmission speed = 10 Gps, average transmission rate = 1 Gps,

12 11 Fig. 4. Proaility distriution of the numer of ursts. Poisson Arrivals, no residual, B min = 16 Kytes c 2 IP P = 5, B min = 0, B max = 16 or 112 Kytes. average packet size 1/ = 500 ytes, and arrival rate during ON period: λ = 2.5 packets/µsec. In this case, c 2 > 0.5 and we use the C 2 fit. Figures 5 (a) and 5 () show the proaility of having k ursts B max = 16 Kytes and the aggregation period is T = 16, 32 µsec respectively. In Figures 5 (c) and 5 (d) we increase the B max to 112 Kytes. There is almost no difference etween the simulation results and the numerical results. 5 Conclusions The urst aggregation process defines to a large extent the urst arrival process to the OBS network. This urst arrival process has not as yet een adequately studied. However, it is important that it is well characterized if we are to understand etter the performance of the OBS network. In this paper, we have otained analytically the proaility distriution of the numer of ursts created y an aggregation algorithm that uses a timer and a minimum and maximum urst size. The analytical results are approximate ut they seem to have good accuracy. Acknowledgements We would like to thank Boldea Otilia for her helpful comments in the IPP analysis. References 1. Xu, L., Perros, H.: Performance analysis of an ingress optical urst switching node. (Sumitted.

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