KCV Kalyanarama Sesha Sayee and Anurag Kumar

Transcription

1 Adaptive Algorithms for Admission of Elastic Sessions in the Internet KCV Kalyanarama Sesha Sayee and Anurag Kumar Abstract In the Internet, the majority of the traffic consists of elastic transfers. Users of elastic applications are generally not sensitive to the end-to-end delay of each packet, but to the time necessary to transfer an entire file, which depends on the number of such flows sharing the network bandwidth. In this paper we propose algorithms for blocking of new flows during overload conditions as a means to achieve a transfer throughput target. We study bandwidth sharing of elastic flows at a bottleneck link. We propose two control techniques, namely, Occupancy Limit (OLC) and Connection Limit (CLC) to implement admission control. We make the observation that by appropriate choice of parameters these controls yield the same performance. We then select OLC to implement admission control as it has implementation advantage over CLC. We propose some estimation based self-tuning algorithms for adaptively determining the connection blocking probabilities. Simulations are used to demonstrate the efficacy of these control algorithms. I. INTRODUCTION The bulk of the traffic in the Internet is due to the transfer of files between computers; such file transfers are generated by application protocols such as ftp and http. Such transfers are elastic in the sense that sources can adapt to the time varying available bandwidth in the network. For an elastic flow, quality of service is manifested essentially by the time it takes to complete the document transfer. This time depends both on the way bandwidth is shared and on the random fluctuations in the number of flows in progress as flows begin and end. Though there are no intrinsic temporal characteristics associated with elastic applications, it is widely accepted that elastic flows do not provide any utility once their resource share falls below some minimum acceptable value. This situation results in lot of ineffective traffic using the network. World wide web users often interpret bad throughputs as lost connections and hence terminate them thereby wasting the resources used in the partial transfer of the document. Also, the reduced bandwidth share increases delays experienced by packets at the gateways resulting in application timeout, and packet retransmissions. This adds to the ineffective traffic carried by the network. To ensure quality of service in case of traffic overload it appears necessary additionally to employ admission control ([3], [4], [8], and []) with flow blocking appearing as a more acceptable quality degradation than diminishing throughput. Instead of accepting a fresh connection in the presence of overload conditions, thus reducing the throughput of ongoing connections, it is better to reject new connections. In the rest of the paper we consider a bottleneck link shared by elastic traffic and then study various ways of implementing admission Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-56002, India. sayee, anurag@ece.iisc.ernet.in control to achieve a target performance. server server Internet II. OUTLINE OF PAPER ISP s leased line Local ISP ISP s controller Fig.. A model for a small Internet service provider (ISP) client client In Section III we introduce the notation to be used in this paper. In Section IV we describe the basic model that we propose to study and discuss ideal bandwidth sharing and its Processor Sharing model. Then we introduce a performance measure for bandwidth sharing known as Average Bandwidth Share. In Section V we look at the choice of blocking new sessions and then study how blocking affects the aforementioned performance measure. Toward this we propose two control strategies, namely, Occupancy Limit (OLC) and Connection Limit (CLC). Finally, in section VI, we come up with a few adaptive algorithms that implement the OLC strategy. Section VII provides some concluding remarks and then we briefly touch upon the users tendency of retrying their rejected connection requests and give a pointer to some work done by the authors on this problem. III. NOTATION The following notation will be used in the rest of the paper. ffl = the arrival rate of transfers ffl N (t) = number of ongoing transfers as a function of time ffl X k = service requirement of k th transfer (data units), assumed to be i.i.d. r.vs with X denoting the general r.v., with mean E(X) ffl W k = transfer completion (sojourn) time of the k th transfer ffl C = bottleneck link capacity ffl = the link utilization = E(X) C IV. THE MODEL AND A PERFORMANCE MEASURE The clients generate file transfer requests (see Fig. ). After the requests reach the server, we can view the files to be queued at the server, waiting as they are gradually transferred to the clients. Assuming a very high speed Internet backbone, we can see that the ISP s leased line forms the bottleneck link. Then effectively files get queued at the bottleneck link. With ideal bandwidth sharing, when there are n

2 files, each file transfers its packets at th of the link rate. The n network link carries the superposition of these rate controlled flows from each file transfer. Thus, considering a fluid model, we can model the file transfer processes, with ideal bandwidth sharing, by a processor sharing (PS) queue model. The PS model assumes ideal bandwidth sharing, and does not explicitly model the crude manner in which TCP attempts to adaptively share the link bandwidth. However ([2]), for small delay-bandwidth product, the processor sharing model captures the TCP performance quite well. As delay-bandwidth product increases the quality of the processor sharing model approximation becomes poor. The PS model needs to be modified to capture large delay-bandwidth products [5], [6], [7] and [3]. For large round trip times(rtt) the modifications proposed in ([]) can be used to model N (t). Since the instantaneous bandwidth available to a connection depends on the number of active flows, and because of random arrivals and departures, the instantaneous throughput experienced by a user is time varying in a random manner. Formally, per session throughput T W is defined as the mean throughput that an individual session gets, i.e., T W = P n lim n! Xk n k= Wk. Though this measure is simple to define, is not analytically tractable. For the rest of this paper we use a throughput measure known as average bandwidth share [] which has the advantage of being simple for analysis. If we let ff denote the average bandwidth share, then for a unit capacity server this measure is given by R t 0 N(u) I fn(u) gdu ff = lim t! R t 0 I fn(u) gdu ; () where the denominator converges to and is included in the expression to compensate for the fraction of time when the server is free, i.e., the fraction of time when there is no customer (and hence no throughput). Assuming that there is a steady state, let ß(n) denote the probability that there are n sessions in the system in steady state. Then we have ff = P ß(n) n=. Assuming ideal bandwidth sharing the steady n state distribution of N (t) is the same as that of M/M/ PS queue with arrival rate, service rate μ and utilization factor = E(X), the steady state probability measure exists for C <, and is given by It is easily seen that, ff() = ß(n) =( ) n X ß(n) n n= = ln ; (2) where we used the notation ff() to explicitly bring out the dependence of the average bandwidth share on. V. AVERAGE BANDWIDTH SHARE WITH BLOCKING Let admission control be used to limit the number of flows in progress to max. Then, the distribution of number of customers N (t) is given by truncating and renormalizing the above geometric distribution. In particular for any load, the probability a new request is blocked is given by PrfN (t) = max g = max ( ) max+. This follows from the insensitivity of M/G/ PS model to the customer file size distribution. Thus for a given blocking probability 0 <fi<,wehave η max fi ln ( fi) max = ln (3) load, β = 0. β = 0.2 β = 0.3 β = 0.4 β = Fig. 2. Maximum number of allowable transfers max vs. load. For a given blocking probability fi, the average bandwidth share can be found. In the average bandwidth share formula we considered () the denominator converges to the load allowed into the queue which is ( fi). If we let ff CLC (; fi) denote the average bandwidth share with a blocking(connection Limit, CLC) probability of fi, then ff CLC (; fi) = X max k ( ) ( fi) k max+ k= ( fi) = ln (4) ( fi) ( fi) Now, we make the important observation from (2) and (4) that ff CLC (; fi) = ff(( fi)) (5) Before we explain the significance of this observation, we first make a few comments on the plots shown in Fig. (2) and Fig. (3) that help us understand some of the available design choices. The plot in Fig. (2) shows how the maximum number of sessions max varies with the offered load,, for various blocking probabilities. The other plot shown in Fig. (3) shows the average bandwidth share, ff CLC (; fi), as a function of the offered load for different blocking probabilities. Comments: From the Figures 2 and 3 we can make the following observations. ffl From Fig. 3 we can observe that for a given traffic load, bandwidth share increases with target blocking probability fi.

3 Average Bandwidth Share, σ CLC (, β) β max = 0.2 β = 0 β = 0. β = 0.4 β = * load, * β = 0 β = 0.2 Fig. 3. Average bandwidth share(ff CLC (; fi)) versus load() for various blocking probabilities. ffl In Fig. 2 we can see that for a fixed fi maximum number of sessions permitted increases with increasing traffic load. ffl When the offered load decreases from 2 to, where 2 >, we see from the curves that we have two options.. One option is to keep the blocking unchanged. Using CLC, this would require us to adjust max downwards (see Fig. 2) so that at the smaller load,, the blocking probability remains the same as with 2. It is then clear from the curves in Fig. 3 that ff will increase as we move along the constant fi curve. 2. On the other hand, users probably would not like to be blocked but would require a minimum throughput. In that case there would be a maximum acceptable fi max that would define the maximum acceptable operating load. For instance, if a desired throughput of (See Fig. 3) was being provided at ß 0:9, at some acceptable blocking of 0.2, and if the load drops to ß 0:72, the same throughput can be provided at a lower blocking, which in this case is zero. That is, if ( ;fi ) and ( 2 ;fi 2 ) can provide a desired throughput ff then ( fi )= 2 ( fi 2 ). This observation says that to achieve a desired ff it is sufficient to control the carried load value to a certain threshold Λ. This observation has the implementation advantage that, it may be processing wise less expensive to measure the carried load, rather than count the number of active connections. This can be seen from how TCP flows are identified in a. A TCP flow is identified by the 4-tuple source and destination IP addresses, and source and destination port numbers. When a handles a fewer TCP flows(for example, an edge connecting end users to an ISP), then it may be processing wise affordable to count active flows by classifying TCP packets by the 4-tuple. If the handles a larger number of flows(for example, a core ) then it is expensive to do the same. Having seen that either of the two control policies can be implemented as the other, for the rest of this paper we concentrate on OLC. In OLC we block the excess traffic once the server occupancy exceeds Λ. Suppose we know that the offered traffic load is (> Λ ), then we need to block a fraction fi = Λ of the offered load. So it is clear to see that fi is well determined by and Λ. Since we allow all customers when the load served by the queue is less than Λ, the concise representation for the desired acceptance probability is ff Λ = fi Λ = min Λ ; : VI. ESTIMATING THE OFFERED LOAD, In the previous section we have seen that to be able to implement Occupancy Limit (OLC) we need to know the offered load at the bottleneck link. In this section we study a linear model to estimate the offered load. For the scenario depicted in Fig. () with blocking new sessions, the offered load and the controlled load at the bottleneck link are as shown in the Fig. (4). The rectangle bars shown in the same figure represent unsent fragments of the files in the server. offered load λ - α α rejected requests controlled load Fig. 4. Figure shows the relation between the offered, controlled and rejected traffic at the bottleneck link To measure link occupancy, time is divided into fixed length intervals of length fi, and the measurements are made over these intervals. The time interval [(n )fi;nfi) will be identified as the nth measurement interval or slot. In any given interval the link occupancy is defined as the fraction of time the link is busy. For example, if c n is the aggregate data carried on the link in the nth measurement interval, then the link occupancy is cn. fic These occupancy measurements are used to compute the offered load estimate which in turn are used in the OLC algorithm to compute the control value. At the end of each of these successive intervals, an adjustment to the control level may be made and a new measurement is begun. We assume that during each measurement interval, the load presented to the link and the control level are fixed. Let the control level in the k th interval, k, the fraction of the offered load accepted into the queue, be denoted by ff k, and the load carried by the link in the k th interval be denoted by y k. We write the measurement in the k th interval as y k = ff k + w k ; < (6) where w k is the measurement noise. For further analysis we would like this noise to have zero mean. Obviously, the larger the fi is the better this approximation will be, but less frequently we will get the chance to adjust the control level. In order to determine suitable values for fi, we studied the random variable w k+ in detail. In this connection we can compute the first two moments of the measurement noise using the regen-

4 erative central limit theorem(theorem 23, Chapter 2, [0]). E(w k+ ) ß 0; 8k E(wk+ 2 ) ß ff k E(X 2 ) fic 2 ; 8k (7) Letting ^ n denote the offered load estimate, we use two algorithms (VI) and (VI) to estimate in the model (6) assuming that the data fy ;:::;y n g, fff 0 ;:::;ff n g and the initial load estimate ^ 0 are given. Least Squares Estimate(infinite memory version): ^ n = P n Pk=0 y k+ffk n k=0 ff2 k P n k=n W y k+ffk P n ; n Least Squares Estimate(finite memory version): ^ n = ; n k=n W ff2 k Once the offered load estimate is known then we can choose the control value as ff n = min Λ ^ n ; ; n. In other words, at each instant k we make an estimate of, and then we choose the control as if the estimate were the true parameter [9]. If ^ n converges to the true parameter, then Λ ; lim k! ff k =min = ff Λ, and such an adaptive control law is known as self-tuning. The following theorem guarantees that least squares estimate for the offered load converges to the true value and hence the control converges to the true control value. Theorem VI.: Let fw k ; F k g be a martingale difference sequence and suppose that sup n E(wk+ 2 jf k)» a:s:. Let fff k g be a process adapted to ff k g, i.e., ff k is F k -measurable for each k, where F k = fffff l ;w l ;l» kg. Then lim ^ n = a.s. and so lim ff k =min Λ ; holds a.s. Proof: For details see the theorem.2.7 of [9]. In the exponential forgetting algorithm, the offered load estimate in any measurement interval is computed as the sum of estimate in the previous time interval plus the estimate calculated from the most recent data. In this algorithm, the constant a, 0 <a<, is called the weighing factor or forgetting factor. By properly choosing a we can emphasize (or deemphasize) the influence of most recent data upon the estimate ^ n+. Exponential Forgetting Estimate: ^ n+ = ( a)^ n + a yn+ ffn The following theorem can be stated with respect to the exponential forgetting algorithm. Theorem VI.2: Let F 0 = ff(ff 0 ) and, for k, F k = ff(y s ;ff s ;s» k). Assume E(w k+ jf k )=0. Then. E^ n!. 2. The rate of convergence of the algorithm is geometric with the convergence ratio of a. 3. The mean square estimation error E( ^ n ) 2 converges to Λ a EX 2. 2 a fic 2 Proof: For details see [2]. To see the effectiveness of these adaptive control algorithms we simulated the M/M/ PS model. In the simulation runs the customer file request distribution is taken to be exponential with mean E(X) = 0 KBytes, which is a commonly used value for average file size in Internet. The bottleneck link capacity is taken to be 2 Mbps, which is a typical value for access link speeds in India. The length of the measurement intervals are choosen such that mean square estimation error of the offered load (7) is of some fixed value. In all the simulation results the offered load estimation error is fixed at 0 4. These results are shown in figures (5), (6), (7) and (8). For the simulation result shown in Fig. (5) offered load is estimated using least squares by incorporating the entire past and a large value of 480s for fi is chosen. In Fig. 6 most recent data is used to estimate the offered load. That is least squares with a window length equal to is used. In general, over such long time scales statistics of the arrival process may change and we would like to respond to such changes to maintain QoS. For that, for the simulation shown in Fig. (7), we used small update times fi =60sand then use a number of such previous measurements to estimate the offered load. Here a window length of 8 is used to estimate the offered load. By this choice of small update times we frequently update the control value estimates and at the same time are able to incorporate sufficient past information to improve accuracy of the offered load and control estimates. For the plot shown in fig. (8) exponential forgetting estimate is used. In all the simulation results shown, offered load and the target carried load are set at and respectively. Hence the desired control ff Λ = min( 0:6 ; ) = 0:75. 0:8 In Fig. (5) the control level responds immediately and converges to 5 where as in Figs. (6) and (7) the control oscillates around ff Λ =0:75. This can be explained as follows: for the plot shown in Fig. (5) the entire past information is used to estimate the control level and as more and more information is used the accuracy of the control estimate improves. For the plot shown in Fig. (6), only the most recent data is used to estimate the control level and this explains for the oscillatory behavior. The same explanation can be used to explain the oscillatory behavior shown in the Fig. (7) Offered Load = Target Load = Update Time = 480s Fig. 5. Infinite memory version of least squares: figure shows the control level and the controlled Load versus time; fi = 480s, E(X) = 0KBytes, =, Λ =.

5 5 Offered Load = Target Load = Update Time = 480s 5 control led load Fig. 6. Figure shows the control level and the controlled Load versus time; fi = 480s, E(X) = 0KBytes, =, Λ =, Window Length, W = Time Fig. 8. Exponential Forgetting Estimate: figure shows the control level and the controlled load verses time; a =0:6, fi =60s, E(X) = 0KBytes, = 0:8, Λ = 0: Offered Load = Target Load = Update Time = 60s user decides to abandon the request. This retrial behavior on users part brings to fore some issues which are hard to model and do the relevant performance analysis, partly due to unpredictable user patience levels. For example, in the presence of user retrial behavior, unlike in the model (6), the offered load, carried load, control and retrial probability are related through a non-linear relationship. An attempt has been made to study these issues in [2]. Very soon we plan to publish this work Fig. 7. Figure shows the control level and the controlled load verses time; fi =60s, E(X) = 0KBytes, = 0:8, Λ = 0:6, Window Length, W =8. VII. CONCLUSIONS We considered blocking new elastic flows on a bottleneck link to achieve a bandwidth performance objective. We proposed two control strategies, namely, CLC and OLC, and then used simple system control approaches to study a few well known adaptive control algorithms to implement OLC. Generally most of the time bottleneck links operate under overload conditions. In this paper we have not discussed control strategies that work in overload conditions. An interesting aspect to admission control comes from users retrial behavior. Generally when a user requesting the download of a file or web page is rejected the user will not abandon the request but will retry after some time hoping that this time the request will get through. If the request is again rejected it may decide either to retry after some time or may withdraw its request forever which generally depends on the user patience level. This retrial process goes on until either the request gets through, or the REFERENCES [] J. W. Roberts and L. Massoulie, Arguments in Favour of Admission for TCP Flows, International Teletraffic Conference 6, Edinburgh, 999 [2] A. A. Kherani and Anurag Kumar and Pinaki Chanda, An Approximate Calculation of Max-Min Fair Throughputs for Non-Persistent Elastic Sessions, Internet Performance Symposium, IEEE Globecom, 200 [3] Anurag Kumar and Malati Hegde and S.V.R. Anand and B.N. Bindu and Dinesh Thirumurthy and A.A. Kherani, Nonintrusive TCP Connection Admission for Bandwidth Management of an Internet Access Link, IEEE Communications Magazine, May 2000 [4] J. W. Roberts and L. Massoulie, Bandwidth Sharing and Admission for Elastic Traffic, ITC Specialist Seminar, Yokohama, October 998 [5] Tian Bu and Don Towsley, Fixed Point Approximations for TCP Behaviour in an AQM Network, Proceedings of ACM SIGMETRICS 200 [6] M. Vojnovic and J.-Y. Le Boudec and C. Boutremans, Global Fairness of Additive Increase and Multiplicative Decrease with Heterogenous Round Trip Times, Proc. IEEE Infocom 2000, 2000 [7] Thomas Bonald and Laurent Massoulie, Impact of Fairness on Internet Performance, Sigmetrics, 200 [8] J. W. Roberts and L. Massoulie, Bandwidth Sharing: Objectives and Algorithms, IEEE Tran. on Networking, 0(3), pp , June 2002 [9] P. R. Kumar and Pravin Varaiya, Stochastic Systems: Estimation, Identification, and Adaptive, Prentice Hall, 986 [0] Ronald W. Wolff, Stochastic Modelling and The Theory of Queues, Prentice Hall, 989 [] A. A. Kherani and Anurag Kumar, Stochastic Models for Throughput Analysis of Randomly Arriving Elastic Flows in the Internet, IEEE Infocom, 2002 [2] KCV Kalyanarama Sesha Sayee, Adaptive Algorithms for Admission of Elastic Sessions in the Internet, MS.c (Engg) Thesis, Indian Institute of Science, 200, Bangalore [3] Pinaki Shankar Chanda, Models for Traffic Engineering of Packet Networks with Non-Persistent Elastic Flows, ME Thesis, Indian Institute of Scince, 200, Bangalore