as the algorithm that will be implemented in Asynchronous Transfer Mode ( ATM ) networks.

ON SIMPLIFIED MODELLING OF THE LEAKY BUCKET M Jennings, R J McEliece, J Murphy andzyu Abstract The Generic Cell Rate Algorithm, or more commonly the leaky bucket, has been standardised as the algorithm that will be implemented in Asynchronous Transfer Mode ( ATM ) networks. To model an ATM network it is important to be able to model this algorithm. However most proposed models are complex and because the leaky bucket is only one element within the network the model should be simple. In this paper we attempt to simplify the model by making some assumptions. There are two mathematical models proposed to achieve this. In the rst we look at the cell scale and although the model is detailed, the approximation is that the token arrival is random. In the second model we look at the token scale where it is possible to have multiple cell arrivals in that period. This model is not as detailed as the rst. What is being investigated is if the real system would act better than these so that they could be used as an upper bound on the performance of the network. The results presented in this `work in progress' paper are only preliminary. 1 Introduction Asynchronous Transfer Mode ( ATM ) is the emerging standard for the future broadband-isdn networks and allows for the trac from users to vary both from one call to another and within the call. As ATM will give guarantees to the users it is important for the network to have a method for describing and controlling the trac ow across the user network interface ( UNI ) 11. The conformance denition for ATM networks has been standardised as the Generic Cell Rate Algorithm ( GCRA ), or leaky bucket, by the ATM Forum 2. This algorithm allows variable bit rate trac to be specied in terms of the mean bit rate and a burst tolerance. This will be used by the network to see if a cell is compliant ornotandmay also be used by the network as the Usage Parameter Control ( UPC ) mechanism. When modelling an ATM network it is important to be able to model this algorithm eectively and easily. The leaky bucket is also proposed within the network in a slightly dierent format and acts as a regulator 6. This tends to increase the performance of the network by smoothing the trac ow. The leaky bucket algorithm is shown in Fig. 1. The operation of the leaky bucket is that a splash is added to the bucket for each incoming cell when the bucket is not full. When the bucket is full cells cannot pass through to the network but the bucket leaks away at a constant rate. M. Jennings and J. Murphy, Electronic Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland. Telephone: +353-1-704-5444, Fax: +353-1-704-5508, email: murphyj@eeng.dcu.ie R. J. McEliece and Z. Yu, Electrical Engineering, 116-81, California Institute of Technology, Pasadena, CA 91125, USA. email: zyu@systems.caltech.edu 19/1

The important parameters to be dened in this system are the leak rate of the bucket, the bucket capacity and the peak cell emission. If the cells cannot pass the leaky bucket they are lost, when the leaky bucket acts as the UPC. If the leaky bucket acts as a regulator and if the cells cannot pass they are stored and wait till they can pass. If there is an innite buer there will be no cell loss in the regulator. Incoming Cells Outgoing Cells Leaky Bucket Figure 1: Leaky Bucket or UPC Algorithm Another way of thinking about the leaky bucket is to imagine that there are tokens kept by the network and as long as the network has a token the cells can pass the leaky bucket. The tokens are generated deterministically every D seconds. The capacity of the bucket is the number of tokens that the network can store and is given by M. If the network has M tokens stored then it will not add another token until a cell passes and another token time has passed by at time t = nd. If there are no tokens available in the bucket then the cell is lost in the UPC version of the leaky bucket or else buered in the regulator version of the leaky bucket. If there are tokens available then there can be no cells waiting in the buer and if there are cells waiting in the buer then there are no tokens available. If we denote the buer size to be B then the state of the system can be described by two variables, the number of tokens available and the number of cells waiting. Therefore the possible states of the system are either tokens available (M 0) (M ; 1 0) (1 0) or no tokens available (0 0) (0 1) (0 2) (0 B) where the rst entry tells the number of tokens available and the second entry tells the number of cells waiting as is shown in Fig. 2. M,0 M 1,0... 2,0 1,0 1 > M tokens available 0,0 0,1... 0,2 0,B 0 > B cells waiting Figure 2: Possible States Of The Leaky Bucket There have been a number of methods proposed to model this algorithm 1 3;5 7;10 12;17.However all require complex models and intensive eort. The reason for this is that there is a mix of deterministic and stochastic processes occurring in the model. This would not be suitable for the simulation of a whole network or indeed for two algorithms in tandem. What is needed is a simplied model of the algorithm that captures the basic operation of it while allowing easy analysis. This is achieved in this paper by making the processes stochastic in one model and in 19/2

the other neglecting the detail of the real system. What is analysed in this paper is the operation of the leaky bucket as the regulator as this is the more general case. The UPC version of the leaky bucket is the same as the regulator except that there is no buer space. This paper is organised as follows. In the next section, Section 2, the rst model, the cell level model, is presented along with the assumptions that were used to make the analysis easy. Then in Section 3 the second model, the token level model is presented along with it's assumptions and drawbacks, and is also solved. There is then a comparison made between the two models and the real system in Section 4. This is done by running some simulations and analysing the results. Finally in Section 5 the conclusions are drawn and the areas for future work is examined. 2 Cell Level Model In this rst model, the cell model, the time unit that is taken is the cell time. This means that the system is examined every cell time, or T seconds, and in that time interval the following are possible : 0 cell arrives { 0 token arrives : ) remain in same state 0 cell arrives { 1 token arrives : ) move down a state 1 cell arrives { 0 token arrives : ) move up a state 1 cell arrives { 1 token arrives : ) remain in same state To make the analysis tractable we make the assumption that the token arrival is random, even though the tokens are generates deterministically every D seconds. If this assumption is allowed the system models as an M/M/1 system for Poisson arrivals and can be solved easily. The cell level model of this system would then be as shown in Fig. 3. lambda...... M,0 M 1,0 0,0 0,1 0,B mu Figure 3: Cell Level Model Of Leaky Bucket The cells arrive in a Poisson fashion with average arrival rate and the probability of the token arrival is then calculated so that the same number of tokens are generated in both the real system and the cell model. Therefore the service rate is 1=D. Using the normal notation of = = the probability of being in any state n is given by : p n = n (1 ; ) 1 ; M+B+1 = D n =0 1 2 M + B Being in state n means that the system is either in state (M ; n 0) if n<m or else in state (0 n; M) ifn>m. It is then possible to nd the probability of loss, which is denoted by P L : P L = (D)M+B (1 ; D) 1 ; (D) M+B+1 or P L = 1 M + B +1 if D =1 19/3

3 Token Level Model In this model the system is only examined every time that a token is generated. Therefore we are only interested in what happens every D seconds. There will be a token generated every D seconds so if there are no cell arrivals then the state of the system goes down one place. There can be a number of cells generated in this D seconds and if so then the state of the system increases by the number of cells generated less one. The state transition diagram would then be as is shown in Fig. 4. M,0 M 1,0... 0,0 0,1... 0,B Figure 4: Token Level Model Of Leaky Bucket The average number of cells that will arrive in this D seconds is given by D and has a Poisson distribution for Poisson arrivals. This model does not capture the detail of the real system but does generate the token in the correct fashion. Denote the probability ofi cell arrivals in D seconds by a i and then we have : ;D (D)i a i = e i! The steady state equation for the system 16 18 is then : p n = p 0 a n + nx j=0 p j+1 a n;j 0 n M + B There may be restrictions on the number of arrivals that are possible in D seconds in which case there would not be Poisson arrivals. It is P possible to solve this by P dening the probability generating function (PGF) of p n as G(z) = 1 1=0 p i z i and A(z) = 1 1=0 a i z i. Using these PGF's and the steady state equation it is possible 16 18 to get : G(z) = (1 ; ) (z ; 1) ze (1;z) ; 1 This corresponds to the PGF for the steady state queue of the M/D/1 system. MacLaurin series to expand this gives the steady state probabilities 18 as : Using the p 0 = (1 ; ) p 1 = (1 ; ) (e ; 1) p n = (1 ; ) nx j=1 (j) (;1) n;j e j n;j (n ; j)! + (j)n;j;1 (n ; j ; 1)! 4 Simulations & Results Having solved the two models mathematically, the next step was to simulate these models and compare them with the real system. Thus the following three systems were simulated on a 19/4

software package called SES/workbench. SES/workbench is an integrated collection of software tools for specifying and evaluating system designs by using discrete event simulation methods. It consists primarily of the following components: SES/design { a graphical editor module for specifying a system design SES/sim { a translation and simulation module for converting the design specications into an executable model SES/scope { an animated simulator that provides the ability to observe and debug an executing simulation model The three models were designed and saved within the same module in SES/design so that each time a simulation was run, results were obtained for each of the three models. Each ofthe simulations models about 12.5 seconds of real time, which is about one million cell times and took approximately 30 minutes to execute. The token model is shown in Fig. 5 as an example. SES/design 2.1 Module: real Index What How Submodel: token_model infor3 token_source token_full l_bucket token_sink tokens cell_source infor4 cell_block cell_sink cell_lost Figure 5: Simulation Of Token Model The problem that was modelled was a 34 Mb/s ATM Link with a single source at a mean rate of 2 Mb/s. The source was modelled by giving a probability of an arrival independently to each cell time, which models the Poisson process. Thirteen simulations were carried out and results were obtained for parameter changes in the capacity of the leaky bucket, the buer size and the ratio which the token generation rate exceeds the mean rate of the source. There were three dierent sets of simulations done. The rst ve simulations held the buer size constant and varied the other two parameters. Then for the next ve simulations the buer size was changed and the same parameters as before were used again. The way in which the parameters were changed was that a near linear decreasing relationship was assumed between the capacity of the leaky bucket and the ratio in which the token generation rate exceeds the mean rate. For example when the capacity of the leaky bucket increased the ratio decreased and vice versa. The last three simulations held the last two parameters constant and varied the buer capacity. The 19/5

parameters that were used for all the simulations are shown in Table 1. As a rst approximation the loss was examined for each of the models. The information regarding the queue length was also collected, but is not analysed here. The simulation results for the cell loss are also shown in Table 1. Table 1: Parameters Used For The Simulations Simulation Maximum Ratio Capacity of the Loss Number Buer Size Leaky Bucket Real Cell Token 1 0.9 9 10.5 12.7 9.9 2 0.95 6 6.8 10.2 6.7 3 5 1 5 4.5 8.7 4.8 4 1.1 3 2.2 6.6 2.5 5 1.2 1 1.8 6.4 2.3 6 0.9 9 10.3 11.7 9.6 7 0.95 6 5.9 8.1 5.5 8 10 1 5 3.0 6.0 3.3 9 1.1 3 0.69 3.1 0.78 10 1.2 1 0.22 1.7 0.34 11 5 2.7 5.3 3.0 12 10 1 12 2.1 4.2 2.4 13 20 1.5 3.0 1.7 These results are plotted in Fig. 6 so that the dierences between the three models can be seen. cell loss real cell token 0.1 0.03 0.01 0.003 1 2 3 4 5 6 7 8 9 10 11 12 13 Simulation Number Figure 6: Results It can bee seen that the simulation results correspond well to the mathematical models that were calculated before. The token model is close to the real system over all the ranges. It can be used as an upper bound on the loss except in the cases where the cell losses are high. The 19/6

cell model does act for an upper bound in these simulations and can be used as that. However in most cases it is over estimating the loss considerably. 5 Conclusions Two models have been proposed to model the leaky bucket in a simpler fashion to it's exact operation. This was done in order to allow either analytical or computer simulation of ATM networks which mighthave anumber of these leaky buckets at the edges and within the network. The models are easier to analyse and model than the exact leaky bucket, however, how good an estimate they are, is still uncertain. So far the token model seems more promising as it give results that are close to the real leaky bucket. However it cannot be used as a bound as it seems to produce more loss in some cases than the real leaky bucket and less in others. The cell model although more inaccurate than the token seems to produce an upper bound on the loss. The queue distribution has yet to be investigated along with more detailed mathematical analysis and computer simulations. This might be able to identify which model, if either, could be used to simplify the modelling. Acknowledgements The second and fourth authors acknowledge the support of a Pacic Bell grant. The rst and third authors acknowledge Prof. Charles McCorkell, Dublin City University, for his continuing encouragement and support of this work. References 1. Anantharam, V and Konstantopoulos, T: \Optimality and Interchangeability of Leaky Buckets in Tandem", April 1995. 2. ATM Forum: \ATM User{Network Interface Specication", Prentice Hall, 1993. 3. Bazanowski, Z and Killat, U: \The Superposition of Cell Streams with Geometrically Distributed Interarrivals in an ATM Multiplexer", IEEE trans. on Comms., Vol. 43, No.2/3/4, February / March / April 1995. 4. Berger, A W: \Performance Analysis of a Rate-Controlled Throttle where Tokens and Jobs Queue", IEEE Journal on selected areas in communications, Vol 9, No. 2, February 1991. 5. Butto, M, Cavallero, E and Tonietti, A: \Eectiveness of the "Leaky Bucket" Policing Mechanism in ATM networks", IEEE Journal on selected areas in communications, Vol 9, No. 3, April 1991, pp 335-342. 6. Cruz, R L: \A Calculus for Network Delay, Part I: Network Elements in Isolation", IEEE Trans. on Information Theory, Vol. 37, No. 1, pp 114{131, Jan. 1991. 7. Dittmann, L, Jacobsen, S B and Moth, K: \Flow Enforcement Algorithms for ATM networks", IEEE Journal on selected areas in communications, Vol 9, No. 3, April 1991, pp 343-350. 19/7

8. Eckberg, A E: \The Single server Queue with Periodic Arrival Process and Deterministic Service Times", IEEE trans. on Comms., Vol 27, No.3, March 1979, pp 556-562. 9. Erimli, B, Murphy J and Murphy, J:\On Worst Case Trac in ATM Networks", Twelfth UK IEE Teletrac Symp., Windsor, UK, 15-17 March 1995. 10. Gallassi, G, Rigolio, G and Verri, L: \Resource Management and Dimensioning in ATM Networks, IEEE Network Magazine, May 1991. 11. Hong, D and Suda, T: \Congestion Control and Prevention in ATM Networks", IEEE Network Magazine, July 1991. 12. Murata, M, Oie, Y, Suda, T and Miyhara, H: \Analysis of a Discrete{Time Single{Server Queue with Bursty Inputs for Trac Control in ATM Networks", IEEE journal on selected areas in communications, Vol. 8, No. 3, April 1990. 13. Ohba, Y, Murata M and Miyahara, H: \Analysis of Interdeparture Processes for Bursty Trac in ATM Networks", IEEE Journal on selected areas in communications, vol. 9, No. 3, April 1991, pp 468-476. 14. Rathgeb, E P: \Modelling and Performance Comparison of Policing Mechanisms for ATM networks", IEEE Journal on selected areas in communications, Vol 9, No. 3, April 1991. 15. Roberts, J W and Virtamo, J T: \The Superposition of Periodic Cell Arrival Streams in an ATM Multiplexer", IEEE trans. on Comms., Vol 39, No.2, February 1991, pp 298-303. 16. Sidi, M, Liu, W, Cidon, I and Gopal, I: \Congestion Control Through Input Rate Regulation", IEEE, Globecom '89. 17. Stallings, W: \Opening the Floodgates (Preventing congestion on ATM networks)", LAN magazine, May 1995. 18. Yu, Z: \The Analysis of the Leaky Bucket and the Statistical Multiplexor", Caltech technical report, 13th September 1995. 19/8