CDV Tolerance τ Traffic Source 1 PHY SAP. Physical Layer Functions S B

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1 University of Wurzburg Institute of Computer Science Research Report Series Dimensioning of a Peak Cell Rate Monitor Algorithm Using Discrete-Time Analysis F. Hubner Report No.59 March 1993 Institute of Computer Science, University of Wurzburg, Am Hubland, D-8700 Wurzburg, Federal Republic of Germany Tel.: +49/931/ , Fax: +49/931/ , huebner@informatik.uni-wuerzburg.dbp.de Abstract This study deals with the monitoring of ATM cells from a CBR source which are subject to cell delay variation introduced by cells from other trac sources. We present an exact solution for the dimensioning of the so-called peak cell rate monitor algorithm dened by the CCITT Draft Recommendation I.371. The solution algorithm consists of a discretetime iteration and allows for the exact determination of probability that the monitor algorithm rejects a cell. Using the cell rejection probability, the parameters of the monitor algorithm can be chosen to meet the desired performance objectives like transparency and ability to detect misbehaving sources. We consider cells from a CBR source which are multiplexed together with background trac. The background trac can vary by its mean and coecient of variation of the cell rate. The cell delay variation of the CBR trac introduced by the background trac is taken into account by the monitor algorithm. The numerical results show that the parameters of the monitor algorithm depend strongly on the coecient of variation of the background trac.

2 1 Introduction The peak cell rate is at present the only parameter for trac and congestion control which is dened by the CCITT (see Draft Rec. I.371 [4]). Each connection has to determine its peak cell rate for connection set-up, charging, routing, etc. Once a connection is established, policing of the determined parameter must be performed to be able to guarantee the desired grade of service for all connections according to their trac contracts. The peak cell rate of an ATM connection is dened as the inverse of the minimum time between the emission of two cells from this connection. Therefore, the cell delay variation (CDV) which leads to cell clumping and dispersion can inuence the peak cell rate from an ATM connection. CDV is introduced by: multiplexing cells from dierent ATM connections usage parameter control (UPC) and network parameter control (NPC) segmentation and reassembly in the AAL other network and protocol functionalities The end-to-end CDV is dened in CCITT Draft Rec. I.35B [3]. Measurement processes for cell clumping and dispersion are described in [3] and [5]. Cell clumping causes problems in terms of network congestion. Cell dispersion causes the necessity for larger buers at the receiving site of the ATM network if the customer equipment has strong delay constraints (e.g. play-out parameters for voice/video). The reference conguration for the denition of peak cell rate and CDV is depicted in Fig.1. CDV Tolerance τ Traffic Source 1 MUX Shaper PHY SAP Physical Layer Functions S B Peak Cell Rate Other CEQ 1/T Functionalities UPC Generating CDV T B Traffic Source N ATM Layer Physical Layer Figure 1: Reference conguration from CCITT Draft Rec. I.371. After multiplexing the trac streams from dierent connections, trac shaping to reduce CDV is performed. In [1], [2], [6] a cell spacer was suggested as trac shaper. Another approach using a spacing policer was presented in [10]. The UPC performs the monitoring 1

3 of the cell streams of each ATM connection. CDV between the physical service access point (PHY SAP) and the T B reference point is introduced by physical layer functions and customer equipment (CEQ) functionalities. A higher peak cell rate of an ATM connection than negotiated can stem from the misbehavior of this connection or from CDV for which the connection is not responsible. Therefore, the policing of the peak cell rate must be performed in such a way that the inuence of CDV is tolerated whereas cells from misbehaving connections are recognized. In contrast to already published investigations, we take also into account the variability of the background trac by its coecient ofvariation (CoV). The numerical results show that the parameters of the peak cell rate monitor algorithm depend strongly on the CoV of the background trac. Moreover, the iterative algorithm developed here gives the exact cell rejection probability for general cell inter-generation time distributions. The paper is organized as follows. In section 2 we describe the investigated peak cell rate monitor algorithm. The analysis for determining the probability that a cell is rejected by the monitor algorithm is presented in section 3. In section 4 we describe the underlying trac scenario for the dimensioning of the monitor algorithm. Numerical results are presented in section 5. 2 Peak cell rate monitor algorithm An algorithm for monitoring the peak cell rate according to CCITT Draft Rec. I.371 [4] is shown in Fig.2. The algorithm is called Virtual Scheduling Algorithm beside of another algorithm which is exactly equivalent to it is called Continuous-State Leaky Bucket Algorithm (see [4]). The algorithm determines if a cell is generated too close to the last cell (indicating that the connection generates cells with a rate which is higher than the negotiated rate) or not. The CDV is taken into account by introducing a cell delay variation tolerance. The time at which the next cell is supposed to be generated is called theoretical arrival time (TAT). The peak cell rate is denoted by 1=T, i.e. the time dierence between two cell generations shall not be smaller than T. The time instant atwhich a cell is generated is denoted by t. Clearly, the TAT for the rst cell which shall be monitored is set to t. Three cases can be distinguished: 1. If a cell is generated later than its TAT, the source generates cells with a rate smaller than the peak cell rate. Therefore, the cell is accepted as a compliant cell (see case (3) in Fig.2) and the TAT for the next cell is set to t + T, i.e. the late generation of this cell does not allow foranearliergeneration of the next cell. 2. If a cell is generated before its TAT but later than TAT-, the cell is generated too early but within the allowed CDV tolerance. Therefore, the cell is also accepted as a compliant cell (case (2) in Fig.2) but the TAT of the next cell is set as if this cell would have been generated at its TAT and not earlier. 2

4 cell generation at time t yes TAT < t? no compliant cell (1) TAT:=t+T no TAT > t + τ? yes compliant cell (2) TAT:=TAT+T non-compliant cell (3) Figure 2: Peak cell rate monitor algorithm. 3. If a cell is generated before the time instant TAT-, the cell is recognized as a noncompliant cell (case (1) in Fig.2) and is rejected 1. The TAT for the next cell is not modied in this case. The algorithm guarantees that cells from a connection enter the ATM network at the T B reference point with a long term rate of at most 1=T. 3 Analysis We derive an exact iterative algorithm to determine the cell rejection probability. The cell rejection probability is then used to decide if the monitor algorithm parameter T and are chosen appropriately to identify compliant/non-compliant cells from well-behaving/ misbehaving sources. The following notation is used: Z ; n Z + n A n discrete random variable just before the generation instant of cell n for the number of slots until cell n was expected to be generated. discrete random variable just after the generation instant of cell n for the number of slots until cell n +1is expected to be generated. discrete random variable for the number of slots until cell n is generated. 1 Cells which are identied as non-compliant cells can optionally be tagged or rejected (see [4]). We consider here only the case of cell rejection. 3

5 The distributions of the discrete-time random variables Z ; n, Z + n, and A n are denoted by z ; n (k), z + n (k), and a n (k) respectively. An example evolution of the random variables Z ; n and Z + n is shown in Fig.3. Z - + n,z n generation of cell no. n n+1 n+2 non-compliant cell A n+1 T+ τ T τ T T T τ τ τ time in slots Z - + n Z n Z - n+1 Figure 3: Z ; n and Z + n process example for <T. Since Z + n represents the number of slots until the next cell is expected to be generated, Z + n is decreased by one each slot if it has a positivevalue. The number of slots until generation of cell number n +1 is A n+1. Each accepted cell increases Z + n by T according to the peak cell rate monitor algorithm (cf. Fig.2). If Z ; n+1 is zero at the generation instant of cell n + 1 (cf. Fig.3), the expected time between the generation instants of cells n and n +1 has expired and cell n + 1 is accepted as a compliant cell. The gray shaded regions in Fig.3 denote the time periods in which generated cells are accepted as compliant cells due to the CDV tolerance (cell n +2in Fig.3). The gray shaded regions begin when Z ; n reaches the value. A cell generation at the beginning of the gray shaded regions can therefore lead to the maximum value of T + for Z + n. Cells which are generated at time instants with Z ; n > (i.e. before the gray shaded regions in Fig.3 begin) are recognized as non-compliant cells. The G [X] =D=1 ; S queueing system (with loss of the complete batch of arrivals if there is not enough space for buering the batch) has the same process evolution as depicted in Fig.3 if its system size is considered. The decrease of Z + n each slot if positive corresponds to the deterministic service time. The increase of Z + n by T when a compliant cell is generated corresponds to a batch arrival with deterministic size T (being an integer multiple of the service time). The maximum value of T + for Z + n corresponds to the maximum system size S +1. Therefore, the evolution of the random variable Z + n can be described by the 4

6 system size evolution in the G [T ] =D=1 ; (T + ; 1) queueing system in which [T ] denotes batch arrivals of deterministic size T. In the following, we propose an iterative algorithm for the computation of the distributions of Z ; n and Z + n (denoted by z ; n (k) and z + n (k)) if the cell generation process is a renewal process and follows a general distribution. Using these distributions, the probability to observe non-compliant cells can be easily derived. The algorithm is based on the algorithm for the computation of the system size distribution in the G [X] =D=1 ; S queueing system proposed by Tran-Gia and Ahmadi in [8]. Z ; n+1 is given by the following eqn. using Z + n : Z ; n+1 = maxf0 Z + n ; A n+1g: (1) This eqn. is driven by the decrease of Z n by one each slot until it reaches zero. Z + n determined by Z ; n in the following way: Z + n = ( Z ; n Z ; n +1 Z ; n + T Z ; n : (2) The rst case corresponds to the generation of a non-compliant cell (rejection) whereas the second case corresponds to the generation of a compliant cell (increase of Z n by T ). According to eqns.(1) and (2), the distributions for Z ; n+1 and Z + n are given by z ; n+1(k) = 0 (z + n (k) a n+1 (;k)) 0 k T + (3) is and z + n (k) = 8 >< 0 0 k z >: ; n (k) +1 k T ; 1 z ; n (k)+z ; n (k ; T ) T k T + : (4) In eqn.(3) denotes the discrete convolution operation and 0 (z(k)) is dened by: 0 (z(k)) = 8 >< >: 0P i=;1 0 k<0 z(k) z(i) k=0 k>0 : (5) If T, z + n (k) isdetermined in dierence to eqn.(4) by: z + n (k) = 8 >< >: 0 0 k T ; 1 z ; n (k ; T ) T k z ; n (k ; T )+z ; n (k) +1 k + T 5 : (6)

7 The equilibrium state distributions z ; (k) and z + (k) (0 k T + ) are derived by an iteration using eqns.(3) and (4) for <T and eqns.(3) and (6) for T as and z ; (k) = lim n!1 z; n (k) (7) z + (k) = lim n!1 z+ n (k): (8) Using the iterative algorithm we derive the complete distribution z ; (k), i.e. we know the probabilities that the expected time until the next cell generation is still k slots (0 k T + ) at the time instant of a cell generation. The rejection probability p r, i.e. the probability to observe a non-compliant cell, (cf. Fig.2, case (3)) is simply given by: p r = +T X k=+1 z ; (k): (9) We denote by p c the probability that a cell is recognized as a compliant cell although the time dierence between its generation instant and the generation instant of the last cell is smaller than T. p c is given by: p c = X k=1 z ; (k): (10) The probability that a cell is generated at its TAT orlaterisdenotedby p a and is simply given as: p a = z ; (0) = 1:0 ; p r ; p c : (11) 4 Trac scenario The performance of the peak cell rate monitor algorithm is determined by the correct detection of cells which are generated by well-behaving/misbehaving sources. The parameters T and of the monitor algorithm must be tuned in order to reject cells generated by awell-behaving source with probability lower than 10 ;9 to guarantee the transparency of the algorithm. On the other hand as many as possible cells from misbehaving sources must be rejected. The dimensioning of the peak cell rate monitor algorithm consists of three phases: 1. The inter-generation time distribution of the cells from the monitored source after the multiplexing process is determined via simulation. 6

8 2. The parameters T and are chosen and the cell inter-generation time distribution from phase 1. is used as a n (k) for the analysis of the cell rejection probability as described in section The rejection probability forms the basis for the decision if the monitor algorithm parameters are chosen appropriately or if they must be changed. In the latter case phase 2. is entered again. To investigate how the parameters must be chosen to meet the performance requirements, we monitor cells from a well-behaving/misbehaving source which are subject to CDV by being multiplexed with background trac (see Fig.4). compliant/ CBR traffic background traffic MUX cell inter-generation time distribution of CBR traffic peak cell rate monitor algorithm non-compliant cells Figure 4: Scenario for dimensioning the peak cell rate monitor algorithm. The decision if the chosen monitor algorithm parameter are chosen appropriately or not is based on the proportion of generated cells which are rejected. Several studies concerning dimensioning and performance of peak cell rate monitor algorithms have been proposed. In [2] a Bernoulli distribution for background trac was used in a simulation study. A Poisson stream as background trac was used in [7]. A number of identical Bernoulli sources have been monitored by the spacer policer after multiplexing in [10]. The drawback of assuming Bernoulli, Binomial, or Poisson distributions as background trac is that the mean of the distribution determines the CoV denitely. But the CoV plays an important role for the introduction of CDV in the monitored cell stream. Therefore, we investigate background trac for which the cell stream inter-generation times follow a negative binomial (Negbin) distribution (cf. [8], [9]). The use of the negative binomial distribution allows the investigation of the inuence of the CoV on the background trac while the mean cell inter-generation time remains xed. A negative binomial distribution for the random variable X which denotes the number of slots i until the generation of the next cell is given as: i + n ; 1 x(i) = p n (1 ; p) i : (12) i The negative binomial distribution can be seen as an n-fold convolution of a non-shifted geometric distribution with parameter p (note that n is a real number). Mean E[X] and 7

9 CoV c X of a negative binomially distributed random variable X determine its parameters p and n: p = E[X]c 2 X ;1 (13) and n = E[X] E[X]c 2 X ; 1 ;1 : (14) E[X] andc X can be chosen independently of each other but must fulll E[X]c 2 X > 1. 5 Numerical results To derive some numerical results, we consider the trac scenario depicted in Fig.4 and we monitor the cells generated by a CBR source. The inter-generation time for the cells from the background trac is assumed to be negative binomially distributed and both trac streams are multiplexed together. We consider dierent multiplexing policies for cells which are generated simultaneously, namely we give cells from the CBR source priority over the cells from the background trac, vice versa, and choose one of the cells uniformly from the total number of simultaneously generated cells. The inter-generation time distribution for the cells from the CBR source after multiplexing is derived by simulation, i.e. we determine the distribution of time between the emissions of cells from the deterministic cell stream in the Negbin + D=D=1 queueing system. Unless noticed otherwise, cells from the CBR source are given priorityover simultaneously generated cells from the background trac for multiplexing. For numerical results we assume that the CBR source generates one cell each d = 24 slots before being multiplexed with the background trac. The background trac generates one cell each E[X] slots in equilibrium and the CoV c X is varied. The resulting inter-generation time distribution of the cells from the CBR source after being multiplexed is shown in Fig.5. Clearly, the peak at 24 is higher and the distribution is more lean with lower CoV c X =1 and lower intensity (E[X] = 4) of the background trac. The four cell inter-generation time distributions are used as distribution a n (k) for the iterative algorithm described in section 3. We assume that the monitored CBR source behaves well, i.e. that a peak cell rate of at least 1=(d slot duration) is negotiated. Therefore, the rejection probability p r should be zero when T and are chosen appropriately. In [2] is was stated that the monitor algorithm parameter T must in some cases be chosen signicantly smaller than d. In Table 1 we show the minimum values of for dierent choices of T to keep p r below 10 ;9. 8

10 probability E[X]=4 C =1 X probability E[X]=4 C =2 X cell inter-generation time in slots cell inter-generation time in slots probability E[X]=2 C =1 X probability E[X]=2 C =2 X cell inter-generation time in slots cell inter-generation time in slots Figure 5: Distribution of cell inter-generation time. d =24,T = E[X] =4,c X =1, E[X] =4,c X =2, E[X] =2,c X =1, E[X] =2,c X =2, Table 1: Dependency of the minimum value for on T. From Table 1 we can conclude that depends not only on the background trac intensity (as already has been observed in [7]) but is also strongly inuenced by the CoV c X. A small decrease of T (i.e. to monitor a higher cell rate) makes it possible to decrease (i.e. to allow smaller CDV) by far while keeping the cell rejection probability constant. The probability that cells are recognized as compliant due to CDV tolerance is p r = 13% for the parameter set used in the last row oftable 1 and T = 20 and increases to p r = 56% for 9

11 T = 23. This numerical result demonstrates the capability of the peak cell rate monitor algorithm to work transparently (while allowing CVD) if dimensioned appropriately. For the parameter set E[X] = 4, c X = 2, T = 21, and = 26 (cf. Table 1) we tested if cells from a misbehaving CBR source (i.e. the cell generation rate of the source is too high) are recognized by the monitoring algorithm. It turned out that a source which generates one cell each d = 23 slots has only a cell rejection probability p r = 3:7 10 ;8. Sources generating cells with a higher rate (d = 22) are rejected with p r 1:1 10 ;6. But sources which violate the trac contract more (d 21) experience a cell rejection probability of more than 10 ;3. The results indicate that misbehaving sources cannot be discovered in all cases, but that cells from sources which considerably violate the trac contract are detected. For the numerical results shown wehave assumed until nowthatsimultaneously generated cells from the CBR source are given priority over background trac cells for multiplexing. In Table 2 we show how the numerical results are altered if we give background trac priority over CBR trac or if cells are chosen uniformly from the total number of simultaneously generated cells and are transmitted in a FCFS manner. d = 24, T = 23, =8,c X =1 E[X] =4 E[X] =2 CBR trac rst p r 10 ;9 p r =5:9 10 ;5 FCFS, uniformly chosen p r =2:4 10 ;5 p r =2:0 10 ;4 background trac rst p r =2:5 10 ;5 p r =4:1 10 ;4 Table 2: Dependency of p r on the multiplexer scheduling strategy. From Table 2 we can conclude that the cell rejection probability is strongly dependent on the scheduling strategy of the multiplexer. The cell rejection probability p r increases if cells from background trac are multiplexed before simultaneously generated cells from the CBR trac. In order to test the inuence of varying and c X on the rejection probability p r,wehave plotted in Fig.6 numerical results for the parameter set d =24,E[X] = 2, and T = 20. The dierent curves (for =0, 5, 20) indicate that chosing larger improves p r. But the dependency of p r on c X shows that only a choice of a very large value for will keep p r below 10 ;9 if c X is about 1.5 or larger. If too many cells from an ATM connection are rejected although the trac contract is fullled, there are two possibilities for tuning the parameters of the monitor algorithm. Either the monitored cell rate can be increased (i.e. T is decreased) or the CDV tolerance is increased. If is increased too much, the number of back-to-back generated cells which are still recognized as compliant cells increases. Therefore, the cell clumping eect arises but the long term cell rate of the policed source can be guaranteed to be not larger than 1=T. If T is decreased the cell clumping eect is avoided if is chosen properly. Cells from a trac source which violates the trac contract by generating cells with a 10

12 rejection probability E-1 E-2 E-3 E-4 τ=0 E-5 E-6 τ=5 E-7 E-8 τ=20 E c x Figure 6: Cell rejection probability in dependence on the CoV of background trac. higher rate than negotiated but is not inuenced by CDV too much are not detected. Therefore, a tradeo between the avoidance of cell clumping and the recognition of trac contract violations exists. Acknowledgement The author would like to thank R. Dittmann for providing some program modules for the iterative algorithm and for helpful discussions during the course of this work. The author also would like to thank Prof. P. Tran-Gia for fruitful discussions. References [1] P. Boyer, Y. Rouaud, M. Servel, Methode et Systeme de Lissage et de Contr^ole de Debit de Communications Temporelles Asynchrones, French Patent, INPI No. 90/00770, January 1990 and European Patent Oce Bulletin 91/30, July

13 [2] P. Boyer, F.M. Guillemin, M.J. Servel, J.-P. Coudreuse, Spacing Cells Protects and Enhances Utilization of ATM Network Links, IEEE Network Magazine, Vol.6, No.5, September 1992, pp [3] CCITT Draft Recommendation I.35B, B-ISDN ATM Layer Cell Transfer Performance, December [4] CCITT Draft Recommendation I.371, Trac Control and Congestion Control in B-ISDN, June [5] A. Gravey, P. Boyer, Cell Delay Variation Specication in ATM Networks, IFIP Workshop TC6, Modelling and Performance Evaluation of ATM Technology, La Martinique, January [6] F.M. Guillemin, P.E. Boyer, L. Romoeuf, The Spacer-Controller: Architecture and First Assessments, Workshop on Broadband Communications, Estoril, Portugal, January 1992, pp [7] F.M. Guillemin, W. Monin, Limitation of Cell Delay Variation in ATM Networks, ICCT, Beijing, China, September [8] P.Tran-Gia, H. Ahmadi, Analysis of a Discrete-Time G [x] =D=1;S Queueing System with Applications in Packet-Switching Systems, INFOCOM 1988, pp [9] P. Tran-Gia, Discrete-Time Analysis of Performance Models in Computer and Communication Systems, 46th Report on Studies in Congestion Theory, University of Stuttgart, [10] E. Wallmeier, T. Worster, The Spacer Policer, an Algorithm for Ecient Peak Bit Rate Control in ATM Networks, ISS, October 1992, paper A

14 Preprint-Reihe Institut fur Informatik Universitat Wurzburg Verantwortlich: Die Vorstande des Institutes fur Informatik. [1] K. Wagner. Bounded query classes. Februar [2] P. Tran-Gia. Application of the discrete transforms in performance modeling and analysis. Februar [3] U. Hertrampf. Relations among mod-classes. Februar [4] K. W. Wagner. Number-of-query hierarchies. Februar [5] E. W. Allender. A note on the power of threshold circuits. Juli [6] P. Tran-Gia und Th. Stock. Approximate performance analysis of the DQDB access protocol. August [7] M. Kowaluk und K. W. Wagner. Die Vektor-Sprache: Einfachste Mittel zur kompakten Beschreibung endlicher Objekte. August [8] M. Kowaluk und K. W. Wagner. Vektor-Reduzierbarkeit. August [9] K. W. Wagner (Herausgeber). 9. Workshop uber Komplexitatstheorie, eziente Algorithmen und Datenstrukturen. November [10] R. Gutbrod. A transformation system for chain code picture languages: Properties and algorithms. Januar [11] Th. Stock und P. Tran-Gia. A discrete-time analysis of the DQDB access protocol with general input trac. Februar [12] E. W. Allender und U. Hertrampf. On the power of uniform families of constant depth threshold circuits. Februar [13] G. Buntrock, L. A. Hemachandra und D. Siefkes. Using inductive counting to simulate nondeterministic computation. April [14] F. Hubner. Analysis of a nite capacity a synchronous multiplexer with periodic sources. Juli [15] G. Buntrock, C. Damm, U. Hertrampf und C. Meinel. Structure and importance of logspace-mod-classes. Juli [16] H. Gold und P. Tran-Gia. Performance analysis of a batch service queue arising out of manufacturing systems modeling. Juli [17] F. Hubner und P. Tran-Gia. Quasi-stationary analysis of a nite capacity asynchronous multiplexer with modulated deterministic input. Juli [18] U. Huckenbeck. Complexity and approximation theoretical properties of rational functions which map two intervals into two other ones. August [19] P. Tran-Gia. Analysis of polling systems with general input process and nite capacity. August [20] C. Friedewald, A. Hieronymus und B. Menzel. WUMPS Wurzburger message passing system. Oktober [21] R. V. Book. On random oracle separations. November [22] Th. Stock. Inuences of multiple priorities on DQDB protocol performance. November [23] P. Tran-Gia und R. Dittmann. Performance analysis of the CRM a-protocol in high-speed networks. Dezember [24] C. Wrathall. Conuence of one-rule Thue systems. [25] O. Gihr und P. Tran-Gia. A layered description of ATM cell trac streams and correlation analysis. Januar [26] H. Gold und F. Hubner. Multi server batch service systems in push and pull operating mode a performance comparison. Juni [27] H. Gold und H. Grob. Performance analysis of a batch service system operating in pull mode. Juli [28] U. Hertrampf. Locally denable acceptance types the three valued case. Juli [29] U. Hertrampf. Locally denable acceptance types for polynomial time machines. Juli [30] Th. Fritsch und W. Mandel. Communication network routing using neural nets { numerical aspects and alternative approaches. Juli

15 [31] H. Vollmer und K. W. Wagner. Classes of counting functions and complexity theoretic operators. August [32] R. V. Book, J. H. Lutz und K. W. Wagner. On complexity classes and algorithmically random languages. August [33] F. Hubner. Queueing analysis of resource dispatching and scheduling in multi-media systems. September [34] H. Gold und G. Bleckert. Analysis of a batch service system with two heterogeneous servers. September [35] H. Vollmer und K. W. Wagner. Complexity of functions versus complexity of sets.oktober [36] F. Hubner. Discrete-time analysis of the output process of an atm multiplexer with periodic input. November [37] P. Tran-Gia und O. Gropp. Structure and performance of neural nets in broadband system admission control. November [38] G. Buntrock und K. Lorys. On growing context-sensitive languages. Januar [39] K. W. Wagner. Alternating machines using partially dened \AND" and \OR". Januar [40] F. Hubner und P. Tran-Gia. An analysis of multi-service systems with trunk reservation mechanisms. April [41] U. Huckenbeck. On a generalization of the bellman-ford-algorithm for acyclic graphs. Mai [42] U. Huckenbeck. Cost-bounded paths in networks of pipes with valves. Mai [43] F. Hubner. Autocorrelation and power density spectrum of atm multiplexer output processes. September [44] F. Hubner und M. Ritter. Multi-service broadband systems with CBR and VBR input trac. Oktober [45] M. Mittler und P. Tran-Gia. Performance of a neural net scheduler used in packet switching interconnection networks. Oktober [46] M. Kowaluk und K. W. Wagner. Vector language: Simple description of hard instances. Oktober [47] B. Menzel und J. Wol von Gudenberg. Kommentierte Syntaxdiagramme fur C++. November [48] D. Emme. A kernel for funtions denable classes and its relations to lowness. November [49] S. Ohring. On dynamic and modular embeddings into hyper de Bruijn networks. November [50] K. Poeck und M. Tins. An intelligent tutoring system for classication problem solving. November [51] K. Poeck und F. Puppe. COKE: Ecient solving of complex assignment problems with the propose-and-exchange method. November [52] Th. Fritsch, M. Mittler und P. Tran-Gia. Articial neural net applications in telecommunication systems. Dezember [53] H. Vollmer und K. W. Wagner. The complexity of nding middle elements. Januar [54] O. Gihr, H. Gold und S. Heilmann. Analysis of machine breakdown models. Januar [55] S. Ohring. Optimal dynamic embeddings of arbitrary trees in de Bruijn networks. Februar [56] M. Mittler. Analysis of two nite queues coupled by a triggering scheduler. Marz [57] J. Albert, F. Duckstein, M. Lautner und B. Menzel. Message-passing auf transputer-systemen. Marz [58] Th. Stock und P. Tran-Gia. Basic concepts and performance of high-speed protocols. Marz [59] F. Hubner. Dimensioning of a peak cell rate monitor algorithm using discrete-time analysis. Marz [60] G. Buntrock und K. Lorys. The variable membership problem: Succinctness versus complexity. April [61] H. Gold und B. Frotschl. Performance analysis of a batch service system working with a combined push/pull control. April [62] H. Vollmer. On dierent reducibility notions for function classes. April [63] S. Ohring und S. K. Das. Folded Petersen Cube Networks: New Competitors for the Hyepercubes. Mai [64] S. Ohring und S. K. Das. Incomplete Hypercubes: Embeddings of Tree-Related Networks. Mai [65] S. Ohring und S. K. Das. Mapping Dynamic Data and Algorithm Structures on Product Networks. Mai [66] F. Hubner und P. Tran-Gia. A Discrete-Time Analysis of Cell Spacing in ATM Systems. Juni [67] R. Dittmann und F. Hubner. Discrete-Time Analysis of a Cyclic Service System with Gated Limited Service. Juni [68] M. Frisch und K. Jucht. Pascalli-P. August [69] G. Buntrock. Growing Context-Sensitive Languages and Automata. September [70] S. Ohring und S. K. Das. Embeddings of Tree-Related Topologies in Hyper Petersen Networks. Oktober

16 [71] S. Ohring und S. K. Das. Optimal Communication Primitives on the Folded Petersen Networks. Oktober [72] O. Rose und M. R. Frater. A Comparison of Models for VBR Video Trac Sources in B-ISDN. Oktober [73] M. Mittler und N. Gerlich. Reducing the Variance of Sojourn Times in Queueing Networks with Overtaking. November [74] P. Tran-Gia. Discrete-Time Analysis Technique and Application to Usage Parameter Control Modelling in ATM Systems. November [75] F. Hubner. Output Process Analysis of the Peak Cell Rate Monitor Algorithm. January