University ofwurzburg. Research Report Series. in ATM Systems. M. Ritter. Institute of Computer Science, University of Wurzburg

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1 University ofwurzburg Institute of Computer Science Research Report Series Performance Analysis of the Dual Cell Spacer in ATM Systems M. Ritter Report No. 95 November 1994 Institute of Computer Science, University of Wurzburg Am Hubland, Wurzburg, Federal Republic of Germany Tel.: +49/931/ , Fax: +49/931/ Abstract In this paper, we develop an analysis of the so-called Dual Spacer. In contrast to a conventional spacer, which shapes a trac stream only according to a given Peak Cell Rate, the Dual Spacer takes into account the Peak Cell Rate as well as the Sustainable Cell Rate with the corresponding Burst Tolerance. By shaping, also according to a Sustainable Cell Rate, the Usage Parameter Control function can monitor a bursty cell stream more eciently. The analysis is carried out in discrete-time domain and performance measures like the cell rejection probability, the cell delay distribution and the inter-departure time distribution are derived. All results are of exact nature. Numerical examples which compare the performance of the Dual Spacer with that of a conventional one show a similar performance in terms of delay and cell loss for relatively small values of the Burst Tolerance. Using our analysis, the couple Sustainable Cell Rate/Burst Tolerance, which is used for trac shaping in case of the Dual Spacer, can be chosen adequately to achieve a given target cell rejection probability or a mean delay, respectively, and to allow the network provider to obtain a maximal multiplexing gain.

2 1 Introduction In ATM networks, resource allocation is performed on basis of a trac contract which is negotiated between the network and the user. One part of this trac contract consists of source trac descriptors like the Peak Cell Rate (PCR) and the Sustainable Cell Rate (SCR) of the connection. As long as the source is well behaving, i.e. the negotiated rates are not exceeded, the network commits to meet certain Grade of Service (GoS) parameters like the Cell Loss Rate (CLR), the Cell Transfer Delay (CTD) or the Cell Delay Variation (CDV), which constitute the other part of the trac contract. A major problem is the selection of adequate source trac descriptor parameters which are used for Usage Parameter Control (UPC) at the User Network Interface (UNI) or the Inter Network Interface (INI). On the one hand, these trac descriptors are dicult to choose because of the burstiness and the unpredictable bandwidth variations of certain trac sources, e.g. video and LAN-to-LAN trac. On the other hand, they must be useful for Connection Admission Control (CAC) and resource allocation. For sources without hard time constraints, i.e. delay insensitive applications, trac shaping can be performed to avoid congestion which may be caused by clusters of cells belonging to one connection. An advantage of shaping is, that it not only limits the traf- c volume but also reduces the burstiness of the trac source considerably. This allows an easier choice of the source trac descriptors. The way ofworking of a trac shaper is to delay certain cells of a given connection such that the inter-departure times of consecutive cells are never smaller than a given value (cf. [3, 6, 15]). This is done at the expense of introducing delay. In general, a maximum delay bound will be xed in the trac contract which leads to the discarding of a cell if it would have to be delayed longer than this amount of time. Each incoming cell is therefore either discarded or buered in the shaper, and re-emitted so that the resulting output cell stream is conforming with the negotiated source trac descriptors. At connection setup, the declaration of the PCR is necessary. However, the SCR can be specied optionally [1, 9]. This must be done in conjunction with a Burst Tolerance (BT), which limits the number of cells that can be sent at PCR. The specication of the SCR may allow the network provider to utilize the network resources more eciently. Until now, only trac shaping according to a given PCR has been considered in the literature, where performance measures like cell loss, delay, and characteristics of the output process have been addressed for various trac models, see e.g. [2, 4, 5, 8, 10]. Such a shaping facility is also often called cell spacer. However, if a SCR is negotiated, a cell stream can be shaped to be conforming with the PCR and the SCR specied in conjunction with the BT. The complexity of such a Dual Spacer would only be slightly higher than that of the conventional spacer treated in the literature until now. Using a Dual Spacer, foragiven connection a PCR and a SCR can be guaranteed. In the following we describe the operation mode of the Dual Spacer. Its basic function is to enforce a minimum cell inter-departure time, aiming at the policing of the PCR. The input cell stream is therefore inuenced in such a way, that the time between cells in the output stream is at least T,if1=T is the PCR. However, if a burst, which is emitted at PCR, is getting too large, then the Dual Spacer has to throttle down the cell emission rate to be conforming with the negotiated SCR 1=T s.themaximum Burst Size (MBS) 1

3 depends on the BT s.thus, a cell is delayed as long as necessary to be conforming with both the PCR and the SCR. Generally, a maximum CTD is declared in the trac contract. Cells which have to be delayed longer than a certain delay bound W are therefore rejected by the Dual Spacer. This bound will be chosen according to the corresponding QoS parameter. In Figure 1, the basic model of the Dual Spacer and related parameters are shown. input cell stream spacer T T s s W output cell stream Figure 1: Basic Dual Spacer model. The remainder of this paper is organized as follows. In Section 2 we present an exact analysis of the Dual Spacer. Performance measures we address are the cell rejection probability, the cell inter-departure time distribution and the cell delay distribution. We assume the cell inter-arrival process to be a renewal process. Numerical examples are provided in Section 3 and the paper concludes with a discussion in Section 4. 2 Analysis The basic idea behind the analysis is to introduce a spacer state denoted by the twodimensional time-dependent random variable U(t) = (P (t) S(t)). Depending on the spacer state U(t) that a cell sees upon arrival, the cell has to be delayed for a specic amount of time, is rejected if the delay boundw would be exceeded, or departs immediately from the spacer. A similar approach has been used in [8] to analyze the conventional spacer. If it is positive, the rst component P (t) describes the amount of time a cell arriving at time t has to be delayed at least to be conforming with the PCR. If P (t) is negative, then the cell can depart the spacer immediately, from the PCR shaping point of view. The second component S(t) corresponds in a similar way to the SCR shaping part. The only dierence is that the BT s must be taken into account. Thus, if S(t) is less or equal than s, the cell can depart immediately. Otherwise, it has to be delayed at least by S(t) ; s to achieve conformance. This holds, of course, only from the SCR shaping point of view. As appropriate for ATM environments, the time is discretized in slots of cell duration. The following notation is used: P ; n P (t) justbefore the arrival instant of cell number n, P + n P (t) justafter the arrival instant of cell number n, 2

4 S ; n S(t) justbefore the arrival instant of cell number n, S + n S(t) justafter the arrival instant of cell number n, A n discrete random variable for the number of slots between the arrival instants of cells n ; 1 and n. Accordingly, the discrete random variables U ; n =(P ; n S ; n ) and U + n =(P + n S + n ) (1) can be dened. The distributions of U ; n and U + n are denoted by u ; n (i j) and u + n (i j), where e.g. u ; n (i j) is dened by ProbfP ; n = i ^ S ; n = jg. Furthermore, we denote the distribution of A n by a(k), since the arrival process is assumed to be a renewal process. A sample evolution of the random variables dened above is depicted in Figure 2. P (t) S(t) arrival of cell number n;3 n;2 n;1 n+3 n n+1 n+2 n+4 S + n s + W S ; n s T s P + n T W P ; n t departure of cell number n;3 n;2 n;1 n n+2 n+3 Figure 2: Sample path of the Dual Spacer state process. Starting with the arrival of cell n ; 3, we have P ; n;3 =0andS ; n;3 < s.thus, cell n ; 3 departs immediately from the spacer and the components P (t) and S(t) are increased by 3

5 T and T s, respectively. Subsequently, they are decremented each slot. Just before the arrival of cell n ; 2, S n;2 ; is still smaller than s but P n;2 ; is now larger than 0. Therefore, cell n;2 must be delayed. Since P n;2 ; <W, cell n;2 is not rejected but delayed by P n;2 ; slots. At the arrival of cell n ; 1, we have 0<P n;1 ; <W and s <S n;1 ; < s + W.Cell n ; 1 is therefore not rejected but delayed by P n;1, ; since P n;1 ; >S n;1 ; ; s. The same holds for cell n, only with the dierence that P n ; <S n ; ; s and thus cell n is delayed by S ; n ; s slots. Just before cell n + 1 arrives, S n+1 ; > s + W. Therefore, cell n +1would have tobedelayed longer than W and is thus rejected. For the cells n +2andn +3we have the same situation as for cells n ; 3andn ; 2, respectively. Finally, cell n +4is rejected. The reason for this is that P n+4 ; is larger than the delay boundw. In the following we present an iterative algorithm to compute u ; n (i j) andu + n (i j) if the cell arrival process follows a general distribution. The analysis is based on the algorithm for the computation of the system size distribution in the G [X] =D=1= ; S queueing system presented in [14]. This algorithm has been used in [7, 11, 13] to investigate UPC functions like the Generic Cell Rate Algorithm (GCRA) and an extension to analyze twodimensional state processes has been presented in [12]. As outlined in the next three subsections, performance measures like the cell rejection probability, the cell inter-departure time distribution, and the cell delay distribution can be derived easily using the limiting distribution u ; (i j). We start with the dependencies between the random variables U ; n+1 =(P ; n+1 S ; n+1) and U + n =(P + n S+ n ): P ; n+1 = P + n ; A n+1 (2) S ; n+1 = S + n ; A n+1 : (3) These equations are driven by the decrease of P (t) ands(t) by one each slot. Based on the equations (2) and (3), the corresponding distribution can be found as: u ; n+1(i j) = 0X k=;1 u + n (i ; k j ; k) a(;k): (4) The computation of U + n =(P + n S + n )fromu ; n =(P ; n S ; n ) is a bit more complicated. For P + n and S + n we get the following: T : P ; n 0 ^ S ; n s + W P + n = 8 > <>: P ; n + T : 0 <P ; n W ^ S ; n s + W P ; n : P ; n >W _ S ; n > s + W (5) S + n = 8 > <>: T s : S ; n 0 ^ P ; n W S ; n + T s : 0 <S ; n s + W ^ P ; n W S ; n : S ; n > s + W _ P ; n >W : (6) 4

6 First, let us focus on P (t), i.e. equation (5). If P n ; 0 and cell n is not rejected due to a delay longer than W resulting from SCR shaping, P + n is set to T since the time interval until the emission of the next cell should be at least T. This corresponds to the rst case in equation (5). For a P n ; larger than 0, the rst component of the spacer state is computed by P + n = P n ; + T, if the delay according to the SCR shaping is smaller than W. The third case reects the arrival of a cell which would have tobedelayed longer than W and thus the spacer state remains unchanged. Analogously, equation (6) describes the computation of the second component S(t). To derive the equations for the corresponding distributions, we have to distinguish the four cases illustrated in Figure 3. Regions with dierent shadings represent dierent ways of calculating u + n (i j) due to acceptance or rejection of cell n. Dashed lines exclude the points from the delimited regions, solid lines include them. For sake of clarity, we decomposed the transition into two steps. The rst step reects a shifting operation which truncates the negative part of the spacer state space, where arriving cells are not rejected. Therefore, the probabilities of states (i j), where i or j are negative, are shifted to those states where i or j are replaced by zero if negative. This transition can be expressed by: u n(i j) = (u ; n (i j)) (7) with the operator () dened as: 0 : i<0 ^ j s + W (u ; n (i j)) = 8>< >: 0 : 0 i W ^ j<0 0P 0P 0P i 0 =;1 i 0 =;1 j 0 =;1 j 0 =;1 u ; n (i0 j) : i =0^ 0 <j s + W 0P u ; n (i 0 j 0 ) : i =0^ j =0 u ; n (i j 0 ) : 0 <i W ^ j =0 u ; n (i j) : otherwise : (8) As second step, i.e. the transition from u n (i j) tou+ n (i j), the resulting region in which cells are not rejected is shifted by thevector (T T s ). This leads to the following equation: 0 : (i j) 2A 1 u + n (i j) = 8><>: u n(i ; T j ; T s ) : (i j) 2A 2 u n(i ; T j ; T s )+u n(i j) : (i j) 2A 3 u n(i j) : otherwise : (9) 5

7 S n S n T s + s + W T s + s + W A3 A3 s + W T s A2 T s s + W A1 A1 0 Pn 0 Pn 0 T W T + W 0 W T T + W Case 1: W T ^ s + W T s Case 2: W<T^ s + W<T s T s + s + W S n T s + s + W S n A3 T s s + W A3 s + W T s A1 A1 0 Pn 0 0 T W T + W 0 W T T + W P n Case 3: W T ^ s + W<T s Case 4: W<T^ s + W T s Figure 3: Transition regions for u n(i) to u + n (i). For case 1, the regions A 1, A 2 and A 3 (cf. Figure 3) are given by: A 1 = f(i j) j (;1 i W ^;1j<T s ) _ (;1 i<t ^;1j s + W )g (10) A 2 = f(i j) j (T i W ^ T s j s + W )g (11) A 3 = f(i j) j (W <i T + W ^ T s j T s + s + W ) _ (T i T + W ^ s + W<j T s + s + W )g: (12) 6

8 For all other cases depicted in Figure 3, these regions are dened as follows: A 1 = f(i j) j (;1 i W ^;1j s + W )g (13) A 2 = A 3 = f(i j) j (T i T + W ^ T s j T s + s + W )g: (15) Using the equations presented above, the limiting distribution of the two-dimensional spacer state distribution u ; (i j) can be obtained iteratively: (14) u ; (i j) = lim n!1 u ; n (i j): (16) This forms the basis to derive the performance measures in the next three subsections. 2.1 Cell rejection analysis The cell rejection probability p r, i.e. the probability that an arriving cell would have to be delayed longer than the delay boundw, can be found as p r =1; WX s+w X i=;1 j=;1 u ; (i j): (17) The rejection probability p r is given by the probability of the area of the spacer state space where an arriving cell has to be delayed longer than W either because of PCR shaping or because of SCR shaping. 2.2 Inter-departure time distribution In this subsection we focus on the cell inter-departure time distribution of the Dual Spacer. The following derivation of the output process is performed in conjunction with a renewal assumption. The probability d(k) to observe a time interval of k slots between two cells departing the spacer consecutively can be given by: d(k) = 1 1 ; p r X (i j)2b k u ; (i j): (18) In this context, the set B k contains those states (i j) where the departure of the previous not rejected cell has been occurred or will occur k slots before the departure of the cell which is currently arriving. 7

9 Due to the PCR shaping, we have a minimum inter-departure interval of T slots and thus for k<t: B k = : (19) To derive the sets B k for k T,wehave to distinguish three regions of the spacer state space, which are depicted in Figure 4 with dierent shading intensities. j T s + s + W s + W T s B Ts s B T B x 0 W T + W i minft ; i T s + s ; jg T Figure 4: Regions for calculating the inter-departure time distribution. If an arriving cell has not to be delayed, then the inter-departure interval for the actual spacer state (i j) is determined by minft ; i T s + s ; jg. This is relevant if(i j) is located in the region B x, cf. Figure 4. If i 0, then the last departure has occurred T ; i slots before. For j s the last departure instant has occurred T s + s ; j time slots before. Since PCR and SCR shaping is performed independently of each other, the inter-departure interval is, in this case, equal to the minimum of these two values. If the spacer state upon cell arrival is located in the dark shaded region BT, the interdeparture interval will be T, since there is no further delay required due to SCR shaping. For the spacer states in the light shaded region BTs, an arriving cell has to be delayed due to SCR shaping. Thus the inter-departure interval is equal to T s. Taking these properties into account, the sets B T and B Ts are given by: and B T = f(i j) j (0 <i W ^;1<j i + s ) _ (minft ; i T s + s ; jg = T )g (20) 8

10 B Ts = f(i j) j (0 <i W ^ i + s <j s + W ) _ (;1 <i 0 ^ s <j s + W ) _ (minft ; i T s + s ; jg = T s )g: (21) For all other values of k, B k contains the following states: B k = f(i j) j minft ; i T s + s ; jg = kg : (22) 2.3 Delay distribution The performance measure we address on in this subsection is the cell delay distribution. This measure allows to investigate the additional delay introduced by the shaping according to a SCR compared to that of a conventional spacer. Since the delay for cells which arrive at time instants where one or both components of the spacer state are negative is the same as for time instants where the corresponding component is equal to 0, we can make use of the shifting operation given in equation (8): u (i j) = (u ; (i j)): (23) Dening sets C k which contain those states (i j) where an arriving cell is delayed by k slots, we can compute the delay distribution w(k) fork =0 ::: W by the following equation: w(k) = X 1 u (i j): (24) 1 ; p r (i j)2c k A cell is delayed by k slots, if the delay due to PCR shaping is equal to k slots and the delay due to SCR is shorter or vice versa. Thus, for 0 k W the sets C k are given by: C k = f(i j) j (i = k ^ i>j; s ) _ (j = k + s ^ i j ; s )g : (25) Because of the maximum delay boundw, the sets C k are empty fork>w. The probability p w for an arriving cell to be delayed can be computed by: p w = WX k=1 w(k): (26) 9

11 delay distribution 3 Numerical results To compare the performance of the Dual Spacer with that of the conventional one, we present somenumerical results. In Figure 5, the delay distributions are drawn for several choices of s. As inter-arrival process we used a negative binomial distribution which allows to vary the mean E A and the coecient ofvariation c A almost independently of each other (E A c 2 A > 1must be fullled). We useameanofe A = 10 slots and the coecient ofvariation is set to c A =1:0. The spacer parameters are T =5,T s = 8 and s is varied from 0 to 150. A maximum delay ofw = 200 slots is tolerated ;1 10 ;2 10 ;3 10 ;4 10 ;5 s = 1 s =100 s =50 s =0 10 ; time in slots Figure 5: Inuence of the BT s on the delay distribution (c A =1:0). As can be observed in Figure 5, the delay distribution is strongly dependent on the choice of s. If s increases, the delay distribution rapidly approaches that of the conventional spacer. For s = 1, both are the same, since for a BT tending to innity the SCR shaping has no eect on the delay of the cells. Therefore, only PCR shaping is decisive. In our case, the distributions for s = 1 and s = 150 can not be distinguished. If we lookat Figure 6 where the coecient ofvariation is now set to c A =2:0, a muchslower approach can be observed, i.e. s must be chosen large to obtain a delay distribution close to that of a conventional spacer. In each curve small steps can be recognized at the left hand side of the distribution as typical for GI=D=1 queueing systems. 10

12 mean delay delay distribution ;1 10 ;2 10 ;3 s =0 s =100 s =200 s = ;4 s = 1 10 ;5 10 ; time in slots Figure 6: Inuence of the BT s on the delay distribution (c A =2:0) c A =0:5 c A =1:0 c A =1:5 c A =2: Figure 7: Asymptotic behavior of the mean delay. Burst Tolerance s 11

13 delay coecient of variation In Figure 7, the mean delay is plotted over s for various choices of c A. This gure shows the asymptotic behavior more clearly. It can be observed that the mean value, and as well the delay distribution, approaches the faster a limiting value the smaller c A is. Thus, for a larger BT the supplementary property of the dual mechanism to reduce the burstiness of the trac stream is getting lost. This depends, however, on the choice of the SCR 1=T s. If the SCR is close to the Average Cell Rate (ACR), the SCR shaping plays a dominant role also for large values of the BT. Another interesting performance measure is the coecient ofvariation c W of the delay introduced by the Dual Spacer. Curves for dierent choices of c A are depicted in Figure 8 in dependence on s. As expected, c W also approaches a limiting value. This is due to the approach of the delay distribution against a limiting distribution as shown in the Figures 5 and c A =0:5 c A =1:5 c A =1:0 c A =2: Burst Tolerance s Figure 8: Asymptotic behavior of the delay coecientofvariation c W. For small values of s the coecient ofvariation c W rst increases and then decreases against a limiting value. The initial increase is more distinctive ifc A is small. The later decrease of the delay coecientofvariation is due to the minor inuence of SCR shaping for large values of s.ifc A is getting larger, the curves get closer to each other. Now, we focus on the inter-departure time distribution of the Dual Spacer. In Figure 9, we use the same inter-arrival process as before with a coecient ofvariation c A =2:0. For s = 0, i.e. no bursts are allowed, the inter-departure time distribution is equal to that of a conventional spacer with minimum inter-cell distance T s.if s is increased, the inter-departure time distribution also approaches a limiting distribution. Like in case 12

14 inter-departure time distribution 1:0 0:8 0:6 0:4 0:2 s =0 s =50 s =150 s = 1 0: time in slots Figure 9: Inuence of s on the inter-departure time distribution. of the delay distribution, this limiting distribution is equal to the inter-departure time distribution of the conventional spacer with a minimum inter-cell distance of T. For all other values of s,two steps can be observed. These steps correspond to the PCR and SCR shaping and are therefore located at i =5andi s = 8 (note that T = 5 and T s = 8). The step due to SCR shaping is getting smaller if s increases or c A decreases. From the network point of view, a small value of the BT may allow a higher utilization of the resources since the trac stream is smoother. However, the cell rejection probability and the cell delay will be increased. Using our analysis, the couple (T s, s ) can be chosen to achieve a given target cell rejection probability or mean delay, respectively, and to allow the network provider to obtain a maximal multiplexing gain at the same time. 4 Conclusion In this paper we presented a discrete-time analysis of the Dual Spacer which shapes an ATM input trac stream to be conforming with a given Peak Cell Rate and a Sustainable Cell Rate in conjunction with the Burst Tolerance. The input trac is assumed to be a renewal process and a maximum delay bound for the spacer is introduced. Using the limiting distribution of the two-dimensional system state distribution, performance measures like the cell rejection probability, the cell inter-departure time distribution and the delay distribution have been derived in closed form. All results are of exact nature. Numerical 13

15 examples are given to show the performance of the Dual Spacer for dierent trac conditions. Furthermore, the results have been compared with those of a conventional spacer, i.e. a spacer which shapes the trac only according to a Peak Cell Rate. From the numerical examples we can conclude, that a performance close to that of the conventional spacer is achieved with a Dual Spacer already for small values of the Burst Tolerance if the Sustainable Cell Rate is chosen adequately. The reason for this is the fast approach of the delay and the inter-departure time distribution against their limiting distributions. Thus, the supplementary properties introduced by the dual mechanism are lost already for small values of the Burst Tolerance. However, if an additional delay can be accepted, the Burst Tolerance can be chosen quite small resulting in a trac stream which is very smooth and therefore favorable from the network point of view. Since the network provider needs to allocate the less resources the smaller the negotiated Burst Tolerance is, a small value of the Burst Tolerance is also protable for the user from the taring point of view. Concerning trac management, the Dual Spacer should be implemented preferably, since it also can emulate a conventional spacer, if necessary, and the complexity of implementation remains almost the same compared to the conventional one. Acknowledgement The author appreciates the support of the Deutsche Bundespost Telekom (Forschungsund Technologiezentrum (FTZ)). References [1] ATM Forum, ATM User-Network Interface Specication, Version 3.0, September [2] F. Bernabei, L. Gratta, M. Listanti, A. Sarghini, Analysis of ON-OFF Source Shaping for ATM Multiplexing, IEEE INFOCOM 1993, pp [3] P. Boyer, F. Guillemin, M. Servel, J.-P. Coudreuse, Spacing Cells Protects and Enhances Utilization of ATM Network Links, IEEENetwork, Vol. 6, No. 5, September 1992, pp [4] F.M. Brochin, A Cell Spacing Device for Congestion Control in ATM Networks, Performance Evaluation, Vol. 16, No. 1, 1992, pp [5] A.I. Elwalid, D. Mitra, Analysis and Design of Rate-Based Congestion Control of High Speed Networks, I: Stochastic Fluid Models, Access Regulation, Queueing Systems 9, 1991, pp [6] F. Guillemin, P. Boyer, L. Romoeuf, The Spacer-Controller: Architecture and First Assessments, Broadband Communications, Portugal, January 1992, pp

16 [7] F. Hubner, Dimensioning of a Peak Cell Rate Monitor Algorithm Using Discrete- Time Analysis, Proceedings of ITC-14, Antibes, France, June 1994, pp [8] F. Hubner, P. Tran-Gia, A Discrete-Time Analysis of Cell Spacing in ATM Systems, University of Wurzburg, Institute of Computer Science, Research Report Series, Report No. 66, June [9] ITU-T Study Group 13, Draft Recommendation I.371, Trac Control and Congestion Control in B-ISDN, March [10] L.K. Reiss, L.F. Merakos, Shaping of Virtual Path Trac for ATM B-ISDN, IEEE INFOCOM 1993, pp [11] M. Ritter, P. Tran-Gia, Performance Analysis of Cell Rate Monitoring Mechanisms in ATM Systems, International Conference on Local and Metropolitan Communication Systems, Kyoto, December, [12] M. Ritter, S. Kornprobst, F. Hubner, Performance Comparison of Source Policing Architectures in ATM Systems, University of Wurzburg, Institute of Computer Science, Research Report Series, Report No. 81, November [13] O. Rose, M. Ritter, MPEG-Video Sources in ATM-Systems A new approach for the dimensioning of policing functions, International Conference on Local and Metropolitan Communication Systems, Kyoto, December, [14] P. Tran-Gia, H. Ahmadi, Analysis of a Discrete-Time G [X] =D=1 ; S Queueing System with Applications in Packet-Switching Systems, IEEE INFOCOM 1988, pp [15] E. Wallmeier, T. Worster, The Spacing Policer, an Algorithm for Ecient Peak Bit Rate Control in ATM Networks, ISS 14, October 1992, paper A

17 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 [31] H. Vollmer und K. W. Wagner. Classes of counting functions and complexity theoretic operators. August

18 [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 [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

19 [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 [76] K. Cronauer. A Criterion to Separate Complexity Classes by Means of Oracles. January [77] M. Ritter. Analysis of the Generic Cell Rate Algorithm Monitoring ON/OFF-Trac. January [78] K. Poeck, D. Fensel, D. Landes und J. Angele. Combining KARL and Congurable Role Limiting Methods for Conguring Elevator Systems. Januar [79] O. Rose. Approximate Analysis of an ATM Multiplexer with MPEG Video Input. Januar [80] A. Schomig. Using Kanban in a Semiconductor Fabrication Environment a Simulation Study. Marz [81] M. Ritter, S. Kornprobst, F. Hubner. Performance Analysis of Source Policing Architectures in ATM Systems. April [82] U. Hertrampf, H. Vollmer und K. W. Wagner. On Balanced vs. Unbalanced Computation Trees. May [83] M. Mittler und A. Schomig. Entwicklung von " Due{Date\{Warteschlangendisziplinen zur Optimierung von Produktionssystemen. Mai [84] U. Hertrampf. Complexity Classes Dened via k-valued Functions. Juli [85] U. Hertrampf. Locally Denable Acceptance: Closure Properties, Associativity, Finiteness. Juli [86] O. Rose, M. R. Frater. Delivery of MPEG Video Services over ATM. August [87] B. Reinhardt. Kritik von Symptomerkennung in einem Hypertext-Dokument. August [88] U. Rothaug, E. Yanenko, K. Leibnitz. Articial Neural Networks Used for Way Optimization in Multi-Head Systems in Application to Electrical Flying Probe Testers. September [89] U. Hertrampf. Finite Acceptance Type Classes. Oktober [90] U. Hertrampf. On Simple Closure Properties of #P. Oktober [91] H. Vollmer und K.W. Wagner. Recursion Theoretic Characterizations of Complexity Classes of Counting Functions. November [92] U. Hinsberger und R. Kolla. Optimal Technology Mapping for Single Output Cells. November [93] W. Noth und R. Kolla. Optimal Synthesis of Fanoutfree Functions. November [94] M. Mittler und R. Muller. Sojourn Time Distribution of the Asymmetric M=M=1==N { System with LCFS-PR Service. November [95] M. Ritter. Performance Analysis of the Dual Cell Spacer in ATM Systems. November