University ofwurzburg. Research Report Series. Monitoring ON/OFFTrac. M. Ritter. Institute of Computer Science, University of Wurzburg,


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1 University ofwurzburg Institute of Computer Science Research Report Series Analysis of the Generic Cell Rate Algorithm Monitoring ON/OFFTrac M. Ritter Report No. 77 January 1994 Institute of Computer Science, University of Wurzburg, Am Hubland, Wurzburg, Federal Republic of Germany Tel.: +49/931/ , Fax: +49/931/ , Abstract This study deals with the enforcement of source parameters in ATM networks. An algorithm for monitoring the cell rate of a connection is the socalled generic cell rate algorithm. At the moment, only peak cell rate policing is standardized, but there are discussions in the standard commissions to introduce a second source parameter, the sustainable cell rate in conjunction with a burst tolerance. These source parameters are dened at the physical layer service access point where ATM connections behave like ON/OFFsources due to the preceding cell spacer. In this paper, we present an algorithm to derive the exact cell rejection probability of the generic cell rate algorithm if ON/OFFsources are monitored. This is done by an iterative algorithm based on a discretetime analysis technique. Using the cell rejection probability, the parameters for the monitor algorithm can be chosen to meet desired performance objectives, e.g. the conformance of a connection. Numerical results show that the choice of these parameters strongly depend on the statistical characteristics of the ON/OFFprocess. Moreover, the inuence of the interburst spacing on the cell rejection probability is pointed out.
2 1 Introduction In the current standardization process of ATM networks the design of the usernetwork interface (UNI) with the incorporated usage parameter control (UPC) plays an important role. The design of this function must be carried out very carefully for the following reasons. On the one hand the UPC function must protect the network from overload due to users disregarding their negotiated trac contract, whereas on the other hand connections of users meeting their trac contract should be aected as little as possible by the UPC function. A conguration which denes the sequence of functional groups and reference points that ATM cells have to pass on their way from a connection endpoint to the public UNI (T B reference point) is illustrated in Figure 1. This conguration serves as a reference conguration in the CCITT Draft Recommendation I.371 [5] and was also adopted by the ATM Forum [1]. Trac Source 1 Trac Source N Connection Endpoint MUX Shaper PHY SAP Physical Layer Functions ATM Layer Physical Layer Equivalent Terminal CDV Tolerance Private UNI Other CEQ Functions Generating S B CDV Public UNI T B UPC GCRA(T,0) GCRA(T,) GCRA(T S, S ) GCRA(T S, S ) Figure 1: Reference conguration from CCITT Draft Recommendation I.371. Figure 1 shows a number of trac streams that are multiplexed. By multiplexing, the original cell streams are perturbed, i.e. cell delay variation (CDV) is introduced. To reduce CDV and to smooth the cell streams oered by the trac sources, trac shaping is performed after the multiplexing stage. As shaping device a cell spacer was suggested in [2], [3] and [8] whereas another approach using a spacer policer was presented in [16]. Passing the shaping device, the multiplexed cell streams enter physical layer functions and other CDV generating functionalities before entering the ATM network. At the public UNI, connections are accepted or rejected according to connection admission control (CAC). If a connection is accepted, it has a contract with the network. In this contract, the user agrees to meet the negotiated trac parameters during the holding time of the connection, whereas the network guarantees a predened quality of service (QOS). To prevent the network from congestion and thereby to guarantee the desired QOS for 1
3 all connections according to their trac contracts, a policing function must be provided at the UNI. In the rst phase of implementation and testing of ATM networks, one of the most important connection parameters will be the peak cell rate (PCR). The policing of this parameter is already standardized by the standardization bodies. Currently, there are discussions in the standard commissions about the introduction of a second source parameter, the sustainable cell rate (SCR). A preliminary denition of this measure can be found in [1] and [6]. The SCR is an upper bound for the possible conforming average cell rate of an ATM connection and can be declared by the user jointly to the PCR. However, this declaration is optionally and not necessary it is not needed e.g. in case of CBR calls. An enforcement of this bound by UPC functions mayallow the network provider to utilize the network resources more eciently, since the user species the future cell ow of his connection more detailed than just by the PCR. The declaration of the SCR can benet the user by possibly reduced charges. To control the PCR and the SCR a suitable monitor algorithm is required. This algorithm has to take care of CDV, which can be introduced by various factors as mentioned above. An algorithm for monitoring the PCR of a connection, according to [5], is the virtual scheduling algorithm (VSA). Beside of this algorithm, there exists another one which is equivalent to it, called continuousstate leaky bucket algorithm (see also [5]). A more general notation for these two equivalent algorithms is generic cell rate algorithm (GCRA), which is used in [1]. For the enforcement of the SCR the GCRA is employed too, as proposed in [1] and [6]. In the literature there exist several investigations dealing with performance studies of the GCRA, see e.g. [7], [9], [10], [11] and [12]. However, an algorithm to compute the exact cell rejection probability of the GCRA was derived only in [11]. This was done by the use of a discretetime analysis technique based on the analysis of the G [X] =D=1 ; S queueing system proposed in [14]. In [12], the results of [11] are extended to compute the interdeparture time distribution from the GCRA. A survey on discretetime analysis techniques and its application to UPC modelling in ATM systems is given in [15]. Concerning the results in [11] and [12], the cell interarrival times of the monitored trac stream are assumed to follow a general distribution. Even though the proposed algorithm can deal with nonrenewal processes, a calculation of the cell rejection probability if ON/OFFtrac streams are monitored can not be done with reasonable computation eort. This type of trac can be observed e.g. after the cell spacing device at the physical layer service access point (PHY SAP), where the cell process of a particular connection is an ON/OFFprocess (see Figure 1). Therefore, we derive in the following an algorithm based on the results in [11] to compute the exact cell rejection probability of the GCRA if an ON/OFFtrac stream is monitored. Using this analysis we can focus on SCR policing and we are able to dimension the trac parameters required for it. The rest of this paper is organized as follows. In Section 2 we rst describe the investigated GCRA. The next section deals with the analysis to determine the exact cell rejection probability of the GCRA. Numerical results are presented in Section 4 and Section 5 concludes the paper. 2
4 2 The generic cell rate algorithm In this section we describe the basic functionality of the GCRA. For sake of simplicity, we focus on the VSA which is depicted in the owchart in Figure 2. First, we will consider only PCR monitoring. cell generation at time t yes TAT <t? no conforming cell (1) TAT :=t + T no TAT >t+? yes conforming cell (2) nonconforming cell (3) TAT :=TAT +T Figure 2: GCRA as virtual scheduling algorithm. The PCR is dened as the inverse of the minimum time T between the emission of two ATM cells from a connection. Thus, the time dierence between two cell arrivals (at the public UNI where the GCRA is employed) shall not be smaller than T. The GCRA determines whether a cell arrives too closely to the last cell, indicating that the ATM connection generates cells with a rate higher than the negotiated rate. Since the original cell stream of the source is altered by various CDV generating functions, CDV hastobetaken into account by the GCRA. This is done by thecdv tolerance. As the algorithm is dependent ont and, we will refer to it as GCRA(T ). The time instant at which the next cell is expected to arrive is called theoretical arrival time (TAT), whereas the actual arrival time of a cell is denoted by t. Of course, the TAT for the rst monitored cell is set to its actual arrival time. At the arrival of a cell, three cases can be distinguished: (1) If a cell arrives later than its TAT, the connection generates cells with a smaller cell rate than the negotiated one. Therefore, the cell is accepted and the TAT for the 3
5 next cell is set to t + T, i.e. the late arrival of this cell does not allow for an earlier arrival of the next cell. (2) If a cell arrives before its TAT but not before TAT{, the cell arrives too early but within the allowed CDV tolerance. Therefore, the cell is also accepted but the TAT of the next cell is set to TAT+T. (3) If a cell is generated before the time instant TAT{, the cell is recognized as a nonconforming cell and is rejected 1. The TAT for the next cell is not modied in this case. If the GCRA(T ) is employed at the public UNI, it is guaranteed that cells from a connection enter the ATM network with a long term rate of at most 1=T. As suggested in [1] and [6], the GCRA is used to monitor the SCR together with the corresponding burst tolerance S. Therefore, the increment parameter T S of the GCRA is the inverse of the SCR and S is the limit parameter. The burst tolerance S is introduced on the one hand to take care of burst scale variations of the cell stream and on the other hand to take into account the CDV introduced by various functions (see Figure 1). In the description of the GCRA the parameters T and must be substituted by T S and S respectively for monitoring the SCR. In this case, the monitor algorithm is referred to as GCRA(T S S ). 3 Performance analysis In this part of the paper we extend the results from [11] by deriving an algorithm to determine the exact cell rejection probability for the GCRA if the monitored trac is of ON/OFFtype. As appropriate for ATM environments, the time is discretized into slots of cell duration. The type of ON/OFFprocess we focus on here is the following. In the ONstates consecutive cells arrive with a xed distance to each other, say d cell slots, whereas there are no cell arrivals in the OFFstates. Furthermore, we assume that all ONstates start with a cell arrival. The lengths of the ON and OFFstates follow a general distribution with a lower bound of 1. In the following, we call these distributions the ON and the OFFstate length distribution respectively. A snapshot of such a process is depicted in Figure 3. ONstate OFFstate ONstate d cell Figure 3: Snapshot of an ON/OFFtrac stream. This type of trac can be observed in ATM networks e.g. after a cell spacing device at the PHY SAP (cf. Figure 1), where the output process from a particular connection is 1 Cells which are identied as nonconforming can optionally be tagged or rejected (see [1], [5]). We consider however only the case of cell rejection. 4 t
6 such an ON/OFFprocess. The parameter d cell corresponds to the spacer peak emission rate 1=T for this connection. Actually, this is not quite correct, since the spacer logically demultiplexes, spaces and then logically multiplexes the cell streams. Thus, after the spacing device there is some CDV introduced due to the multiplexing. However, if we focus like here on SCR policing, which is a longterm measure, this eect will be negligible. For the analysis part we consider the number of slots until a new cell is expected to arrive and use for this the timedependent random variable Z(t) (cf. [11]). Specically, we use the following notation: A OF F n A ON n Z ; OF F n k Z + OF F n k Z ; ON n k Z + ON n k discrete random variable for the length of the nth OFFstate in number of slots. discrete random variable for the length of the nth ONstate in number of slots. Z(t) just before the beginning of the kth slot in the nth OFFstate. Z(t) just after the beginning of the kth slot in the nth OFFstate. Z(t) just before the beginning of the kth slot in the nth ONstate. Z(t) just after the beginning of the kth slot in the nth ONstate. The distributions of the discretetime random variables A OF F n and A ON n are denoted by a OF F n (i) anda ON n (i), respectively. If the lengths of the ON and OFF states follow a renewal process, the distributions a OF F n (i) anda ON n (i) are independent ofn. Therefore, we simply use a OF F (i) anda ON (i) to denote the ON respectively OFFstate length distributions. Furthermore, the complementary cumulative probability distributions of these two random variables are denoted by F c A OFF (i) andf c A ON (i). For the distributions of the system state random variables Z ; OF F n k, Z+ OF F n k, Z; ON n k and Z + ON n k, we use the terms z ; OF F n k(i), z + OF F n k(i), z ; ON n k(i) andz + ON n k(i), respectively. An example scenario illustrating the evolution of these random variables is depicted in Figure 4. Now, we propose an iterative algorithm for the computation of the distributions of Z ;, OF F n k Z+, Z; and OF F n k ON n k Z+. Using these distributions, essentially z; (i), ON n k ON n k the probability to notice a nonconforming cell can be derived in closedform. For the nth OFFstate, the random variable Z + is equal to Z; OF F n k OF F n k, since there are no cell arrivals in the OFFstates, i.e.: Z + OF F n k = Z; OF F n k for k =0 ::: 1: (1) Z + is given by Z; ON n k ON n k in the following way: ( Z + = Z ; ON n k : Z ; ON n k > ON n k Z ; ON n k + (k) T : Z ; ON n k for k =0 ::: 1 (2) 5
7 A ON n;1 A OF F n A ON n A OF F n+1 d cell t Z ; OF F n 0 Z ; OF F n 1 Z ; ON n 0 Z ; ON n 1 Z ; OF F n+1 0 Z ; OF F n A OFF n Z ; ON n A ON n Z + OF F n 0 Z+ OF F n 1 Z + ON n 0 Z + ON n 1 Z + OF F n+1 0 Figure 4: Example scenario illustrating the evolution of the random variables. if the nth ONstate is considered. Here, (k) corresponds to the deterministic cell arrival process in the ONstates and is dened as (k) = ( 1 : k mod dcell =0 0 : k mod d cell 6=0 : (3) Z ; OF F n k+1 and Z; ON n k+1 are derived by the next two equations. These equations are driven by the decrease of Z ; OF F n, respectively Z ON n, ; byoneeachslotuntil they reach zero. Z ; OF F n k+1 =maxf0 Z+ OF F n k ; 1g for k =1 ::: 1 (4) Z ; ON n k+1 = maxf0 Z + ON n k ; 1g for k =1 ::: 1 (5) The system state random variables just before the switching instant to the (n + 1)th OFF respectively nth ONstate are given by Z ; OF F n+1 0 = Z ; ON n A ON n (6) for the switching to the OFFstate and by Z ; ON n 0 = Z; OF F n A OF F n (7) for the switching to the ONstate. For < T, the corresponding distributions for Z + OF F n k be derived according to the equations (1) to (7) by and Z + ON n k (k =0 ::: 1) can z + OF F n k(i) =z ; OF F n k(i) for i =0 ::: T + (8) and for (k) = 0, we obtain z + ON n k (i) as z + ON n k(i) =z ; ON n k(i) for i =0 ::: T + : (9) 6
8 If (k) = 1, i.e. the case of a cell arrival, the distribution z + ON n k (i) is computed by z + ON n k (i) = 8 > < >: 0 : 0 i z ; ON n k(i) : <i<t z ; ON n k(i)+z ; ON n k(i ; T ) : T i T + : (10) For T equation (10) changes to z + ON n k(i) = 8 >< >: 0 : 0 i<t z ; ON n k (i ; T ) : T i z ; ON n k (i ; T )+z; ON n k (i) : <i + T : (11) Since the system state is decreased by one each slot, the distributions z ; OF F n k+1 (i) and z ON n k+1(i) ; can be calculated for k =0 ::: 1 by z ; OF F n k+1 (i) = 8 > < >: z + OF F n k(0) + z + OF F n k(1) : i =0 z + OF F n k (i +1) : 0 <i<t+ 0 : T + (12) and z ; ON n k+1(i) = 8 >< >: z + (0) + ON n k z+ ON n k (1) : i =0 z + ON n k (i +1) : 0 <i<t+ 0 : T + : (13) After the computation of these distributions, we obtain the system state distributions just before the switching instant to the(n + 1)th OFF respectively nth ONstate by the following two equations: and z ; OF F n+1 0(i) = z ; ON n 0 (i) = 1X k=1 1X k=1 a ON (k) z ; ON n k (i) for i =0 ::: T + (14) a OF F (k) z ; OF F n k (i) for i =0 ::: T + : (15) The distributions for the dierent slots in the states before are therefore multiplied with the probability for the occurrence of a state with the corresponding length. Using equations (8) to (15) iteratively, the system state distributions in equilibrium z ; ON k(i) (k =0 ::: 1) can be derived by z ; ON k(i) = lim n!1 z; ON n k(i) for i =0 ::: T + : (16) 7
9 From the distribution z ; ON k (i) we can easily compute the probabilities p(k) that a cell arriving at the kth slot in an ONstate is rejected, cf. [11]: p(k) = T X+ i=+1 z ; ON k (i) for k =0 ::: 1: (17) The probability to notice a nonconforming cell can be obtained in closedform by the use of the probability p(k). Therefore, we have only to consider the slots where cells can arrive and weight the probability p(k) by the probability for the occurrence of such slots, i.e. the corresponding value of the complementary cumulative probability distribution F c A ON. After normalization, we arrive at the cell rejection probability p r : 1P k=1 p r = (k) F c A ON (k) p(k) 1P k=1 (k) F c A ON (k) 4 Numerical results : (18) For the numerical examples in this section we focus on SCR policing. Figure 5 shows the cell rejection probability of the GCRA in dependence on the burst tolerance S for ON/OFFstate lengths following three dierent distributions. The cell distance in the ONstate is assumed to be d cell = 5 and the minimum length of an OFFstate is 4 slots. Thus, we consider a trac stream which is conforming to GCRA(5 0), i.e. spaced with T = 5. Wehave chosen a geometric, a binomial and a uniform distribution. The mean lengths of the ON and OFFstates are set to 50 slots and the maximum burst size (MBS), i.e. the maximum length of an ONstate is equal to 100 slots. To guarantee the MBS for the geometric distribution, we have cut the distribution at this bound and normalized it afterwards. The mean value for the geometric distribution is therefore slightly, however neglectable, lower than 50 slots. For each of these distributions, Figure 5 shows curves for T S = 10, which corresponds to the mean cell rate of the monitored sources, and T S = 9. It can be observed, that the choice of the burst tolerance S to achieve a desired cell rejection probability is strongly dependent on the distribution of the state lengths. In case of a binomial distribution and T S = 10 a cell rejection probability oflessthanabout0:05 can not be obtained, even if S is set to signicantly higher values. Before we take a look at further numerical results, we give some brief comments on the context between the MBS and the conformance of cell streams. The maximum size of a burst that can be transmitted at PCR being still in conformance with the GCRA(T S S ) is determined by the SCR 1=T S and by the burst tolerance S. This size, expressed in number of slots, is given by (cf. [1]) 2 S MBS = 1+ T (19) T S ; T 2 bxc denotes the largest integer less than or equal to x. 8
10 cell rejection probability ;1 10 ;2 T S =10 geometric binomial uniform T S =10 T S =10 10 ;3 T S =9 T S =9 T S =9 10 ; burst tolerance S Figure 5: Cell rejection probability for dierent ON/OFF distributions. where 1=T is the PCR of the cell stream. Note that S applies at the PHY SAP of the equivalent terminal (cf. Figure 1). Thus, the MBS in conjunction with T and T S is used to determine the burst tolerance S. However, sources which generate bursts with maximum conforming burst size and with arbitrary spacing between them would not be generally conforming with the GCRA(T S S ). In order for a burst of this size to be conforming, the cell stream needs to be idle long enough for the state of the GCRA to become zero prior to this burst. For that reason, the appropriate choice of S depends on the minimum spacing between consecutive bursts as well as on the MBS. If the minimum spacing between bursts with intercell spacing T is T I at the PHY SAP, then the cell ow is conforming with GCRA(T S S )if1=t S and S are chosen at least large enough to satisfy the following equation (cf. [1]): $ min(ti ; T S S ) MBS = 1+ T S ; T %! T (20) To show the inuence of the minimum spacing between bursts, curves for dierent choices 9
11 cell rejection probability of S are drawn in Figure 6. We used geometric distributed lengths of the ON and OFFstates with the same parameters as before. The minimum spacing between the bursts is varied by shifting the original distribution. To achieve a constant mean length, the mean values of the original distributions are chosen appropriately before the shifting operation. The cell distance is again d cell =5andwehave chosen T S =10,which is equal to the corresponding mean cell rate ;1 10 ;2 10 ;3 10 ;4 10 ;5 S =100 S =200 S =300 S = ;6 S = ;7 10 ;8 10 ;9 S =600 S = minimum spacing between bursts in slots Figure 6: Cell rejection probability fort S =10vs. minimum interburst spacing. Figure 6 shows, that for this example a cell rejection probability in the area of 10 ;9 can only be achieved if the minimum spacing between bursts is almost equal to the mean of the OFFstate lengths and S is set quite large. This implies a nearly deterministic OFFstate length distribution. From equation (19), we get S 500 for a MBS of 100 slots. To satisfy equation (20), the minimum interburst spacing to achieve conformance must be about 500 slots. This, however, is not possible since the mean length of an OFFstate is equal to 50 slots. Another possibility to dimension the parameters T S and S for a given cell stream is by using a SCR higher than the mean cell rate. Results for the example above witht S =9 are depicted in Figure 7. Now, with S = 500 a cell rejection probability of10 ;9 can be attained for a minimum 10
12 cell rejection probability ;1 10 ;2 10 ;3 10 ;4 10 ;5 S =200 S = ;6 10 ;7 10 ;8 10 ;9 S =300 S =400 S = minimum spacing between bursts in slots Figure 7: Cell rejection probability fort S =9vs. minimum interburst spacing. interburst spacing of approximately 20 slots. However, the original intention of the SCR to describe the long term cell rate of a connection is lost if T S is decreased, i.e. the SCR is increased. Therefore, increasing S should be preferred instead of decreasing T S,if possible. Our last example will show that it is not always possible to achieve a desired cell rejection probability by increasing S. For this purpose, we use a uniform distribution for the lengths of the ON respectively OFFstates with a mean of 50 slots in both cases. The MBS is again set to 100 slots and the cell spacing in the ONstates is d cell =5. If we look at Figure 8, we can observe that for T S = 10, what corresponds to a SCR equal to the mean cell rate, a cell rejection probability in the area of 10 ;9 can not be obtained this holds also if the minimum interburst spacing reaches its maximum value and if S is chosen quite large. For T S = 10, the cell rejection probability hardly decreases if the minimum spacing between bursts in increased. To achieve a lower rejection probability, the SCR must be chosen larger than the mean cell rate. Curves for T S = 9 are shown in Figure 8. This, however, leads to a less eective enforcement of the trac source from the network point of view. 11
13 cell rejection probability ;1 10 ;2 10 ;3 10 ;4 T S =10 S =300 T S =10 S =700 T S =9 S = ;5 10 ;6 T S =9 S = ;7 T S =9 S = ;8 10 ; minimum spacing between bursts in slots Figure 8: ON/OFFstate lengths following a uniform distribution. 5 Concluding remarks In this paper we have presented an analysis method to determine the exact cell rejection probability of the GCRA if the monitored source is an ON/OFFsource. The developed iterative algorithm operates in discretetime domain and allows to consider general distributed lengths of ON and OFFstates. In ATM networks, cell streams of this type can be observed at the PHY SAP, where the SCR is dened. Our analysis method is suitable for investigating the performance of SCR policing under dierent trac conditions. Numerical results show that the cell rejection probability is strongly dependent on the type of state length distribution. The maximum size of a burst plays an important role, too. Applying our algorithm, the SCR and the corresponding burst tolerance can be dimensioned for given trac sources and performance objectives. In further studies we will try to introduce the eect of CDV in our source model, i.e. the cells are not equally spaced if the trac process is in the ONstate. Using such amodel would allow for investigating SCR policing at the public UNI. 12
14 Acknowledgement The author would like to thank F. Hubner for stimulating discussions and Prof. P. Tran Gia for reviewing the manuscript. The support of the Deutsche Bundespost Telekom is highly appreciated. References [1] ATM Forum, UserNetwork Interface Specication 3.0, September [2] 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 [3] 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 [4] CCITT Draft Recommendation I.356, BISDN ATM Layer Cell Transfer Performance, March [5] CCITT Draft Recommendation I.371, Trac Control and Congestion Control in BISDN, June [6] CCITT Study Group XVIII Contribution D.2373, A Proposal for a Denition of a Sustainable Cell Rate Trac Descriptor, January [7] A. Gravey, P. Boyer, Cell Delay Variation Specication in ATM Networks, IFIP Workshop TC6, Modelling and Performance Evaluation of ATM Technology, La Martinique, January [8] F.M. Guillemin, P.E. Boyer, L. Romoeuf, The SpacerController: Architecture and First Assessments, Workshop on Broadband Communications, Estoril, Portugal, January 1992, pp [9] F.M. Guillemin, W. Monin, Limitation of Cell Delay Variation in ATM Networks, ICCT, Beijing, China, September [10] F.M. Guillemin, W. Monin, Management of Cell Delay Variation in ATM Networks, GLOBECOM 1992, pp [11] F. Hubner, Dimensioning of a Peak Cell Rate Monitor Algorithm Using Discrete Time Analysis, to appear in Proceedings of 14th ITC, Antibes, France, June [12] F. Hubner, Output Process Analysis of the Peak Cell Rate Monitor Algorithm, University of Wurzburg, Institute of Computer Science, Research Report Series, Report No. 75, January
15 [13] F. Hubner, P. TranGia, A DiscreteTime Analysis of Cell Spacing in ATM Systems, University of Wurzburg, Institute of Computer Science, Research Report Series, Report No. 66, June [14] P.TranGia, H. Ahmadi, Analysis of a DiscreteTime G [x] =D=1;S Queueing System with Applications in PacketSwitching Systems, INFOCOM 1988, pp [15] P. TranGia, Discretetime analysis technique and application to usage parameter control modelling in ATM systems, Invited contribution to the 8th Australian Teletrac Research Seminar, Melbourne, December [16] E. Wallmeier, T. Worster, The Spacing Policer, an Algorithm for Ecient Peak Bit Rate Control in ATM Networks, ISS 14, October 1992, paper A
16 PreprintReihe Institut fur Informatik Universitat Wurzburg Verantwortlich: Die Vorstande des Institutes fur Informatik. [1] K. Wagner. Bounded query classes. Februar [2] P. TranGia. Application of the discrete transforms in performance modeling and analysis. Februar [3] U. Hertrampf. Relations among modclasses. Februar [4] K. W. Wagner. Numberofquery hierarchies. Februar [5] E. W. Allender. A note on the power of threshold circuits. Juli [6] P. TranGia und Th. Stock. Approximate performance analysis of the DQDB access protocol. August [7] M. Kowaluk und K. W. Wagner. Die VektorSprache: Einfachste Mittel zur kompakten Beschreibung endlicher Objekte. August [8] M. Kowaluk und K. W. Wagner. VektorReduzierbarkeit. 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. TranGia. A discretetime 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 logspacemodclasses. Juli [16] H. Gold und P. TranGia. Performance analysis of a batch service queue arising out of manufacturing systems modeling. Juli [17] F. Hubner und P. TranGia. Quasistationary 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. TranGia. 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. TranGia und R. Dittmann. Performance analysis of the CRM aprotocol in highspeed networks. Dezember [24] C. Wrathall. Conuence of onerule Thue systems. [25] O. Gihr und P. TranGia. 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
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