University of W urzburg Institute of Computer Science Research Report Series Performance Analysis of the Dual Cell Spacer in ATM Systems
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University of Wurzburg
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, 97074 Wurzburg, Federal Republic of Germany
Tel.: +49/931/8885512, Fax: +49/931/8884601
e-mail: ritter@informatik.uni-wuerzburg.d400.de
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.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 sha-
ping 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 of working 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
speci ed 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 speci cation 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 speci ed 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, for a given 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 , if 1=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=Ts . The Maximum Burst Size (MBS)
1depends 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.
spacer
input cell stream output cell stream
T Ts s
W
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 two-
dimensional 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 speci c
amount of time, is rejected if the delay bound W would be exceeded, or departs immedia-
tely 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:
Pn; P (t) just before the arrival instant of cell number n,
Pn+ P (t) just after the arrival instant of cell number n,
2Sn; S (t) just before the arrival instant of cell number n,
Sn+ S (t) just after the arrival instant of cell number n,
An discrete random variable for the number of slots between the
arrival instants of cells n ; 1 and n.
Accordingly, the discrete random variables
Un; = (Pn; Sn;) and Un+ = (Pn+ Sn+) (1)
can be de ned. The distributions of Un; and Un+ are denoted by u;n (i j ) and u+n (i j ),
where e.g. u;n (i j ) is de ned by ProbfPn; = i ^ Sn; = j g. Furthermore, we denote the
distribution of An by a(k), since the arrival process is assumed to be a renewal process.
A sample evolution of the random variables de ned above is depicted in Figure 2.
arrival of cell number n +3
n ;3 n;2 n;1 n n +1 n +2 n +4
P (t) S (t)
Sn+
Sn;
s +W
s
Pn+
Ts
T
Pn;
W
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 Pn;;3 = 0 and Sn;;3 < s . Thus, cell n ; 3
departs immediately from the spacer and the components P (t) and S (t) are increased by
3T and Ts, respectively. Subsequently, they are decremented each slot. Just before the
arrival of cell n ; 2, Sn;2 is still smaller than s but Pn;;2 is now larger than 0. Therefore,
;
cell n ; 2 must be delayed. Since Pn;;2 < W , cell n ; 2 is not rejected but delayed by Pn;;2
slots. At the arrival of cell n ; 1, we have 0 < Pn;;1 < W and s < Sn;;1 < s + W . Cell
n ; 1 is therefore not rejected but delayed by Pn;;1 , since Pn;;1 > Sn;;1 ; s. The same
holds for cell n, only with the dierence that Pn; < Sn; ; s and thus cell n is delayed by
Sn; ; s slots. Just before cell n + 1 arrives, Sn;+1 > s + W . Therefore, cell n + 1 would
have to be delayed longer than W and is thus rejected. For the cells n + 2 and n + 3 we
have the same situation as for cells n ; 3 and n ; 2, respectively. Finally, cell n + 4 is
rejected. The reason for this is that Pn;+4 is larger than the delay bound W .
In the following we present an iterative algorithm to compute u;n (i j ) and u+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 GX ]=D=1= ; S queueing sy-
stem 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 two-
dimensional state processes has been presented in 12]. As outlined in the next three sub-
sections, 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 Un;+1 = (Pn;+1 Sn;+1) and
Un+ = (Pn+ Sn+ ):
Pn;+1 = Pn+ ; An+1 (2)
Sn;+1 = Sn+ ; An+1 : (3)
These equations are driven by the decrease of P (t) and S (t) by one each slot. Based on
the equations (2) and (3), the corresponding distribution can be found as:
X
0
u;n+1 (i j ) = u+n (i ; k j ; k) a(;k): (4)
k=;1
The computation of Un+ = (Pn+ Sn+) from Un; = (Pn; Sn; ) is a bit more complicated. For
Pn+ and Sn+ we get the following:
8>
>> T : Pn; 0 ^ Sn; s + W
<
Pn+ = > Pn; + T : 0 < Pn; W ^ Sn; s + W (5)
>> ;
: Pn : Pn; > W _ Sn; > s + W
8>
>> Ts : Sn; 0 ^ Pn; W
<
Sn+ = > Sn; + Ts : 0 < Sn; s + W ^ Pn; W : (6)
>> ;
: Sn : Sn; > s + W _ Pn; > W
4First, let us focus on P (t), i.e. equation (5). If Pn; 0 and cell n is not rejected due to a
delay longer than W resulting from SCR shaping, Pn+ 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 Pn; larger than 0, the rst component of the spacer state is
computed by Pn+ = Pn; + 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 to be delayed 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:
un (i j ) = (u;n (i j )) (7)
with the operator () de ned as:
8
>> 0 : i < 0 ^ j s + W
>>
>> 0 : 0iW ^ j > P0
>< u;n (i0 j ) : i = 0 ^ 0 < j s + W
;
(un (i j )) = > P0 P0 ; 0 0
0
i = ;1 : (8)
>> i =;1 j =;1 un (i j ) : i=0 ^ j =0
>> P0
0 0
>> u;(i j 0) : 0> j =;1 n
: u;n (i j )
0
: otherwise
As second step, i.e. the transition from un(i j ) to u+n (i j ), the resulting region in which
cells are not rejected is shifted by the vector (T Ts). This leads to the following equation:
8>
>> 0 : (i j ) 2 A1
>< u (i ; T j ; Ts) : (i j ) 2 A2
u+n (i j ) = > n : (9)
>> un(i ; T j ; Ts) + un(i j ) : (i j ) 2 A3
>:
un(i j ) : otherwise
5Sn Sn
Ts + s +W Ts + s + W
A3 A3
s + W Ts
A2
Ts s +W
A1 A1
0 Pn 0 Pn
0 T W T +W 0 W T T +W
Case 1: W T ^ s + W Ts Case 2: W < T ^ s + W < Ts
Sn Sn
Ts + s +W Ts + s + W
A3
Ts s +W A3
s + W Ts A1
A1
0 Pn 0 Pn
0 T W T +W 0 W T T +W
Case 3: W T ^ s + W < Ts Case 4: W < T ^ s + W Ts
Figure 3: Transition regions for un(i) to u+n (i).
For case 1, the regions A1, A2 and A3 (cf. Figure 3) are given by:
A1 = f(i j ) j (;1 i W ^ ;1 j < Ts) _
(;1 i < T ^ ;1 j s + W )g (10)
A2 = f(i j ) j (T i W ^ Ts j s + W )g (11)
A3 = f(i j ) j (W < i T + W ^ Ts j Ts + s + W ) _
(T i T + W ^ s + W < j Ts + s + W )g: (12)
6For all other cases depicted in Figure 3, these regions are de ned as follows:
A1 = f(i j ) j (;1 i W ^ ;1 j s + W )g (13)
A2 = (14)
A3 = f(i j ) j (T i T + W ^ Ts j Ts + 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:
u; (i j ) = nlim ;
!1 un (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 pr , i.e. the probability that an arriving cell would have to
be delayed longer than the delay bound W , can be found as
X
W sX
+W
pr = 1 ; u; (i j ): (17)
i=;1 j =;1
The rejection probability pr 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:
X ;
d(k) = 1 ;1 p u (i j ): (18)
r (ij )2Bk
In this context, the set Bk 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.
7Due to the PCR shaping, we have a minimum inter-departure interval of T slots and thus
for k < T :
Bk = : (19)
To derive the sets Bk for k T , we have to distinguish three regions of the spacer state
space, which are depicted in Figure 4 with dierent shading intensities.
j
Ts + s + W
s +W
Ts BTs
s
BT
Bx
0 i
W T +W
minfT ; i Ts + s ; j g 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 Ts + s ; j g. This is relevant if (i j ) is
located in the region Bx, 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 Ts + 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 inter-
departure 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 Ts.
Taking these properties into account, the sets BT and BTs are given by:
BT = f(i j ) j (0 < i W ^ ;1 < j i + s ) _
(minfT ; i Ts + s ; j g = T )g (20)
and
8BTs = f(i j ) j (0 < i W ^ i + s < j s + W ) _
(;1 < i 0 ^ s < j s + W ) _
(minfT ; i Ts + s ; j g = Ts)g: (21)
For all other values of k, Bk contains the following states:
Bk = f(i j ) j minfT ; i Ts + s ; j g = kg : (22)
2.3 Delay distribution
The performance measure we address on in this subsection is the cell delay distribu-
tion. 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)
De ning sets Ck which contain those states (i j ) where an arriving cell is delayed by k
slots, we can compute the delay distribution w(k) for k = 0 : : : W by the following
equation:
X
w(k) = 1 ;1 p u (i j ): (24)
r (ij )2Ck
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 Ck are given by:
Ck = f(i j ) j (i = k ^ i > j ; s) _ (j = k + s ^ i j ; s )g : (25)
Because of the maximum delay bound W , the sets Ck are empty for k > W . The proba-
bility pw for an arriving cell to be delayed can be computed by:
X
W
pw = w(k): (26)
k=1
93 Numerical results
To compare the performance of the Dual Spacer with that of the conventional one, we
present some numerical 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 EA and the coecient of variation cA almost independently of
each other (EA c2A > 1 must be ful lled). We use a mean of EA = 10 slots and the
coecient of variation is set to cA = 1:0. The spacer parameters are T = 5, Ts = 8 and
s is varied from 0 to 150. A maximum delay of W = 200 slots is tolerated.
100
delay distribution
10;1
10;2
10;3
10;4
10;5 s = 1 s = 100 s = 50 s = 0
10;6
0 50 100 150 200
time in slots
Figure 5: Inuence of the BT s on the delay distribution (cA = 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 in nity 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 look at
Figure 6 where the coecient of variation is now set to cA = 2:0, a much slower 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.
10100
delay distribution
10;1 s = 0
s = 100
10;2
s = 200
10;3 s = 300
10;4 s = 1
10;5
10;6
0 50 100 150 200
time in slots
Figure 6: Inuence of the BT s on the delay distribution (cA = 2:0).
50
mean delay
cA = 0:5
40 cA = 1:0
cA = 1:5
30 cA = 2:0
20
10
0
0 50 100 150 200 250
Burst Tolerance s
Figure 7: Asymptotic behavior of the mean delay.
11In Figure 7, the mean delay is plotted over s for various choices of cA. 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 cA 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=Ts.
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 of variation cW of the delay
introduced by the Dual Spacer. Curves for dierent choices of cA are depicted in Figure
8 in dependence on s. As expected, cW 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 6.
4
delay coecient of variation
3
2
1
cA = 0:5 cA = 1:5
cA = 1:0 cA = 2:0
0
0 50 100 150 200 250
Burst Tolerance s
Figure 8: Asymptotic behavior of the delay coecient of variation cW .
For small values of s the coecient of variation cW rst increases and then decreases
against a limiting value. The initial increase is more distinctive if cA is small. The later
decrease of the delay coecient of variation is due to the minor inuence of SCR shaping
for large values of s. If cA 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 of variation cA = 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 Ts. If s is increased, the
inter-departure time distribution also approaches a limiting distribution. Like in case
121:0
inter-departure time distribution
0:8
0:6
s = 0
0:4 s = 50
s = 150
0:2 s = 1
0:0
0 10 20 30 40
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 = 5 and is = 8 (note that T = 5 and
Ts = 8). The step due to SCR shaping is getting smaller if s increases or cA 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 (Ts,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 limi-
ting 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
13examples are given to show the performance of the Dual Spacer for dierent trac condi-
tions. 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
pro table 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 implemen-
tation remains almost the same compared to the conventional one.
Acknowledgement
The author appreciates the support of the Deutsche Bundespost Telekom (Forschungs-
und Technologiezentrum (FTZ)).
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14] P. Tran-Gia, H. Ahmadi, Analysis of a Discrete-Time GX ]=D=1 ; S Queueing
System with Applications in Packet-Switching Systems, IEEE INFOCOM 1988,
pp. 861-870.
15] E. Wallmeier, T. Worster, The Spacing Policer, an Algorithm for Ecient Peak Bit
Rate Control in ATM Networks, ISS 14, October 1992, paper A5.5.
15Preprint-Reihe
Institut fur Informatik
Universitat Wurzburg
Verantwortlich: Die Vorstande des Institutes fur Informatik.
1] K. Wagner. Bounded query classes. Februar 1989.
2] P. Tran-Gia. Application of the discrete transforms in performance modeling and analysis. Februar 1989.
3] U. Hertrampf. Relations among mod-classes. Februar 1989.
4] K. W. Wagner. Number-of-query hierarchies. Februar 1989.
5] E. W. Allender. A note on the power of threshold circuits. Juli 1989.
6] P. Tran-Gia und Th. Stock. Approximate performance analysis of the DQDB access protocol. August 1989.
7] M. Kowaluk und K. W. Wagner. Die Vektor-Sprache: Einfachste Mittel zur kompakten Beschreibung endlicher
Objekte. August 1989.
8] M. Kowaluk und K. W. Wagner. Vektor-Reduzierbarkeit. August 1989.
9] K. W. Wagner (Herausgeber). 9. Workshop uber Komplexitatstheorie, eziente Algorithmen und Datenstrukturen.
November 1989.
10] R. Gutbrod. A transformation system for chain code picture languages: Properties and algorithms. Januar 1990.
11] Th. Stock und P. Tran-Gia. A discrete-timeanalysis of the DQDB access protocol with general input trac. Februar
1990.
12] E. W. Allender und U. Hertrampf. On the power of uniform families of constant depth threshold circuits. Februar
1990.
13] G. Buntrock, L. A. Hemachandra und D. Siefkes. Using inductive counting to simulate nondeterministic computa-
tion. April 1990.
14] F. Hubner. Analysis of a nite capacity a synchronous multiplexer with periodic sources. Juli 1990.
15] G. Buntrock, C. Damm, U. Hertrampf und C. Meinel. Structure and importance of logspace-MOD-classes. Juli
1990.
16] H. Gold und P. Tran-Gia. Performance analysis of a batch service queue arising out of manufacturing systems
modeling. Juli 1990.
17] F. Hubner und P. Tran-Gia. Quasi-stationary analysis of a nite capacity asynchronous multiplexer with modulated
deterministic input. Juli 1990.
18] U. Huckenbeck. Complexity and approximation theoretical properties of rational functions which map two intervals
into two other ones. August 1990.
19] P. Tran-Gia. Analysis of polling systems with general input process and nite capacity. August 1990.
20] C. Friedewald, A. Hieronymus und B. Menzel. WUMPS Wurzburger message passing system. Oktober 1990.
21] R. V. Book. On random oracle separations. November 1990.
22] Th. Stock. In uences of multiple priorities on DQDB protocol performance. November 1990.
23] P. Tran-Gia und R. Dittmann. Performance analysis of the CRM a-protocol in high-speed networks. Dezember
1990.
24] C. Wrathall. Con uence of one-rule Thue systems.
25] O. Gihr und P. Tran-Gia. A layered description of ATM cell trac streams and correlation analysis. Januar 1991.
26] H. Gold und F. Hubner. Multi server batch service systems in push and pull operating mode | a performance
comparison. Juni 1991.
27] H. Gold und H. Grob. Performance analysis of a batch service system operating in pull mode. Juli 1991.
28] U. Hertrampf. Locally denable acceptance types|the three valued case. Juli 1991.
29] U. Hertrampf. Locally denable acceptance types for polynomial time machines. Juli 1991.
30] Th. Fritsch und W. Mandel. Communication network routing using neural nets { numerical aspects and alternative
approaches. Juli 1991.
31] H. Vollmer und K. W. Wagner. Classes of counting functions and complexity theoretic operators. August 1991.
1632] R. V. Book, J. H. Lutz und K. W. Wagner. On complexity classes and algorithmically random languages. August
1991.
33] F. Hubner. Queueing analysis of resource dispatching and scheduling in multi-media systems. September 1991.
34] H. Gold und G. Bleckert. Analysis of a batch service system with two heterogeneous servers. September 1991.
35] H. Vollmer und K. W. Wagner. Complexity of functions versus complexity of sets. Oktober 1991.
36] F. Hubner. Discrete-time analysis of the output process of an atm multiplexer with periodic input. November 1991.
37] P. Tran-Gia und O. Gropp. Structure and performance of neural nets in broadband system admission control.
November 1991.
38] G. Buntrock und K. Lorys. On growing context-sensitive languages. Januar 1992.
39] K. W. Wagner. Alternating machines using partially dened \AND" and \OR". Januar 1992.
40] F. Hubner und P. Tran-Gia. An analysis of multi-service systems with trunk reservation mechanisms. April 1992.
41] U. Huckenbeck. On a generalization of the bellman-ford-algorithm for acyclic graphs. Mai 1992.
42] U. Huckenbeck. Cost-bounded paths in networks of pipes with valves. Mai 1992.
43] F. Hubner. Autocorrelation and power density spectrum of atm multiplexer output processes. September 1992.
44] F. Hubner und M. Ritter. Multi-service broadband systems with CBR and VBR input trac. Oktober 1992.
45] M. Mittler und P. Tran-Gia. Performance of a neural net scheduler used in packet switching interconnection
networks. Oktober 1992.
46] M. Kowaluk und K. W. Wagner. Vector language: Simple description of hard instances. Oktober 1992.
47] B. Menzel und J. Wol von Gudenberg. Kommentierte Syntaxdiagramme fur C++. November 1992.
48] D. Emme. A kernel for funtions denable classes and its relations to lowness. November 1992.
49] S. Ohring. On dynamic and modular embeddings into hyper de Bruijn networks. November 1992.
50] K. Poeck und M. Tins. An intelligent tutoring system for classication problem solving. November 1992.
51] K. Poeck und F. Puppe. COKE: Ecient solving of complex assignment problems with the propose-and-exchange
method. November 1992.
52] Th. Fritsch, M. Mittler und P. Tran-Gia. Articial neural net applications in telecommunication systems. Dezember
1992.
53] H. Vollmer und K. W. Wagner. The complexity of nding middle elements. Januar 1993.
54] O. Gihr, H. Gold und S. Heilmann. Analysis of machine breakdown models. Januar 1993.
55] S. Ohring. Optimal dynamic embeddings of arbitrary trees in de Bruijn networks. Februar 1993.
56] M. Mittler. Analysis of two nite queues coupled by a triggering scheduler. Marz 1993.
57] J. Albert, F. Duckstein, M. Lautner und B. Menzel. Message-passing auf transputer-systemen. Marz 1993.
58] Th. Stock und P. Tran-Gia. Basic concepts and performance of high-speed protocols. Marz 1993.
59] F. Hubner. Dimensioning of a peak cell rate monitor algorithm using discrete-time analysis. Marz 1993.
60] G. Buntrock und K. Lorys. The variable membership problem: Succinctness versus complexity. April 1993.
61] H. Gold und B. Frotschl. Performance analysis of a batch service system working with a combined push/pull control.
April 1993.
62] H. Vollmer. On dierent reducibility notions for function classes. April 1993.
63] S. O hring und S.K. Das. Folded Petersen Cube Networks: New Competitors for the Hyepercubes. Mai 1993.
64] S. O hring und S.K. Das. Incomplete Hypercubes: Embeddings of Tree-Related Networks. Mai 1993.
65] S. Ohring und S.K. Das. Mapping Dynamic Data and Algorithm Structures on Product Networks. Mai 1993.
66] F. Hubner und P. Tran-Gia. A Discrete-Time Analysis of Cell Spacing in ATM Systems. Juni 1993.
67] R. Dittmann und F. Hubner. Discrete-Time Analysis of a Cyclic Service System with Gated Limited Service. Juni
1993.
68] M. Frisch und K. Jucht. Pascalli-P. August 1993.
69] G. Buntrock. Growing Context-Sensitive Languages and Automata. September 1993.
70] S. O hring und S.K. Das. Embeddings of Tree-Related Topologies in Hyper Petersen Networks. Oktober 1993.
71] S. Ohring und S.K. Das. Optimal Communication Primitives on the Folded Petersen Networks. Oktober 1993.
72] O. Rose und M. R. Frater. A Comparison of Models for VBR Video Trac Sources in B-ISDN. Oktober 1993.
1773] M. Mittler und N. Gerlich. Reducing the Variance of Sojourn Times in Queueing Networks with Overtaking.
November 1993.
74] P. Tran-Gia. Discrete-Time Analysis Technique and Application to Usage Parameter Control Modelling in ATM
Systems. November 1993.
75] F. Hubner. Output Process Analysis of the Peak Cell Rate Monitor Algorithm. January 1994.
76] K. Cronauer. A Criterion to Separate Complexity Classes by Means of Oracles. January 1994.
77] M. Ritter. Analysis of the Generic Cell Rate Algorithm Monitoring ON/OFF-Trac. January 1994.
78] K. Poeck, D. Fensel, D. Landes und J. Angele. Combining KARL and Congurable Role Limiting Methods for
Conguring Elevator Systems. Januar 1994.
79] O. Rose. Approximate Analysis of an ATM Multiplexer with MPEG Video Input. Januar 1994.
80] A. Schomig. Using Kanban in a Semiconductor Fabrication Environment | a Simulation Study. Marz 1994.
81] M. Ritter, S. Kornprobst, F. Hubner. Performance Analysis of Source Policing Architectures in ATM Systems.
April 1994.
82] U. Hertrampf, H. Vollmer und K. W. Wagner. On Balanced vs. Unbalanced Computation Trees. May 1994.
83] M. Mittler und A. Schomig. Entwicklung von "Due{Date\{Warteschlangendisziplinen zur Optimierung von Pro-
duktionssystemen. Mai 1994.
84] U. Hertrampf. Complexity Classes Dened via k-valued Functions. Juli 1994.
85] U. Hertrampf. Locally Denable Acceptance: Closure Properties, Associativity, Finiteness. Juli 1994.
86] O. Rose, M. R. Frater. Delivery of MPEG Video Services over ATM. August 1994.
87] B. Reinhardt. Kritik von Symptomerkennung in einem Hypertext-Dokument. August 1994.
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 1994.
89] U. Hertrampf. Finite Acceptance Type Classes. Oktober 1994.
90] U. Hertrampf. On Simple Closure Properties of #P. Oktober 1994.
91] H. Vollmer und K.W. Wagner. Recursion Theoretic Characterizations of Complexity Classes of Counting Functions.
November 1994.
92] U. Hinsberger und R. Kolla. Optimal Technology Mapping for Single Output Cells. November 1994.
93] W. Noth und R. Kolla. Optimal Synthesis of Fanoutfree Functions. November 1994.
94] M. Mittler und R. Muller. Sojourn Time Distribution of the Asymmetric M=M=1==N { System with LCFS-PR
Service. November 1994.
95] M. Ritter. Performance Analysis of the Dual Cell Spacer in ATM Systems. November 1994.
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