Predicted Crankback by statistical measures
Raviv Brueller and Ilia Koifman
Supervised by Dr. Ofer Hadar
Technion, Haifa 32000
Israel
abstract - When an ATM node discovers that it cannot continue the set-up of a virtual
channel under the requested QoS, it initiates a backtracking procedure called
"crankback". A new technique to improve the crankback mechanism is proposed based
on statistical measures that are distributed trough the network. Under the proposed
scheme, nodes check during the connection admission control procedure the probability
that the accumulative parameter, such as delay of a virtual channel will be admitted over
the entire designated route. If this is not the case, crankback is initiated even before a
certain QoS parameter is exceeded. By using our algorithm it is possible to reduce the
set-up time for creating a new connection, and to increase the efficiency of using the
network resources. The efficiency of the algorithm is increased for situations of large
variability with the probability density function of the delay function at the nodes a long
the transmission route. In this paper we focus on the Gaussian distribution. The main
advantage of the proposed scheme is that it lowers the set-up delay and the processing
and communication load imposed by signaling messages that establish unused portions of
VCs.
1.
Introduction.
ATM (Asynchronous Transfer Mode) is a connection oriented, cell-based transport
service designed to carry a wide variety of applications. As a connection oriented
technology, before information is transferred from a source terminal through the ATM
network to a destination terminal, a virtual connection has to be set. This connection,
generally referred to as virtual channel (VC), assigned for the duration of the connection,
is traveled by the cells, belonging to the specific session, on their way from the source to
the destination. In addition, ATM guarantees performance requirements of applications
by letting them define certain parameters representing the quality of service (QoS) they
expect to receive from the network.
Therefore, when setting a connection between a source node and a destination node, the
ATM needs to find a route that fulfills the requested QoS. When a node finds out that it
cannot continue such a set-up process, it initiates a backtracking procedure called
“crankback”. In a hierarchical network like the ATM, this node sends a RELEASE
message to the last node that has made a routing decision, encouraging it to search an
alternate route that would answer the requested QoS. If an alternative route cannot be
found, the roll back process continues recursively, and the former node that has made a
routing decision, closer to the source, tries to compute an alternate route.
One of the objectives the ATM is expected to meet is a very efficient utilization of
network resources. Hence, it is desirable to reduce the amount of time consumed during
the set-up procedure as well as the load in the network. These improvements can be
achieved by realizing as early as possible that a crankback is about to take place.
This paper presents a crankback mechanism, which is based on statistic rather than
deterministic analysis. The idea is that every node along the designated route advertises
statistic information concerning the cost of crossing it. The advertised statistic
information is then used in each step of the set-up procedure for deciding whether a
crankback is predicted.
By invoking the crankback mechanism even before resources have fully been consumed
has the important advantage of reducing the crankback overhead. The overhead caused
by the set-up includes the delay related to the call set-up and the bandwidth as well as the
buffers allocated but not used during the set-up attempt. By reducing the time period that
elapses between receipts of a SET-UP (forwarded) and a RELEASE message
(backwarded) it is possible to prevent these resources from being wasted.
The ATM consists of a hierarchy of subnetworks called domains. These domains
advertise only a summary of their internal structure, as proposed by the ATM forum
PNNI standard. However, In this paper we try not to restrict ourselves to a specific
topology, and therefore we describe the statistic mechanism in a general way, which can
easily be implemented in any hierarchy, as discussed in [2].
The rest of the paper is organized as follows: In Section 2 we describe the mechanism for
crankback prediction when a single route from source to destination is considered. In
Section 3 we show examples of simulations, in which we implemented the algorithm for
predicted crankback, bring the results obtained in these simulations, and discuss their
significance. In Section 4 we explain how the algorithm for predicted crankback is
extended so it can be used when we can choose among multiple routes leading from
source to destination. In this section we also present results for simulations, which show
the advantages involved in using the scheme we propose, by comparing a technique for
set-up based on our idea with a method where crankback is not used. In Section 5 we
present our general conclusions and remarks.
2.
The Predicted Crankback Mechanism: Single Route Case.
In this section we present the statistic crankback mechanism in the case of a single path
between source and destination. Every node along the designated route from the source to
the destination advertises the probability density function (pdf) for its crossing cost. The
pdf statistics parameters can be advertised by the PNNI topology advertisement
mechanism. For independent delay pdfs, the end-to-end distribution function can be
derived by convolution. The correlation coefficient of traffic on successive network links
was calculated in [3]. While some correlation exists, the degree of which seems to be
small, and in view of Kleinrock’s assumption developed in [4] the independence
assumption appears to be valid in cases where the target connection occupies only a small
fraction of the total link capacity.
The work in [3] uses the accumulation algorithm in order to compute the end-to end cell
transfer delay (CTD). In this section we suggest to use the accumulative CTD function
for deriving the crankback probability at each stage of the set-up process. We propose an
algorithm that evaluates the probability for reaching the destination, using no more than
the specified quota allocated for the set-up of the connection.
In what follows we derive the statistical condition for crankback based on the pdf
advertised by every node along the route. We consider a route that contains N
consequent nodes. The allocated quota for crossing this route, namely  , can be
determined based on the algorithm introduced in [2]. Using accumulative pdf, we derive
the predicted crankback probability at each node along the route. If the calculated
probability is higher than some threshold, predicted crankback is invoked. We start by
defining the following terms:

Pdf (j) - the probability density function advertised by the j'th node along the
designated route. This node will be referred to as node j.

E(pdf(j)) and VAR(pdf(j)) - the average and variance of the pdf(j).

 i and VARi are the average and the ?standard deviation of the pdfi .

  j  - The total amount of delay spent for crossing node j.

 - the threshold probability that according to which we should determine whether
or not to invoke predicted crankback.

 1 j   kj 1 k  - the actual accumulative delay from the source of the
considered route to the j'th one. Hence,    1 j  indicates the quota left for
crossing the rest of the route after crossing its j'th node.
Based on the independence assumption, the proposed algorithm uses convolution
in order to derive at each node the distribution function of the accumulative delay for
the rest of the route. To this end, we define for every j, 1  j  N
pdf 1 N   pdf  j   pdf  j  1    pdf N .
(1)
The method for deciding whether a crankback should be invoked at each node along the
path is described in Figure (1). The average value of pdf 1 N  in each stage after
crossing the  j  1' th node is given by:
N
E  pdf 1 N    E  pdf k .
(2)
k j
The solid line in Figure 1 represents the actual Quota left for crossing the rest of the
route, namely    1 j , whereas the dashed line represents the center of the
accumulative pdf. The probability, P_fail(j), for consuming more than the remaining
quota which would lead to crankback is given by the area bounded by the solid -line and
pdf 1 N  . Hence, the following holds:
P _ fail  j  

 pdf  j  N  x dx.
  1 j 1
(3)
For symmetrical pdf function this probability is larger than 0.5 when the solid-line is
located left to the central line. Crankback prediction is invoked before node j is traversed
if
P _ fail  j   
(4)
holds. As discussed later, selecting the right value of  has a great impact on the success
of the algorithm.
Eq. 4 holds for a general distribution function. However, the calculation can be simplified
for a Gaussian distribution, in which case the pdf can be derived analytically due to the
assumption that the delays in successive links along the route are independent. With this
assumption, the variance of the accumulative pdf can be represented as the sum of all
variances, namely
N
VAR pdf  j  N    VAR pdf k .
k j
(5)
By replacing the general pdf in Eq. 4 with the Gaussian distribution function and using
Eq. 5 we get:
P _ fail  j /  1  j  1 
 x  E  pdf  j  N 2 


exp

 2VAR pdf  j  N   dx 
2  VAR pdf  j  N    1 j 1


1

    1  j  1  E  pdf  j  N  
.
 1  
2  VAR pdf  j  N 


(6)
The results for Eq. 6 can be derived from common probabilistic tables of the
complementary error function.
1
3
2
N E pdf (1), VAR pdf 1
N
j
N E pdf ( j), VAR pdf  j 
...
N E pdf ( N ), VAR pdf N 
N E pdf ( j  1 N , VAR pdf ( j  1 N 
N
 N
N   i ,   i ,
 i  j 1
i  j 1





P_fail(j)
j
  D j     di
i 1
Fig. 1 : Derivation of the probability of predicted crankback at each node.
As we approach to the end of the route the probability density function N becomes
narrower (since the components convolved are fewer) and the uncertainty about the total
delay reduces as well.
4.5
x 10
-3
left quota
4
3.5
3
P
pdf(j…N)
2.5
2
1.5
P_fail(j)
1
0.5
0
60
70
80
90
100
Delay
110
120
130
140
Figure 2: Deriving the probability for routing failure from the accumulative pdf.
Normal crankback
P_fail(j)
1
Threshold
0.8
0.6
crankback
gain
0.4
Predicted crankback
0.2
0
0
2
4
Node number j
6
8
10
(a) Predicted crankback is successfully invoked
P_fail(j)
1
Threshold
0.8
0.6
0.4
0.2
0
0
2
4
6
Node number j
(b) No predicted crankback
8
10
P_fail(j)
Threshold
False_prediction
0
0
2
4
6
8
10
Node number j
(c) Predicted crankback is unnecessarily invoked (false prediction).
Figure 3: Three typical representative examples for deriving the crankback decision: (a)
true predicted crankback (b) true decision for not performing crankback and (c) false
predicted crankback.
Figure 3 depicts three representative cases: the case where predicted crankback is
invoked successfully (a), where predicted crankback is not invoked (b), and where it is
unnecessarily invoked (c). In these figures, the calculated value of P_fail(j), is presented
as a function of the node’s relative position along the path. The threshold described in
Figure 3(a) demonstrates a case of successful predicted crankback that benefits in saving
the need to traverse nodes 7 and 8 before the ordinary crankback would have been
invoked. Figure 3(b) presents the case where the procedure makes a correct decision not
to invoke crankback, because P_fail(j), for j = 1…10 is smaller than the threshold  .
Finally, Figure 3(c) depicts the case of false prediction (at node 8).
The last case describes a scenario where despite the fact that before crossing the 8’Th
node the probability of failure exceeds the predefined threshold, a connection could have
been established by continuing to traverse the nodes. This kind of false alarms is inherent
to this method due to the statistical nature of the process, and we will try to minimize
their relative occurrence.
Statistical Analysis of the Failure Probability
In the last section three representative examples for the possible results obtained from the
analysis were presented. It is possible to predict the success of the algorithm by applying
a statistical analysis to all the ensemble functions of the random process. The probability
of failure as a function of the node along the route can be viewed as a random process
and a problem where two possible scenarios can be defined.
In this section we calculate the probability of success of our algorithm, as a function of
the threshold value. We derive the probability of detection, as a function of the
parameters in the problem. We are interested in deriving the probability of error and the
probability of success of our algorithm.
In any non-trivial hypothesis-testing problem, there is a non-zero probability of making
error. In our problem, a “false alarm” occurs when according to the algorithm, we decide
to invoke predicted crankback, when in fact, it is unnecessary because eventually, the
need to activate normal crankback does not arise. In other words, although the probability
of failure exceeds the threshold at some stage, if we continue to traverse the remaining
nodes, we would be able to reach the destination, and establish the connection. the total
delay of the all route is bellow the allocated delay at that case. The other possible error is
a “miss”. This term is used to describe the situation where in a case of normal crankback,
there was no earlier decision to invoke the predicted crankback. In general, when
designing a binary hypothesis test, one must trade off false alarms and misses. A
conservative test may have a very low probability of creating a false alarm (Pfa) but this
desirable trait is almost always accompanied by the undesirable trait of a high probability
of miss (Pm). Conversely, a test that has a low Pm usually has a high Pa. For any threshold
value  , Pfa and Pm are given by:

1
p fa     pdf real _ crank  x dx
2 Th
p miss   
Th
1
pdf no _ crank  x dx
2 
Analytical Results and Random Process Analysis:
In this section we present the derivation of an analytical calculation for the probability of
detection. The idea behind this analysis is the following: We will use the Gaussian
distribution for representing the delay function at each node. In this case there is an
analytical solution for the probability of failure, which can be expressed by:
P _ fail  j  
  x  E  pdf  j  N 2
exp 
 2VAR pdf  j  N 
2  VAR pdf  j  N    1 j 1


1

    1  j  1  E  pdf  j  N  
.
 1  
2  VAR pdf  j  N 




2
 x 

N j
Pcrank j 
 exp 
2VARN  j  i T
 2  VARN  j
j

1

  Q  E N K 

 i  T j 
 1

N j
  2 erfc
 2  VARN  j



2  VARN  K
2  






 dx 


R
f y   
VARN  K
VARK
2

1
1
 R 

R
exp 
,
2
2


2
8




0    1
The last approximation given is correct for cases of small area under the Gaussian
distribution function. It can be shown that there is a fair agreement between the analytical
results and the numerical results.
One of the reasons for using the Gaussian distribution is the central limit theorem imbues
the Gaussian distribution with special significance in the probability theory. Loosely
stated, it holds that an appropriately normalized sum of independent random variables has
a distribution tending to the Gaussian distribution, as the number of summed variables
becomes large.
In simulations Pcrank_1 the value of the probability of cranking back, obviously depends on
the quota that was allocated for crossing the route. For the simulation related to
proceeding figures, Pcrank_1 is equal to 0.5 at the first node, and this is due to the fact that
the poll-line and the central point of the accumulative pdf function are coincident at the
this node. The value of 0.5 is actually the lower value for the threshold in this case
because the uncertainty about the crankback gets its maximum value at the first node. In
general, we can say that the threshold should be higher than the value of Pcrank_1 and
smaller than 1. Figure (8) illustrates it.
We can define two PDF’s, describing the histograms of the attempts to establish a
connection: one representing the attempts in which a predicted crankback was invoked,
and one representing the attempts in which a predicted crankback was not invoked. We
observe that as we approach the end of the designated route, these PDFs are becoming
more separable. As a result it is easier to distinguish between them, and predict the way
in which such an attempt would end up. Therefore, the performance of the algorithm
improves, meaning a higher probability of detection (Pd) and a smaller probability of
false alarm (Pfa) and miss detection (Pmiss). On the other hand, as the decision for
predicted crankback is invoked in a node that is close to the destination, the gain from
prediction crankback is reduced. The corresponding histograms are plotted in figure ().
It is expected to see that as the probability density function for the delay is wider the gain
from the statistical prediction is increased.
Probability density function (pdf)
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
(a) The two pdf’s functions, pdfcrancback and pdfuncranckback at node No. 4
0.9
0.8
Probability density function (pdf)
0.7
0.6
0.5
0.4
0.3
pdfuncranckback
Pdf cranckback
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
S
(b) The two pdf’s functions, pdfcrancback and pdfuncranckback at node No. 8
Figure 4: Two examples of ensemble functions representing the probabilities of failure in
a route that consists of 10 nodes.
Threshold
pdf uncrancback
pdf crancback
Pd
Pmiss
Pfa
Fig. 5: Deriving the following probabilities: probability of detection (Pd), probability of
false alarm (Pfa) and probability of miss detection (Pmiss).
Average node for predicted
cranckback
Average node for normal
cranckback
1
Probability
0.9
Pd
0.8
Pmiss
0.7
Average gain= 4.25 nodes
0.6
0.5
0.4
Pfa
0.3
0.2
0.1
0
0
2
4
6
8
10
Node number
Fig. 7: The probability results as function of the node number
9
8
7
Number of nodes gains
6
5
4
3
2
0.5
0.6
0.7
Threshold ()
0.8
0.9
1
Fig. 8: The number of gained nodes as function of the threshold value () – an example.
Single Route Case Simulations
We have conducted an experiment simulation, which consists of 10 nodes, each with the
same type of delay distribution function. We focused our attention on the Gaussian
distribution.
For the Gaussian distributions, the average delay at each node was randomly and
uniformly selected between the values of 100 and 200 delay units, and the variance of the
delay at each node was selected uniformly in the range between 30 and 40.
The additive metric in our simulation is the delay, and the allocated quota for crossing the
route is also a parameter that has a great influence on the results of the simulations.
Naturally, as the amount of quota increases, the probability of invoking crankback
decreases due to the fact that we allow “wasting” more delay a long the designated route.
We started our simulations allocating 1400 time units, and then examined how the results
change as we reduce the allocated time quota to 1300, 1200, 1100 and 1000 time units.
The thresholds for predicted crankback invocation we imposed were of two kinds:
constant thresholds and variable thresholds. In contrast to a constant threshold, where the
threshold is the same for each node along the route, a variable threshold is generally not
the same for different nodes. The constant thresholds we examined were 0.81, 0.82, 0.83
and 0.84. As variable thresholds we chose to examine decaying thresholds.
The
motivation for this decision was the observation that as we approach the end of the route
it is easier to estimate if a set-up attempt would end successfully or not. The variable
thresholds we imposed lie between the values of 0.84 (At the first node) and 0.81 (At the
last node along the route).
In each run we computed the average number of nodes that crossing them was saved and
examined how this number was affected by replacing the initial parameters. Due to early
detection of the need to crank back, the average number of nodes that we cross before
cranking back, in case that the crankback is regarded as a false alarm, the percentage of
attempts which were declared as “false alarms”.
Single Route Case Simulations Results
The results of the simulations are shown in Tables 1(a)-(d).
Mean Saved
Mean Unused
Transitions
Transitions
Const: 0.79
2.87
1.02
Const: 0.80
2.73
1.10
Const: 0.81
2.34
1.28
Const: 0.82
1.92
1.45
Const: 0.83
1.76
1.73
Const: 0.84
1.42
2.05
Dyn: Dec1
2.86
0.91
Dyn: Dec2
2.32
1.21
Dyn: Dec3
2.40
1.21
(a) Allocated time quota: 1400 time units
Mean Saved
Mean Unused
Transitions
Transitions
Const: 0.79
5.33
0.37
Const: 0.80
5.12
0.38
Const: 0.81
4.43
0.39
Const: 0.82
4.03
0.43
Const: 0.83
3.5
0.56
Const: 0.84
2.51
0.72
Dyn: Dec1
4.51
0.68
Dyn: Dec2
3.95
0.68
Dyn: Dec3
5.20
0.63
(b) Allocated time quota: 1300 time units
Mean Saved
Mean Unused
Transitions
Transitions
Const: 0.79
7.32
0.06
Const: 0.80
7.21
0.10
Const: 0.81
7.02
0.14
Const: 0.82
6.71
0.28
Const: 0.83
5.63
0.30
Const: 0.84
4.76
0.36
Dyn: Dec1
6.76
0.49
Dyn: Dec2
6.76
0.52
Dyn: Dec3
7.02
0.43
(c) Allocated time quota: 1200 time units
Mean Saved
Mean Unused
Transitions
Transitions
Const: 0.79
7.91
0.02
Const: 0.80
7.85
0.04
Const: 0.81
7.69
0.05
Const: 0.82
7.35
0.10
Const: 0.83
6.45
0.11
Const: 0.84
6.34
0.13
Dyn: Dec1
7.37
0.06
Dyn: Dec2
7.45
0.07
Dyn: Dec3
7.52
0.06
(d) Allocated time quota: 1100 time units
Table 1: The results obtained for allocated for different time quotas. The dynamic
thresholds used are: Dec1 = [0.84,0.84,0.84,0.82,0.82,0.82,0.82,0.81,0.81,0.81]
Dec2 = [0.83,0.83,0.83,0.82,0.82,0.82,0.82,0.81,0.81,0.81]
Dec3 = [0.82,0.82,0.82,0.82,0.82,0.81,0.81,0.81,0.81,0.81]
Analysis of the Single Route Case
Coming to analyze the results, an important conclusion, which arises, is concerned with
the threshold that can be imposed. As the threshold value decreases, the probability of
true detection is increased for all nodes, making it is possible to gain more benefits, by
deciding on crankback earlier and though saving more unnecessary nodes to cross.
However, decreasing the threshold causes more false predictions. As we can notice, the
average number of saved nodes is increased, as well as the number of nodes, which have
been traversed, but crossing them did not yield any benefit, due to false alarm. Hence,
there a tradeoff between early detection of true crankback and increasing the probability
of false prediction exists. The thresholds we used around the value of 0.8 were shown to
be a good compromise between the competing goals of increasing the probability of true
detection and lowering the value of false detection. In this context, we can see that the
idea of using decaying thresholds proved itself, especially when the amount of time
allocated for the set-up is relative small.
In general, we can say that the advantages obtained by using our algorithm become more
noticeable in cases where the uncertainty about the delay in each node increases, and as
the time quota becomes smaller. As we can deduce, when the amount of time allocated
for the set-up of the connection should almost certainly suffice for establishing a
connection, the need for predicted crankback becomes negligible. When the amount of
time that is allocated for establishing the connection is not very large, our algorithm
proves to be efficient in foreseeing that the need for crankback would arise.
4. The Predicted Crankbak Mechanism: Multi Route Case.
In this section we describe how the algorithm for crankback prediction based on
statistical measures is extended when there are a few possible routes between the source
node and the destination node. We assume the existance of more than one path from the
source to the destination. The different routes are mutual exclusive. This means that we
should choose the route in which we will try to set up the connection in an optimal way.
Once we chose a specific route we start travelling it. After crossing each node, a decision
whether to proceed in this route towards the destination or crank back to the source is
neccessary. A crankback should be activated when the current route is no longer optimal.
In this case, after cranking back to the source node, we should start traverse the new
optimal route.
We present here a general algorithm for the multi-route case. The algorithm was
developed as a generalization to the case of a single route, and is illustrated in fig (). We
assume the existance of m possible routes {1 ,  2 ,...,  m } . The nodes belonging to route
 i are denoted by v1i , v2i ,..., vni i . As in the single route case, every node advertises the
pdf for its crossing cost in terms of delay. As before, the statistics can be advertised by
the PNNI topology advertisement mechanism, and we use them in the algorithm during
the set up procedure to determine the actions to be performed.
We denote the following terms:

pdf (v ij ) - the probability density functionadvertised by the j’th node on the i’th
optional route.

 i ( j ) - the total amount of delay spent for crossing node j at the (currently
processed) route i.

 i ( j ) - the cost of the set-up “rolling back”, if the crankback was performed at
the j’th node on the i’th optional route.

 ' - time, allocated for the VC set-up.

 - time slot, available at the current moment, to finish the VC set-up.

U (t ) - specified utility function, which represents the desirability of the
successful VC set-up exactly by the time t.
Definition 1 Denote the expected utility of successful crossing nodes v1 ...vn within the
time slot t by:
t
EU t (v1 ...v n ) 
 pdf (v ...v
1
n
)( x)  U ( x)dx

For a utility function, which is a hard-deadline, boolean step function, which represents
only outcomes of success or failure, with a deadline T, the utility function can be
expressed as:
min( t ,T )
EU t (v1 ,..., v n ) 
 pdf (v ,..., v
1
n
)( x)dx

Definition 2 Given a set of m possible routes R = {1 ,  2 ,...,  m } , denote by   the
route  i  R
s.t.
t j  i t '  t ( pdf (v1i ,..., vni i )(t )  0  pdf (v1j ,..., vnj j )(t ' )  0)
The convolution pdf(x) of the route   will be denoted by pdf  (x).
Informally,   is the route with the smallest lower bound on the pdf function.
The algorithm for crankback prediction in face of multiple routes receives as input the
Optional routes R = {1 ,  2 ,...,  m } from source to destination, and its objective is a VC
set-up with statistically minimal delay.
The steps that describe the algorithm can be summarized as shown in Fig.9:
Algorithm Crankback
Input:
Optional
routes R = {1 ,  2 ,...,  m } from source to destination;
Multi-Route
Case
Simulations
Available time slot  ' .
Output:
VC setup with statistically minimal delay.
1. Initiate the remain time slot    '.
2. Delete from R all routes, pdf’s of which have no overlap with pdf  in (, ).
3. Choose the route  k  max EU
T
( v1i ,..., vni i )
(R ).
4. Start from the beginning: j=1; while (v kj  destination) and (  0) .
5. Cross the node v kj and update the remain time slot      k ( j )1 .
6. Delete from R each route  i , s.t. EU k ( j ) (v1i ,..., vni i )  0 .
7. Choose the route  l  max EU
i
i
  k ( j ) ( v1 ,..., vni
)
( R \ k ) .
if EUT (v kj1 ,..., vnk1 )  EU k ( j ) (v1l ,.., vnl l ) then
8. Continue with the current route: j++.
else
9. Roll back to the source node and update the remain time:      k ( j ) .
10. Switch to the chosen alternative path: k=l.
11. Start from the beginning: j=1.
if v kj  destination) then
return VC  i .
return set-up fail.
Fig. 9: Crankback Algorithm
Let us consider a topology of 10 possible routes between the source and the destination.
Each route consists of 10 nodes. Each node has a Gaussian pdf representing the delay
needed to cross it. The pdf has a mean selected uniformly between 100 and 200 time
units, and a variance, which is selected at random between the values of 20 and 40. For
each connection set-up an initial time quota is allocated. When crossing each node, a
value is randomly selected, according to the node’s pdf, to represent the amount of time
units needed for crossing the node. When cranking backwards to the source node, a very
small amount of time (one time unit in our simulations) is required to cross any node.
This choice we made here represents the fact that in real networks, the cost of crossing a
node in a direction opposite to that of the set-up is usually negligible.
The implementation of the multi-route algorithm in this case simplifies to a scheme
involvnig the following steps:
1. Derive the end-to-end crossing time pdf for each route, by convoluting the pdfs of
the nodes along the route.
2. Calculate the probability of failure in the connection set-up for each route,,
according to its end-to-end pdf, as in the single route case.
3. Choose the route with the smallest probability of failure, and cross its first node,
and reduce the amount of time needed to cross that node from the initial time
quota.
4. Calculate the optimal route according to the probability of failure in travelling
towards the destination. All routes should be examined, while taking in
consideration the time needed to roll back to the source. Check if the current
route is still optimal. If it still is, cross its next node, and reduce the time needed
for crossing the node from the current time quota. If it became suboptimal,
crankback to the source node, cross the first node in the new optimal route, and
reduce the time needed for crossing this node.
5. Repeat Step 4 until either the destination is reached, in which case the set-up
procedure has succeeded, or the time quota has fully been consumed, in which
case the attempt to set up a connection ended in a failure/failed.
In contrast to the single-route case, here there is no need to choose a threshold, since the
routes are compared against each other.
In order to evaluate the benefits achieved by using this method, we compared the results
obtained in this way with the results achieved by applying a scheme, in which after
calculating the a-priori optimal route (As in the beginning of the proposed algorithm) we
traverse this route, no matter what the delays selected in the nodes along the route are,
until the destination is reached, or the time quota is fully consumed. With this approach
there is a relative large chance of traversing a suboptimal route, and the crankback
procedure will not be activated.
To compare between these two approaches, we applied both of them in the same
situations: same nodes’ pdfs and same initial quota. We examined the ratio of the
attempts to establish a connection, which ended successfully to the total number of
attempts. We further checked the average amount of time needed to complete the
establishment of the connection, in both methods.
Quantitative and qualitative evaluations were done.
Multi Route Case Simulations Results
The results of the simulations are shown in Table 2:
Crankback
No-Crankback
Crankback:
No Crankback:
Mechanism
Mechanism
Mean Time
Mean Time
Success Ratio
Used for Set-up
Used for Set-up
Success Ratio
Quota=1500
1
0.57
1329.9
1370.4
Quota=1400
1
0.52
1240.1
1286.7
Quota=1300
1
0.45
1219.3
1236.8
Quota=1200
1
0.4
1145.7
1170.8
Quota=1100
1
0.3
1132.4
1156.2
Table 2: Multi Route case results
Multi Route Case Analysis
As we can clearly see in the results presented in table [], the algorithm we implemented
achieved good results. Even when a quota of 1500 time units, which is the mean of the
end-to-end pdf is allocated, our algorithm achieves much better results as compared to the
algorithm where no crankback is activated. Furthermore, when a smaller amount of time
is allocated for the set-up procedure, the advantage of our algorithm can clearly be seen,
as it ranks in performance above the method of no cranking back: The rate of success in
establishing a connection, when our algorithm for crankback is implemented, is
significantly higher that the corresponding rate where the predicted crankback procedure
is not implemented.
Final Conclusions and Remarks:
We have introduced a new scheme for improving the crankback mechanism in ATM
networks. Our algorithm is based on statistical measures for invoking an early crankback,
prior to crossing the limits of network resources, but in a situation in which we can
predict that the resources are going to fully be consumed with a relative small chance for
completing the connection’s establishment. In this paper we focus on an accumulative
measure, which is the end-to-end delay. The decision for crankback is made on the basis
of hop by hop decision: in each stage the probability of failure is derived and and tested
against a predefined threshold, provided that only the travelled route is a possible route
for a set-up, or against the probabilities of failures of other possible paths, in face of
multiple routes. This comparison is needed in order to determine whether a crankback
should be activated. We have shown various numerical, as well as analytical results.
The simulations we ran show a significant improvement as compared to the ordinary
approach, especially when the resources are limited, in which case the amount of time
allocated for establishment of the connection is quite small. The methods based on
statistical measures, which we implemented, are found to be most fruitful in preventing
waste of network’s resources. In particular, this kind of research can be helpful for
improving the network utilization during the set-up connection in ATM networks.
References:
1. ATM Forum PNNI SWG 94-0471R13. ATM Forum PNNI Draft Specifications,
March 1996.
2. E. Felstaine, R. Cohen, and O. Hadar, “Crankback Prediction in Hierarchical
ATM Networks”, INFOCOM, 1999, pp. 671-679, 1999.
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cdv.
Technical Report 95-0556, ATM Forum, 1995.
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1976.
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1994.
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R. Haendel, M. N. Huber and S. Schroeder, ATM Networks – Concepts,
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