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Deterministic Formalization of the Processing Gain for Reducing MAI in Multi-Channel Communications System

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Abstract

Multiuser detectors suffer from their relatively higher computational complexity that prevents widespread use of this technique. In addition, one of the main characteristics of multichannel communications that can severely degrade the performance is the inconsistent values of processing gain (PG) that result in high multiple access interference (MAI). However, if we could lower the complexity of multiuser detectors and produce better values of PG, most of the CDMA systems would likely take advantage of this technique in terms of increased system capacity and a better data rate .This paper presents a deterministic formalization of the PG for a wireless multi-channel DS-CDMA system The proposed deterministic formalization demonstrates that how the reduced BER could be used to achieve reasonable values of PG by which unwanted signals can be suppressed relative to the desired signal at the receiving end. The performance measure adopted in this paper is the achievable bit rate for a fixed probability of error (10<sup>-7</sup>).
Abstract
Null message algorithm (NMA) is one of the efficient
conservative time management algorithms that use null
messages to provide synchronization between the
logical processes (LPs) in a parallel discrete event
simulation (PDES) system. However, the performance
of a PDES system could be severely degraded if a
large number of null messages need to be generated by
LPs to avoid deadlock. In this paper, we present a
mathematical model based on the quantitative criteria
specified in [12] to optimize the performance of NMA
by reducing the null message traffic. Moreover, the
proposed mathematical model can be used to
approximate the optimal values of some critical
parameters such as frequency of transmission,
Lookahead (L) values, and the variance of null
message elimination. In addition, the performance
analysis of the proposed mathematical model
incorporates both uniform and non-uniform
distribution of L values across multiple output lines of
an LP. Our simulation and numerical analysis suggest
that an optimal NMA offers better scalability in PDES
system if it is used with the proper selection of critical
parameters.
Keywords— Conservative distributed simulation, discrete
event, null messages, parallel and distributed systems.
1. Introduction
While there has been much research evaluating the
performance of conservative NMA in terms of message
transmission overhead and processor idle time, there
has been comparatively little work devoted to
suggesting any potential optimization for the NMA.
This paper presents a mathematical model based on the
quantitative criteria specified in [12] to optimize the
performance of NMA by minimizing the null message
transmission across each LP.
In PDES systems, the distributed discrete events
need to be tightly synchronized with each other in order
to produce the correct results. However, if these
discrete events are not properly synchronized, the
performance of a PDES environment may degrade
significantly [2]. Time management algorithms are,
therefore, required to ensure that the execution of a
PDES is properly synchronized. Two main classes of
time management algorithms are optimistic and
conservative. In optimistic time management
algorithm, errors are detected and recovered at run
time. However, the performance of optimistic
synchronization protocols is mainly dependent on the
transmission delay [13]. On the other hand, in
conservative PDES, each LP processes events strictly
in time stamp order. Since all LPs do not have a
consistent view of the state of the entire system, LPs
must exchange information to determine when events
are safe to process [1, 3].
Although, much research has been done to evaluate
the performance of conservative NMA for
inefficiencies and overhead [3, 12], none of them
suggest any potential optimization for the NMA.
Reference [12] proposed a quantitative criterion that
incorporates many critical parameters relevant to the
performance of NMA. It has been shown that the
selection of values for several critical parameters such
as the values for Lookahead (L), null message ratio
(NMR), and frequency of transmission plays an
important role in the generation of null messages [12].
If these values are not properly chosen by a simulation
designer, the result will be an excessive number of null
messages across each LP. This situation gets more
severe when the NMA needs to run to perform a
detailed logistics simulation in a distributed
environment to simulate a huge amount of data [9].
This paper presents a mathematical model based on
the quantitative criteria specified in [12] to optimize
the performance of NMA by reducing the null message
Reducing Null Message Traffic in Large Parallel and Distributed Systems
Syed S. Rizvi and Khaled M. Elleithy
Computer Science and Engineering Department
University of Bridgeport,
Bridgeport, CT 06601
{srizvi, elleithy}@bridgeport.edu
Aasia Riasat
Department of Computer Science
Institute of Business Management
Karachi, Pakistan 78100
aasia.riasat@iobm.edu.pk
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978-1-4244-2703-1/08/$25.00 ©2008 IEEE
traffic. The reduction in the null message traffic
significantly improves the performance of a PDES
system by both minimizing the transmission overhead
and maintaining a consistent parallelization. Moreover,
the proposed mathematical model can be used to
approximate the optimal values of some critical
parameters such as frequency of transmission, L values,
and the variance of null message elimination. These
optimal values can be further used to eliminate
unnecessary generation of null messages across the
LPs. In addition, the performance analysis of the
proposed mathematical model incorporates both
uniform and non-uniform distribution of L values
across multiple output lines of an LP. Our simulation
and numerical analysis suggest that an optimal NMA
offers better scalability in PDES system if it is used
with the proper selection of critical parameters
The rest of the paper is organized as follows. Section
2 provides an overview of the conservative
synchronization protocols. Section 3 presents the
proposed mathematical model based on the quantitative
criteria specified in [12]. Section 4 discusses the
potential optimizations in the NMA based on the
proposed mathematical model. Section 5 presents a
performance analysis for both the proposed
mathematical model and the optimizations for NMA.
Finally, Section 6 concludes the paper.
2. Related work
Event synchronization is an essential part of parallel
simulation. In general, synchronization protocols can
be categorized into two different families: conservative
and optimistic. Conservative protocols fundamentally
maintain causality in event execution by strictly
disallowing the processing of events out of timestamp
order. The main problems faced in conservative
algorithms are overcoming deadlock and guaranteeing
the steady progress of simulation time. Examples of
conservative mechanisms include Chandy, Misra and
Byrant's NMP [6], and Peacock, Manning, and Wong
[11] avoided deadlock through null messages. The
primary problem associated with null messages is that
if their timestamps are chosen inappropriately, the
simulation becomes choked with null messages and
performance suffers. Some intelligent approaches to
null message generation include generation on demand
[8], and generation after a time-out [5]. Some earlier
research on discrete event simulation has focused on
variants of null message protocol (NMP, with the
objective of reducing the high null message overhead.
For instance, Bain and Scott [4] attempt to simplify the
communication topology to resolve the problem of
transmitting redundant null messages due to low
Lookahead cycles. Other recent developments [10]
have focused on incorporating knowledge about the LP
into the synchronization algorithms. Cota and Sargent
[7] focused on the skew in simulation time between
different LPs by exploiting knowledge about the LPs
and the topology of the interconnections.
Although earlier work has aimed to optimize the
performance of the NMA by proposing the variants of
the NMP [3, 4, 8, 10, 12], it has not addressed reducing
the exchange of null messages that is caused by
improper selection of the parameters.
The principal problem with the NMA is that it uses
only the current simulation time of each LP and the L
value to predict the minimum time stamp of messages it
can generate in the future [12]. These messages with
the minimum time stamp are then used to avoid
deadlock. As a result, if one of the important
parameters such as the L value is chosen poorly, the
performance will degrade significantly due to an
excessive number of null messages. However, the
prediction of minimum time stamps of messages can be
improved by understanding the relationship between
the time stamp and the L value [12].
3. Mathematical model for NMA
A PDES environment involves synchronization
overhead which is added due to the distributed nature
of simulation. With NMA, this overhead is mainly
associated with the transmission of null messages.
Therefore, when comparing the performance of a
PDES environment that uses NMA with the
performance of sequential execution, the message
overhead can make a significant performance
difference between the two approaches. Before
presenting a proposed mathematical model, it is worth
mentioning some of our key assumptions.
We assume that the value of L may change during
the execution of a Lookahead period. However, the
values of L can not instantaneously be reduced.
Initially, a constant event arrival or job intensity rate
is assumed for each participating LP in the
simulation. However, for the sake of experimental
verifications, we also consider the non-uniform
distribution of L values across multiple output lines
of an LP.
For the frequency of message transmission, we
assume that all messages are equally distributed
among the LPs. For the proposed mathematical
model, we assume that we have n number of LPs in
the simulation where all LPs are connected with each
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other by means of highly reliable mesh networks
topology. Also, each LP maintains n-1 input and
output links for both input and output neighboring
LPs, respectively.
3.1. Definition of system parameters
All model variables, along with their definition, are
listed in Table 1. Based on the concept of NMA, we
assume that each LP maintains two clock times, one for
each of its input and output neighbors as shown in
Figure 1. One is the minimum receiving time (MRT) for
the input neighbor LP whereas the second is the
minimum sending time (MST) for the output neighbor
LP. The MRT represents an earliest time when an LP
can receive an event message from one of its input
neighboring LPs, where as the MST represents an
earliest time by which an LP can send a message to one
of its neighboring LP. The performance (P) of a
conservative distributed simulation environment mainly
depends on the amount of computation required for
processing an event per second. In addition, the event
arrival rate (
ρ
) represents the number of events that
occur per second (in practice, events occur per
simulation second). Unlike performance, the
parameter
ρ
mainly depends on the model. Lookahead
(L) is measured in seconds. Frequency of transmission
(
T
F
) is the frequency of sending a message from one
LP to another. In addition, TNull represents the
timestamp of a null message sent from one LP to other
LPs.
In order to measure the performance, it is
imperative to consider one parameter that can compute
simulation time advancement (STA). The STA can be
defined as a ratio of performance to event arrival rate.
This relationship can be expressed as:
ρ
= (1)
The value of MRT is updated by the time stamp of a
null message coming from other neighboring LPs on
one of the input links of a receiving LP. Any event
scheduled by an LP must have a timestamp at least as
large as the LP’s simulation time clock [1]. This
requirement is also referred as the local causality
constraint requirement. To strictly follow this
requirement, a large number of null messages can be
transmitted by LPs before the non null-messages can be
processed. This large message overhead may degrade
the performance of a conservative distributed
simulation. It is, therefore, worth computing the ratio of
null messages to the total messages transmitted among
LPs. The NMR can be simply defined as the ratio of
total number of null messages to total messages where
total messages include both null and event messages.
Mathematically, it can be expressed as follows:
Total Number of Null Messages
NMR Total Messages
= (2)
3.2. The proposed mathematical model
First, we present a mathematical model based on the
quantitative criteria specified in [12]. In addition, the
proposed mathematical model is also based on the
internal architecture of an LP as shown in Fig. 1. The
architecture for m number of LPs is shown in Fig. 2.
Using the quantitative criteria defined in [12], we can
approximate the advancement in the simulation time as
a ratio of performance to event arrival rate. This leads
us to the following mathematical expression of the
relative speed for simulation advancement:
( )
{
}
Msg
P T E
s (3)
Taking this into account, we can give the following
hypothesis for approximating the number of null
messages transmitted per LP: “If we assume that we
have an average value of L which associates with one
of the output lines of an LP, then P can be
approximated as”:
(
)
1
Msg
P E L
(4)
Combining (3) and (4) yields the estimated number
of null messages transmitted per LP that has only one
output line as shown in (5).
( ) ( )
(
)
( )
1 1
Msg S Msg S
LP
Null E L T E T L
(5)
Furthermore, if we assume that we have O number of
output lines attached with each LP with the uniform
distribution of L value on each output line, then (5) can
Table 1
System parameter definition
Parameter Definition
P
Computation required for processing an event
per second
ρ
Event arrival rate (events per second)
MRT
Minimum receiving time
MST
Minimum sending time
L
Lookahead
STA
Simulation time advancement
F
T
Frequency of transmission
T
Null
Timestamp of a null message
T
S
Current simulation of a LP
T
Total
Total simulation time in seconds
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be further generalized for O number of output lines per
LP as follows:
( ) ( )
(
)
( )
Msg S Msg S
LP
Null E O L T E T O L
(6)
It should be noted that (6) represents total number of
null messages transmitted per LP via O number of
output lines to the neighboring LPs. If we assume that
we have m number of total LPs present in a system
where each LP has O number of output lines, then this
allows us to extend (6) and generalize it for m number
of LPs present in a distributed simulation as shown in
(7). It can be seen that (7) gives total number of null
messages exchange among all LPs.
( ) ( )
(
)
( )
Msg S Msg S
m LP
Null E O L m T E T O L m
(7)
where the term
O L
in (7) shows a uniform distribution
of L value for O number of output lines.
The assumption of uniform distribution of
Lookahead among O output lines of an LP simplifies
the procedure for computing the number of null
messages transmitted per LP to other neighboring LPs.
However, the values for L may change during the
execution of a Lookahead period that makes the
uniform distribution assumption of Lookahead a little
unrealistic. This argument leads us to the fact that we
should also account a non-uniform distribution method
for Lookahead where each output line of an LP can
have a different value of L. We can rewrite (6) as:
( )
( )( )
( )
1 1
1
O O
Msg i S Msg S i
LP
i i
Null E L T E T L
= =
∑ ∑
(8)
It should be noted that (8) represents the total
number of null messages transmitted per LP to other
neighboring LPs.. If we assume that the model is
partitioned into m number of total LPs where each LP
can have at most O number of output lines, this allows
us to extend (8) for m number of LPs.
( )
( )( )
( )
1 1 1 1
1
m O m O
Msg hi S Msg S hi
m LP
hi h i
Null E L T E T L
== = =
∑ ∑∑
(9)
It can be evident that (9) gives the total number of
null messages exchange among all LPs.
4. Performance optimization of NMA
In this section, we introduce two different ways to
optimize the performance of NMA. We first implement
the concept of frequency of transmission described in
[12] to minimize the exchange of null messages across
the LP. Secondly, we present the new concept of
variance that works with the frequency of transmission
to avoid the unnecessary generation of null messages
and consequently minimize the overall synchronization
overhead. For both concepts, we derive a closed form
mathematical expression that can be used to evaluate
the performance of NMA in the presence of deadlock
situation.
4.1. Frequency of transmission
Transmission of null-messages on each occurrence
of an event results in unnecessary generation of null
messages that causes an increase in the synchronization
overhead. We believe, instead of sending null message
after every one event, it should be transmitted with
respect to a certain frequency of transmission. This
frequency of transmission (FT) is a fixed amount of
time and it should be measured in simulation second
per second. Recall (2) and (4), number of events
processed per second per LP can be equated from both
equations. This yields the following approximation for
FT in term of L value.
Figure 1. Internal architecture of an LP
Figure 2. m number of LPs with I number of
input queues and O number of output queues
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( )
2 1 2
Msg T Msg T
E L F E L L F
(10)
Substituting the value of (10) into (5), we get,
( )
1 1
2 2
S
S
LP
T T
Msg Msg
T
Null E T
E
F F
 
   
 
   
   
 
   
 
(11)
Equation (11) can be generalized for O number of
output lines per LP when the number of null messages
is assumed to generate with a certain frequency of
transmission as shown in (12).
( )
2 2
S
Msg S
LP
Msg
T T
TO O
Null E T
E
F F
 
 
   
 
   
 
 
(12)
Equation (12) gives an estimated number of null
messages transmitted by an LP that has O number of
output lines where each line carry an equal percentage
of the L value in terms of a fixed frequency of
transmission per output line. In addition, if we assume
that the system consists of m number of total LPs where
each LP has a fixed number of output lines, then (12)
can be further extended for m number of LPs.
( )
( )
( ) ( )
%
2 2
S
Msg T S T
m LP
Msg
T
T
Null E m O F T m O F
E
where F L
 
× ×
 
 
 

 
(13)
where the denominator of (13) (i.e.,
T
F
O) represents
a uniform rate of null message transmission per output
line. Based on (13), we can conclude that a non
uniformity in null message algorithm results a non
linear generation of null messages. An expression can
be derived for O number of output lines where each
line may carry a different value of FT
( )
1 1
%
1
2 2
O O
Msg S
S
LP
i i
Msg
Ti Ti
Ti i
ET
Null T
E
F F
where F L
= =
   
 
   
 
   
 
   

∑ ∑
 
(14)
Furthermore, (14) can be further extended and
generalized for m number of LPs where each LP can
have at most O number of output lines.
( ) ( )
( )
( )
( )
( ) ( )
1 1
%
1 1
2
1
2
m O
Msg S Msg
m LP T ki
k i
m O
S
T ki ki
k i T ki
Null E F T E
T where F L
F
= =
= =
 
  
 
 
(15)
4.2. Variance for null message elimination
Also, in this scenario, it is essential to cancel out the
unnecessary generation of null messages. Variance
represents the probability of cancellation of
unnecessary null messages. The value of variance may
exist between 0 and 1. It should also be subtracted from
1, so that we can show that increase in variance causes
a decrease in the over all null messages where as a
decrease in variance results an increase in null
messages. If we consider variance as 0, then it should
give us the same results that we could achieve with out
using variance. In order to reflect the variance of null
message cancellation, we can rewrite (13) for m
number of LPs with the uniform distribution of null
message transmission per output:
( )
( )
( )
%
2(1 )
(1 ) 0 1
2
S
Msg T
m LP
Msg
S T
T
T
Null E m O F E
O
T m where F L and
F
σ
σ σ
 
× −
 
 
 
 
× ≤ <
 
 
(16)
where
σ
represents probability of null message
cancellation.
The same concept of null message cancellation can
be implemented with a simulation model where the L
values are non-uniformly distributed among O number
of output lines. This leads us to the following
modification in (16):
( ) ( )
( )
( )
( )
1 1
1 1
2 (1 )
1 2 (1 ) 0 1
m O
S
Msg T
m LP ki
k i Msg
m O
S T ki
k i
T
Null E F E
T F where
σ
σ σ
= =
= =
 
 
 
 
≤ <
(17)
5. Performance analysis of NMA
For the sake of performance analysis, we simulate 5
different cases. The system is modeled in C++.
5.1. Multiple output lines per LP
Using (6), Fig.3 shows the null message transmission
with the following simulation parameters: simulation
time (Ts) = 500 sec, L is uniformly distributed per
output line. The number of output line may vary from 0
to 8 for all results as show in Fig.3. Simulation results
of Fig. 3 presents a comparison of null message
transmission per LP versus multiple output lines.
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5.2. Multiple LPs with multiple output lines
per LP
We assume that we have multiple LPs with O output
lines (fixed per LP). Let the output lines per LP is 4
with the (Ts) of 500 sec. Using (7), Fig.4 shows the
null message transmission with the following
simulation parameters: Ts = 500 sec, L is uniformly
distributed per output lines, the output lines are
assumed to be fixed for each LP (O = 4). The numbers
of LPs are varied from 1 to 10 as show in Fig.4.
5.3. Multiple output lines per LP with non-
uniform distribution of Lookahead
For this simulation, we assume that we have single
LP that has O number of output lines where each
output line of an LP can have different value of L.
Using (8), Fig.5 shows the null message transmission
with the following simulation parameters: Ts = 500 sec,
L is non-uniformly distributed per output lines. The
numbers of output lines may vary from 1 to 10 as show
in Fig.5. Also, it should be noted that the value of L is
chosen randomly within the range of 0 to 1 and
assigned to each output line at run time. This random
selection may control the generation of unnecessary
null messages as long as the value is chosen
appropriately.
5.4. Multiple LPs with multiple fixed output
lines
For this simulation, we assume that we have multiple
LPs that can have fixed number of output lines where
each line of an LP can have different value of L. Using
(9), Fig.6 shows the null message transmission with the
following simulation parameters: Ts = 500 sec, L is
non-uniformly distributed per output lines. The
numbers of LPs are varied from 1 to 20. Also, it should
be noted that the value of m and O are both varying
quantity for this particular scenario. In harmony with
our expectations, the number of null messages
increases due to an increase in number of LPs.
However, this increase in null messages is limited and
controlled due to the random behavior of Lookahead.
This can also be considered as irregular networks due
to the non uniform distribution.
2 3 4 5 6 7 8
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Output lines (O) Per LP
Null Message Transmission Per LP
Null(LP) with L=0.2
Null(LP) with L=0.4
Null(LP) with L=0.6
Null(LP) with L=0.7
Figure 3. Multiple output lines per LP versus
null message transmission per LP
2 3 4 5 6 7 8 9 10
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Number Of LPs
Null Message Transmission
Null(m-LP) with L=0.2
Null(m-LP) with L=0.3
Null(m-LP) with L=0.4
Null(m-LP) with L=0.6
Null(m-LP) with L=0.7
Figure 4. Multiple LPs with fixed output lines
per LP versus null message transmission
2 4 6 8 10 12 14 16 18 20
400
600
800
1000
1200
1400
1600
1800
2000
Number Of output lines (O) per LP
Null Message Transmission per LP
Null Messages where 0<L<1 (100 Runs)
Null Messages where 0<L<1 (300 Runs)
Null Messages where 0<L<1 (500 Runs)
Null Messages where 0<L<1 (700 Runs)
Figure 5. Multiple output lines per LP with
non-uniform distribution of L value
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6. Conclusion
We have proposed a mathematical model to predict
the optimum values of critical parameters that have
great impact on the performance of NMA. The derived
properties of the proposed mathematical model account
for the cases when the NMA would send too many null
messages. The proposed mathematical model provides
a quick and practical way for simulation designers to
predict whether a simulation model has potential to
perform well under NMA in a given simulation
environment by giving the approximate optimal values
of the critical parameters. We have experimentally
verified that if critical parameters, specifically the L
value, are chosen intelligently, we can limit the
transmission of null messages among the LPs and
consequently improve the performance of NMA in a
distributed simulation environment.
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2 4 6 8 10 12 14 16 18 20
600
700
800
900
1000
1100
1200
1300
1400
1500
Number Of LPs
Null Message Transmission
4 output lines per LP where 0<L<1 (100 Runs)
6 output lines per LP where 0<L<1 (300 Runs)
8 output lines per LP where 0<L<1 (500 Runs)
10 output lines per LP where 0<L<1 (700 Runs)
Figure 6. Multiple LPs and multiple fixed output
lines with non-uniform distribution of L versus
null message transmission
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