A SIMULATION STUDY FOR OPTIMIZING THE

PERFORMANCE OF SEMI-LAYER DELTA NETWORKS

Eleftherios Stergiou

and John Garofalakis

2, 3

Department of Informatics and Telecommunications Technology, Technological Educational Institute of Epirus, Arta, Greece

Department of Computer Engineering and Informatics, University of Patras, Patras, Greece

Research Academic Computer Technology Institute, Rion, Patras, Greece

Keywords: Delta network, Banyan networks, Performance evaluation, Buffer, Multilayer multistage interconnection

networks.

Abstract: In this paper, a semi-layer multistage delta network is presented and exemplified considering various values

of buffer size by using simulation. The proposed network configurations are evaluated and compared with

each other. A performance evaluation was conducted via our simulator assuming uniform conditions and

arrivals of Bernoulli type. Performance statistics were collected for the two most important performance

indicators of the network that is throughput and packet latency. From this study emerges the appropriate

configuration of single and semi-layer delta networks in terms of buffer size. The evaluation methodology

can be applied to several network configurations, providing the basis for a fair comparison, and the

necessary data for network engineering to optimize the performance of semi-layer delta networks.

1 INTRODUCTION

Multistage Interconnection Networks (MINs) are

used for interconnecting processors in parallel

systems and to ensure efficient internetworking

(Suet, 2004). The advantages that they have, include

their ability to route multiple communication tasks

concurrently, as well as their low cost/performance

ratio. Banyan MINs are MINs which have the

property of the existence of one, and only one path

between each source and destination. On the other

hand, the non-banyan interconnection networks are

more expensive and more complex to manage.

This paper is a study of the performance

optimization of semi-layer multistage delta

networks. Delta networks are a subclass of banyan

networks. The Delta networks properties are

explained in the next section.

Performance evaluation methods for delta

networks (or in general banyan networks) mainly

include analytical methods, Petri nets modelling and

simulation.

Analytical methods are considered in general to be

complex. Nevertheless, they have been extensively

used by some researchers. Most of the MIN analysis

focuses on uniform traffic (i.e. packages) coming to

a network with an equal probability of reaching any

output (Hsiao and Chen, 1991), (Bouras et al.,

1987). On the other hand, there are numerous non-

uniform traffic patterns in real applications that

require special treatment. One such non-uniform

approximation can be seen in (Tutsch and Hommel,

2002). Other typical analytical studies of a MIN’s

performance are exemplified by various studies

(Garofalakis and Stergiou, 2008), (Bouras et al.,

1987), (Garofalakis and Spirakis, 1990).

Petri nets serving as MIN modelling methods

have also been employed. The (German, 2000),

(Haas, 2002) and (Linderman, 1998) are examples of

such approaches. Petri nets methods are also

considered complex. When there is an interest in

more realistic results, simulations are used.

Simulations allow flexibility in network parameters,

making it possible to analyze the network with

different communication patterns. Examples of such

approaches are (Vasiliadis at al., 2006), (Vasiliadis

et al., 2007), (Vasiliadis et al., 2008). All the above

cited studies involve single layer multistage

interconnection networks (SiLMINs).

Dietmar Tutsch and his group (Tutsch and

Hommel, 2008) introduced multilayer multistage

interconnection networks (MLMINs). Firstly, they

demonstrated that the single layer MINs show a high

saturation when the packets population is increased

257

Stergiou E. and Garofalakis J..

A SIMULATION STUDY FOR OPTIMIZING THE PERFORMANCE OF SEMI-LAYER DELTA NETWORKS.

DOI: 10.5220/0003598202570265

In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2011), pages

257-265

ISBN: 978-989-8425-78-2

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

dramatically. The MLMINs were developed mainly

to meet the need for efficient handling of multicast

traffic (Tutsch, 2006). MLMINs are more suitable

fabrics for modern traffic as well as on-line

multimedia applications, which are increasing in

importance.

The main weakness of the MLMIN architecture

is attributed to the exponentially growing number of

layers as the stages increase, which leads to higher

costs. If we try to reduce the number of layers then

hardware complexity is reduced and, therefore, so is

the overall cost of the fabric.

Semi-Layer MINs (SeLMINs) are special cases

of the multi-layer MIN. SeLMINs are defined

(Garofalakis and Stergiou, 2009), (Garofalakis and

Stergiou, Oct 2010) as a multilayer MIN which

consists of two segments. The second segment must

keep the levels growth fixed and equal to the Switch

Element (SE) size. The second segment of the MIN

is an unblocked segment. Figure 2 illustrates

examples of two SeLMIN cases in 2D view, which

have two and four layers, respectively.

When the layers of a SeLMIN are Delta type

multistage networks, we have semi-layer multistage

Delta type networks, which are the kind of networks

being studied here.

These multistage fabrics are devices which can

be constructed using a finite buffer size. However,

the main question which arises is: what is the

suitable buffer size in each case of traffic? This work

tries to provide an answer to this question.

Hence, the main goal of this paper is to evaluate

the performance of semi-layer delta type networks

assuming the offered load is of unicast type, for

different buffer size constructions. Ultimately, the

objective is to determine the buffer size which

optimizes throughput and packet latency.

Performance evaluation was conducted through

simulation, considering uniform traffic conditions.

Metrics were collected for the two major important

network performance factors, which is throughput

and packet latency.

The remainder of this paper is organized as

follows: in section II, a brief analysis of a semi-layer

delta network, which is the main research subject, is

presented. Subsequently, in section III, the

performance criteria and parameters that are related

to the above network schemes, are presented.

Section IV reveals the results of our simulation-

based performance analysis, examining the effect

that the buffer size has on overall network

performance. Finally, section V provides concluding

remarks.

2 DEFINITION OF MULTILAYER

DELTA NETWORKS

A typical multistage (

NxN

) MIN is constructed by

log

parallel stages of (

cxc

) Switch

Elements (SEs), where c is the degree of the SEs.

Each stage contains

)/( cN SEs. Hence, the total

number of SEs of a MIN is equal to

NcN

log)/(

⋅

. Thus, there are

)log( NNO

⋅

interconnections between all the

stages, in contrast to the crossbar network that has

)(

NO links. Also, a MIN is distinguishable from

the others if we know, except of its topology, the

switching techniques and the routing algorithm used.

The fabrics examined here use the store and forward

switching technique and shuffle perfectly as a

routing algorithm. The routing is performed in a

pipeline manner, which means the routing process

occurs in every stage, in parallel.

The whole network operates “synchronously”,

which means that the time cycles refer to global

clock ticks. The network clock consists of two

phases. In the first phase, the queues are serviced

and then any new packets are received.

Moreover, each MIN operates under the following

assumptions:

 The service time of the output queues at each

switch is assumed to be fixed and equal to the

network cycle time.

 The traffic feeding the first stage of the MIN

switch follows a Bernoulli type distribution, so

the arrivals are considered independent from

each other. If (

k ) is the random variable

denoting the count of arrivals of packets at the

end of a network cycle on a queue of a

SE at the first stage of the MIN, the formula is

(Garofalakis, 2008):

⎪

⎩

⎪

⎨

⎧

≤≤

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

otherwise

kcfor

ckc

,,0

0,1

)1(

(1)

Where

)1(

,ck

x : depicts the probability of ( c ) packets

accepted in an arbitrary first stage queue with in

general (

k ) inputs at an arbitrary time cycle.

However, usually the under study systems have

k inputs, hence the c can be 0, 1 or 2 at the

most.

Also

depicts the probability of packets arrivals in

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

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258

an arbitrary input of a random first stage queue of

the switch system at an arbitrary time cycle.

All the packets are considered to have identical fixed

sizes.

 Any arrived packet at the first stage is lost if the

relevant buffer of the SE is full.

 Each queue uses the FIFO policy for all output

ports.

 Any packet will be blocked at a stage, if the

destination buffer at the next stage is full.

 At the last stage, output links of the MIN

signify that there is no blocking.

All packet conflicts are randomly resolved and the

routing logic at each switch is fair.

2.1 Delta Networks Property

Delta networks were proposed by Patel (Patel.

1981). Delta networks which belong to banyan

property networks, are usually used to connect a

significant number of processors in a multiprocessor

system.

In general, delta networks are constructed by

xcc Switch Elements (SE) (Figure 1). Let’s

consider

o an output of a random SE,

where

1,...,,1,0

−= cj . If an input of a SE in

stage is connected to an output of another SE in

stage

)1( −i , then all the other inputs must be

connected to outputs

o of the same index

of SE

in the previous stage.

cc ×

…………………….

….

Stage 1

Stage 2 Stage nShuffle Shuffle

−c

−

−c

−

I n p u t P o r t s

O u t p u t P o r t s

Figure 1: The general structure of a delta network.

For a banyan MIN of size N and degree c ,

which is denoted as network (

cN, ), suppose that

the switch’s inputs and outputs are presented by

in the form of

d ,

d ,…,

1−c

d . If the inputs and

outputs of the SEs in the networks have the same

indexes, then digits

d of all inputs of a switch must

be equal.

The above described mathematical translation is

deemed a delta property. All the interconnection

networks which have this characteristic are said to

possess the delta property.

All the SEs in any delta network contains

digitally controlled crossbars. Digitally controlled

SEs are controlled by a sequence of bits that hold all

the packets which have to traverse through the MIN.

In delta networks this sequence of bits represents the

packet destination.

Our study case considers symmetrical SEs with

ccc

, given that it is very common in MINs

systems.

2.2 Semi-layer MINs

Semi-layer MINs are a subclass of MLMINs which

consist of two distinct segments (Figure 2).

The front segment (first stages) of the MIN contains

only one layer which employs a backpressure

blocking mechanism. Replication at the first

segment is not recommended. It is a key challenge to

keep the overall cost of such fabric at low levels.

The second segment encompasses the rest of the

construction. The second segment is the multilayer

segment of a MIN (a full fan-out), which is free of

blocking. If we consider the SEs of second segment

to be represented by

cc ×

, then the SeLMINs of the

second segment keep the level growing at a fixed

rate and equal to

c . According to (Tutsch &

Hommel, 1997), the SE’s outputs in the last stage

are multiplexed. In this case, if either the multiplexer

or the data sink do not have enough capacity to

absorb the packets, then at this point blocking can

occur. However, in this study it is assumed that

multiplexers (data sinks) have adequate capacity.

The main drawback of MLMINs is their high cost,

owing to their complexity. Semi-MLMINs were

introduced as a better trade-off between cost and

performance of the multistage fabric, when the

traffic demands are raised to very high levels.

In a SeLMIN (Figure 2), let

L represent the

number of single layer stages and let

L be the

number of stages that have full layer growth which

can also service multicast traffic without blocking.

Hence,

MLSL

LLL

A SIMULATION STUDY FOR OPTIMIZING THE PERFORMANCE OF SEMI-LAYER DELTA NETWORKS

259

Single Layer interconnection network

Semi-Layer interconnection network (4 Layers)

1 234 56 78

Outputs

Inputs

256x256 Delta networks

Semi-Layer interconnection network (2 Layers)

1 234 56 78

Inputs

Outputs

Figure 2: Literal views of 8 layer delta networks (SiLMIN,

and SeLMINs).

For a given

L the total number of Layers ( NoL )

in the second segment is

cNoL = , where c is

the number of inputs per SE (e.g., in the case of 2x4

switches,

c is equal to two).

Due to their appealing performance/cost ratio, the

SeLMINs are expected to play an important role in

the future regarding the overall performance of

internet interconnections, parallel systems and grid

systems.

2.3 Semi-layer MINs with Delta Type

Property

Semi-layer delta networks are multilayer fabrics

where all the layers are maintained in a delta

multistage network, keeping the same permutation

pattern. Throughout this study, a performance

investigation has been employed, exploring typical

semi-layer delta type networks.

Our case study considers SiLMIN with

stages, and SeLMINs with

8=L

stages and

2=NoL and 4, respectively. In addition, we

assume that the under study semi-layer MINs use

typical

22× SEs in the first segment and 42

SEs

in the second segment.

3 METRICS & METHODOLOGY

FOR PERFORMANCE

EVALUATION OF

SEMI-LAYER DELTA

NETWORKS

This study will present results of the performance of

SiLMINs and SeLMINs when they service exclusive

unicast traffic. The basic performance metrics used are:

Average throughput of a single layer delta

network (

Th ): Average throughput of a delta

interconnection network is defined as the number of

packets delivered to their destination per time cycle.

Formally,

Th can be defined as:

∑

∞→

)(

lim

(2)

where

)(i

denotes the number of packets that

reach their destination during the

time interval.

Using simulations, the throughput is calculated as

the number of packets that arrived at their

destinations over a certain multitude of trials.

Average throughput of semi-layer delta

network (

)(out

Th ): If we consider the throughput of

the first segment of a SeLMIN as (

Th ), then the

total SeLMIN’s throughput (at the fan-out output)

can easily be calculated as follows: In the case of

unicast traffic the formula is:

)()( SLout

ThTh = .

In the case of unicast and multicast traffic the

expression is:

()

SLout

wThTh +⋅= 1

)()(

where

is the ratio of multicast traffic (see

(Garofalakis and Stergiou, 2009), (Garofalakis and

Stergiou, Oct 2010)) for a definition of

w ).

Normalized throughput of single and semi-

layer delta network (

Th ): Normalized throughput

of the delta network (

Th ) is the ratio of the

average throughput over the network size

Formally,

Th can be defined as:

NThTh

SLN

: in the case of a single layer

MIN and

NThTh

outN

)(

: in the case of SeLMINs

Average packet latency of a single layer delta

network (

D ): The packet latency of a delta

network is defined as the number of time units

needed for all packets of a permutation to arrive at

their destinations. Formally,

D can be defined as:

∑

∞→

)(

lim

(3)

where

n depicts the total number of packets

accepted by destinations in

time intervals and

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

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260

)(itΔ represents the total number of network cycles

that an arbitrary

packet needs in order to arrive

at its destination.

)(itΔ

, includes the total number

of network cycles for a packet waiting at any stage

and the total number of network cycles the same

packet needs to remain in active transmission mode

until it reaches its destination.

The network latency is directly related to the

maximum multitude of time cycles needed to route a

certain number of packets to their destinations via

permutations.

Average packet latency of a SeLMIN delta

network (

out

D ): Assuming that the packet latency

of the first segment is

D , then the packet delay of

a SeLMIN

out

D can be expressed as:

MLSLout

LDD += .

This occurs because in the second segment the

packets don’t suffer from contentions, so the delay

to traverse the second (

) stage of the fan-out is

exactly equal to (

L ) time cycles.

Normalized latency of a single and semi-layer

delta network (

D ): Normalized packet latency

D of a delta network is the ratio of the average

packet latency

D over the minimum packet delay

which is considered as equal to

L number of time

cycles. Formally,

D can be expressed by:

In case of single layer MIN:

LDD

SLN

In case of SeLMIN:

LDD

outN

A unique indicator for performance

evaluation of multilayer networks

From the initial experiments it became apparent that

the values of MIN's throughput and the values of

packet latency are inversely proportional to each

other for various values of buffer size.

Nevertheless, the optimal solution is to have high

throughput rates and low values of packet latency.

Hence, it is interesting to have a general evaluation

using only one factor. The factor must reveal the

better overall performance, that is, the first factor

maximized and the second factor minimized

simultaneously. So, this demanding overall

performance factor is defined based on the

correlation of the two individual performance

factors. Because the individual factors have different

measurement units and ranges, it is necessary to

normalize them to obtain a common reference value

domain. We call this factor the Combined

Performance Factor (CPF) which is expressed by

the following formula (Garofalakis and Stergiou,

March 2010):

⎟

⎠

⎞

⎜

⎝

⎛

ThCPF

(4)

In any multi-criteria decision-making problem,

however, the importance of each criterion is a design

problem. Therefore, when it is of interest to assign a

weight (in terms of its importance in the network) to

each separate metric, then the above formula can be

replaced by:

DTh

DNTh

DTh

wThw

wwCPF

⎟

⎠

⎞

⎜

⎝

⎛

⋅+

),(

(5)

where

w ,

w are the corresponding weights of

the normalized system’s parameters: normalized

throughput and normalized packet latency.

According to this equation, when the

Th metrics

become larger and/or the

metrics become

smaller, the CPF becomes larger. The reference

value domain of CPF ranges from 0 to 1.

The main condition which must be satisfied when

the CPF factor is applied, is the assumption

that

≠

D . Besides this, all the measured factors

must be calculated and manipulated as inter-

individual metrics.

Hence, as the CPF becomes higher, the

performance of the MIN is considered to have been

improved.

Here we limit our study to two performance

evaluation factors knowing that the proposed

methodology is general, and that it is available to

add additional factors chosen to evaluate the

performance of a MIN.

Consequently, the following parameters affect

the above performance aspects of multistage delta

networks.

 Network size

L , where NL

log= , is the

number of stages in a (

NN × ) multistage

delta network. In our study it is assumed

that

256

N , thus

8256log

==L

 Offered load (

) is the steady-state fixed

probability of packet arrivals at each queue on

inputs. In our study,

is assumed to be

=0.10, 0.20 … 0.50, 0.60 … 1.

 Buffer size (

b ) is the maximum number of

A SIMULATION STUDY FOR OPTIMIZING THE PERFORMANCE OF SEMI-LAYER DELTA NETWORKS

261

packets that an input buffer of a SE has the

ability to hold. In our study,

b is assumed to

b =1, 2, 3 and 4. In addition to those values

of buffer size, we chose constructions with

higher values of buffer size that are considered

to be extremely expensive fabrics yet not as

good in performance. This happens because the

cost of multilayer delta type fabrics is an

exponential function of the buffer size.

4 SIMULATION AND

PERFORMANCE RESULTS

4.1 Simulation

Here we estimate the performance of multilayer

delta networks using simulations. We are interested

)( NN × multilayer delta networks that consist

)22( × and )42( × SEs, using internal queues.

We developed a general simulator for SeLMINs that

was capable of handling several switch types and

load conditions which work at the packet level. The

simulator was programmed in C++ and is capable of

running various configuration schemas. In building

the simulator, every

)22(

and )42( × SE was

modelled by two buffered queues. Each buffer

operates according to FCFS principle. All the

packets are forwarded by the store and forward

mechanism and in each time slot, they are forwarded

by at most one stage. Cases of packet contention, are

solved randomly with equal probability.

We use as input parameters, the probability of

packet arrivals, the buffer length, the number of

inputs/outputs ports, the number of stages and the

number of layers.

Metrics such as throughput and packet latency

are gathered at the output of the system. The

simulation needs at least

10 iterations (clock

cycles) in order to ensure that the system operates in

steady-state operating condition.

4.2 Results

Figure 3 shows the normalized throughput of an 8-

stage SiLMIN and SeLMIN versus the probability of

packet arrivals for MIN’s buffer size

=b 1, 2, 3 and

4 when the offered load is exclusively of unicast

type. The dot-dashed curves depict results for

SiLMINs, while the solid curves illustrate results for

SeLMIN with 4 layers for buffer size

1, 2, 3 and

4, respectively.

From Figure 3 it becomes apparent that the larger

the values of the buffer size in MINs, the greater the

value of the MIN’s throughput.

256x256 delta networks

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(p)~ Probability of packet arrivals

Normalized throughput

BS=4-SeLMIN(NoL=4)

BS=3-SeLMIN(NoL=4)

BS=2-SeLMIN(NoL=4)

BS=1-SeLMIN(NoL=4)

BS=4-Si LMIN

BS=3-Si LMIN

BS=2-Si LMIN

BS=1-Si LMIN

Figure 3: Normalized throughputs vs. probability of packet

arrivals for an 8-stage delta SiLMINs and SeLMINs with

4 layers.

Also, we can notice that the throughput of

SeLMINS – here 4 layer constructions - have higher

values of throughput compared with the

corresponding, in terms of number of stages and

buffer size, single layer MINs. In addition, for

offered load

7.0≥p , throughput stabilization can

be observed in the system due to the high value of

blockings that takes place in the system.

Figure 4 represents the values of normalized

packet latency of 8-stage SiLMINs and SeLMINs

versus the probability of packets arrivals on the

inputs for MINs with buffer size

1, 2, 3 and 4

when the offered load is exclusively of unicast type.

The dot-dashed and solid curves depict results for

SiLMINs and SeLMINs (NoL=4), respectively,

when buffer ranges from 1 to 4.

From Figure 4 it can be seen that the Semi-layer

MINs with 4 delta type layers, and with a single

buffer size, achieve the best values (lower) of packet

latency in comparison to the corresponding

SeLMINs with higher values of buffer size. In

addition, in the single layer MINs, the packets delay

increases sharply, especially for high values of

offered load, as compared to the corresponding

SeLMINs in terms of buffer size. So, it is obvious

that as the buffer size is increased, the packet delay

also deteriorates (values become higher).

The SiLMINs maintain low values (

5.1

≤

D )

of packet delay when the offered load is

5.0

≤

p .

On the other hand, the same packet delay values are

achieved when the offered load is

6.0≤p . This

gain which the SeLMINs fabrics have over the

SiLMINs, is owing to the exploitation of the

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

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262

256X256 delta networks

1.5

2.5

3.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(p)~ Probability of packet arrivals

Normalized packet latency

BS=4-Si LMIN

BS=4-SeLMIN(NoL=4)

BS=3-Si LMIN

BS=3-SeLMIN(NoL=4)

BS=2-Si LMIN

BS=2-SeLMIN(NoL=4)

BS=1-Si LMIN

BS=1-SeLMIN(NoL=4)

Figure 4: Normalized packet latency vs. probability of

packet arrivals for 8-stage delta type SiLMINs and

SeLMINs with 4 layers.

additional layers at the last stages, which on one

hand provide routes to packets, and on the other

eliminates the phenomenon of packet collisions, thus

improving the packets’ speed, as they move to the

outputs.

By observing Figures 3 and 4 it is obvious that

the two performance indicators (throughput and

packet delay) are contrary to each other. For a given

MIN’s configuration, when the buffer size is

increased, the throughput follows incrementally

while the packets delay deteriorates. Hence, to

evaluate the system by one general performance

indicator we use the CPF factor which has been

defined above.

4.3 Simulator Validation

To validate our simulator, a single-layer, single

buffer and 6-stage MIN is modelled assuming the

offered load on inputs is of unicast type. The results

that are obtained by our simulations are compared

with the corresponding results reported in other

works of the literature. So, in the case of unicast

traffic, Figure 5 depicts the normalized throughput

versus the offered load on 64x64 MIN inputs for

buffer sizes 1 and 2.

The results of this simulation which include

Figure 5 curves: ‘BS=1 Our Simulation’ and ‘BS=2

Our Simulation’, are almost identical with the results

reported in (Garofalakis and Stergiou, 2008), which

comes from an analytical method.

In addition, results presented by Theimer’s

model in (Theimer et al., 1991) for 64x64 MIN with

b=1, notably showed that the two curves (our

simulation and Theimer’s model) are almost in

complete agreement with each other. On the other

hand, Mun’s model (Mun and Yoon, 1994) (curve:

BS=1 Mun’s and Yoon’s model) deviates

diagrammatically from the other models.

single and double

buffered 64X64 MIN

0.2

0.3

0.4

0.5

0.6

0.7

0.20.40.60.8 1

(p)- probability of arrivals on inputs

Normalized throughput

BS=2 Youn's model

BS=2 Mun's model

BS=2 Our simulation

BS=1 Jenq's model

BS=1 Mun's model

BS=1 Theimer's mode

BS=1 Our simulation

Figure 5: Normalized throughput of a 64x64 MIN vs. the

probability of packets arrivals for b=1 and 2 from various

models.

In the schema herein, it was found that the results

of our simulation for buffer size 2 are in agreement

with the results reported in Mun’s model (Mun and

Yoon, 1994) (curves: BS=1 and BS=2 from Mun’s

model), while the Yoon’s model (Yoon et al., 1990)

(BS=2 from Yoon’s model) deviates significantly.

All the foregoing validates the results from our

simulations.

4.4 Throughput and Latency CPF

Figure 6 shows the Combined Performance Factor

(CPF) for 256x256 Semi-layer MINs with 4 layers

versus the probability of packet arrivals when the

total offered load is of unicast type. Figure 6

illustrated the CPF indicator for fabrics with buffer

sizes equal to 1, 2, 3 and 4, respectively.

For buffer size

, the value of CPF is low

owing to the high packet delay values. It shows

better behaviour but very near the fabrics

with

3,2

b . By looking at Figure 6 it is obvious

that the best performance is achieved when the 4

layer SeLMIN has a buffer size equal to 1. This

happens because the delay of packets is significantly

reduced.

Moreover, the performance of a MIN can be

applied and tailored to the needs that a specific type

of load demands.

0.75

0.8

0.85

0.9

0.95

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(p)~Probability of packet arrivals

CPF(1,1)

BS=1

BS=2

BS=3

BS=4

Figure 6: CPF of an 8 stage semi-layer delta MIN vs.

probability of packets arrivals.

A SIMULATION STUDY FOR OPTIMIZING THE PERFORMANCE OF SEMI-LAYER DELTA NETWORKS

263

Figure 7 illustrates the Combined Performance

Factor (CPF) in 8-stage and 4 layer MINs for cases

of applications traffic in which it is necessary to

have extra low prices of packet delay. Therefore, the

calculation of a general CPF indicator considers the

packets delay factor with a weight of 2.

Figure 7 shows that the 4 layer Delta network

with buffer size equal to 1 provides the best

performance. On the other hand, the general

performance indicator (CPF) deteriorates as buffer

sizes increases.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(p)~Probability of packet arrivals

CPF(1,2)

BS=1

BS=2

BS=3

BS=4

Figure 7: CPF of an 8 stage semi-layer delta MIN vs.

probability of packets arrivals.

Figures 6 and 7 reveal that the single buffered

SeLMINs are more suitable devices for applications

which demand low values of packets latency and

jitter when considering jitter as a variation of

packets latency. Hence, e.g. applications like

streaming media of voice tracking devices present

better attributes when they are constructed by single

buffers. Contrary to this, cases which require high

throughput rates and are indifferent to the

information’s time transmission, are rather rare.

Finally, the main finding of this study remains that

the single buffered SeLMINs constructions present

optimum performance behavior in terms of

throughput and latency, compared to the

corresponding SeLMINs with higher values of

buffer size. This performance behavior of SeLMINs

is strengthened when it comes to service applications

that require small values of latency or jitter.

Also, the single buffered SeLMINs present as

better performance as many number of layers they

have for a given network size

. Also, they have an

earlier point in starting the layer replication and thus

eliminating the backpressure phenomenon.

This indication remains interesting as it is known

that the SiLMINs give their optimum performance -

according to the existing literature - when the buffer

size is equal to 2.

In addition, this SeLMINs’ finding leads to the

following observation: In their construction it is not

necessary to use large values of buffer size which

would ultimately increase the cost of their

manufacturing.

5 CONCLUSIONS

In this paper we studied Delta networks of

SeLMINs, which is a possible performance

improving strategy for Delta MINs. We present also,

an evaluation and comparison methodology of

MINs. This approach was applied on Delta type

SeLMINs and Delta type SiLMINs.

It is obvious that the delta type SeLMINs seem

to be more powerful but this is due to a higher

complexity, relatively speaking, than delta type

SiLMINs. However, in the literature there is a lack

of studies relevant to multi or semi layer MINs.

It is noteworthy that the predictions of the

simulations are validated in marginal cases by

existing related works in the literature.

The findings of this study can be utilized by

MINs designers to optimally configure their

networks.

The methodology presented herein is to be used

in future work in order to estimate the improvement

in performance of Delta networks when servicing

unicast and multicast traffic. Future work will also

focus on studying other load patterns where there is

hotspot and burst type of traffic. Additional work

will also examine the MIN’s performance under

different selection algorithms.

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