INVESTIGATING MULTILAYER OMEGA-TYPE NETWORKS OPERATING WITH THE CUT-THROUGH TECHNIQUE UNDER UNIFORM OR HOTSPOT TRAFFIC CONDITIONS

International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.5, September 2021
DOI: 10.5121/ijcnc.2021.13506 89
INVESTIGATING MULTILAYER OMEGA-TYPE
NETWORKS OPERATING WITH THE CUT-THROUGH
TECHNIQUE UNDER UNIFORM OR
HOTSPOT TRAFFIC CONDITIONS
Eleftherios Stergiou1
, John Garofalakis2
,
Dimitrios Liarokapis1
and Spiridoula Margariti1
1
Department of Informatics & Telecommunications, University of Ioannina, Arta, Greece
2
Department of Communication Engineering and Informatics, University of Patras and
Computer Technology Institute and Press-
“DIOPHANTUS” (CTI) Patras, Patra, 26504, Greece
ABSTRACT
The continuous increase in the complexity of data networks has motivated the development of more
effective Multistage Interconnection Networks (MINs) as important factors in providing higher data
transfer rates in various switching divisions. In this paper, semi-layer omega-class networks operating
with a cut-through forwarding technique are chosen as test-bed subjects for detailed evaluation, and this
network architecture is modelled, inspected, and simulated. The results are examined for relevant single-
layer omega networks operating with cut-through or ‘store and forward’ forwarding techniques. Two
series of experiments are carried out: one concerns the case of uniform traffic, while the other is related to
hotspot traffic. The results quantify the way in which this network outperforms the corresponding single-
layer network architectures for the same network size and buffer size. Furthermore, the effects of the
dimensions of the switch elements and their corresponding reliability on the overall interconnection system
are investigated, and the complexity and the relevant cost are examined. The data yielded by this
investigation can be valuable to MIN engineers and can allow them to achieve more productive networks
with lower overall implementation costs.
KEYWORDS
Omega network, Performance evaluation, Cut-through, Semi-layer multistage networks, Uniform traffic,
Hotspot traffic, Banyan networks.
1. INTRODUCTION
Multistage interconnection networks (MINs) are devices used in datacentres or supercomputers to
ensure the satisfactory quality of networking flows [1-4] and can be implemented at a relatively
low cost. MIN fabrics consist of parallel stages formed of small-scale Switching Elements (SEs)
in a parallel arrangement. Each stage is preceded (or followed) by linked states called
permutations. Banyan MINs are devices characterised by the banyan property, meaning that from
an arbitrary input to a specified output, there exists a unique path. Omega-type MINs are a
specific form of banyan MINs and are self-routing equipments with the banyan characteristics
[5]. The ‘perfect shuffle’ mechanism (operating mechanism) is used in omega-type networks.
The term self-routing (forwarding algorithm) means that when an arbitrary SE receives a packet
at one of its inputs, it can send it to the designated output based on the destination address of the
packet. This forwarding method functions by shifting the bits of the destination address to the

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left. Figure 1 shows a common view of an omega-type network, and this is described in
more detail below. The form of the links (permutations) between stages stays the same for all
omega-type network stages (see Fig. 1). This is another key feature of these networks.
Multi-layer type of MINs (MLMINs) are recently developed fabrics that can ameliorate the
performance of conventional MINs. Semi-layer MINs (SeLMINs) are a specific type of MLMINs
which have been fully investigated [6-9] (Figs. 3 and 4). These researchers explored the
behaviour of SeLMINs operating via the common ‘Store and Forward’ (SaF) mechanism. Figure
3 shows how a level branches to form two levels. It can be seen that in the stage at which
branching takes place, 2x4 SEs are used. Forwarding mechanisms: In addition to the acclaimed
SaF forwarding method, more advanced techniques may also be used to transmit data to the
target, like as the wormhole and cut-through (C-T) techniques. A particular combination of a
network topology and forwarding method creates an individual environment that requires specific
research.
Uniform traffic: Most network surveys have been carried out based on the assumption that the
load follows a uniform distribution. This means that the probability of a packet appearing at each
input of the fabric is the same. Applying a balanced load helps us to understand the behaviour of
a device. Although the use of uniform traffic remains a very useful study tool, mainly due to its
symmetry, real-world traffic distributions are highly non-uniform.
Figure 1: A representative topology of an omega-type semi-layer multistage network
Hotspot traffic: Most data packets originating from contemporary applications and arriving at
MINs have a non-uniform pattern of traffic. In the MIN architecture studied here, hotspot-type
traffic was examined as a case of particular importance and was obtained as a scenario for
investigation. In general, hotspot traffic is a pattern of data packets in which the destination of a
certain proportion of the packets is a single outlet, called a hotspot. This type of traffic is
common, for instance when a server has been set up and multiple clients repeatedly access it to
obtain data and services, or where miscellaneous network nodes are connected through trunk
ports.
Fig. 2 shows the traffic pattern in an omega-type multistage system with only one hotspot point,
called output 2: this output is named the hotspot as it collects a higher share of the total MIN
traffic than the rest of the outputs. More widely, if we signify as the

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the probability that a packet coming at input port has output port as its destination, then
for all and for all : . Hence, any input port are targeted to
a single hotspot point that accepts a higher share of the load.
.
Figure 2: Α snapshot of hotspot traffic in an omega-type MIN
In hotspot-type traffic, the SEs of the MIN can be subdivided into two parts, Hot-Spot Traffic
(HS-T) and No-T, where HS-T represents SEs that accept and serve hotspot traffic, and No-T
(normal traffic) indicates those that accept only normal traffic and are free of hotspot load.
Related work
Although the C-T packet routing method has recently been investigated [10-11], but those studies
mainly concerned torus interconnection networks, which are different from omega-type
networks. In [12] have presented a brief study of omega networks working via the C-T
technique. In [13] researched hotspot traffic using an OMNeT++ simulator, but this traffic has
been implemented to torus-type networks, while in [14] hotspot traffic was investigated using the
SaF forwarding mechanism in conventional single-layer MINs (SiMINs). In [15] also
investigated a scenario involving delta-formed networks working via the C-T technique when the
arriving load was hotspot-type traffic, and in [16] carried out a recent study investigating the
issue of the reliability of MINs. Even though all the above studies are related to data forwarding
techniques, they were applied only to classical MIN architectures. Due to the wide range of
different types of interconnection fabric, many issues remain open. In this paper, a specific
scenario is selected for thorough investigation.
Selecting a testbed fabric for investigation: With the aim of investigating a high-performance
fabric that can meet the current demands on contemporary nettings, semi-layer omega-type MINs
re been chosen as a basic testbed, and the C-T forwarding method was applied. We then
investigated the behavior of current fabrics under conditions of uniform or hotspot traffic.
Contributions of this work: This investigation extends existing research with regard to the
following points:
For uniform-type traffic: The throughput is estimated for fabrics with 2x2 SEs as basic units. This
performance indicator is quantified demonstrating that the C-T technique is superior in
performance and in handling HS traffic compared to the SaF one. The same indications were

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observed for fabrics using 4x4 SEs as basic units. The latency of a packet is estimated for the
above MIN architectures.
For hotspot-type traffic: The throughput metric is estimated for single- and semi-layer omega
formed networks with different buffer sizes.
Overview of the Results: A comparison of results is presented for networks of the same size
operating with the SaF and C-T forwarding methods. This comparison gives a quantitative
estimate of the extent to which the C-T forwarding technique outperforms SaF. The advantage of
this approach is revealed in terms of the reliability factor for the fabrics under study. All of the
results from the experiments indicate that the current fabric is a very productive device that can
provide reliability and high performance to network nodes.
The rest of this paper is structured as follows: Section 2 gives details of the omega-formed and
semi-layer MINs, and the C-T forwarding mechanism. Section 3 presents definitions for
fundamental performance metrics. Section 4 explains the simulators used and the results of the
experiments are portrayed. In the end, Section 5 briefly summarises the work in this paper, with
conclusions and directions for future work.
2. PROPERTIES OF OMEGA-TYPE NETWORKS
2.1. Omega-type MINs
An ordinary multistage ( ) MIN is constructed from maximum parallel stages of
( ) SEs, where represents the size of the SE and each stage consists of SEs. Thus,
the overall number of SEs in a SiMIN is calculated equal to . Hence, there exist
interconnections between all the stages, unlike in a crossbar network, that
has links. These are self-routing networks and are distinguished by the fact that there
exists only a unique route from each inlet to each outlet (the banyan attribute), as illustrated in [1-
4]. The ‘perfect shuffle’ routing method is used by omega networks, in which routing is done by
shifting the bits of the destination tag to the left [2]. A version of this method (algorithm) is also
utilized by other omega-formed fabrics. Typically, in generalized cube-type interconnection
networks, the used routing- tag is produced based on source and destination labels. An additional
discreet feature of omega-type networks is the symmetry of the links that exists between the
stages.
Figure 3: Creating a simple double-layer structure using 2x4 switch elements
in a stage before branching into layers.

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A typical configuration for an Omega-type network is illustrated in Fig. 1 and explained
below. The entire multistage fabric works synchronously, which means that the time-cycle is
based on a global clock-tick. The time-cycle of the fabric consists of two distinct phases (time
periods): during the first phase, the queues forward the packets (service), while in the second
phase, new packets are accepted by queues.
2.2. C-T forwarding in omega MINs
The omega-type MIN studied here is assumed to operate in accordance with the following
conditions:
1. At any input, the packets' arrival is considered to follows Poisson distribution. The
probability of an arrival at each input of a MIN within an arbitrary clock-cycle is independent
of the probabilities for the other inputs.
2. All packets arriving at the MIN have the same size.
3. The service-time of the output queues in any switch is constant and is equal to the network-
time. Each network cycle is long enough that the server of the SE can designate routing ways
and transmit a packet via the stages to output using the C-T technique.
4. Each switch input can receive at most one packet in any time cycle.
5. The whole fabric operates in a synchronous manner, meaning that in each network-time, the
access of the next outbound link is checked. If it is accessible, a packet starts moving; it goes
through as many stages as is feasible (C-T forwarding) and ends up in the most remote buffer
that is technically reachable.
6. In the preamble to any time-cycle of the MIN, all SEs transmit backpressure signals to their
corresponding upstream SEs to advertise the status of their buffer.
7. A C-T mechanism’s operation (packet movement through buffers) is performed when (a) this
is demanded by routing; (b) the intermediate join-links are not busy, and (c) adequate buffer
space is accessible.
8. The First Input, First Output (FIFO) service method is applied by all the buffers in the
system.
9. In any time-cycle, only one packet at most is permitted to pass through an output-link of a
SE.
10. The arrival of packets is considered to take place at the end of each time cycle. In other
words, the systems process the first elements in the queue before accepting new packets.
11. If the buffers of the first stage are full, but new packets are arriving at these buffers, then the
new packets are lost (buffer overflow).
12. Any packet at any stage will be blocked if the corresponding buffer at the next stage do not
have free space.
13. Conflicts between packets are resolved by selecting randomly the packet to proceed, which
can also be considered a fair policy.
14. The output links of the MIN (the links of the last stage) are considered to be non-blocking
links. This is because we only wish to study the performance of this device.
15. Here, is a random variable that depicts the number of packets coming at the end of a
network-cycle at any buffer of a SE at the first stage of the MIN, and we use the
formula provided by Stergiou in [6]. In this approach, the arrival of packets follows a
binomial distribution
(1)

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where represents the fixed-probability of a packet being generated by a processor in each time-
cycle. represents the probability of ( ) packets being accepted by a first-stage buffer with an
average of ( ) inputs at any time step. In general, the fabrics under study have SEs with
inputs, and less often with . In addition to the topologies of MINs, other characteristics
make them important as fabrics. For example, the switching techniques and routing mechanisms
that they employ make their behaviour extremely unusual. The fabrics investigated here apply a
C-T switching-method and a ‘perfect shuffle’ algorithm of routing.
2.3. Semi-layer MINs
SeLMINs belong in a particular category of MLMINs. Their most important characteristic is that
they are composed by two distinct segments, where the first segment is a single-layer MIN and
the second is a fully developed multi-layer module (Fig. 4(ii)). The first segment of the MIN is
made of one exclusive layer that is working via a backpressure blocking mechanism (and hence
does not allow for packet loss) and which has a low cost of production. The second module is a
full 'fan-out' section, which also utilizes a backpressure mechanism but is free of blocking, thus
facilitating the output of the packets.
In an arbitrary semi-layer type of MIN (Fig. 4ii), let depicts the number of single-layer stages
and let represent the number of stages that have full layer development and may therefore
service multicast traffic without blocking. We can therefore write:
(2)
If the fabric generally consists of SEs then the last-stage of the single-layer segment
(referred to here as the 'replicated' stage) is generally manufactured using for l = 1, 2, …
and the second segment will have a replication degree (or number of layers) equal to .
A similar development could apply for case the fabric consists of 4x4 SEs yielding number
of layers.
Figure 4: Two six-stage omega MINs: (i) a single-layer MIN;
(ii) a semi layer MIN (with two-layer fan-out).

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If the total length of the second segment is a value of , then the total number of layers ( )
in the full fan-out is equal to , where represents the inputs’ number per SE (for
instance, for 2×4 switches, is equal to two). As reported in [7], the outlets of the SEs in the last
stage of a multi-layer MIN are multiplexed. Hence, if either the data sinks or the multiplexer has
insufficient capacity to absorb the packets, blocking may be observed. Nevertheless, in this study,
we assume that the multiplexers at the data sinks have the necessary capacity.
The primary disadvantage of MLMINs is their higher value of cost owing to their increased
complexity. The semi-layer type of MINs were originally used due to their balance between
performance and cost, and in cases there are increased traffic requirements to be met. Owing to
their excellent performance/cost ratio, semi-layer MINs are anticipated to become widely used in
the future in Internet interconnection nodes and parallel systems.
2.4. Semi-layer MINs incorporating an omega-type permutation
When the schema of the links between the stages of a SeLMIN is kept the same with an omega-
type permutation, then the fabric becomes a semi-layer omega network (omega-type SeLMIN).
In this paper, this special type of MIN is selected for a detailed analysis and study of its
performance under various traffic conditions. More specifically, this paper considers SiMINs and
SeLMINs with 10 and five stages, with a double-layered fan-out at the end. In addition, it is
assumed that the SemiLMINs use SEs in the stage where the replication starts and typical
SEs in the rest of the stages. Furthermore, an alternative form of SeLMINs is investigated
that utilises SEs in the stage where the replication starts and SEs in the other stages.
2.5. Cut-through forwarding mechanism
The C-T mechanism is a forwarding method that can be implemented in miscellaneous networks
(including omega-type fabrics), in which the packets (or the frames) may be transferred to their
destination points immediately after the routing tags has been examined, without expecting the
rest of data to be completely accepted [10-11,16]. The C-T forwarding method generally provides
low-latency performance, and operates as follows:
 The same clock is utilized for all networks SEs. .
 All the SE buffers should have the least a whole packet free space, as happens in the SaF
forwarding mechanism.
 When a message has arrived, it takes one-time unit for its header (if it avoids being blocked)
to be placed in the MIN internal input buffer.
 Forthwith after a packet has s arrived in a MIN, rather than expecting for the whole packet to
be completely placed, the header continues moving forward through the next stages (‘cut-
through’); the routing decision has already been taken, and the relevant channel is working
normally (viz without blocking).
 Each package is divided into flits, creating a chain of flits following the head of the packet.
The chain of flits is forwarded in a snake-line fashion.
 Each flit of a packet may store when it arrives at a stage, or it also cuts through to the
subsequent stage if the next link is verified as being free, following the same route of the
packet’s header.
 Forthwith after a packet’s header has been accepted at a stage, if the header can no longer
proceed (because of the busy channels), then the packet must be fully received before it is
retransmitted. In this situation, all the subsequent data flits are attempting to release their
channels which they have been occupy so far. Otherwise, the packet tries to accumulate as
close as possible to the packet’s header.

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 If there are no packets' conflicts along a route, each packet will be successfully
pipelined through a series of sequential stages, in the form of a loose chain
consisting of data flits.
 Throughout any C-T transfer operation, any buffer of the stages along the specific routing
pathway is closed to another parallel requirement for transmission.
 The time cycle of the pipelined packets in the C-T method is defined based on the maximum
value of the switching time period and the inter-stage time delay once the header arrives at
a final target.
 It is supposed that there are no staging delays for the cutting through method until a packet is
blocked by a preceding one.
The C-T forwarding technique is credible for packet error handling in which the recovery time of
incorrectly targeted packets is restricted since the recovery process can be started more quickly
than in the SaF forwarding method.
Proposal: In an equilibrated MIN operating with C-T forwarding, if represents the average
number of hops (or average span) and portrays the probability of waiting at a randomly
selected channel (i.e. with no cut-through), then the average number (the mean) of cut-throughs
in a multistage fabric can be portrayed by the following formula:
(3)
In summary, the C-T forwarding method is an elegant and efficient method but is expensive
compared with the SaF or wormhole methods that also can be applied in MINs. The C-T
forwarding modal is currently utilized in a wide range of switch fabrics, for example in the Cisco
Nexus 5000 series of switches [17], the Cisco Nexus 5548P switch [18], and the Juniper
QFX3500 switch [19].
2.6. Transition graph for the queue state
The state transition of a queue in an arbitrary fabric containing a 2×2 SE constructed with buffer-
size is illustrated in Fig. 5. In each intermediate stage , we consider that the following
two states will take place in one clock-unit:
is the probability of a buffer containing no packets, and
)
(i
u is the probability that a buffer is being held by a packet.
represents the probability that the next queue, positioned in stage ( ), is blocked. The
value of m falls into the values' scale: . This says that the next
( ) consecutive queues are unaffected of blocking facilitating the transition, and
only the queue of stage m is blocked.
In addition, , , and represent the transitions of zero, one, or two
packets, respectively, from the prior stage to stage . The value of is in the range:
. These parameters are adjusted to appropriate values for every stage of the fabric.
The clock cycle in a MIN contains two time periods:
 the first time-period, where flow control information is transferred via the network from
the last stage to the first,

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 the second time-period, where packets flow from their current stages towards all available
subsequent stages, in accordance with the current networking conditions.
Fig. 5 shows the state-transition graph for a single-buffered queue of an SE in the C-T forwarding
method. In this graph, every arrow represents the transition and may be accompanied with
formulas which indicates the policy of each possible transition.
Figure 5: State-transition graph for a single-buffer ( ) model of a fabric utilising the C-T forwarding
method, at an arbitrary stage ( ).
The relationships shown in Fig. 5 allow for a roughly analytical approach to the forwarding
method used in the system. The state of queue contains two sub-states:
 a blocked route and
 a C-T transition.
Likewise, the state of the other queue, , also contains two sub-states: an empty state and a C-T
transition.
Boundary conditions of the system: (i) The necessity for the foregoing MIN’s stage is , since
there is no foregoing stage, and the probability of an arrival is assumed to be equal ; and (ii)
the demand for the last stage is that every packet at an outlet of the last stage is assumed to
leave in each case with no difficulty. Hence, the blocking probability is equal to zero .
3. DEFINING THE RELEVANT METRICS
We will show performance outcomes for semi-layer and single-layer omega-type multistage
networks when these fabrics are handling unicast and hotspot types of traffic. Additional metrics
are also used for evaluation, such as the reliability metric.
3.1. Performance metrics
The main performance metrics used in this study are as follows:

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The average throughput of a single- and semi-layer omega-network ( ): This metric is
calculated as the number of packets per time cycle coming out of the fiber’s outputs. Formally,
can be defined as:
, (4)
where represents the number of packets that come to their destination at the time-interval.
This is an indication of how efficiently the network is utilised. Here, the throughput is measured
using simulations, and this metric is estimated based on the packets' quantity that arrived at their
target over a given number of trials.
Note: The average throughput of a semi-layer omega-type network ( ) can be assumed to be
, where represents the throughput of the first part of a SeLMIN. That happens
because no blocking is seen in the second (multi-layer) segment.
The normalised throughput of single- and semi-layer omega networks ( ): This is
determined as a fraction of the average throughput over the network size N. Typically, is
defined by the formula: .
The average packet latency of a single-layer omega network
): This is determined as the number of time-cycles required for all packets of a
permutation to reach their final targets. More formal, can be denoted as:
, (5)
where represents the total number of packets obtained by the sinks within time-intervals and
is the total number of network-time slots required by a randomly packet to reach its final
target. contains both time periods: the total number of network-time cycles for a packet
waiting at any stage, and the total number of network-time cycles required for the packet to
involved in real transmission until it reaches at its target. The packet’s latency in a MIN is
associated to the overall number of hops (time-cycles) required to distribute a given number of
packets to their final targets via links of fabric.
Note: The average packet latency of a semi-layer omega-type MIN is expressed as . If is
the packet latency of the first segment, then the packet latency of a SeLMIN can be
expressed as . This arises because the packets in the second segment of fabric do not
suffer from conflicts.
The normalised packet latency of a single- and semi-layer omega-type network ( ): In
general, this is the ratio of the average packet’s latency to the lowest packet’s latency,
which is equal to time-cycles. Formally, can be expressed as .
3.2. Additional evaluation metrics
Reliability: The reliability of a MIN may be determined as its capacity to handle all possible
unanticipated situations. The reliability of a random path (an arbitrary input–output pathway) is
dependent on all the intermediary SEs, since the failure of any SE involves the non-success
circumstance (e.g., disaster) of the current routing pathway. The concept of terminal reliability is

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utilized here to illustrate the above metric and can be defined as the probability of existing at least
one failure-free pathway between a given inlet – outlet pair. The term point-to-point reliability
may also be used for the same metric.
Terminal reliabilities of SE structures in series and parallel
 Estimation of terminal reliability for SEs in series: For n SEs SEs in sequence, each of them
with reliability (where the specification of is given by industry), then the overall
reliability or terminal reliability of this structural system may calculate using the formula:
. (6)
 Estimation of terminal reliability for SEs in parallel: Supposing we have n SEs in a parallel
layout, then the total reliability of this structural system may calculate with formula:
. (7)
 Estimation of terminal reliability for a semi-layer MIN: Regarding the case of a SeLMIN, the
overall terminal reliability for an arbitrary pathway may estimate with the formula:
. (8)
Without losing the generality, and taking the reliability of any SE to be equal to , Equation (6)
can be rewritten as:
(9)
where S is the span of a single-layer module and is the overall number of fan-out layer’s
multiplication. In the present case, the estimations assumed at the reliability of any SE was
equal to . For a SE, the reliability was assumed to be . The worst case of a MIN that has
the banyan property and that is implemented using SEs arises more rarely, due to the high
cost involved.
4. SIMULATIONS AND RESULTS
4.1. Simulations
The performance of multi-layer omega-type networks is evaluated in this research utilizing
simulations. The focus of the discussion is on multi-layer omega-type fabrics composed
of and SEs accompanied with and SEs, which are implemented
in stages with the multiplication of layers. A specific simulator was created for SeLMINs that
could handle several types of switches and various load conditions at the data packet scale. This
simulator tool was implemented in the C++ language and could run various configurations of
networks.
When building the simulator, each SE was represented by two or additional un-shared buffered
queues. Each buffer operated on a First Come, First Served (FCFS) basis. The C-T mechanism
was utilised to transmit all packets; that is to say, the packets had the potential to be proceed
ahead by one or additional stages within a given time cycle. Cases of conflict between packets
were randomly resolved with the same chance. Within the simulator, the size of the buffer, the

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number of inlets and outlets, the probability of packets arriving, the total number of stages, and
the total number of layers were originally determined as input parameters. Ultimately, the
simulations were executed at the packet level, based on the assumptions that all packets had a
fixed size and were transmitted within time spans of equal length. Performance metrics such as
throughput and packet latency were assessed at the output of the model. To ensure that the
simulator was running in a stable equilibrium state, the first iterations (clock cycles) of each
simulation were disregarded.
We have not observed any other limitations or deficiencies regarding the simulation software for
this study. However, we need to point out that the simulation software can be used only for
Omega-type MINs, and it needs to be modified if there is a need to simulate a different
architecture.
4.2. Results
a) Uniform traffic
Fig. 6 shows the throughput of the normalized 10-stage SeLMIN and the corresponding
equivalent SiMIN correlated with the load on the MIN inputs. All the test-bed MINs examined
here had a buffer size of one, and all the SEs were 2x2 SEs, except those that are in the branching
stage. The last stage of the SeLMIN architecture contained a simple two-layer fan-out.
It can be seen from Fig. 6 that the C-T forwarding mechanism provides higher throughput values
which are shown by the first two bars in yellow and orange colour respectively compared to the
corresponding omega-networks that use the SaF mechanism depicted by the last two bars per
offered load value. The second and fourth bars show the results from SiMINs using the C-T and
SaF forwarding mechanisms, correspondingly. It can be seen in Fig.6 at the rightmost bundle of
bars that for an offered load of 100%, the SeLMIN with the C-T technique provides approx. 20%
(0.8-0.6≈0.2) extra throughput compared to the SeLMIN of equivalent size using SaF.
Figure 6: Normalised throughput correlated with load on a 10-stage omega-network operating via the C-T
or SaF forwarding methods, correspondingly.
A similar improvement is observed in comparing the SiMIN and SeLMIN fabrics. Furthermore,
when the arrived load is lower than ~40%, the type of forwarding technique and architecture do
not matter. From the same graph, the normalized throughput’s values divergence owed to

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architecture, and for the C-T method, this starts when the arrived load is approximately ~40% as
shown by the first two bars in the corresponding bundle. On the other side the divergence of the
throughput values owed to architecture for the SaF case starts a little bit earlier at an offered load
of 40% as can be observed in the third and fourth bars.
Fig. 7 shows the normalised packet latency values for the 10-stage (1024x1024) single- and semi-
layer omega-type networks when C-T and classic SaF data forward strategies are utilised. We
take as an example the case in which the offered load is equal to 10%. When the SaF mechanism
is used by the omega network, the packet normalized latency is close to one which actually
means a latency of 10 time-cycles, as this is the total number of stages of the current device. For
the C-T mechanism, the corresponding normalized latency is 0.1, meaning that it takes about one
time-cycle for a packet to arrive at the output of the device, assuming that the MIN is free of
packets and no collisions are presented. The figure clearly demonstrates the superiority of the C-
T forwarding technique over the SaF tactic, in terms of packet latency.
Figure 7: Normalised latency for various arrived load on a 1024x1024 single- and semi-layer omega-
network operating via the C-T or SaF forwarding methods, correspondingly.
Subsequently, except for the 10-stage (1024x1024) omega-networks that included 2×2 and 2×4
SEs, further simulation trials were also conducted for 5-stage (32x32) single- and semi-layer
omega-type MIN, which were constructed of and SEs. Several MΙΝs with low-
latency data transfer have been proposed, such as the scheme in [20]. However, these proposals
have certain weaknesses in terms of routing. In addition, most of the contemporary low-latency
MIN architectures are more appropriate or are tailored to multiprocessor reconfigurable devices.
Fig. 8 shows the normalized throughput from a five-stage SeLMIN and from the analogous
SiMIN in terms of fabric size, versus the arrived load at the inputs. These MINs are also
considered to be consisting by one-sized buffers. The second and fourth bars which are shown in
blue and yellow respectively, indicate the outcome of the SiMINs, while the first and third bars
depicted by grey and orange respectively, represent the results from the SeLMINs with the two-
layered fan-out at the end of the fabric.
Fig. 8 shows that a SeLMIN with the C-T mechanism produces higher values of throughput,
although this gain is less significant. This conclusion is also supported by the results in Fig. 6.

102
Figure 8: Normalised throughput versus load for five-stage single- and semi-layer omega-networks
operating via the SaF or C-T forwarding methods. Each SE used here has four inputs.
A further basic conclusion is that the use of larger numbers of SEs does not provide significant
performance improvements, while at the same time, the cost is greatly increased.
Validation of simulator: The quality of the simulator was checked for certain boundary
conditions. More specifically, some of the simulations were validated in particular instances of
single-layer MINs (without a fan-out module). The data from these experiments are illustrated in
Figs. 6 and 7, by the second and fourth bars. These data exactly coincide with the results reported
in Figs. 5 and 6 in [21], which are related to SiΜΙΝs with the C-T forwarding method.
b) Hotspot traffic
In addition to experiments with uniform traffic at the inputs, another series of experiments was
carried out with hotspot traffic. This type of traffic is the most common in everyday life since
web servers and other client-server-type applications create this type of load. It is important to
produce some traffic statistics to understand the upper limits created by the aggregation of flow at
hotspots due to the corresponding traffic requirements. As in the previous series of experiments
which was concerning uniform traffic, the performance evaluation was carried out when the
operation of the MIN was in a stable condition.
Fig. 9 shows the normalized throughput for a single-layer omega-type network with size equal to
six, meaning that the MIN block has 64 inputs and 64 outputs ‘working’ with the C-T or SaF
techniques with different buffer sizes, for arrived traffic load which has a hotspot ratio (h) of
10%. The three first bars, depicted as grey, yellow, and blue respectively, show devices
‘working’ with the C-T method, and it can be infered that a relatively minor amelioration in the
throughput metric is achieved when the buffer size increases. For example, if a maximum offered
load with a hotspot traffic ratio equal to ~10%, the normalised throughput is 39% if the system
has a buffer size of one. For the same traffic conditions but with a buffer size of 10, the
normalised throughput is 41.6% indicating a gain of just 2.6%.
This gain is insignificant in relation to the cost of the hardware required. In other words,
increasing the buffer size does not considerably meliorate the performance of the devices, and
hence it is not recommended.

103
Figure 9: Normalized throughput of single-layer omega-MINs ‘working’ with the C-T or SaF
method versus arrived load, which has a 10% hotspot ratio and for buffer-sizes of 1, 2, and 10.
However, the usage of the C-T packet’s forwarding technique markedly ameliorates the rate of
outcoming traffic by approximately 26%, compared to the SaF method for an omega-type MIN
with a buffer size of one.
Fig. 10 shows the normalised throughput for single- layer and semi-layer omega-networks with a
network size of six, ‘working’ via the C-T and SaF methods and with different buffer-sizes,
versus the various values of the offered packet traffic which has a hotspot ratio (h) of ~10%. The
SeLMINs investigated in the current study have a double-layer fan-out at the end, i.e., .
Moreover, Fig.10 shows that increasing the buffer size does not significantly ameliorate the
throughput, while the analogous hardware cost increases to a high level. In addition, it is again
shown that the C-T forwarding operation significantly enhances the MIN’s throughput compared
to the SaF method. In addition, Fig. 10 shows that the SeLMINs are more productive than the
analogous SiMINs in terms of the network diameter. More specifically, for an omega-type semi-
layer network (SeLMIN) with a network scope of six stages, a buffer size of one and ‘working’
via the C-T method, for a full offered load, the normalized throughput is 39%.
Table 1: Values of normalized throughput for semi-layer and
single-layer MINs ‘working’ via the SaF and C-T methods
Offered
load
SemiLMIN,
C-T, b=1
SemiLMIN,
C-T, b=2
SemiLMIN,
C-T, b=10
SiMIN, SaF,
b=1
0 0 0 0 0
0.1 0.1 0.1 0.1 0.09
0.2 0.16 0.16 0.16 0.12
0.3 0.21 0.21 0.21 0.125
0.4 0.25 0.25 0.25 0.13
0.5 0.28 0.28 0.285 0.13
0.6 0.31 0.312 0.315 0.13
0.7 0.335 0.34 0.345 0.13
0.8 0.36 0.365 0.37 0.13
0.9 0.38 0.385 0.395 0.13
1 0.39 0.4 0.416 0.13

104
For the corresponding single-layer MIN operating via the SaF forwarding method, the
normalised throughput is 13%. The first bar in orange represents the results for a SiMIN with the
SaF method. The second and the third bars in grey and yellow respectively, represent results for
SiLMINs that use the C-T forwarding method and they differ in buffer size. The fourth and fifth
bars in light and dark blue respectively, depict results of SemiLMIN that operates via C-T
mechanism but differ in buffer size. Table 1 shows the values of the normalized throughput for
SemiLMINs and SiMINs using the C-T and SaF mechanisms, respectively.
The normalized throughput for a SiMIN with b = 10 and full offered load is only 1.6% better than
the corresponding architecture with b = 2. This is confirmed by looking at the last row in Table 1,
where the corresponding difference is: 0.416−0.4 = 0.016. It can be seen from the graph that the
SeLMIN with buffer size and the C-T method gives a 26% improvement in throughput
(0.39–0.13=0.26) compared to the corresponding SiMIN with .
Figure 10: Normalised throughput versus offered packet traffic having a hotspot portion of 10%, for single-
and semi-layer MINs ‘working’ via the SaF and C-T methods and constructed with 1, 2 or 10 buffer sizes.
Fig. 11 depicts the normalized latency versus arrived load on a 6-stage single- and semi-layer
omega-network operating via the C-T forwarding methods, respectively versus offered load
which have a hotspot portion equal to ~10%. In addition, Fig. 11 depicts how an architecture
which operates via C-T method with small buffer size prevails in terms of packet latency. A 6-
stage SeLMIN with buffer size equal to 1 that operates via C-T mechanism presents approx. 3
times less packet latency compared with an equivalent SeLMIN accommodated with buffer size
equal to 10, when the offered load is full. That is confirmed by estimating the difference in values
of the last two bars in Fig. 11. i.e., 4.2-0.79=3,41.
Also, the same SiLMIN consisting of buffer-size equal to 1 that operates via C-T mechanism
presents approx. 4 times less packet latency compared with an equivalent SiMIN equipped with
buffer size equal to 10 for full offered load. That is confirmed by estimating the difference in
values of the last two bars in Fig. 11, i.e., 5.2-0.79=4,41.

105
Figure 11: Normalised latency versus offered packet load on a 6-stage single- and semi-layer omega-
network operating via the C-T forwarding methods, respectively.
The above performance investigation yields quantitative results for the improvement in the
efficiency of the MINs due to the multi-layer architecture. Furthermore, is shows that C-T
forwarding method provides further increases in the performance of the fabric. A combination of
the two techniques gives the best solution, and it seems that this solution works properly even for
hotspot traffic.
In addition to the performance evaluation, other aspects of this system were investigated, for
example the complexity, the cost, and the reliability factor.
c) Complexity and cost analysis
The key problem in the construction of a MIN is the cost since a performing MIN with a very
high cost might not worth it. The MIN’s cost is associated to the complexity of the system and
the SEs. Here, we follow the approach which is presented in the studies [22, 23] where is carried
out a cost analysis.
The complexity of a MIN depends on the number of SEs involved. An SiMIN consists of
SEs. Furthermore, an SeLMIN with a double -layer fan-out in the
latter stage is made up of SEs in each single-layer segment, and a number
of SEs in the fan-out module.
Table 2: Cost functions for omega-type MINs
Network type Cost function
SiMIN
SemLMIN
(
SemiLIN
(
The complexity of a MIN directly correlates with the number of input-links connected to the
switch together with the sum of cross-points that included in a switch. Hence, a SE has an

106
overall cost weight of four units, ; a SE has a cost weight of ;
and a SE has an overall cost of twelve units .
On the basis of complexity, the functions of cost can be computed, considering the individual
cost units of every SE, mentioned as weighting items. The functions of cost for some typical
omega-type MINs are reported in Table 2.
Fig. 12 shows the units of cost for typical omega multistage networks versus network size (N).
From this diagram, it can be seen that for any MIN with network size (total stages ≤ 7),
the cost remains quite low, regardless of the architecture, and there are insignificant differences
due to the architecture used.
Figure 12: Units of cost versus network size for single- and semi-layer MINs
However, as the network size increases, the cost also increases and can become an important
factor for large network sizes. The greatest deviation is observed for a network of size Ν = 1024.
This deviation is about 4,000 units of cost, representing a burden of 20% in relation to the overall
cost. As a result of the cost examination of MINs, it can be concluded that industrial users should
not be dissuaded by the cost aspect. The cost is significant only for sizable networks, and in these
instances, the differences in the costs of various architectures (for equivalent network sizes) are
relatively small.
d) Reliability of the fabric
The terminal reliability of a semi-layer omega-MIN with double-layer fan-out in the latter stage
( ) versus the simple reliability of a switch-unit is shown in Fig. 13. The simple (or
elementary) reliability of a switch-unit is a construction feature and is given by the manufacturer.
Theoretically, the elementary reliability ranges between 0.9 and 1.

107
Figure 13: Overall terminal-reliability of semi-layer omega-MINs with and different network
dimensions, constructed with two-input and four-input SEs, versus the simple reliability of a typical SEs.
The fabrics considered here rely on two-input ( ) or four-input ( ) SEs, forming
fabrics with various network diameters. Fig. 13 shows that SeLMINs based on two-input SEs
have higher values for the overall terminal reliability (as shown by the first three bars) compared
to corresponding fabrics built with four-input SEs (as shown by the last three bars). The first bar
(shown in yellow) represents the overall reliability of a SeLMIN with 64x64 inputs/outputs and
constructed with 2x2 SEs. Similarly, the next two bars (green and brown) show the overall
reliability of the corresponding 256x256 and 1024x1024 SeLMINs. The remaining bars represent
interconnection systems based on 4x4 SEs. It is therefore demonstrated with numbers that the
bigger the size of the networks, the lower the total terminal-reliability. Ιn summary, it can be
inferred from our experiments that semi-layer omega-MINs operating via the C-T technique and
based on two-input SEs are more effective in terms of their operation than MINs of equivalent
dimensions that utilise an SaF technique or that are constructed with higher numbers of SEs.
Although various reliability studies have been carried out in this area, these studies deal with
specific MIN architectures. For example, a reliability analysis was carried out by Prakash et al.
[24], but their work referred to a shuffle-exchange gamma logical neighbourhood interconnection
network (SEGLNIN). In the same way as these authors, we developed a specific approach to
estimating the reliability of the fabrics investigated here. Because the real production of the fabric
requires significant time and ηιγη cost, this study offers a practical way of calculating the
reliability in advance before the actual fabrication.
e) General Comments
Besides the methods of packet forwarding mentioned earlier, there are new studies appearing in
the literature such as the one by Birmpilis et al [25], regarding applications where parallelization
is critical. According to this study, packets transmitted through such networks can be interrupted
using buffers in order to maximize network usage and minimize the time required for all
messages to reach their destination. Also, in order to improve the performance of the network
nodes there are new proposed architectures in the literature [26].
The results we have seen so far rely on devices that rely on classical semiconductor memories.
Recently, a novel type of memory has been proposed in [27] which can achieve data access
speeds on the scale of 10 Gbps. This new form of memory is very promising in terms of allowing

108
interconnection networks to overcome their existing limitations and operate at ultra-fast data
rates. That innovation will therefore reignite attention in new architectural structures of MINs.
5. CONCLUSION
In this study, semi-layer omega interconnection-type networks that use the C-T forwarding
technique were examined. The outcomes yielded from these experiments were compared with
SiMINs with similar network properties that utilize the classical SaF forwarding technique. Two
series of experiments were carried out: the first involved a uniform traffic profile, while the other
dealt with hotspot traffic. It is apparent that the omega-type SeLMINs are more efficient and
robust, despite their slightly higher complexity, in proportional terms, compared to SiMINs. It
should also be mentioned that the data derived from the simulations were validated in some
marginal cases based on available relevant pre-existing investigations.
The findings of this investigation can be utilized by MIN engineers to build more efficient
interconnection networks and configure them appropriately. MIN designers must consider and
evaluate the quality of service that they desire to achieve. The omega-type MINs investigated
here operate at the ns scale due to their conventional electronic memories.
The current investigation can offer an exemplary approach to study additional traffic patterns
such as bursty-type traffic. In addition, other research scenarios could evaluate the performance
of similar network architectures utilizing modern routing techniques. We plan to investigate
further the application of the cut-through technique in Tandem Multistage Networks.
CONFLICT OF INTEREST
The author declares no conflict of interest.
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