Analytical model for performance evaluation of Multilayer Multistage Interconnection Networks servicing unicast and multicast traffic by partial multicast operation

Performance Evaluation 67 (2010) 959–976

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Performance Evaluation journal homepage: www.elsevier.com/locate/peva

Analytical model for performance evaluation of Multilayer Multistage Interconnection Networks servicing unicast and multicast traffic by partial multicast operation John Garofalakis a,b,∗ , Eleftherios Stergiou c a

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

b

Research Academic Computer Technology Institute, Rio Patras, 26500, Greece

c

Department of Information Technology and Telecommunications ATEI of Epirus, Arta,47100, Greece

article

info

Article history: Received 10 July 2009 Received in revised form 1 April 2010 Accepted 8 June 2010 Available online 30 June 2010 Keywords: Multistage interconnection networks Analytical model Performance analysis Multicast Partial Multilayer MINs Throughput Blocking

abstract Nowadays, since the proportion of multicast traffic has increased compared to that of unicast traffic, the need for Multilayer Multistage Interconnection Layers Networks (MLNINs) has become more intense. In this paper a thorough evaluation of the performance of MLMINs using an analytical model is presented, as such an evaluation has not previously been developed. The multicasting policy that is used by MLMIN queues is the ‘‘partial multicast’’ and all the MLMINs studied use the ‘‘Cell Replication While Routing’’ (CRWR) technique. The performance model was applied under different offered loads to various network size MLMINs supporting various proportions of unicast and multicast traffic. The results have been confirmed in some marginal cases by existing work and the study reveals quantitatively the improvement in the performance metrics of MLMINs compared to the corresponding single-layer MINs. The findings of this paper are important as they could be useful in building optimum networks regarding their performance. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Multistage Interconnection Networks (MINs) provide efficient communication resources between network components in an appealing cost/performance relationship. Also, MINs are proposed to connect a large number of processors in parallel systems, providing satisfactory routing and efficient multiple communication tasks concurrently as main benefits. Multistage Interconnection Networks (MINs) with the Banyan property [1–3] are networks where a unique path from an input to an output exists. In order to handle QoS requirements by MINs when they have to service various types of traffic, one approach is to assign a higher service priority to real-time traffic (e.g. video or voice delivery traffic) and a lower priority to non-real-time traffic (such as file transfer traffic). Such QoS strategies have been proposed and studied in [4–6]. However, a different strategy is necessary when we have to meet QoS requirements with MINs when they support multicast traffic in order to service a lot of data rapidly. This study focuses on transferring unicast and multicast traffic via multilayer MINs in an efficient manner [3]. 1.1. Prior work Some prior surveys and analytical techniques relating to single-layer MINs give us important help at the beginning of our work. In [1,2,7], Bouras et al. analyzed single-layer Banyan networks with finite buffers, providing the solution of the steady

∗

Corresponding author at: Department of Computer Engineering and Informatics, University of Patras, Greece. Tel.: +30 2610 994935. E-mail address: [email protected] (J. Garofalakis).

0166-5316/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2010.06.001

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state distribution of the first stage. They also approximated the solution for the subsequent stages, and presented the exact solution for all stages of MINs with unbuffered switches. In [8], Bouras et al. extended their study and provided a solution regarding the queuing delay in different types of MINs. The classic MINs remain interesting as long as they are improved continuously. In the work of Tse [9], there is an in-depth analysis of router requirements by switch fabrics. The router’s taxonomy, according to this overall survey, proves to be the cause of variation among currently existing switching systems. Some new studies regarding multicast traffic deal with new types of MIN structures. A new effort for constructing better inter-processing communication was proposed by Sharma et al. in [10], who also developed an algorithm for computing the cost according to the MINs complexity. In [11] a new multistage interconnection network was proposed for multicast traffic with its multicast routing algorithm. The surveys of Shabtai et al. [6], Garofalakis and Stergiou [12], and Tutsch and Hommel [13] are typical studies of performance evaluation carried out by applying analytical methods to multistage networks. In particular, Shabtai et al. [6] developed an analytical model for solving performance evaluation of MINs in terms of priority, using a Markovian approach. Also, Tutsch and Hommel [13] present a novel analytical method regarding single priority MINs. In addition to multicast traffic, Tutsch et al. [14], Tutsch and Hommel [15] provide additional views on performance aspects of MINS in special cases, using simulations. In [14], a cut-through switching technique for forward packet switching is studied. In [15], different sizes of SEs are used as the basis for a case study that compares performances. In addition to the above, Garofalakis and Stergiou [16] have developed a performance evaluation model for single-layer MINs, applying an analytical method. Nevertheless, all the above-mentioned papers are related to single-layer MINs. By studying all of the above work, we can draw the conclusion that single-layer MINs tend to become saturated quickly under broadcast and multicast traffic. In order to overcome this problem, the replication of the whole MIN network or certain stages of it, has been proposed, leading to MLMINs. Thus in 2003, Tutsch and Hommel [17] introduced the multilayer MIN. That proved to be a milestone in the evolution of the construction of MINs. These multilayer configurations will enhance performance factors favorably, making them an inevitable component of today’s network development. There are already some switches on the market that accommodate multicast traffic possibilities; for example, Cisco has build the CRC-1 router [18], which is in fact a typical multistage switching device, consisting of three stages and operating as a self-routed device. 1.2. This work The performance prediction of multilayer MINs has not been studied sufficiently by analytical models so far. This paper deals with single- and multilayer MINs which support unicast and multicast traffic acting internally with ‘‘partial multicast’’ operation, and focuses on the performance of multilayer MINs under this traffic pattern. The analytical model describes the discrete time behavior of an arbitrary queue within the system. In our analysis, a utilization formula of an arbitrary MIN queue is extracted, and then factored into the rest of the calculation process. A recursive convergent algorithm is included in the calculation process by this novel analytical method. This new analytical model for performance evaluation is developed assuming uniform traffic and Bernoulli type arrivals at the MIN’s inputs. Moreover, for our experiments, a simulator was developed and used to confirm the analytic results. The following summarizes the points made in this paper:

• A Switch Element (SE) architecture which acts with ‘‘partial multicast’’ transmission mode is introduced. • A performance evaluation of MINs is carried out, based on a novel analytical model. By applying this analytical model, an approximate estimate can be made of the MIN’s performance metrics. This analytical method converges very fast and gives accurate results in a small number of iterations. The analytical method compares favorably to simulations which involve more time-consuming process (as is normally the case with simulation experiments). • The current analytical framework is easily adapted and applied to multilayer MINs. This approach is very useful in studying QoS issues in communications links that simultaneously support unicast and multicast data. Thus, the results could be useful in some industrial areas or to network designers. The findings will help to configure MIN construction in MIN engineering, in order to meet the MIN’s performance and cost requirements better under the anticipated QoS specification and traffic load pattern. • The findings from both the analytical model and simulation experiments were found to be in close agreement. The difference between the two was smaller than 2%, which is evidence of the accuracy of our analytical method. Organization. The paper is organized as follows: Section 2 presents in brief the description and operation of MINs which support multicast traffic and apply ‘‘partial’’ operation. In Section 3, the multilayer MINs are explained. Section 4 presents some operational definitions and lemmas related to internal queues. In Section 5, the analytical method for single-layer MINs is presented. Based on this analysis, approximate analytical formulas were extracted. In Section 6, the applied convergence algorithm used is presented. In Section 7 the major performance metrics are illustrated. Section 8 contains some of the results of our analytical approximation scheme for different network size single and multilayer MINs with

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Fig. 1. An N × N single-layer L-stages MIN constructed by 2 × 2 SEs supporting multicast traffic.

various multicast ratios of traffic. Finally, Section 9 concludes with a short summary and a discussion of future work on the subject. 2. Description of MINs supporting unicast and multicast traffic A typical N × N MIN is constructed by L = logk N stages of k × k Switching Elements (SEs) where k is the degree of the SEs. Let (i) depict an arbitrary number of stages, where (i) can be escalated from 1 to L. Generally, each SE consists of k-input and k-output ports. In the fabric, there are exactly (N /k) SEs at each stage, so the total number of SEs of a MIN is (N /k) ∗ logk N. There are (N ∗ logk N ) interconnections among all stages, as opposed to the crossbar network which requires O(N 2 ) SEs and links (Fig. 1). In the remainder of this section, the case of one-layer MINs that are constituted by switch elements that have FIFO type buffers of size (b) in front of each switch input is presented. Then in Section 3, the case of the corresponding multilayer MINs that are mainly studied here is presented. In this paper, all the single and multilayer MINs studied, include buffers which have the ability to service both types of packets: unicast or multicast. Moreover, the backpressure blocking mechanism operates in this fabric, while the packets move through the MIN. 2.1. Packet routing An internal clock synchronizes the packet routing process that is performed at every stage in a parallel manner. An arbitrarily arriving packet is able to enter the network if the buffer of the first stage has available space to accept it. Any packet that enters the MIN fabric receives a Routing Address (RA). This address is used by the packet in order to reach the output destination of the MIN. In addition, as well as the above RA that is obtained when the packets enters the MIN, a second address (like a ‘mask’) is also received. This second address is called the Multicast Tree Number (MTN). Both types of addresses (RA and MTN) have as many bits as the number of stages in the MIN. Each bit is assigned to each distinct stage. In our study of MINs, it is considered that any queue in the system has the ability to handle unicast or multicast transmission modes. When there is a packet in a stage of the fabric, then the corresponding bit of the packet’s MTN address determines the mode of operation by the queue server. Thus, if the specific bit of the MTN address is equal to ‘0’ then the queue’s server acts in unicast mode and when the specific bit is equal to ‘1’ then the queue’s server operates in multicast mode. This multicasting routing technique is called Cell Replication While Routing (CRWR) [19]. In every time slot, all queue servers check the type of packet that is at the head of the queue. If the packet is a multicast type packet (which is determined by checking the MTN address), the queue server duplicates it (for 2 × 2 SEs) and tries to send the two packets to suitable queues (according to their routing tags) of the successive stage each time. If the packet is of unicast type, then the queue server only attempts to send one packet to the successive stage queue following its routing tag (RA). The RA and MTN addresses can produce all desired multicast output combinations, if they are accompanied by a suitable mechanism to be applied at the last stage, e.g., an N-bit mask which allows the nth output (if the nth bit of the mask is 1), to transmit the packet.

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Fig. 2. Modelling operation of a queue (unicast, full, and partial multicast transmission modes).

2.2. Basic system assumptions Continuing our study, the following points are accepted as prerequisites which must be satisfied by our studied systems: 1. Each switch input at each time slot is able to accept a packet. 2. In all inputs, the same packet load is offered. 3. The traffic feeding the first stage of the MIN switch follows a Bernoulli type distribution. So, the probabilities of packets arriving at the inputs of a MIN within a clock cycle are fixed and independent of each other. 4. All packets that arrive in the MIN have identical fixed sizes. 5. Any packet arriving at the first stage is lost if the relevant buffer of the corresponding SE is full. 6. Uniform traffic is considered. Thus, the destination addresses of the packets are distributed uniformly over all output links of the MIN. 7. It is assumed that the buffer length (b) does not include the server; for example, a single buffered switch is denoted as b = 1. 8. It is assumed that arrivals happen at the end of each cycle. Thus, first the queue is served and then any new packets are received. 9. It is assumed that all the conflicts are randomly resolved. Thus the routing logic at each switch is assumed to be fair. 10. All the queues of the MIN are able to operate in unicast or multicast modes of transmission. 11. Routing is performed in a pipeline manner. That means that the routing process occurs at every stage in parallel, as all the queues are synchronized by the same internal clocking. 12. The output links of the MIN signify that there is no blocking at the last stage. 2.3. Modelling the unicast and the multicast operation of queues Any queue in the network has the ability to operate in either unicast or multicast mode. In Fig. 2, the first line schemas are modelling the operation of a processor in unicast mode of transmission. The second line and the third line schemas illustrate the modelling of the multicast mode of the queue’s operation. In particular, the two last lines of the schemas shown in Fig. 2 depict the queues’ corresponding operations in full and partial multicast modes respectively. Unicast mode operation: when a queue operates in unicast mode, the queue server can only send a packet to the destination queue of the successive stage (Fig. 2, first line of schemas). This transmission is achieved if at least one buffer position is free, that is, the destination queue was not full and blocked at the previous network cycle and at least one buffer position still remains free, after a possible conflict resolution procedure at the current network cycle has taken place. The multicast operation mode of the queue can be categorized into full and partial multicast transmission modes. Full multicast transmission mode: when a queue operates in ‘‘full multicast’’ transmission mode, a queue server is capable of sending two packet copies only when both destination buffers are available. When neither destination buffer is available, the packet is blocked totally, and no packet copy is forwarded to the available destination (Fig. 2, second line of schemas). If one destination buffer has space, while the adjoining destination is blocked, then a partial multicast mode operation takes place, which is explained below. Partial multicast mode transmission: when one of the destination queues (of successive stages) is blocked, while the other destination queue has free space, the queue server sends a copy of multicast packet to the successive queue that has free space and also holds this packets expecting service in a subsequent cycle (Fig. 2, third line schemas). This remaining packet behaves like a unicast type packet in terms of its transmission in a subsequent cycle.

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However, in this paper, when we use the term ‘‘partial-multicast operation’’, which is also called ‘‘partial-multicast transmission’’, it means that the multicast operation which is employed, uses both full and partial operation mechanisms depending on the availability of the next stage’s queues. 3. Multilayer and semi-layer MINs All the single-layer MINs (SiLMINs) can be considered as a special class of multilayer MINs. However, as well as this special group of MINs, a special group of MLMINs can be considered as SeLMINs, defined in Section 3.2. 3.1. Multilayer MINs (MLMINs) Multicast operation multiplies the number of packets since each packet is delivered to many destination outputs. Thus, each multicast packet is multiplied during its route through the MIN. Because of this, replication routing provides that the greater the stage number, the greater the number of packets. Therefore, additional switching power is needed in the later stages compared to the first ones, due to the fact that the number of packets being serviced at those stages is very high. In order to achieve better management of multicast traffic, MLMINs were introduced. The operation and the structure of multilayer MLMINs takes into consideration the above multicast packet density and operates in order to eliminate the possibility of blocking. Tutsch and Hommel [17] have defined three ‘‘description factors’’ to give a concrete description of MLMIN structure. These three factors are: 1. Start replication factor Gs , where Gs ∈ N ∗ . This factor denotes the stage number in which the replication starts. 2. Growth factor GF , where GF ∈ N ∗ . This factor denotes the number of layers in which the MIN is developed at each stage and in each SE. 3. Layer limit factor GL , where GL ∈ N ∗ . This factor denotes the number from which further layer replication stops. According to Tutsch and Hommel [17], when the MIN replicates the number of layers in a stage and the growth is equal to the number of inputs per switch, this structure ensures that no internal blocking occurs in all SEs, even if all SE inputs broadcast their packets to all SE outputs. More concretely, when the number of layers in each stage is multiplied according to the growth factor GF = k, (where k is the number of inputs per switch) the backpressure phenomenon does not appear in the fabric. However, the main weakness of the MLMIN architecture is attributed to the exponential growth in the number of layers as the stages increase. As the number of network inputs increases, the numbers of stages and layers increase as well. If we try to reduce the number of layers, then the hardware complexity is reduced and therefore so is the overall cost of the fabric. To achieve this we have two options: One is to start the replication at a later stage and the other is to stop further layer replication at a given number of layers. The high cost of MLMINs due to their complexity is their major drawback. On the other hand, semi-layer MLMINs were introduced as a good trade-off between cost and performance of the multistage fabric. 3.2. Definition of semi-layer MINs (SeLMINs) The Semi-Layer MINs (SeLMINs) are a special class of MLMINs. They consist of two segments. The front segment (first stages) of the MIN is a single-layer part of the MIN which employs a backpressure blocking mechanism, while the next segment is the multilayer construction of a MIN (a full fan-out) which is free of blocking. In addition, the SeLMINs must satisfy a second specification. According to this, SeLMINs are the fabrics which at the second segment keep the GF factor fixed and equal to k, and the replication starts from a stage that indicates the GS factor, until the last stage of the MIN, where k is the number of inputs per SE. The second specification makes the second segment of the MIN unblocked. In Fig. 3, the first schema illustrates an example of a SeLMIN case in 2D view, whereas the second schema on the same figure shows an MLMIN which does not belong to the SeLMIN class. If the GF factor is kept constant at the last stages, the corresponding SeLMINs’ fabrics avoid the high density of packets that appear at those stages of the MIN. SeLMINs are expected to play a decisive role in the future regarding the overall performance of Internet interconnections, parallel systems, and grid systems. 4. Operational parameters and lemmas of evaluated MINs 4.1. Definitions and lemmas 1. The probability of offered load on MIN’s inputs is depicted by (p): In our experiment we used p = 0.2, 0.4, 0.6, 0.8 and 1.0. This offered load is distinguished in unicast and multicast class of load.

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Fig. 3. Lateral view of an SLMIN type and a non-SeLMIN type of MLMIN with L = 4 stages.

2. Multicast ratio (w) of packets, denotes the probability that a packet arriving to a particular i-stage SE where i = 1, . . . , L, has its ith bit of its multicast mask MTN set to 1, effectively expressing the probability that this SE will do a broadcast transmission by copying this packet to both its output links. In this paper the w factor is considered to be fixed at all SEs. 3. Queue Utilization (u(i) ). As u(i) we define the steady state probability that a particular output server of stage (i) of the k × k switch network is busy. An output server is busy either because it is serving a packet, or because it is blocked. This is the utilization in steady state of an output buffer of stage (i) of the k × k switch network. The queue utilization is distinguished (i) (i) in unicast uu and multicast um type of utilization. (i) The unicast utilization uu depicts the steady state probability that a particular output server of stage (i) is busy by unicast (i) type packets. Similarly, um depicts the busy states of queue by multicast type packets. Thus in each queue of (i) stage we have: u(i) = u(ui) + u(mi) . 

(1) (i)

(i)

4. The service probability of queue (pserv ). As pserv we define the steady state probability that a particular output server of stage (i) of the k × k switch network is serving a packet. In addition we can write that: (i)

(i)

(i)

i) p(serv = pserv(u) + pserv(fm) + pserv(pm)

(i)

(i)

(i)

where pserv(u) , pserv(fm) , pserv(pm) denotes the probability of packets serviced by unicast, full multicast and partial multicast transmission mode respectively.  (i) (i) 5. The blocking probability of queue (pb ). As pb we define the steady state probability that a particular output server of stage (i) of the k × k switch network is blocked. Hence, we can write: i) u(i) = p(serv + p(bi) .

(2) (L)

And obviously for the last stage: pb = 0. (i)



(i)

6. The probability of an empty queue (p0 ). As p0 we define the steady state probability that a particular output buffer of (i)

stage (i) of the k × k switch network is empty. Obviously, p0 = 1 − u(i) . 7. The aggregate probabilities of all states in a finite-buffered queue is: (i)

p0 + u(i) = 1.



(3)

8. The arrival process of packets at the output queues of the network arbitrary stage (i) : x(i) is given by a binomial distribution bin(k, u(i−1) /k) where u(i−1) is the utilization of the previous stage (i − 1). In our case study, the degree of SEs k is assumed to be 2, while the boundary condition for utilization is considered to be u0 = p, where p is the offered load at inputs of MIN.  Lemma 1. Relating blocking probabilities with last stage utilization in a single-layer MIN. In the single-layer MIN with backpressure blocking, for all stages except the last one, the probability of blocking in stage (i), where i = 1, . . . , (L − 1), depends on the utilization of the current stage (i) and the utilization of the last stage, according to the following formula. (i) pb = u(i) −

u(L)

(1 + w)(L−i)

.

(4)

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Proof. The following operational argument is used: Let’s consider a MIN and assume that (NT ) total number of packets during time (T ) have been serviced from the last stage (L), which were not lost upon entering stage 1. Due to the uniform distribution of the offered load, all outputs are considered equal. On the other hand, due to the multicast attitude in an arbitrary stage, there are w new copy packets (where w is the ratio of multicasting packets in this stage as it is defined in 4.1). Based on the above, the existing total packets population can be calculated in all stages. NT If in stage L, NT total packets are serviced, then in stage (L − 1), (1+w) total packets are serviced. Going back the same way, we can calculate the number of serviced packets. So, for all stages i = 1, 2, . . . , L during time T we have as total serviced population the following: NT

(1 + w)

L−1

,

NT

(1 + w)

L−2

,...,

NT

(1 + w)

2

,

NT

(1 + w)1

, NT .

Due to the homogeneity for an arbitrary queue of stage (i), the total number of serviced packets is given by: x(i) =

1 N

·

NT

(1 + w)(L−i)

,

(5)

where N is the total number of the input ports in the stage (i) of the MIN. After that, the service probabilities for all stages during the time period (T ) are: 1) p(serv =





NT

·

N ·T

1

(1 + w)

L) , . . . , p(serv = L−1





NT N ·T

.

(6)

Moreover, it is assumed that there was no blocking in the last stage, thus: x(L) NT L) u(L) = p(serv = . = N ·T T

(7)

By using formulas (2), (5), (6) and (7) we obtained (4).



Lemma 2. Evaluation of unicast, full multicast and partial multicast type of utilization in an arbitrary queue of MIN. In an arbitrary queue of a MIN, the full, partial and unicast types of utilization can be expressed as following:

w

(i)

 · u(i)    1 + pb + w · pb    (i+1)   2 · w · p b (i) (i) · u upm = . (i+1) (i+1) 1 + pb + w · pb      (1 − w) · (1 + p(bi+1) ) (i)    u(ui) = · u (i+1) (i+1) 1 + pb + w · pb ufm =

(i+1)

(i+1)

(8)

Proof. In an arbitrary system queue, for each type of service the following equations can be written: (i) (i+1) pserv(u) = u(ui) · (1 − pb ) (i)

(i)

(i+1) 2

pserv(fm) = ufm · (1 − pb (i)

(9)

(i)

)

(i+1)

pserv(pm) = 2 · ufm · (1 − pb

(10) (i) ) · p(bi+1) + upm · (1 − p(bi+1) )

(11)

where: (i) (i) (i) (i) (i) (i) (i+1) uu , ufm and upm are explained in Section 4.1(3), pserv(u) , pserv(fm) and pserv(pm) are explained in Section 4.1(4), pb is the (i+1)

probability of a queue at (i + 1)th stage to be blocked. 1 − pb is the probability of a queue at (i + 1)th stage to have space for accepting packets. (1 − p(bi+1) )2 is the joint probability that both successive queues at (i + 1)th stage have space for accepting packets. (i+1)

(i+1)

2 · (1 − pb ) · pb is the joint probability that one of the successive queues at (i + 1)th stage has to be blocked, while the other queue – the adjoin queue – has space for accepting packets. The partial utilization can be written: (i)

(i+1)

i) u(pm = 2 · ufm · (1 − pb

(i) ) · p(bi+1) + upm · p(bi+1)

(12)

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(12) implies that (i)

(i+1)

i) u(pm = 2 · ufm · pb

(13)

and subsequently from (12) and (13) it is implied that (i)

(i)

(i+1)

pserv(pm) = 4 · ufm · (1 − pb

) · p(bi+1) .

(14)

Consequently, the aggregate multicast-service probability of a packet can be expressed by: (i)

(i)

(i)

pserv(m) = pserv(fm) +

pserv(pm) 2

= u(fmi) · (1 − p(bi+1) )2 + 2 · u(fmi) · (1 − p(bi+1) ) · p(bi+1)

i) (i) (i+1) ⇔ p(serv ) · (1 + p(bi+1) ). (m) = ufm · (1 − pb

(15)

Furthermore, according to our test bed multicasting scenario and (9), (15) we infer that (i)

pserv(u) (i)

=

pserv(m)

⇔ u(ui) =

i−1) (1 − w) · p(serv

(i−1)

w · pserv

(i)

⇔

uu (i)

(i+1)

ufm · (1 + pb

)

=

(1 − w) w

(1 − w) · (1 + p(bi+1) ) · u(fmi) . w

(16) (i)

(i)

(i)

Consequently, from (13), (16) and formula u(i) = uu + ufm + upm the expressions (8) of Lemma 2 are inferred.



5. Analytical approximation for the single buffered, single-layer multicast service MIN consisting of (2 × 2) SEs In this paper, a solution for MINs with 2 × 2 SEs and buffer size 1 is given, which easily can be extended to other cases with buffer size greater than 1. Extension of our solution to MINs with SEs of more than two inputs (e.g. 3 × 3, 4 × 4) will require some additional analytic effort. Any queue in the fabric operates either as Be/G/1/1 or Be/G/2/1, depending on the type of packet that it has to service. 5.1. Queue’s state transition diagram Here the state transition diagrams of a MIN’s single buffered queue when it services unicast and multicast types of packets by ‘partial mode’, is presented (Fig. 4). Consider the following three states as basic buffer states: (i) (p0 ): Buffer is empty of packets. (i)

(uu ): Buffer has a packet (normal or blocked) that expects a unicast mode of transmission. (i) (ufm ): Buffer holds a packet (normal or blocked) that expects a full multicast mode of transmission, if it is available. However, if the full multicast type of service is not achieved for any reason, a partial multicast type of transmission can be tried, if it is available. (i) (upm ): Buffer holds a packet (normal or blocked) that expects a partial type of service. The state transition diagram is formed as follows: (i) (i) (i) The basic queue states uu , ufm and upm contain their relevant blocked sub-cases and incorporate the normal and blocked sub-states. The blocked sub-states can also be illustrated in detail. Nevertheless, for our analysis, the state diagram that is pictured in Fig. 4 is sufficient. (i) Also in Fig. 4, the transition probabilities starting from state p0 , are: (i)

1. x2,1u,1m is the probability of a unicast and a multicast packet entering the queue, where the corresponding bit of Multicast Tree Number MTN was set to 0 for the packet that will follow a unicast transmission, while that bit was set 1 for the multicast packet. Also, (i)

(i)

(i)

(i)

2. x2,2u and x2,2m are the probabilities of two unicast and multicast packets entering the queue respectively. 3. x2,1u and x2,1m are the probabilities of only one packet, unicast or multicast, entering the queue respectively. (i) 4. x2,0 is the probability of no incoming packet at all in this specific queue. 5.2. Basic analysis for queue utilization The following is an analysis of a single buffered MIN (b = 1) which consists of 2 × 2 SEs (k = 2).

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Fig. 4. State transition diagram of the single buffer MIN’s model for queues servicing unicast and multicast packets by partial multicast operation.

For the steady state probabilities of stage (i) queues and based on Lemma 2 in [2], the following set of equations are written: (i)

(i)

(i)

i) ) · x2(i,)0 + (uu(i) + u(pm ) · (1 − p(bi)+1 ) · x(2i,)0

(i+1) 2

(i)

p0 = p0 · x2,0 + ufm · (1 − pb (i)

p0 + u(i) = 1,

) (17)

(i+1) 2

) is the joint probability that both successive queues at (i + 1)th stage are blocked. the probability of not having packet arrivals from the previous (i + 1)th stage (defined in Section 4.1(8)). pb : is the probability of a queue at (i + 1)th stage being blocked. (i+1) : is the probability of a queue at (i + 1)th stage being empty. 1 − pb

where (1 − pb (i) x2,0 : is (i+1)

Solving the above set of Eqs. (17) in conjunction with expression 8, the solution for the total utilization of an i-stage queue, is as follows: 1

u(i) = 1+

(i+1) 2 ) )

(1−(pb (i+1)

(1+pb

(i+1)

+w·pb

)

·

(i) x2,0 (i) (1−x2,0 )

(i+1)

.

(18)

(i)

pb is expressed by Lemma 1 and x2,0 is expressed according to the definition (8) in Section 4.1. The above formula (18) is a recursive function of utilization depending on the utilization of the previous, current, successive and last stages. This leads us to use a relevant convergent method in order to find the steady state solution for the utilization of all stages. Boundary conditions:

• The requirement for the preceding stage i = 0. Since there is no preceding stage, the probability of packet arrivals to the inputs is (p). Thus, u(0) = p. • The requirement for the last stage i = L: A packet at an output port of the last stage can always proceed. However, buffers (L+1) L) in the SEs of the last stage cannot be in the blocked state. Thus: pb = 0 ⇔ u(L) = p(serv . Distinct cases: The general formula (18) has two possible distinct cases: (i) When the multicasting traffic ratio is w = 0 (all the traffic is considered as full unicast traffic), then: u(ui) =

(i)

1 − x2,0 (i)

(i+1)

1 − x2,0 · pb

.

(19)

The above formula confirms the formula for the case of exclusively unicast traffic in [12]. (ii) When the multicast traffic ratio is equal to w = 1, which denotes that all traffic considered is multicast (e.g., broadcast), then: 1

u(mi) = 1+

(i+1) 2 (1−(pb ) ) (i+1) (1+2·pb )

·

(i) x2,0 (i) 1−x2,0

.

(20)

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6. Applying the convergence algorithm Using fixed-point iteration (ε < 10−4 ) over the state utilization, a steady state is reached from which the performance metrics of interest are determined. Let [m] u(i) be the value of u(i) , during the mth iteration of the following algorithm: Algorithm m := 0 /*Initial PHASE A */ Initialize [0]u(i) /* for all stages i = 1, . . . , L*/ /* End of PHASE A */ REPEAT m := m + 1 /* Start of PHASE B (Forward Calculations) */ Calculate [m] u(1) /* stage i = 1*/ FOR i = 2 TO L − 1 DO BEGIN Calculate [m] u(i) /* stage i = 2, . . . , (L − 1)*/ END FOR /* stage i = L*/ Calculate [m] u(L) /* End of PHASE B (Forward Calculations) */ m := m + 1 /* Start of PHASE A (Backward Calculations) */ Calculate [m] u(L) /* stage i = L*/ FOR i = L − 1 DOWNTO 2 DO BEGIN Calculate [m] u(i) /* stages i = (L − 1), . . . , 2*/ END FOR Calculate [m] u(1) /* stage i = 1*/ /* End of PHASE A (Backward Calculations) */ UNTIL ([m] u(i) − [m−1] u(i) ) < ε for all stages i = 1 to L Set u(i) to the values of [m] u(i) for all stages i = 1 to L.

7. Metrics for single- and semi-layer MINs 7.1. Metrics for single-layer MIN 7.1.1. Average and normalized throughput for SiLMINs Average throughput Thavg is the average number of packets accepted by all destinations per network cycle. Normalized throughput ThN is the ratio of the average throughput Thavg to the number of network outputs N. Furthermore, the normalized throughput is equal to the utilization of the last stage queues, since the last stage’s queues are never blocked. Formally, ThN is defined as: ThN = u(L) .

(21)

7.1.2. Average and normalized packet latency for SiLMINs In a single-layer MIN, average latency Davg is the average time which a packet spends passing through this MIN. On the other hand, the utilization depicts the time that the packets remain in the buffers. Thus, the following formula gives the average packet latency for a single buffered SiLMIN as a function of utilization per stage, L P

Davg =

u(i) · (1 + w)(L−i)

i =1

u(L)

,

(22)

where (1 + w)(L−i) is the replication factor of the system. The normalized latency DNorm can be defined by the ratio of the average latency Davg of packets, to the minimum delay that a packet needs to traverse the SiLMIN without any blocking. This minimum packet delay depends on the number of stages that have a SiLMIN. The normalized latency is expressed by: DNorm = Davg /L.

(23)

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7.2. Metrics for semi-layer MIN 7.2.1. Throughput for semi-layer MINs In a SeLMIN, let LSL represent the number of single-layer stages with multicast operations and let LML be the number of stages that have full layer growth, which can also service multicast traffic without blocking. Thus, L = LSL + LML . If LML is known, then the total number of layers in the second segment is GL = kLML , where k is the number of inputs per SE (e.g., in the case of 2 × 2 SEs, k is equal to two). According to Sections 5 and 6, by running the algorithm, among the other metrics, the utilization of the last stage can be obtained. Owing to the last stage never being blocked, this utilization is the normalized throughput of the fabric. Moreover, if the throughput value of a single-layer segment of SeLMIN is known, then the total SeLMIN throughput (at the fan-out output) can easily be calculated (it is not blocked), by the following formula: Th(out) = Th(SL) · (1 + w)LML

(24)

where Th(SL) is the throughput at the end of the single-layer segment of the SeLMIN and Th(out) is the throughput at the output of the SeLMIN fabric. 7.2.2. Average and normalized packet latency for semi-layer MINs Average latency Davg is the average time a packet spends passing through the MIN. On the other hand, utilization depicts the time that the packets remain in the buffers. Thus, the average packet latency on a SeLMIN is calculated as the sum of packet delays in the blocked and non-blocked segments of a SeLMIN fabric. This average packet latency is given by the following formula: Davg = D(SL)avg + D(ML)avg

(25)

where D(SL)avg represents the average packet delay on the single layer, first and blocked segment of the SeLMIN and D(ML)avg represents the average packet delay on the multilayer, second and unblocked segment of the SeLMIN. The average packet latency D(SL)avg for a single-layer segment of a single buffered SeLMIN is given as a function of utilization per stage by the formula: LSL P

D(SL)avg =

u(i) · (1 + w)(LSL −i)

i =1

u(LSL )

,

(26)

where (1 + w)(LSL −i) is defined as the replication factor of the system. The average packet latency D(ML)avg of the second segment of the SeLMIN (the fan out) is: D(ML)avg = LML

(27)

where LML depicts the number of stages of the second segment (the fan out) of a MIN. Because there is no blocking in the second segment of the MIN, the packet delay is equal to the number of stages. Consequently, the normalized latency DNorm on a SeLMIN is also calculated by: DNorm = Davg /L.

(28)

8. Performance results We compared our analytical method against simulation. This allowed observing the anticipated trade-off between speed of execution of the analysis and accuracy of simulation. Both calculation methods – the numerical convergent method based on analysis and the classic simulation – have been conducted for this study using C++ programming. Both are able to operate under different configuration set-ups. In addition, both methods implement the ‘partial multicast’ operation as a multicasting transmission technique. For our experiments, we considered single-layer MINs with L = 4, 6, 8, 10 stages, and single buffer. The exact experiment setups are described in each experiment case. We observed that our analytical method converges in at most 63 iterations, while the simulation needs at least 104 iterations. As input parameters of an arbitrary MIN set-up, the following are considered: the number of stages which determine the network size, the offered load on the inputs, and the multicast traffic pattern. The two methods provide results in close agreement, as confirmed by all the performance experiments. The differences between the values of all performance metrics by the two methods are less than 2% for various distributions of offered loads and various multicast traffic ratios. The following sections focus on some of our findings. 8.1. Results of single-layer MINs servicing unicast and multicast traffic (by partial operation) Both analytical and simulation methods are applied to single-buffered, single-layer MINs (SiLMINs) which employ a backpressure mechanism and act with ‘‘partial multicast’’ transmission policy. According to partial transmission, a packet

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Fig. 5. Normalized throughputs vs. probability of packet arrivals on inputs of six-stage SiLMIN for various multicast ratios.

can be serviced either fully at both destinations, or partially, being sent to one destination and the remaining in the current queue expecting service during the next time cycle. Some results are presented below: 8.1.1. Normalized throughput of a six-stage single-layer MIN for various multicasting ratios Fig. 5 illustrates the normalized throughput of a single-buffered, six-stage SiLMIN versus the probability of packet arrivals on inputs for various multicast ratios. The offered load consists of unicast and multicast packets and its multicast ratio w is equal to 0, 0.1, 0.2, 0.5, and 0.6. In the diagram, the solid curves SM-w = a% and dot curves AM-w = a% depict the total normalized throughput of a six-stage SiLMIN. This was evaluated by simulation experiments, the analytical model for various multicast ratios of traffic (a% = w ) and the SiLMIN operation applying the partial mode of packet transmission. The total normalized throughput increased considerably with higher multicast ratios due to the multiplication of packets. Also, as can be noted for high values of w (w > 0.2), the network reaches its peak of throughput even with very small values of offered load (approx. 10%). Some validation points in Fig. 5 compared with the results from previous related works are mentioned in Section 8.1.4. Those validations confirm the accuracy of the results in Fig. 5. In order to shift the point of saturation in higher values of offered loads, it is proposed to use multilayer constructions in the final stages. 8.1.2. Probability of packet loss on a single-layer MIN servicing unicast and multicast traffic Having calculated the normalized throughput of the fabric, we can then calculate the probability of accepted packets on the input, as the following formula shows: paccepted = u(L) /(1 + w)L .

(29)

The above formula is valid because all the packets that enter the system are not lost. Then, the packet loss on the fabric’s input is expressed by: plost = p − paccepted .

(30)

Fig. 6 shows the normalized total packet loss vs. probability of packet arrivals on inputs of the six-stage SiLMIN for multicast ratios w = 0, 0.1, 0.2, 0.5 and 0.6. The solid and dot curves depict results emanating from simulation experiments and the analytical method, respectively. In Fig. 6, the normalized packet loss for a SiLMIN (here a 64 × 64 network size MIN) becomes greater as the multicast ratio rises. This happens because the total number of packets is increased in the fabric due to the multicast operation. This increases the backpressure blocking which has as a consequence the increment of the lost packets upon entry to the SiLMIN. It is also worth mentioning that the packet loss when w ≥ 0.6 remains stable and it is equal to the lost packet amount of the broadcast operation case, where w = 1.0. That happens because the SiLMIN is almost saturated even when the multicast ratio (w ) reaches the value of 0.6. 8.1.3. Packet latency of a single-layer MIN servicing unicast and multicast traffic Fig. 7 depicts the normalized packet latency vs. the probability of packet arrivals on inputs. In the diagram, the dot curves AM-L = i and solid curves SM-L = i depict the values of normalized packet latency of an i-stage SiLMIN, as they have been estimated by the analytical model and simulation experiments respectively, where i = 4, 6 and 8. In Fig. 7 it is clear that the normalized packet latency increases with bigger networks, due to the considerable multiplication of packets, while stage-by-stage multicasting operations maintain a fixed ratio. A general conclusion that can be inferred from all the above results is that the performance of SiLMINs operating with partial transmission mode (when, e.g., serving multicast traffic on the internet in parallel systems or included in grid

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Fig. 6. Normalized packet loss vs. probability of packet arrivals on inputs of a six-stage SiLMIN with various multicast ratios.

Fig. 7. Normalized packet latency vs. probability of packet arrivals on inputs of an i-stage SiLMIN where i = 4, 6, 8 with traffic multicast ratio w = 0.5.

computing systems) shows saturation even for very low values of multicast traffic ratios. This commits the resources of the networks as the percentage of multicast traffic increases considerably. 8.1.4. Validation of the results 1. Figs. 5–7 clearly show that the results of both methods (approximate analytic and simulation) are in close agreement (the differences are less than 2%) with each other. That reveals that both methods give accurate results. Nevertheless, the current analytical method is faster at determining the saturation point of the system. 2. In Fig. 5, when all the traffic on a six-stage SiLMIN input is of unicast type, the normalized throughput which is estimated by both our methods (Fig. 5: AM-w = 0% and SM-w = 0% curves) is in close agreement with the corresponding results reported by Shabtai and his colleagues in [6, Fig. 13], where they have studied a six-stage MIN which works with single priority. 3. In Fig. 5, for the same marginal case of exclusively unicast traffic, the results (Fig. 5: AM-w = 0% and SM-w = 0% curves) coincide with our results reported in [12, Fig. 5]. In this paper [12], a similar analytical model is studied for SiLMINs which exclusively supports unicast traffic. The results, which are depicted by relevant curves in [12, Fig. 5], are found to be more accurate compared with performance statistics obtained by older study models in the literature. More precisely, in [12, Fig. 5] it is proved that, in the exclusively unicast case of traffic, our analytical model for single-buffered SiLMINs gives more accurate results than some classical models like those of Jenq [20], Mun and Youn [21]. This finding has been clearly presented in [12, Fig. 5] and this is additional evidence regarding the validation of our analytical approach. 4. Finally, in the same Fig. 5, for w = 0.5, the saturation is avoided only up to very small offered loads (p < 0.1), such as the Tutsch’s and Hommel Model reported in [15, Fig. 8]. Tutsch and Hommel in [15] also apply a ‘partial’ method as multicast queue server operation, but they have a slightly different assumption. According to them (Tutsch’s and Hommel Model approach), all possible combinations of destination addresses for each packet entering the network were equally distributed. Nevertheless, these matching results can be considered as another indirect verification of the accuracy of our results. 8.2. Results of semi-layer MINs servicing unicast and multicast traffic (by partial operation) In terms of improving the performance value, semi-layer MINs have been introduced to overcome the restriction of the very low throughput saturation point and the corresponding considerable values of packet delay that are evident even in very small values of multicast ratio. Applying the above analytical method to SeLMINs in order to evaluate performance, we derive accurate results in a timely manner. In the next subsections, some results are illustrated for single-buffered SeLMINs which employ a backpressure mechanism and operate with a ‘partial multicast’ transmission mechanism.

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Fig. 8. Normalized throughput vs. probability of packet arrivals on semi-layer MIN with L = 5 stages and LSL = 3 and LML = 2.

Fig. 9. Normalized throughput vs. probability of packet arrivals on semi-layer MIN with L stages, GF = 2 and Gs = 3, 4, 5, 6 and 7.

8.2.1. Normalized throughput of a five-stage semi-layer MIN for various multicasting ratios In Fig. 8, the dot curves AM-L = 5 (w = 0.x) represent the normalized throughput of a single-buffered five-stage SeLMIN calculated by the analytical method, versus probability of packet arrivals on inputs. The specific five-stage SeLMIN set-up is described by the following set of factors: GS = 3, GF = 2 and GL = 4. Similarly, in the same Fig. 8, the collateral solid curves SM-L = 5 (w = 0.x) also depict the normalized throughput of the same SeLMIN set-up, under the same values of probability of packet arrivals calculated by simulation. The study cases of multicast ratio (w ) for both methods are considered as equal to 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0. From the data depicted in Fig. 8, it is concluded that for a given SeLMIN set-up and a specific offered load, the normalized throughput increases as the traffic multicast ratio (w ) is increased. This is obvious because the increment of traffic multicast ratio multiplies the packets in the switch fabric. Also, from Fig. 8 it is inferred that it is useless to increase the load on the inputs of fabrics beyond the value of about 50% of the full load. That means when the values of offered load becomes greater than ∼50% of the full load, a throughput saturation point appears. It is clear that this packet saturation of the system is due to the large amount of packets in the system. This saturation causes an inability in the system to lead the packets quickly to the outputs. It is also worth noting that in all cases of SeLIMINs, the limit of throughput saturation has been shifted to the higher value in comparison with the relevant limit of corresponding SiLMIN throughput saturation. In this study, for example, the saturation edge in the SiLMIN appears at 10% offered load (Fig. 5), whereas the corresponding saturation edge in a corresponding SeLMIN has shifted to approx. 40%–60% (Fig. 8). A throughput gain of about 30%–50% is observed in our study, as can be seen by associating the curves in Figs. 5 and 8. The obtained throughput gain of the SeLMINs, when they act with ‘partial multicast’ transmission, is one of the main reasons that makes them more efficient interconnection fabrics in servicing unicast and multicast traffic. Conversely, the corresponding – in terms of the number of stages – SiLMINs provide very low value of throughput, which means they are unable in practice to support networks with real multicast traffic. 8.2.2. Normalized throughput of semi-layer MINs with various start replication factors (GS ) Fig. 9 depicts the normalized throughput of five different eight-stage SeLMIN set-ups, when multicast ratio is w = 0.2 versus the probability of packet arrivals on inputs. The five SeLMIN set-ups of single-buffered eight-stage SeLMINs are denoted by the following sets of factors: GS = 3, 4, 5, 6 and 7, the collateral factor is GL = 32, 16, 8, 4, 2 respectively, while the factor GF remains fixed and equal to two. The dot curves AM-Gs = a% depict results which are obtained by the analytical model, whereas the solid curves SM-Gs = a% show results that are obtained by simulation.

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Fig. 10. Probability of lost packets vs. probability of packet arrivals on semi-layer, eight-stage MINs with multicast ratio w = 0.2.

From Fig. 9 it is concluded that when a SeLMIN incorporates more levels, the relevant values of normalized throughput are improved considerably. SeLMINs with a high number of levels mean the GL factor has high values, whereas the corresponding factor Gs has low values and the replication starts from ‘earlier’ stages—close to the inputs. On the other hand, when the number of parallel layers declines (low value of GL factor), then the size of the SeLMIN is minimised, and the throughput decreases also. Thus, when the replication starts in later stages, the normalized throughput follows a gradual declination. The SiLMIN = 8 curve depicts results from a single-buffered eight-stage SiLMIN that appears to have the smallest values of normalized throughput compared with all the eight-stage SeLMIN set-ups. The conclusions inferred from Fig. 9 are additional evidence which show that, the larger the multilayer degree a SeLMIN has, the higher the levels of throughput values obtained by it for a given offered load are. 8.2.3. Packet loss on semi-layer MINs servicing unicast and multicast traffic By applying the algorithm to the single-layer segment of a multicast MIN, we obtain the utilization of the first LSL number of stages (first segment) in a SeLMIN. The stage with i = LSL is a stage which is never blocked and it is also connected to the second unblocked segment (the fan out) of the SeLMIN. Therefore, the utilization of the LSL stage, is the throughput of the first segment of a SeLMIN and this can be calculated by the analytical method. Then, on the basis of this finding, the probability of accepted packets on the SeLMIN inputs can be quantified by the formula: uLSL

(1 + w)LSL

.

(31)

Consequently, the packet loss probability on the SeLMIN’s fabric inputs, can be expressed as: plost = p −

uLSL

(1 + w)LSL

.

(32)

Fig. 10 shows the change of the loss packet probability value in eight-stage SeLMIN fabric inputs in correlation with the probability of packet arrivals, when the multicast ratio (w ) of traffic is fixed and equal to 0.2. The dot curves AM-Gs = a and the solid curves SM-Gs = a represent results which are obtained by the analytical method and simulation respectively. Moreover, the curve SiLMIN-L = 8 shows the case of an eight-stage SiLMIN fabric when the ratio (w ) of multicast traffic is also equal to 0.2. From Fig. 10 it is concluded that a SeLMIN with many layers (e.g. GS = 3 (GL = 32)—replication starts just after the third stage), has the ability to accept bigger bulk (fewer lost) of packets on inputs as it has better performance than a corresponding SeLMIN with a smaller number of layers (e.g. GS = 6(GL = 4)—starts in a later stage). This large number of packets is present, because the more layers there are, the more alternative routes the packets have to choose from, in order to arrive at their destination outputs. Those alternative routes on the second segment of the SeLMIN, operate as unblocked packet ‘‘free ways’’ in the switch system. In addition, it is worth noting that a dilated SeLMIN has the ability to absorb most of the traffic. For example, (Fig. 10) a SeLMIN with L = 8, GS = 4 GF = 2 and GL = 32 layers, and less than 20% value of traffic on its inputs with multicast ratio equal to 20%, accepts all the traffic for serve. On SeLMINs, the blocking phenomenon is limited only in the first stages. In Fig. 10, it is clear that the worst case regarding the lost packets is the SiLMIN case, which presents the highest probability of lost packets in comparison with all the corresponding SeLMIN set-ups under the same traffic conditions. 8.2.4. Blocking probability per stage of a semi-layer MIN servicing unicast and multicast traffic Fig. 11 depicts in detail the blocking probability per stage on single and semi-layer five-stage MINs when full load is offered and the fixed multicast ratio (w ) is equal to 0.2. In Fig. 11 it can be observed that the single-buffered five-stage SiLMIN has a gradual reduction of blocking probability as we move from the first to the last stage.

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Fig. 11. Blocking probability per stage in 5-stages single and semi-layer MINs with full offered load (p = 1) and multicast ratio w = 0.2.

Fig. 12. Normalized packet latency vs. probability of packet arrivals on five-stage SeLMIN inputs when the multicast ratio w = 0.2. The three SeLMINs are: (GS , GF , GL ) = (2, 2, 8), (3, 2, 4), (4, 2, 2) respectively.

In the last stage of SiLMIN there is no blocking of packets, as expected. On the other hand, all the corresponding SeLMINs have obvious lower blocking probabilities values in comparison with the corresponding SiLMIN. This blocking in the case of the SeLMINs also appears only in the first stages. As the SeLMIN has more levels in its earlier stages, the blocking probability is reduced. Thus, for example, the SeLMIN with factors GS , GF , GL = 2, 2, 8 appears to be blocked only in the first stage and this blocking value is very low (in our case, three times less than in the corresponding SiLMIN), as can be seen in Fig. 11. 8.2.5. Packet latency of a semi-layer MIN Fig. 12 illustrates the normalized packet latency data obtained from the analytical method and simulation (dot curves and solid curves respectively) for three different single-buffered five-stage SeLMIN set-ups in correlation with the probability of packet arrivals on inputs when the multicast ratio (w ) is equal to 0.2. The three distinct five-stage SeLMIN set-ups are denoted by the following sets of GS , GF , GL factors: (2, 2, 8), (3, 2, 4) and (4, 2, 2) respectively. The above diagram clearly verifies that when the start replication factor GS of SeLMINs has low values, then the packet latency is also kept to a low level of values. Fig. 12 shows that for a given value of offered load, the packet latency decreases in proportion to the number of SeLMIN layers. This is because for the max size of SeLMINs with many parallel levels, the blocking phenomenon is limited to the first few stages of the SeLMIN while the rest of the stages operate without blocking. This analytical method can determine exactly the value of packet latency for a given SeLMIN set-up. Moreover, Fig. 12 shows that the packet delay is increased as the offered load increases. It is also noticeable that the value of the average packet latency of a given SeLMIN set-up is remarkably low compared with the corresponding – in terms of the number of stages – SiLMIN. This reduction in the value of packet delay observed in SeLMINs, is another advantage (the other is the throughput) that makes them more efficient devices in servicing mixed types of traffic (multicast and unicast). The approach presented here, can help to estimate all the performance indicators which are related to the application of SeLMIN fabrics in real network environments. Nowadays, to satisfy new networking requirements, a special QoS set-up is needed by each type of traffic. Thus, for example, VoIP and voice trunking have particular requirements concerning QoS demands. In such types of applications, low-level values of packet delay and jitter are the basic factors required to be serviced by single buffered MINs in contrast

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with corresponding finite buffered fabrics. Moreover, when voice data are needed to be delivered in a multicasting manner, then single-buffer SeLMINs are the most suitable choice for this service. However, a disadvantage of the SeLMINs in comparison with the corresponding SiLMINs, is their complexity, which rises exponentially when they are inflated by the addition of more parallel layers. Complexity is a quantitative term related to the construction cost. Sandeep Sharma and colleagues in [10] give an algorithm for computing the cost of MINs according to their complexity. This is applied only to the single-layer MINs that are used in parallel processing. The evaluation of a SeLMIN, whether it is hardware or software or both, needs a full study of the cost of designing and implementing it. Therefore, the cost appreciation has to be approached in terms of space and time. It is obvious that this study will help in the determination of overall performance of SiLMINs and SeLMINs in terms of unicast and multicast traffic. 9. Conclusion MINs with the Banyan property are proposed to connect a large number of processors to establish a multiprocessor system, and contrast with non-Banyan MINs, which are in general more expensive and more complex to control. The multilayer MINs have evolved from Banyan type switches mainly due to the increment of multicast and broadcast types of traffic and because these types of traffic have increased due to new applications such as video and voice delivery. Significant advantages of multilayer MINs include their low cost/performance ratio and their ability to route multiple communication tasks concurrently. In this paper, a new analytical method and simulation to evaluate the performance of multilayer MINs under different offered loads is developed, thus filling the gap in the research that existed in this field. A large number of experiments were carried out utilizing this analytical model, and all the relevant performance statistics were determined. The experiments were performed using various loads and various multicast ratios of traffic on different SiLMIN and SeLMIN setups, using the ‘‘partial multicast’’ transmission policy. This analytical model converges in a small number of iterations. All the findings obtained by the analytic model are also validated by simulation. Beside this, the predictions of this approximate analytical framework in some marginal cases are validated by previous related work. The phenomenon of throughput saturation on single-layer MINs was estimated and it is revealed as significant, chiefly when they have to service multicast traffic. In contrast, when using semi-layer MINs, this throughput saturation point moved towards more high value levels of offered load. Also, it is likewise observed that the semi-layer MINs present lower packet latency in comparison with the corresponding single-layer MINs. Thus, both major metrics in terms of throughput and latency show that the semi-layer MIN fabrics facilitate service of unicast and multicast traffic in a more effective manner than singlelayer MINs. For the design and construction of an optimum semi-layer MIN in terms of performance, a proper setup should have a balance between cost requirements and performance behavior under anticipated traffic load and quality of service specifications. Therefore, the findings of this paper can be used by MIN designers to optimally configure their networks. In the future we will focus on studying the performance evaluation of multilayer MINs when they have to deliver unicast and multicast traffic simultaneously with priority patterns. This scenario is a service requirement of modern networks (e.g. grids). References [1] C. Bouras, J. Garofalakis, P. Spirakis, V. Triantafillou, A general performance model for multistage interconnection networks, in: Euro-Par’97, August 25–29. [2] C. Bouras, J. Garofalakis, P. Spirakis, V. Triantafillou, An analytical performance model for multistage interconnection networks with finite, infinite and zero length Buffers, Performance Evaluation 34 (1998) 169–182. [3] Peng Zhu, Hideya Yoshiuchi, Satoshi Yoshizawa, QoS-aware multicast for internet video applications, in: Packet Video 2007, Volume, Issue, 12–13 November, 2007, pp. 46–51. [4] D.C. Vasiliadis, G.E. Rizos, C. Vassilakis, E. Glavas, Performance evaluation of two-priority network schema for single-buffered delta network, in: Proceedings of the 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC’07, 2007. [5] D.C. Vasiliadis, G.E. Rizos, C. Vassilakis, E. Glavas, Routing and performance analysis of double-buffered omega networks supporting multi-class priority traffic, in: Procs. of the Third International Conference on Systems and Networks Communications, ICSNC 2008, Malta, October 2008. [6] G. Shabtai, I. Cidon, M. Sidi, Two priority buffered multistage interconnection networks, Journal of High Speed Networks (2006) 131–155. [7] J. Garofalakis, P. Spirakis, The performance of multistage interconnections networks with finite buffers, in: Proc. ACM SIGMETRICS Conf., Short Paper, 1990. [8] C. Bouras, J. Garofalakis, P. Spirakis, V. Triantafillou, Queuing delays in differed multistage interconnection networks, in: Proc. 1987 ACM Simetrics Conf., Banff, Alberta, Canada, May 11–14, 1987, pp. 111–121. [9] Elizabeth Suet Hing Tse, Switch fabric architecture analysis for scalable b-directional reconfiguration IP router, Journal of Systems Architecture: the EUROMICRO Journal 50 (1) (2004) 35–60. [10] Sandeep Sharma, K.S. Kahlon, P.K. Bansal, Kawaljeet Singh, Irregular class of multistage interconnection network in parallel processing, Journal of Computer Science 4 (3) (2008) 220–224. [11] Dong Li, Mei Ming, Bo Fu, New multistage interconnection network for multicast, Communications, 2003, APCC 2003, in: The 9th Asia–Pacific Conference on vol. 3, Issue, 21–24 September, 2003, pp. 993–997. [12] John Garofalakis, El. Stergiou, An analytical performance model for multistage interconnection networks with blocking, in: IEEE Sixth Annual Conference on Communication Networks and Services Research, CNSR2008, Halifax, Nova Scotia, Canada, May 5–8, 2008. [13] D. Tutsch, G. Hommel, Generating systems of equations for performance evaluation of buffered multistage interconnection networks, Journal of Parallel and Distributed Computing 62 (2) (2002) 228–240. [14] Dietmar Tutsch, Marcus Brenner, Gunter Hommel, Performance analysis of multistage interconnection networks in case of cut-through switching and multicasting, in: Proceedings of the High Performance Computing Symposium 2000, HPC 2000, Washington, DC, SCS, April 2000, pp. 377–382.

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[15] D. Tutsch, G. Hommel, Comparing switch and buffer sizes of multistage interconnection networks in case of multicast traffic, in: Procs. of the High Performance Computing Symposium, HPC’02, 2002. [16] J. Garofalakis, E. Stergiou, Performance evaluation for multistage interconnection networks servicing multicast traffic, in: The First International Conference on Advances in Future Internet, AFIN 2009, IEEE Press, Athens, Greece, 2009. [17] D. Tutsch, G. Hommel, MLMIN: a multicore processor and parallelcomputer network topology for multicast, Computers and Operations Research (C&OR) Journal 35 (12) (2008) 3807–3821. [18] Cisco systems, 2004. http://newsroom.cisco.com/dlls/2004/next_generation_networks_and_the_cisco_carrier_routing_system_overview.pdf. [19] D. Tutsch, M. Brenner, MINSimulate—a multistage interconnection network simulator, in: 17th European Simulation Multiconference: Foundations for Successful Modelling & Simulation, ESM’03, Nottingham, UK, 2003, pp. 211–216. [20] Y.-C. Jenq, Performance analysis of a packet switch based on single—buffered banyan networks, IEEE Journal on Selected Areas in Communications SAS-1 (6) (1983) 1014–1021. [21] H. Mun, H.Y. Youn, Performance analysis of finite buffered multistage interconnection networks, IEEE Transactions on Computers 43 (2) (1994) 153–161.

John Garofalakis (http://athos.cti.gr/garofalakis/index_en.htm) is Associate Professor at the Department of Computer Engineering and Informatics, University of Patras, Greece, and Director of the applied research department ‘‘Telematics Center’’, of the Research Academic Computer Technology Institute (RACTI). He is responsible and scientific coordinator of several recent European and national IT and Telematics Projects (ICT, INTERREG, etc.). His publications include more than 120 articles in refereed International Journals and Conferences. His research interests include Web and Mobile Technologies, Performance Analysis of Computer Systems, Computer Networks and Telematics, Distributed Computer Systems, Queuing Theory.

Eleftherios Stergiou is lecturer in the department of Information Technology and Telecommunications, at Epirus Institute of Technology in Greece since 2000. He is also a research fellow at the University of Patras. He received the B.S. degree in electrical engineering from NTUA, Athens Greece, and he finished his postgraduate studies at the computer science department of the University of Sheffield (1998). His research interests on performance evaluation of networks integrate by publishing papers in international journals. Among these interests, computing analytical methods, interconnection networks, parallel and distributed systems, high-speed networks, are included. Mr E. Stergiou is member of IEEE Computer Society.