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Dependency Analysis of Message Packet Queues in

Interconnection Networks with Faults

F. Safaei

* #

, A. Khonsari

† *

, M. Analoui

‡

, A. Dadlani

* †

*

IPM, School of Computer Science, Tehran, Iran

#

Faculty of Electrical and Computer Engineering, Shahid Beheshti University, Tehran, Iran

†

Electrical and Computer Engineering Department, University of Tehran, Tehran, Iran

‡

Department of Computer Engineering, Iran University of Science and Technology, Tehran, Iran

{safaei, ak, a.dadlani}@ipm.ir, analoui@iust.ac.ir

Abstract— Many contemporary communication networks carry

different types of traffics with specific service characteristics of

their own. The arrival process of packets outsourced from every

single source node is approximated to a Poisson arrival process.

Mostly, in network performance models, the exponential nature

of the inter-arrival time in fault-free networks is determined by

the dependency between inter-arrival times. But, in most of the

communication environments, in addition to minimizing packet

delays and maximizing the network throughput, continuous

functionality in the presence of faulty components has become a

major issue. On such basis, the dependency phenomenon between

consecutive service times as well as between service times and

inter-arrival times for packet queues in interconnection networks

and in vicinity of the faulty components can be of great

importance. In this paper, we analyze the effect of such

dependencies in packet queues through simulation experiments.

We also study the behavior of an M/G/1 queue with Poisson

processes in face of faults. This study can be used to justify the

predicted packet delays obtained from the analytical models

under diverse traffic patterns and various network conditions

and prove beneficial by enlightening the limitations of network

analytical approaches in using approximation methods for

evaluating network of queues.

Keywords-Network of queues; interconnection networks; fault-

tolerance; independence assumption.

I. INTRODUCTION

Effective analytical models are necessary for predicting the

behavior of large networks to help weigh the cost-performance

trade-offs of various routing algorithms. However, in order to

make such analytical models tractable, they require certain

simplifying assumptions about the underlying interconnection

networks and application traffic patterns. In fact, analytical

model is a simplified and abstract form of a real-world

problem, such that if abstracted in a proper manner, would

result in a useful approximation of the actual problem or at

least, a part of it. Although such assumptions can degrade the

accuracy of the performance evaluation, they are accepted in

most cases as results obtained through such simplified models

are yet applicable. In the past few years, several analytical

models have been proposed that evaluate the performance of

routing algorithms in interconnection networks with and

without faults [1-4]. One of the most significant results

commonly inferred from such researches is the derivation of an

accurate expression to estimate the average latency experienced

by a message packet while passing through the network.

Calculating such latency mandates the introduction of one of

the most important hypothesis known as Kleinrock’s

independence assumption [5], without which derivation an

analytical model would be close to impossible. Such a

hypothesis allows analytical models to analyze and evaluate

each communication link separately. The independence

assumption has been investigated in the context of packet

switching [6], but inter-node dependencies play a potentially

larger role in other switching mechanisms. In this paper, to

study the independence assumption, and the phenomena it

makes, we characterize the impact of packet routing on

network performance through simulation experiments.

Although the results obtained through simulations verify the

basic trends in the analytical models, inter-node dependencies

consistently cause the analytical models to overestimate or

underestimate actual network performance, depending on the

network workload. Further experiments demonstrate the fact

that the routing algorithms, switching methods, network

topologies and application workloads common in modern

parallel computers could exacerbate such effects by limiting

the mixing of traffic from different incoming communication

channels.

II. NETWORK OF QUEUES

In order to provide a framework for an end-to-end

performance evaluation of routing algorithms and switching

methods, we introduce a basic queuing model in this section,

which shall be referenced throughout the paper. This model has

been used widely to derive several analytical models in

interconnection networks [1-4]. In such reported works, the

basic model decomposes the interconnection network into a

collection of independent communication links and queues.

The various routing schemes and switching mechanisms are

differ in how they affect the blocking probability of virtual

channels, which, in turn, depends on the number of candidate

output channels that any packet can consider at each node in its

route. In the queuing network under study, it is assumed that

messages have an arbitrary (but known) length or service

distribution. However, the arrival process will be taken to be

Poisson, a single server is assumed, and the queue buffer size is

taken to be infinite. Such a queue is called M/G/1 queue using

Kendall’s notation [6]. With regard to the uniformity nature of

the traffic within the network, the packet arrival process

present the same statistical behavior throughout all the network

channels and the packets are destined uniformly to other

surviving nodes in the network. The M/G/1 queuing model can

represent the service queue at each communication link, if the

queue size is not restricted and the inter-arrival times are

exponentially distributed. In this framework, each source node

generates packets as a Poisson process with rate

λ

packets per

cycle, which results in a stream of packets with exponentially

distributed inter-arrival times with mean

1/

λ

.

With sufficient mixing of traffic from different traffic

sources, Kleinrock’s independence assumption enables

analytical models to decouple the service time and inter-arrival

distributions at each node. Calculating the service time

distribution is a difficult task due to occurrence of blocking; the

service time is not only a function of the packet length, but also

depends upon the waiting time (blocking time) for all

communication channels along the path [1-4, 7, 8]. Fortunately,

the independence assumption as well as experimental

observations allows us to derive a suitable approximation for

estimating the service time distribution for a communication

link. This assumption permits the analytical approaches to

model the network channels as independent general servers

which allow a message to choose a new length (service time)

when routed from one node to another. Under this hypothesis, a

mixture of sources with Poisson arrivals would result in a

Poissonian output stream for every communication link. Each

communication link, in turn, forms the traffic source for an

input channel to an adjacent node. With sufficient

randomization of packet routes, the Poisson traffic yields a

product-form solution that separately analyzes each link on a

packet’s route using Jackson’s theorem available in the

literature [6]. Therefore, the long (short) inter-arrival time

would not imply a long (short) service time for the arriving

packet in the adapted switching technique. With Poisson

process arrivals and Kleinrock’s independence assumption, the

process on a single communication link can be expressed easily

an M/G/1 queuing system, a system that has been well studied

in the literature [1-4, 7, 8].

III. INTER-NODE DEPENDENCY IN INTERCONNECTION

NETWROK QUEUES

The dependency between inter-arrival and service times in

a queue was first stated as an important event in

communication networks and then, later on, studied thoroughly

by Kleinrock [5]. Kleinrock considered a queuing network

model of a communication network, in which messages pass

through a series of queues on their way from source to

destination. Since the same message visits each queue on the

path, the service times of each message at the successive

queues it visits will typically be positively correlated and may

even be identical (identical transmission rates). In addition, if

the service times of two serial queues are dependent, then the

inter-arrival times and the service times of the second queue

would also be dependent. If these two service times are

relatively long, then the inter-arrival time as well as the service

time of the second queue would also be lengthy. In cases where

such dependencies exist, the delays experienced by packets in

the second queue would tend to decrease. Kleinrock observed

that with a proper mixture of traffics, the influence of such

dependencies can be taken to be insignificant and almost

neglected.

IV. INDEPENDENCE ASSUMPTION

Results obtained from proposed analytical models show

that the complexities and problems experienced in solving

generic communication networks arise due to the constant

length allocated to each packet. Such allocation is the main

cause of a phenomenon known as the dependency between

inter-arrival times and length of adjacent packets traversing

through a path in the network. In fact, neglecting this

dependency, would simplify the proposed analytical models for

evaluating computer network performance to a great extent.

Once again, consider the hypothesis of independency between

the inter-arrival time and length of a packet when entering the

network from an external source. We shall show that this

hypothesis of independency can be precise in describing the

applied foreign traffics for some interconnection networks.

Knowing that the external packet source comprises of a large

number of nodes, each of which generate packets at different

rates, the inter-arrival times and the length of packets generated

by each node are dependent on each other. This means that

requires a finite amount of time to generate a packet and this

time period strictly depends on the length of the packet. But,

the total inter-arrival time and the length of the packets

generated by all the nodes bring about the independency

phenomenon as the length of any packet generated by any

assumed node is completely independent of the arrival times of

the packets generated by other nodes.

A similar situation exists for the internal traffic of many

practical communication networks. That is, there is, in general,

more than one channel delivering messages into any particular

node (in addition to those messages arriving from the external

source feeding this node). Further, there is, in general, more

than one channel (assuming the presence of virtual channels)

that transmitting messages out of this node. Fortunately, to

assess the behavior of fault-tolerant routing algorithms for

analytical purposes, this multiplicity of paths to/from each node

considerably reduces the dependency between inter-arrival

times and lengths of messages as they enter various channels

(queues) within the network. In this paper, we aim at

presenting proves and reasons for the occurrence of the

independency phenomenon based on results obtained through

simulations. By assuming the following hypothesis, the

hypothesis of independency can be implied in our analytical

model:

Definition (Independence assumption): Every time some

node in a given network receives a message, a new length, say

ν

, based on the PDF function is chosen as follows

( )

p e

λν

ν λ

−

=

It is obvious that this condition is not in consistency with

the actual condition prevailing in operational communication

networks. Nevertheless, the obtained mathematical results

result in a model that describes the packet delay behavior seen

in most real-world networks in a very precise manner. In the

following section, we provide proves that justify our

aforementioned claim.

V. SIMULATION AND DISCUSSIONS

In spite of the various evaluation proves available for

generic networks, we still are in search of solutions for various

problems expected in the structure of a generic interconnection

network. In this section, we shall present a simple reasoning on

why the assumption of independency provides a relaxed form

of the actual problem. We shall prove correctness and precision

of this hypothesis with the help of simulation, about which has

been referred in [7, 8].

Firstly, we shall use the results obtained through simulation

to illustrate the affect and delay experienced by a message

traveling in a random network channel. The simulator abides

by the assumptions made in [1-4, 7, 8]. A network cycle is

defined in terms of the transmission time of a flit from one

router to another. The message arrival is taken to be a Poisson

process with a rate of

λ

messages per cycle. The message

length is considered to be constant and the average message

delay is defined in terms of the average of the time at which a

message is generated to the time at which the last message flit

reaches the local processing element (PE) at the destination

node. Several experiments have been conducted for networks

with variant sizes and packets with variant lengths. For the sake

of simplicity, the inter-arrival time distributions on an arbitrary

communication link have been obtained for a 2-D torus of

dimensions 8×8 and 16×16 with message length,

32

M

=

flits, failure rates of 0% and 10%,

5

V

=

virtual

channels per each physical channel. Figures 1 and 2 depict the

results obtained through simulation experiments.

The result of every simulation model depends on the input

data set. In the simulation of the queuing systems presented in

this paper, the input data are the distributions of time between

arrivals and of service times. The various types of distributions

are introduced in statistics. However, coming up with

appropriate distributions with input data is a major task from

the standpoint of time and resource requirements. In order to

design a valid analytical model from the input data, the first

step of every evaluation based on simulation results is

gathering suitable test data. However, the second step involves

identifying a probability distribution typically begins with the

development of a frequency distribution, or histogram, of the

data. Given the histogram and a structural knowledge of the

process, a family of distributions is chosen. Finally, the last

step engages the validation of the hypothesis which states that

the chosen distribution is a good approximation of the data.

Analyst usually repeats the above procedure for the gathered

data. Each of the aforementioned steps is shown in Figures 1

and 2. Additionally, we discuss about data regarding related

variables.

Though histogram provides an estimate of the probability

density function, is not a suitable tool for evaluating the fit of

the chosen distribution. When there is a small number of data

points, say 30 or fewer, a histogram can be rather ragged [9]. In

addition, our observations of the fit depend on the widths of the

histogram intervals. But, even if the intervals are chosen

appropriately, grouping data into cells can make it difficult to

compare a histogram to a probability density function. The Q-

Q (Quantile-Quantile) plot is an apt tool for evaluating

distribution fit which is not affected by such issues [9]. The

tool introduced in this section is used to compare the predicted

cumulative function of the gathered data with that of the

discrete random variable in 2-D 8×8 and 16×16 torus networks

with message length

32

M

=

flits, random node failure rates

of 0% and 10%, and

5

V

=

virtual channels per physical

channel through diagrams shown in Figures 1 and 2. Data

analysis softwares mostly include stand-alone tools for

generating Q-Q plots, especially for exponential and normal

distributions. The Q-Q plot is used to compare two data

instances and measure their likeliness. If X is a random variable

with CDF F, then q- quantile of X is some value

γ

such that

(

)

(

)

for 0 1

F P X q q

γ γ

= ≤ = < <

When F has an inverse, we have

( )

1

F q

γ

−

=

Now, let

{

}

, 1,2, ,

i

x i n

=

…

be an instance of data taken by

X. We denote the observations by

{

}

, 1,2, ,

j

y j n

=

…

and

order from the smallest to the largest such that

1 2

n

y y y

≤ ≤ ≤

…

. We let j denote the ranking number.

Thus,

1

=

j

denotes the smallest observation and

nj

=

specifies the largest. The Q-Q plot picturized in Figures 1 and 2

are based on the fact that y

j

is an estimate of the

(

)

0.5 /

j n

−

quantile of the random variable X. In other words,

(

)

1

0.5

j

j

y F

n

−

−

≈

Now suppose that we have chosen a distribution with

cumulative distribution function F as a possible representation

of the distribution of variable X. If F is a member of an

appropriate family of distributions, then a plot of y

j

versus

( )

(

)

1

0.5 /

F j n

−

−

will be approximately a straight line. If F

is chosen from an appropriate family of distributions and also

has proper parameter values, then the above mentioned line

will have a slope 1. On the other hand, if the assumed

distribution is inappropriate, the points will deviated from a

straight line.

It is difficult to claim that the data are represented correctly

by the exponential function based on the independence

assumption from Figures 1 and 2. But, as can be observed

clearly, a straight line is obtained in the Q-Q plot given in

Figures 1 and 2 that supports the exponential distribution

assumption. It should be noticed that while evaluating the Q-Q

plot, the arranged values are not independent of each other and

are in fact, ranked. Thus, if a point occurs above the straight

line, it is possible to have the next point lying above the line

too, but it is not liable to have the points scattered around the

line. The variance of the extreme points (the smallest and

largest values) is much larger than those in the middle and the

differences are mostly seen at the extremes. The linearity of the

points in the middle is far more important than that at the

extremes. In this paper, we have plotted

n

points,

(

)

(

)

(

)

0.5 1.5 0.5

, , , , , ,

n n n

n

X X X

n n n

−

…

and have compared them with exponential cumulative function

to present their probability similarities. As obvious from

Figures 1 and 2, a set of constant points in the plot are located

almost along a straight line. In fact, we have replaced the

problem of comparing the exponential cumulative function

with that of specifying a straight line. The approach described

in this paper is based on the comparison of values taken by the

assumed random variable and the experimented random

variable so as to match the two assumed and observed

cumulative functions. By plotting all the points, we obtain a

straight line passing through the origin and a slope of 45

degrees. The probability diagram allows us to check whether

the cumulative distribution function obtained through

experiments meets with the exponential cumulative function or

not. In this scenario, we have assumed a specific probability

distribution such as exponential and have estimated the

parameters of above mentioned distribution on the basis of the

gathered information. The simulation results prove that the

exponential cumulative function is the best approximation that

matches with the experimental cumulative function and show

that the inter-arrival time distribution at each outgoing link,

specifically under heavy traffic load, is similar to that of an

exponential distribution (e. g. a Poisson process). However, at

light traffic loads, the simulation results show some spikes in

Figures 1 and 2, which when ignored, the inter-arrival time

distribution shows a closer resemblance to an exponential

distribution. Since the effect of blocking probability makes no

sense at light traffic loads, we believe that the Poisson arrival

approximation is reasonable.

VI. CONCLUSIONS

In this paper, we aimed at presenting the essential

complexity and intractability of the mathematical analysis of

interconnection networks in the presence of faulty components.

It is worth mentioning that even simpler class of networks like

tandem may result in relatively limited analytical outcomes.

Over here, we showed that the main source of such complexity

is the dependency between the inter-arrival times and the

lengths of internal message traffic. But, relying on the results

obtained through simulations, we showed that by introducing

the independence assumption, this dependency can be ignored

to some extend, thus resulting in analytical models that can

estimate the average message latency existing in

interconnection networks to a rational precision. We have also

shown that our observations, with regard to Burke’s theorem,

allows us to consider each communication channel (or node)

existing in the network to be independent in our analytical

models. Such considerations prove beneficial in describing

mathematical equations and relations prevailing in

interconnection networks. As future works, we shall study the

effects of various channel capacity allocations, different

routing algorithms and several underlying topological

structures. We believe that such considerations would be

further simplified by using the independence assumption.

REFERENCES

[1] M. Ould-Khaoua, “A performance model for Duato’s adaptive routing

algorithm in k-ary n-cubes,” IEEE Transaction Computers, vol. 48, no.

12, pp. 1–8, 1999.

[2] J. Draper, and J. Ghosh, “A comprehensive analytical model for

wormhole routing in multicomputer systems,” Journal of Parallel and

Distribted Computing, vol. 32, pp. 202–214, 1994.

[3] Y. Boura, “Design and analysis of routing schemes and routers for

wormhole-routed mesh architectures,” Ph.D Dissertation, Department

of Computer Science ad Engineering, Penn State University, 1995.

[4] H. Sarbazi-Azad, “Performance analysis of wormhole routing in

Interconnection Networks,” Ph.D Thesis, Computing Science

Department, Glasgow University, 2001.

[5] L. Kleinrock, “Communication Nets: Stochastic Message Flow and

Delay,” New York, McGraw-Hill, 1964.

[6] L. Kleinrock, “Queueing systems,” vol. 1, John Wiley and Sons, 1975.

[7] F. Safaei, A. Khonsari, M. Fathy, and M. Ould-Khaoua, “Performance

anlysis of fault-tolerant routing algorithm in wormhole-switched

interconnections,” The Journalof Supercomputing, vol. 41, no. 3, pp.

215-245, 2007.

[8] F. Safaei, A. Khonsari, M. Fathy, and M. Ould-Khaoua,

“Communication delay analysis of fault-tolerant pipelined circuit

switchig in torus,” Journal of Computer and System Sciences, 2007.

[9] J. Banks, J. S. Carson, B. L. Nelson, and D. M. Nicol, “Discrete-event

simulation,” 4

th

Ed., Pearson Prentice-Hall, 2005.

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Fig 1 Inter-arrival time distributions provided by simulation experiments for a single communication link in a 2-D

8×8 torus network using wormhole switching, message length

32

M

=

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10%, and

5

V

=

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16×16 Torus, V=5, M=32, Failure Rate= 0%, λ=0.0001

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16×16 Torus, V=5, M=32, Failure Rate= 0%, λ=0.0001

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16×16 Torus, V=5, M=32, Failure Rate= 0%, λ=0.02

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For Evaluation Purposes Only

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16×16 Torus, V=5, M=32, Failure Rate= 0%, λ=0.02

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16×16 Torus, V=5, M=32, Failure Rate= 10%, λ=0.02

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16×16 Torus, V=5, M=32, Failure Rate= 10%, λ=0.02

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16×16 Torus,V=5, M=32, Failure Rate= 10%, λ=0.0055

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Fig 2 Inter-arrival time distributions provided by simulation experiments for a single communication link in a 2-D

16×16 torus network using wormhole switching, message length

32

M

=

flits, random node failure rates of 0%

and 10%, and

5

V

=

virtual channels per physical channel; The corresponding Q-Q plot for each case is shown.