Optimization of Mean and Variance of End-to-EndDelay in Interconnected Networks
By
Manu Khanna
CENTER FOR COMMUNICATIONS AND SIGNAL PROCESSING
Department of Electrical and Computer EngineeringNorth Carolina State University
Raleigh, NC 27695-7914
December 1984
CCSP-TR-84/21
ABSTRACT
KHANNA, MANU. Optimization of mean and variance of end-
to-end delay in interconnected networks. (Under thef
direction of ARNE A. NILSSON).
In this research different strategies to optimize
mean and variance of end-to-end delay in interconnected
networks, where messages are passed between two networks
of identical architecture through an intermediate network
of different architecture, have been investigated. In
particular two schemes 'strip-and-pad' and 'pad-and-pass'
have been compared on the basis of end-to-end delay
incurred. Analytical modeling has been incorporated to
find the trade-oifs between the two schemes. Simulation
techniques are used as a tool to validate the analytical
models. The choice of either 'strip-and-pad' or 'pad
and-pass' has been found to be a function of processing
times at gateways, mean message lengths to be taken care
of and the number of hops to be traversed through the
intermediate network. Parallel to Kleinrock's
'independence assumption' the concept of 'partial
independence' has been introduced.
11
BIOGRAPHY
Manu Khanna was born in Chandigarh, India on October 16,
1960. He received his S.Se. Engg. from Punjab Engineer
ing College, Chandigarh, in Electronics & Electrical Com
munications, in 1982. After working for about half an
year with Uniscans & Sanies Ltd., he came to the U.S.A. to
pursue his Master's degree at North Carolina State Univer
sity, Raleigh.
His major area of interest is Computer Communica
tions.
111
ACKNOWLEDGEMENT
I wish to express my heartiest gratitude to Dr. Arne A.
Nilsson, Chairman of the Advisory Committee, for his
invaluable guidance, advice and persistent encouragement
at every step of this work. My special thanks are also
extended to Dr. Harry G. Perras for his constructive cri
ticism and the many papers he provided me for this study.
I would also like to extend my thanks to Dr Dharma P.
Agrawal for his encouragement during this work.
Finally, I would like to thank the Center for Commun
ications and Signal Processing for making available the
computing facility and the funds for this research.
1v
TABLE 0 f CONTENTS
1.
2.
t nt r oduce i on ••••...••••..•••••••••••.••••..•...
1.1 Computer Networks -- motivation •••••••.•.•
1.2 Network Interconnection -- some issues ••.•
1.3 Problem Definition •••••••••••..••.••.•••••
Review of Some Existing Tools •••••••••••..••...
2.1 Kleinrock's Independence Assumption .••••••
2.2 J.W. Wong's Representation of a
1
2
5
14
19
20
Netwo r k ..••..••••••.•••••••••.•••.....•..• 26
2.2.1 Model - some assumptions and
notation •••••••••••••..•••••••....• 26
2.2.2 Summary of Results •.•.•.•..•....•.• 27
3.
4.
Verifying Wong's Analysis ••••••••••••••••••••••
3.1 Validation of Wong's Model ••••.•.•..•....•
3.2 Conclusion ••••••••••••.•••.•••.•••••••••.•
Modeling the Network •••••••••••••••••••••••••••
4.1 Some Assumpt ions ••••••••••••••••••••••••••
4.2 A Tentative Model •••.•••••••••••••••••••.•
4.3 Network-Modeling based on Wong's
30
31
69
70
71
7S
Ana1ysis •••••••••••••••••••••••••••••••••• 81
5. Towards a Refined Model •••••••••.••••••• •••••••
5.1 Verification of Partial Independence
for ./D/1 type gateways and 2 classes of
messages ••.•.•...••..••.•••••..•.•...•••.•
99
100
v
5.2 Comparison of the two Paths •••..•.•.••••.• 124
6 • Fina 1 Mode 1 ••••.•..•••••••••••••••••••••••••••• 141
6.1 Verification of Partial Independence for
D+M type of message distribution .•.•..••.. 142
6.2 Comparison of path 1 and path 2 •..••..•..• 154
7 • Can c 1us ion ••••••••••••••••••••••••••••••••••••• 170
7 • 1 Ove rv i ew •••••••••••••••••••••••••••••••••• 170
7 • 2 Fu t ure Wot: k ••••••••••••••••••••••••••••••• 173
8. List of References •.....•.••...•...•...•..•.... 176
CHAPTER 1
Introduction
'Analytical Engine', the world's first general
purpose digital computer was developed nearly a century
and half ago by Charles Babbage. He also built a
prototype of the world's first special purpose digital
computer, which he called Difference Engine, in 1812 - ten
years after the invention of the steamboat!
We have come a long way since then. Mechanics of
Babbage's computer have been replaced by electronics.
Most of the modern ideas, many of which were first
conceived by Babbage(and remained just that because of the
state of the art at that time), have been made possible by
the tremendous breakthroughs in solid state Physics and
other sciences.
And so continues the man's quest for more, more and
more!
2
1.1. Comouter Networks -- motivation:t
Maximum utilization of resources has always been the
major area to be explored. In 1960 this led to the
multiprogramming and time sharing systems. And then began
an era of computer networks. The motivation for having
computer networks is many-fold. The first objective is to
have resource and load sharing among computer sites ~hich
may be situated thousands of miles apart. The tenn
'resource' may imply specialized hardware, software or
databases. In a computer network it becomes possible for
a user to access these resources housed at any of the
connected sites. Big organizations which have branches
allover the world can exchange information within a
matter of seconds.
A second goal is to provide high reliability by
having alternate sources of power (computing). If a
computer system breaks down at some site, its users can be
accommodated somewhere else until the service is =estored.
Another factor which influenced the evolution of
computer networks ~as the dramatic fall in processi~g
costs as compared to communication costs. Prior to 1970,
computers were relatively expensive as opposed to
communication facilities. So in some applications data
were collected at different sites and sent to a central
office for processing. But nov it is very cheap to
3
provide a computer at every individual site, and so data
can be processed right where it is collected.
Consequently, only occasional exchange of information
takes place, thus saving in communication costs.
Apart from above, the computer networks have emerged
as a strong medium of communication. Tanenbaum [3] claims
that it will have a strong impact on the society of the
future. Home-banking, automated newspaper, computer aided
education, teleconferencing will be commonplace. It will
then truly be an age of information. 'The information
revolution may change society as much as the industrial
revolution did.'
ARP~~ET, ETHERNET, TELENET, TYMNET, TRANSPAC,
DATAPAC, EURONET are a few of the examples of networks
working at local, national or international level.
A step ahead in the same direction is the
interconnection of computer networks themselves (the topic
on which the present thesis is based). Basically, the
motivation for interconnecting computer networks is the
same as for having computer networks, namely, to make the
best use of resources existing in different computer
networks, and exchange of information among remotely
situated sites. As more and more computer networks come
into existence, interest in interconnecting them will also
increase. Also operators of local area networks have
4
vital interest in having access to long haul networks.
Local area networks, in general use different t~ansmission
media(e.g. coaxial cable or optical fiber), to optimize on
delay and cost. Long hauls on the other hand use leased
line or satellite channel. So these two types of
networks, employing different technologies, are there to
exist, and so is the desire to interconnect them.
The legal, technical and political issues, concerning
interconnection of computer networks or communication
net~orks have been described at length in [1] and [2]. In
~hat follows we will give a brief account of technical
issues involved in interconnection of networks.
1 . 2 . Network Interconnec~ion --~ issues:
5
As a simple example, suppose net~orks A and B are to
be interconnec~ed, where A and B may be similar or
dissimilar net~orks.
fig 1.1: A simple ne~~ork interconnection
7he interface bet~een them may be though: of as a black
box, which makes messages originating in net~ork A,
presentable to network B, and vice versa. This black-box
has been called 'gateway' by INWG (Internetwork Working
Group) of IFIP (International Fedration of Information
Processing). A gateway may be an actual processor
6
connected to both networks, or may be additional software
implemented in existing processors on one or both
networks. It may be noted that while designing a gateway
it is of utmost importance to keep in view the
independence of the individual networks. That is, if the
interconnection demands drastic changes in the hardware,
software or protocols of the existing networks it is in no
wayan acceptable proposition
There are a number of issues which must be looked
i~to before any interconnection can be designed. The
important among them are the following:
1) level of interconnection;
2) addressing and ~outing;
3) flow and congestion control;
4) accounting;
5) access control:
6) internet services;
The level of interconnection refers to at which level
in the hierarchy of network layers the individual networks
will be connected. T~o identical networks may be
connected at packet level. In this case gatevay consists
of only software routines which may provide readdressing
and accounting functions. Each node or user will see the
interconnected network as a single network but vith
modified add~ess space.
7
If the networks to be interconnected are dissimilar
the interconnection may be made at network layer level.
The function of the gateway now is to make the packets
originating in one network, presentable to the other
network by attaching appropriate headers and trailers
along with the addressing information. The individual
networks are supposed to take care of the higher level
protocols. The interface may provide 'datagram' or
'virtual circuit' interface.
Another alternative for interconnection is protocol
translation at the gateways. In certain instances when
the protocols of one network cannot be modified, protocol
translation is the only choice if interconnection is to
take place. But this becomes really difficult and
impracticable if the interconnection has to be done over
several networks.
The second important issue concerning
interconnections is addressing and routing. Addressing
and routing are in fact very closely related to each
other. An address tells where to go and routing tells how
to reach there. There are different approaches to the
problem of addressing and routing. An internet packet may
be wrapped in the header and trailer of local protocols.
Subsequently, at the gateway it is extracted and passed on
to the next network after examining the internet header
and t:-ailer,
a
~hich contain the required addressing
information.
/,
Local- Internee Local-ne: DATA !nt:ernet nee
headerheader r::-ailer trailer
Local-nee: text
t ,..... !~~e~-net packe: e~bedded in local nee ?ackec
~h:s app~oach has the advantage in that i: prese~ves the
':~cepencence' of the incividual networks.
In another approach the add=ess space may be modified
and an individual. user may di~ect~1 acdress the
ces:ina~ion i~ a hie~a~chical fashion as in t:aditional
tele;hone sys~ems. A pa~t of the acc:-ess ~ill tell the
local ne~~o~k ~o be accessed and the rest ~ill tell the
destination in that particular network.
Exic:t.Qc:aL-ner: Desr:inac:ion
gateway adcir!ss in Local-~ec
Address :ieLd
fig 1.3: Hierarcbi~al Addressing
9
A nice thing about hierarchical addressing is that a
source does not have to specify the whole route to
destination, and so it may not know about the routing
policies in intermediate networks.
A nice account of different addressing techniques has
been given by Carl Sunshine in [1].
Flow and Congestion control:
Flow control implies the regulation of traffic from
source-to-destination. Congestion control refers to the
procedures by which network resources such as bufferspace,
channel bandwidth, CPU capacity etc. are protected from
overload conditions, generated by 'all' sources of traffic
in a network. In general, successful operation of flow
control for all source destination pairs may not imply
that the system is uncongested.
Flow and congestion control problem is very complex
for individual networks even, and becomes all the more
complex for interconnections. The networks connected to a
gateway may have highly varying speeds (e.g. local area
networks connected to long haul) and may be dealing
differen~ly with congestion produced in them (e.g. some
networks may just drop the congested packets, while others
may rule out the dropping out of packets and stop entry of
new packets in the network to overcome congestion). All
this adds to the complexity of the problem and makes it
10
tough to deal with.
Functions like accounting and access control are also
supposed to be performed by the gateways. Accounting is
necessary for billing and revenue purposes. Access
control is a mechanism by which networks can prevent
traffic entering or leaving them. e.g. a public network
may allow some networks to be connected only during a
certain part of the day.
Another problem to be taken care of is that of
segmentation and reassembly. Each network imposes some
maximum size on its packets which may be governed by
factors like hardware (e.g. the width of a TOM
transmission slot), operating system (e.g. all buffers are
512 bytes), compliance with some national standarcs or a
need to prevent one packet from holding the channel too
long [4]. The need for segmentation and reassembly arises
vhen relatively large packets are transmitted over
subnetwo~ks which support relatively smaller packets.
Segmentation may be done at the source before a packet is
entered in the net~ork, or it may be done at the gateway
before a packet is passed to the next network which does
not support the packet size as carried by the original
packet. It may be possible to reassemble the packets at a
point prior to the destination, although the best
mechanism to do this is still a research topic.
11
Two different approaches to the reassembly of packets
can be found. The first approach called 'transparent
fragmentation' requires reassembly of packets as soon as
they leave the network for which they were fragmented. In
the second approach reassembly is done at the destination.
The second approach requires reassembly processing only at
the destination. But the messages travels in the form of
many small pieces, thus resulting in overheads in terms of
headers and trailers for each separate piece. In the
former approach overheads show up in the form of
reassembly to be done at every exit gateway (fig. 1.4).
Thus the two approaches have their pros and cons, and
before anyone can s~ttle on one of these, a great deal of
research is required to be done.
llicket I
o
[3J 00J Pacl<et I G 1 ~
0°o
fig 1.4:
oo
illr Packet 1---CSJo
(i) Transparent fragmentation
o°0_.-. .. _~ .... -DO(]
(ii) Non-transparent fragmentation
Oifferent fragmentation approaches
., Pacl(et I
o00
00
o
......"-l
13
A proposed solution to handle the problem of
segmentation and reassembly is to define some global limit
of packet size, which must be supported by all networks.
The choice of this limit may be difficult and it may not
find favor with the operators of the individual networks.
Other relavant work in the area of interconnection of
networks can be found in [5,7,9].
Finally, it is worthwhile to point out that efforts
are being made to standardize the interconnection
procedures, although not much success has come by so far.
X.7S, recommendations of the CCITT is one such standard
[6]. It was first defined in 1978 for the interconnection
of X.25 networks.
14
1.1. Problem Definition:
We have described above the concept of
interconnection and the related issues. Of this wide
gamut of problems and issues, we have taken up a very
specific p~oblem for our research. It is one of
optimizing delay and its variance in a configuration of
network interconnections given below. (fig. 1.5)
NetworkA
NetworkB
Net\lorkAt
fig 1.5: Interconnection of networks A, A' (identicalarchitecture> and network 8 (different architecture)
~
U1
16
Networks A and A' are identical in the sense that
they support the same set of protocols. Network B is
different from network A or A' • Gateways G1
and G2
interconnect A and S, and 8 and A' , respectively.
A packet 'm' originated in network A is to be taken
to network At (identical :0 network A) via network B. (The
same holds for a packet originated in A' and destined for
network A. But, as is obvious, the two cases are exactly
same. So from now onward we will concentrate only on the
first case i.e. traversal of a packet from network A to At
The interconnection of networks is at Network layer
level. The packet 'm' has to be provided with headers and
trailers in accordance with the protocols of the network
in ~hich it may be at a particular instant. Thus, a~ the
source (in A or At) packet 'm', before entering the
network, is appended with header and trailer as requi~ed
in network A. The packet 'm' traverses its path th~ough
the network A and reaches the gateway G1•
At this moment
gateway G1
accounting,
will take
verifying
ca:-e
access
of usual
rights
func~ions
etc. aut the most
important function of gateway G1
is to make packet 'm'
suitable for entry in network S by providing it with
header and trailer as per ~equirements of protocols in
that netW'ork. 3efore gateway G1
can do this it has two
17
alternatives before it:
i)Embed the packet 'm' directly in header
and trailer for network B. Subsequently at
gateway G2,
the original packet will be
extracted by stripping off the header and
trailer of network S. This extracted
packet can then be passed on to the network
A' without any further action (because this
packet started with the header/trailer of
network A/A' and retained them throughout
its journey in network B).
ii) As a second alternative, at gateway Gl
,
header and trailer of network A are
stripped off from the packet 'm' and new
header and trailer are attached as per
requirements of network B. At gateway G2
just the reverse action takes place. The
header and trailer of network B are
stripped off and those of network A' have
to be attached to packet 'm' (because 'm'
did not retain its original header and
trailer with which it started unlike in
approach 1).
Now, obviously, we are confronted with the problem of
selecting out of these two approaches the one which gives
18
better performance with regards to overall delay and its
variance. If we take first approach, the packet length
that traverses network B is comparatively larger (implies
more delay) but processing times at gateways are
comparatively less (implies less delay). On the other
hand, with approach2, we have more processing times at
gateways but less packet length to carry through network
B. The relative effects of these two factors, i.e. packet
length and processing times, dictate which approach to
take. Their effect in turn, is controlled by va~iables
like gateway traffic, internal traffic of the networks,
numbe~ of hops traversed through the network E etc.
The pr~sent research is based on finding the
t~adeoffs between the two approaches i.e. which one :s
better in relation to overall delay and variance of delay,
and under what environment (where environment may be
defined to mean the values of parameters like gateway
traffic, traffic internal to the networks, gate~ay
?rocessing times, packet length to be encountered in two
approaches and number of hops traversed through network
E).
19
CHAPTER 2
Review of Some Existing Tools
In the last chapter, we described the motivation for
interconnection of communication networks and the related
issues. In this chapter we take time out to treat two
very important subjects on which the analytical treatment
of our problem is based. The first one is Kleinrock's
Independence Assumption. The importance of independence
assumption may be judged from the fact that it is the
single most important factor which makes the application
of 'Queueing Theory' practicable to communication
networks. Without it no analytical treatment of any
communication network is tractable.
The second subject to be discussed in this chapter is
a model of communication network as suggested by J.W. Wong
[14].
20
2.1. Kleinrock'! rnde~endence Assumotion[lO):
Consider a simple tandem of t~o service stations ~ith
exponential service time dist~ibution, and infinite
\Waiting room. Customers enter at one end and afte~
~eceiving se~vice at both stations leave the system. The
interar:-ival time of customers is exponen~ial.ly
cist~:bu:ec(with mean such that uti~ization of ei~her
customer does not exceed 1). !n:erarrival time, service
ti~e at fi~st serve~ and service time at se~ond se~ve~ are
a" independen~ random variables i.e. ~no'Jledge of
i~~era~~ival ~ime of a custcme~ does no~ dete~ine its
service ~e~uirement at first se~vice station. Simila:-ly,
service re~uirement at second station is not dete~~ined by
:he se~~ice re~u:~emen~ a~ fi~s: station. The study of
such ~ueueing networks ~as :irs~ unde~taken by R.R.?
Jackson [ll]. He discoverec :ha~ the joi~t p~obabil:~y 0:hav:ng xl cus~ome~s in fi~st queue a~d x2 custorne~s i~
~ 0- --~ 0fig 2.1: 7andem of t~o queues ~i~h exponencial se~vice time
disc~ibucion and poisson arrivals.
21
second queue is given by
( 2 . 1 )
where P(xl) and P(x2) are the probabilities of having
xl and x2 customers in respective queues if they had
Poisson inputs at a rate equal to the external input.
Also the symbol '*' here means multiplication and not
convolution.
The result implies not only the independence of the
states of the two systems but also that the second system
has the same distribution as it would have if the input to
~hat system were Poisson with the same intensity as
external input. This in turn poses the question: is the
output process from a MIMII queue in fact Poisson? The
issue was settled by Burke [13] and Reich [12]. They
discovered that the output of a MIMic is indeed Poisson.
The above result put R.R.P. Jackson's result on more
general and firm footing. The joint state distribution
for a tandem consisting of k-exponential service stations
having c(i) servers can now be given as
( 2 . 2 )
where P(xi) is the state probability of the i-th
queue if it were having Poisson arrivals with the same
rate as the external arrivals.
22
Now let us turn our attention to the behavior of a
communication network. In a communication network it is
quite reasonable to assume that interarrival times for
messages for a given source destination pair have
exponential distribution. Similarly, the message length
distribution can be well approximated as exponential. The
servers ccrrespond to the transmission links between
adjacent nodes and service time is the time required to
transmit a given message of specific length on a link
having specific capacity. (For the time being we ignore
processing times at nodes, and the propagation delays
involved). 2.2 shows a specific route through the
network. Messages enter at the node A and
destination as node 8.
have their
At first glance, the above path seems to resemble the
'tandem' discussed earlier. But, as we know, a message
preserves its length as it t=averses the net~ork. So at
all nodes, except the first one, interarrival times a~e
dependent upon the service requirement. Consequently, the
given path cannot be t~eated as a tandem of JacKson's
networks. This 'dependency' ~akes analytical treat~ent of
even the simplest communication net~orks almost
intractable. The only exception where an exact solution
exists for a tandem is when the constant sized messages
are under consideration Yith no interfering traffic on any
23
\, , , intermediate nodes
fig 2.2:
node (18,19].
A path through the network
A great deal of research has gone into getting around
the problem of 'dependency'. In his doctoral thesis,
Kleinrock [10] first gave the 'Independence Assumption',
according to which at every node a 'new' length is
selected from the same exponential distribution. Under
this assumption, we can treat each individual link
independently as M/M/l queue and find associated delays
without much efforts. But now the question arises: How
far the independence assumption' is justified or ho~
closely we are approximating the behavior of the network
by invoking 'independence assumption' ?
24
In our earlier disc~s~:on we assumed that a single
traffic s~ream ar~ives at a given node and depa~ts as a
single traffic stream. 9ut in any real life net~ork, on
any link traffic ente~s f~om multiple sources and leaves
:or multiple destinations. Thus ~henever a message is
se:ec:ec for t~ansmission, though its length has some
depe~dency on the ar~ival st~eam to whic~ it belongs, the
dependence on t~e overall a~~ival proc~ss tencs to :ace
o~t.
Simulations were run by KleinrocK for- va=ious
c~nf:'gu=at.icr:s to ve:-ify the reascnability cf
'independence assumption' in approximating the ~e~~ork
delay [lOj. !t was discovered that ~henever the~e is a
mul:i?licity of tra:::c streams enteri~g anc leaving a
l:nk, :nce?endence assumption gives fai~ly good results up
depar::ng :ra::ic s:~earns
~
~en~e~i~g traffic s:reams
fig 2.3: A typical arri~al node l~ ~he nec~o~~ ~i~~ mul:iar~ival and ~ul~i-de?a~:ure screams.
25
to a link utilization of around 0.7, if applied to
calculate delay. (In a later chapter the validity of
'independence assumption' to calculate variance will be
dealt with).
We close our discussion about the 'independence
assumption' with the final comments that we can invoke it
with confidence for any network of 'moderate connectivity'
(where 'moderate connectivity' implies that most of the
nodes have more than one traffic streams entering and more
than one traffic st~eams leaving).
26
2.2. J.W. Wong'! ReDresentation of a Network:
Eased on the famous ECMP theorem [15,16,17] and
Kleinrock's independence assumption, J.W. Wong [14] have
suggested a model for calculating mean delay and higher
moments of delay in a network.
2.2.1. Model - some assumctions and notation:
The delay experienced by a message is assumed to be
ccmprised of queueing time and data transfer time on the
transmission links or channels. The processing times at
nodes and the propagation delays are assumed to be
negligible.
Let M be the number of channels and c·1 be the
capacity of channels i~1,2,3 •• M. Each of these M channels
can be represented as a single server queue. The queueing
discipline at each of the queues is FCFS. The chan~els
are ass~~ed to be error-free and buffer space is unlimited
i.e. infinite waiting room at each queue.
Messages are classified according to source-
destination pair. A message belongs to class (s,d) if it
originates at source '5' and has destination 'd'. Thus
there are total of R=N(N-l) classes in a net~ork of N
nodes. The message classes can be numbered f~om 1 to R.
The a~~ival process of class 'r' ~essages from outside the
27
network is assumed to be Poisson with rate 7(r). The
message length for all classes is assumed to be
exponentially distributed with mean l/~. Thus data
transmission time of all messages at node ' i ' is
exponentially distributed with mean l/~C .. Both fixed and1
random routing have been treated in the original paper
[14], but here we concentrate only on fixed routing. In
fixed routing a unique path exists for each message class
'r', denoted by a(r).
2.2.2. Summary of Results:
let Air (i=1,2 ..•.M;r=l,2 ..... R) be the mean arrival
rate of class 'r' messages to cha~nel 'i', which can be
given as:
Air = 7(r)
=0
if channel i,a(r)
otherwise ( 2 • 3 )
Similarly, we can define p.lr'
as utilization of
channel 'i' by class 'r' messages,
p.lr
= A. / ( J.LC. )lr 1
( 2 .4)
The total utilization of channel 'it can then be
written as:
RPi:S 1: Pir
r::l
28
( 2 .5)
Pi < 1 for all channels if no saturation is to take
place.
Let t (x) be the probability density function of ther
end-to-end delay of class 'r' message.
Laplace transfo~ is given by:
*Then T (5), itsr
( 2 • 6)
The above result is a direct consequence of our
ability to trea~ delay at each channel independently
because of the 'independence assumption' and the output 0:a M/M/l queue being identical to the input process.
*T (5) can easily be inverted using par~ial fractionsr
technique. Mean and variance of delay can also be
*obtained directly from T (5):r
and
( 2 • 7 )
= 1: 1
it:a(r)f ILC (l-p )1 2L~ i i J
( 2 .8)
29
In our analysis of the problem we have taken the
above model as a guideline. But this model is very
idealistic in the sense that it allows only exponentially
distributed message length and no other service
distribution is assumed to exist on a source to
destination path. As ~e know, message length is rather a
combination of fixed-length header (trailer) plus a
randomly chosen length (which may be approximated as
exponentially distributed). Also we may encounter some
servers with constant service time, as we go from one
network to the other. Thereby, we will have to deviate
from Wong's model.
30
CHAPTER 3
Verifying Wong's Analysis
It was pointed out in the last chapter that
validation of 'independence assumption' if used to compute
delay has been established beyond doubt. The model
proposed by Wong (14] also incorporates 'independence
assumption' and can be used .to calculate even higher
moments of delay. In this chapter our stress ~ill be to
verify the reasonability of Wong's model when applied to
compute the higher moments of delay. As a by-product ~e
will also be able to ve:ify the 'independence assumption'
for computing mean value of delay.
31
3.1. Validation of Wong'~ Model:
For our purpose we need to compare the values of
higher moments of end-to-end delay as obtained by two
different methods viz. by invoking 'independence
assumption' and by not invoking 'independence assumption'.
Computations using 'independence assumptions' are rather
trivial and can easily be carried out. But when we turn
to 'non-independence' case, it is not so easy to handle,
and the only feasible tool available is 'simulation'. The
simulation model used to carry out the above computations
is briefly explained below.
We know that the reason that independence assumption
is a good approximation is the entry and departure of
multiple streams of traffic at all nodes. We can
represent a specific path as a tandem of service stations
(fig. 3.1). Messages originating at A are to be carried
to B over this path. The traffic from A to B is referred
as 'internal' or 'tandem' traffic. The traffic streams
which are not a part of A-to-B traffic can be ~epresented
as a single traffic stream (because of their Poisson
nature) called external traffic. External traffic
enters a particular node and leaves it without going over
to any other node in the tandem. (refer fig. 3.1) The part
of non A-to-B traffic which does go to the nodes other
than the one where it entered in the tandem, can be
32
assumed to be a part of 'external traffic' for those
nodes. The message length distribution is exponential
with equal mean for tandem and external traffic.
Configurations (i) and (ii) given overleaf are identical.
We have simulated configuration (ii)8 We also assume that
at all nodes we have infinite buffer space and there is no
balking. The se~vice time consists of only t~ansmission
~ime on ~he links between the nodes.
A --~~ B
33
fig 3.1:
(ii)
Representacion of a path in the net~ork.
34
The simulation program has the capability to simulate
the required path vith or withou~ independence assumption,
but it was used to collect data only for 'non-independence
assumption'. For 'independence assumption' the analytical
model was used to avoid excessive CPU time.
Several runs were made for 'independence' and 'non-
independence' cases by varying external traffic and
internal t:-affic. In the discussion to folloW' ·Je assum.e
throughout that our- tandem path consists of three nodes
only. All links have a capacity of 9600 bps. Data 'Jere
collected for- mean and variance of delay at individual
nodes as TJell as for end-to-end case. The detailed
discussion for diffe~ent runs is given belo~:
(i) The exte~nal tra:fic .as maintained at 3000 bps.
Internal t~a!fic varied from 125 bps to 3500 bps. Hence,
the range of internal traffic varies f~om a point ~here
internal traffic is negligible compared to the exte~nal
traffic, to the point where internal traffic is comparable
to the external t~affic. The data collected were for mean
of end-to-end delay, variance 0: delay at individual nodes
or links and variance of end-to-end delay (~eferred as
varianceE). A comparison of these val~es for
'independence' and 'non-independence' has been depicted i~
the table 3.1 and fig. 3.2 to 3.7
35
The following points may be established from the
collected data:
(a) The 'independence assumption' for delay is quite
reasonable and computation by two different methods
match pretty closely.
(b) variance of delay at individual nodes, computed
using 'independence assumption', also seems to be a
good approximation as long as tandem traffic is not
very high in comparison to external traffic.
(c) Although variance of delay at individual nodes,
in t~o cases, is almost equal at low tandem
utilization, the variance of end-to-end delay has an
error ranging up to 50% at these low values of tandem
utilization.
36
Table 3 • 1 : Comoarison of independence and non-independence (exter-nal traffic 3000bps)
,I Ena-to-endtanaem
DelayVar'. a~
traffic individual I Yare
(bits)(ms) nodes (ms sms ) I (ms*ms)
Iy I N t y I N I y I N I
I I I , I I
125 2321 2321
59631 5921 178891 335921250 2361 241
162001 6386 186001 3715°1
3i5 2411 243 1 64521 6451 193561 36947,500 2461 253
167191 7051 201571 39049,
251 1t
625 2561 7003i 6962 21009i 385741750 256 1 2641 7305
17411 21915
1414631
875 262 1 2721 76281
7720 228841
4245611000 268 1 2i61 7972
18117 23916
1438521
I2aa
l I, , -"'.-27~1 83401 8551 250201 443371
__ i.:;:)
300 11250 2801 87341 9197 262021 50252,1375 2871 299 1
91571 9397 274711 5075511500 2941 319l 96121 10178 288361 55008,
! t
30303\1625 302' 3171 10101i 10341 5500411750 309 1 3231 10628
110401 31884
1543641
1875 318 1 3261 111981
10476 335941
5604812000 326 1 3461 11815
112125 35445
1637051
I , I
2125 335 3531
124841 12170 37452' 6l~3512250 345 369
1132121 14028 396361 725741
2375 353 3641
140051 12214 420151 6457312500 366 388, 148721 14330 446161 72803,
!
2625 377 4051 15822i 15105 47~66i 8459512750 390 4101 16866
116031 50598
1778061
2875 403 4211 180171
16842 540511
8166113000 417 4381 1929°1 17872 57870
1893901, t
3125 432 4561
207031 17895 621u91 91~5913250 448 454
1222771 18053 668311 89~6~
3375 "*65 4871
240371 21342 721::1 105314 1
3500 484 4981 260151 2280-* 780451 110460:
~Y=Independence Assumption invoked;N=Non-i~dependence.
.-"'\.
37
oo...,
Legend:C -> independenceo -> non-independence
o:::-L-----.,.-----r-----....,..-----,.------,c
fig 3.2:
D.l a.J 0.3 o.~ 0.5
tanden utilizationComparison of 'independence' and 'non-independence' fo~a path shown in fig 3.1.
38
:r..,...
Nr..
........"--c
\Jr =.- "':~.O
D
•L.ot: \I!
...." c:,>.0r:
~ C Cc CCCcO ~ cC C a
C ~ c C c eCce CCae C a
"!J-----.,.-----r----~----,__---..,ca
fig 3.3:
D.l Q.J 0.3 o.~ 0.5
t~nden utilizationDelay compu:ed under independence has been no~ali:edwith ~espect to delay found by non-i~dependence.
39
o~N
111NN
~
CO~ 0
cr• ...,
"l-E.. U'I
"U1~
e"D0c 0
0r N-'
L Legend:~
> o -> independencea -> non-independence
inaJ
a~+-----....------~---~----..,..------,
0.5
fig 3.4:
....
Nus.o
lit
C.,..----.,....----,~---~----------
40
fig
a
3.5:
0.1 a.J 0.3
tanden utilizationVariance computed unde~
normalized with respect toindependence.
0.1' 0.5
independence ~as
variance foundbeen
by non-
41
InLegend:
M C -> independence...a non-independence->
&It......
"0... UI• ~
"DE
• U1p) ...E
vlJJ
1)
LJc
IIID &11....L~
>
U'IIf'
~-l-:=-----r-----r----~-----r------,a
fig 3.6:
D.l O.J 0.3 o.~ 0.5
~ande~ utilization
Comparison of 'independence' and 'non-independence' forend-to-end variance.
42
ro.
UIIII,
A 0
"\)D..~ ...
D ~
&.
0LaeC~ ...LI.IY:
&) 00eI
I- rD :-:> c
,.'".
c cC
C a aCeQ acacrzc c a
c
c
c
IIIr4o.,..----......---.....----~----...------.
D.l a.J 0.3 0.1
tancen utilization0.5
fig 3.7: End-to-end variance computed unde~ independence hasbeen normalized with respect to delay found by nonindependence.
43
(ii) Another set of 'simulation-runs' were made for
external traffic of 6000 bps at each node. Internal
traffic was varied from 125 bps to 2000 bps The results
obtained are shown in table 3.2 and fig. 3.8 to 3.13, and
are very much like the previous case.
44
Table 3.2: Comparison of indeoendence and non-independence (external traffic 6000bps)
Delay(ms)
N I!
a124218435°192877199486,
110486'117227112560811494961
I
2197401268066131638412802151
153S05,171513,180481,236023,"
621091668311721111780451
End-to-endVar.
(ms*ms)
1922761
1
219138,252048
12929681
84: 7411
1
923371010011110946
1
1224361135807115149il1700671
N
224-ii23180238352666829000320003330039641
43000459004637563662
55663741938100078956
y I1
Var. at Iindividual I
nodes (ms*ms) I,
408121452691504991566891
207031222i71240371260151
28247 1
130779133667,
369821
640921
1
730461840161976561
N 1,
5161539155115911
6311
432,451
14651
486r
767185018861931
1
y II
4321448146514841
6061638167417141
I
125250375500
625750875
1000
11251250137515001625175018752000
tandemt:affic
(bits)
·Y=Inde~endence Assumotion invoked;N=Non-lncepencence. ·
45
orr
~
• 0KI ,..
E~
Legend:C -> independenceo -> non-independence
c:.U1
0.2D.DS 0.1 O.!S
tande~ utilization
Compariso~ of 'independence' and 'non-independence' fora path shoYn in fig 3.1.
ao..L.---~---.,..----r----r----~~ 0.2S
fig 3.8:
46
N·
a:I~ 0LJ ....
'oJD....
...... CI
UI ~ w c ae rr C a c a.. · c aD 0 0e~
N~·
LC:
0-4-----~----,----...,.----_r_---___,
fig 3.9:
D.05 a.l o,~ 0.2 O.2~
tanden utilizationDelay compuced under independence has been normalizedwich respect to delay found by non-independence.
0P"1....
Legend:o -> independence0 -> non-independence
U"0-
r:c:..,.-I QI( aT
""I)faIt
illP) Lae...."
l)
0CIJ 0
&II....LJ;I,.
U1",
~+-~---""-----.,r------r-----,------,
47
D
fig 3.10:
D.DS D.l 0.15 0.2 0.25
t3nden utilizationComparison of 'independence' and 'non-independence'.
48
Nfit...
CN
I...~-0 =e Q aI
D ~ C
C1I
CE ct, UI a ce ~ c Cc I C D~ C'-'.
CC
CC Cc: ~IJ ~... Q!..a:>
I"t~
I
0
UI
o~----..------"'!'-----,..----------.a D.a~ a.l OILS 0.2t~nden utilization
fig 3.11: Variance computed undernormalized with respect toindependence.
independence has beenvariance found by ~on-
49
00m
~
0 /10
LaN
I,,IIt
0;/&0~ 0
N• N
"""1:1~E
• Q
IP'J IZI-4
E 0'V"
UJ
IL)0u 0c 0
J.J :r )1~....Legend:L
~
r/~ C -> independence)-
a -> non-independence0Q...
~/
~
o...l-...Q;=---...-----,r-----,.-----,-------,u:a'D 0.250.05 a.l O.L~ 0.2
tande~ utilization
fig 3.12: Comparison of 'independence' and 'non-independence' forend-to-end variance.
50
llIIlJ.c
UJ&JCC :faD ,..
.... Q
l-II>
cC a
C ec
c
c
C a
a
c
e
ao
D.QS a.l o.~~ 0.2
tanden utilizstiona
U'lJ- ...- -,. ..,.. -r- --,
o O.~5
fig 3.13: End-to-~nd variance computed unde~ independence hasbeen normali%ed with respect to delay found by ncnindependence.
51
(a) 'Independence Assumption' is again found to be
reasonable over most of the range of tandem
utilization for calculating mean end-to-end delay.
(b) Variance of delay at individual nodes behaves in
a manner similar to the last run.
(c) Variance of end-to-end delay, again, does not
seem to be a good approximation with independence
assumption. Even in the range where variance at
individual nodes is almost same in two cases,
varianceE may be in error by as much as 25%.
This, really, is an intriguing situation. On one
hand ~e find close match between 'independence' and 'non
independence' variance at individual nodes, (implying that
it is reasonable to assume independence), at the other
hand we may get an error even up to 50% when the same
assumption is invoked to calculate the variance of end
to-end delay. Or put in other words, 'independence
assumption' seems to work at individual nodes but not when
the system (tandem path) is considered as a whole. So
there must be some factor which causes some kind of
dependence among different nodes. And this factor cannot
be anything else but the length of tandem messages, which
is kept same all through the tandem. But then the
question arises how do we get around this factor ~hile
52
calculating variance at individual nodes ?
Eefo~e ~e ans~er this question, another system which
shows some~ha~ similar behavior is considered below. (:ig.
3 .14 )
.twt : ./M/l ./M/l
fig 3.14: Tandem of three ex?onen~ial queues wit~ Poissonar~ivals and ucili:acion approaching :ero.
~he exte~nal input to the sys~em is Poisson and a"
se~vice s~a~ions have exponentially dis~=ibuted service
time ~i:h ~ean equal to l/~. Also ass~~e ~~e util:zat:on
of the serve~s is close to zero. So any time a c~s~ome~
comes it will almost neve~ wait to get into service. The
service ~equir~ment of the cus~cmers stays cons:ant as
they go from one server to the next. Hence, this
situation is parallel to 'non-independence' case. The
mean delay at any service station is l/~ and its variance
is clea~ly 1/~2 Sut what about the mean and variance of
end-to-end delay ?
53
Now, if xl is random variable representing the
service requirement of individual customers, then mean of
end-to-end delay is given by, mean[3x ] = 3x and variance1 1
is given by var[3xl
J = 9var[xl
J , . rather than 3var[xl
J.
Hence, although, by using 'independence' we can correctly
find the mean and variance of delay at individual service
stations, it really does not help when we turn to end-to-
end delay.
Coming back to our original system, ~e find that
because of external traffic, the overall conditions like
queue length, mean delay, variance of delay at individual
nodes are controlled by external traffic. So when a
tandem customer arrives at a node, the above measures of
performance for it can be well approximated by
'independence assumption'. But once a tandem customer has
entered service, its service time is not at all under the
control of external traffic and it will stay 'same'
(assuming equal capacity of all links on the path) at the
subsequent nodes.
~~1
r~
><
t~
><
t~
><
fN
><
1'-- 1
..~
CQQ.
eou
eft::e....
..~.......,
54
55
Here we introduce the term 'partial independence' to mean
that only delay encountered in queues (excluding service
time) at individual nodes are independent random
variables. The original 'independence assumption' will be
referred as 'complete independence'. Consider below, a
system consisting of three nodes only. Random variables
x. are waiting times or delays as shown in the fig. 3.151
As argued earlier, if, the queue characteristics are
controlled by external traffic we can consider Xl' x3' xs,
to be independent random variables. But x2' x4' xs' are
not independent random variables (because the message
length remains same). Under these conditions, the mean of
end-to-end delay is given by mean[xl+x3+xS+3x2] i.e. the
same as with complete 'independence'.
given by
The variance is
Based on above argument varianceE was calculated for
different traffic conditions. A comparison of 'non-
independence' , 'complete independence' and 'partial
independence'
3.3 and 3.4.
is given in fig. 3.16 to 3.19, and tables
56
Table 3.3: Comcarison of comlete,partial and non-independence cases (external traffic3000bps)
N 1r
3359213715°136947139049,
4~3371
50252150755155008,
845951778061816611893901,
61435ii2:7~1
64573172803,
3857';1414631424561438521
I
550041543641560481637051
I
91~59\
894641la53l~
1':"0460:
p II
7838St831061883871943201
34i651348761356311364321
637~311
66875\70328,74146
1
537281559111582911608291
41296142479143748145111146578:4816°14987°15172°1
37284138191
139159140192,
y jI
Ena-to-endYare
(ms*ms>
25020j262021274711288361
621091668311721111780451
47~661150598
15405115787°1
21009 1
121915122884,
239161
178891186001193561201571
t
37452139636[420151446161
30303131884
1335941354451
N
59216386645170516962741177208117
178951805321j4222804
15105160311684217872
855191979397
l0178
12170140281221414330
1034110401104i612125
y I,
83~OI
873419157196121
I
70031730517628,
79721
59631620016452167191
207031222771240371260151
15822:16866
11801711929°1
Var. at Iindividual I
nodes (ms sms ) 1
I
124841132121140051148721
10101i10628
1111981118151
3531
3691
3641
388 1
3171323132613461,
2501264127212761
I288'300 1
299 1
319l
2321
241 1
243 1
253:
y 1I
1Delay I
(ms) J
IN I,
27~i
280129712941302:
3351345135513661
I
432144814651484
1
2321236124112461
251'
125250375500
2125225023752500
3125325033753500
2625275028753000
1625175018752000
1125125013iS1500
625750875
1000
* Y=Complete independence: P:Partial independence:N=Non-independence.
57
Legend:C -> complece independenceo -> non-independence~ -> partial independence
"o--41!!• rr
...,J)
e
• U1l. ,...e
vlLJ
1)
DI:r: ~....
:1-t------r-----.,...------,-----...----.....D.l D.2 0.3 o.~
tanden utilization0.5
fig 3.16: Comparison of 'independence', 'non-independence' and'partial independence' as used to find end-co-endvariance (external traffic 3000 bps).
58
Legend:C -> complete independence~ -> par:ial independence
a
cc cC
C ca cC a D CC CD C c a
c
,..It!.
...L&.i ~
1.) <:)
IJc~
L. rD r> c
v0
~l~,..lorQ .. J... .. ~
"~.A
•'"
4,A~
III ~ ..... ~ ~
"'" Q .. .. .£~ 4-C ..~U
~.... AD,...'" .. CD P-
6.
C)
1- C0
C cc.....,C CCC
'".....~ .....----......----,.....---......-----,,-----..
0.1 Q.J 0.3 O.~
~~ndc" utilizationo.s
fig 3.17: End-to-end variance compuced underindependence' and 'partial independen~e'
normalized with respect to va~iance foundindependence'.
'compLetehas beenby 'non-
59
Table 3.4: Comoarison of comlete,partial and non-independence cases (external traffic6000bps)
tandeml IDelayVar. at I End-to-end I
trai- I I individual I Var. 1fie I (ms) I nodes (ms sms ) I (ms*ms) I
(bits) Iy I
I I II ;
I 1I N I y N Y P N II 1
4321
, 1 I I125
I
432\ 20703 22447 621091 783851 81242,250 4481 451 1 22277 23180 668311 831061 84356,375 4651 465 1 24037 23835 721111 883871 928771500 4841 486
126015 26668 780451 943201 994861
t 1 !
1010161625 504 1 5161 28247 29000 84741i 1104861750 526' 5391 30779 32000 92337 108612
11172271
875 550 1 5511 33667 33300 1010011 1172781
12560811000 577 1 5911 36982 39641 110946
1
1272231
14949611
6311
153805:1125 6061 40812 43000 1224361 13871311250 6381 652
1 45269 45900 1358071 1520841 171513,1375 6741 672 1 50499 46375 1514971 1677721 180481,1500 7141 754 1
56689 63662 1700671 1863441 236023,I1625 760 1 7671 640921 59663 1922761 2085531 21974611750 811 1 8501 73046
1
74193 2191381
2354141
26806611875 870 1 8861 84016
181000 252048
1268324
1
31638412000 938 1 931
197656
1
78956 2929681
3092451
2802151I
*Y=Complete independence: P=Partial independence:N=Non-independence.
60
~#0
o~.1/
o /aI ~!I./. Legend:
~C -> ~omplete independence
. 0 -> non-independence4 -> partial independence
~/
~oJ....c::....--..----..,------..,r------,-------,UI
oom
C'U2N
J!0 e::-..-4 N• N
~
rte• Q
~CD~
E,~
a.&J
"0c: =D ~....1-~
>0Q,..
a D.a~ a.l O.lS o.~ o.=~
tandcn utilization
fig 3.18: Comparison of 'independence', 'non-independence' and'par:ial independence' as used to find end-eo-endvariance (external traffic 6000 bps).
,.. Legend:N
.... C -> complete independence~ -> partial independence
...J:.....c......
~ =L' rrP:' . ...... 0 A
-- ~ ~ AU Jj 4 ..E ~
'- •0 &Q ~
~II! CC I
.,J 0 ~ .a-D
~ 0c []Lj C C C C ~
U 0 Cc ~ c CI:J r: c..... Q
CL.[J')0
t'4~
0
61
fig
a D.O~ 0.1 0.15 0.2
t~~den utilization
3.19: End-to-end variance computed underindependence' and 'partial independence'no~malized with respect to variance foundindependence' •
'completehas beenby 'non-
62
The following points can clearly be established from
these curves:
(a) At low tandem utilization, 'partial independence'
gives results ve~y close to simulation data for 'non-
independence' • In this range of low tandem
utilization the error in varianceE may be brought down
from 50% to almost nil.
(b) At higher tandem utilization, the curves for
'partial independence' and 'non-independence' start
drifting apart, 'non-indepencence' curve being always
above the 'partial independence' curve. (This poses a
little problem because we are underestimating the
varianceE at these points. This can be overcome by
incorporating an appropriate safety-factor). The
region where 'partial independence' and 'non-
independence' curves a:-e not close by, is, in fact th.e
region where 'independence assumption' in general
starts fading out and even variance at individual
nodes is not well approximated by 'complete
independence' assumption.
At this point of our discussion, we should seek a
modification to the results obtained by Wong (14]. As
described in chapter 2, if tr(x) be the probability
density function of end-to-end delay of class 'r'
*messages, and Tr (s), be its Laplace Transform i.e.
63
= n s + ~C.(l-p.)ita(r) 1 1
( 3 • 1 )
The above result is under 'complete independence'
assumption. From this we can find the Laplace Transform
of delay encountered waiting in queue only ( i . e.
excluding service time), at all the nodes on path a(r).
*Let this Laplace transform be denoted by Tr q (5). Then
,uC. (l-p . ) s+,u.C.T *(s) = n 1 1 * 1 (3.2)
r q i 1: a ( r ) S + J.LC i ( 1-Pi) ,ue i
*Let T (s) denotes the Laplace transform for thers
*service times at all the nodes on path a(r). Then T (s)rs
is given by:
xli -s L c}
T *(s) = E[e l i e a I r ) iJ]rs
=( 3 .3)
*Now we can write Tr (5) as
* * *Tr
(5) = Tr q (5) X Tr 5 (5)
( JJ.C. ( 1-P • ) s +~C · 1= I n 1 1 * 11
lit: a ( r ) 5 + ~C i ( 1-P i ) ~c i J
{3 • 4 }
64
X t'
11Is. 1: C l + ~
i e a t r ) iJ
End-to-end delay and its variance can be calculated,
for class 'r' messages, along the same lines as:
and
1= . 1: ( ) J.'C. ( 1-p . )
1£ a r 1 1
(
= t I 1
i c a ( r ) t r~C. ( 1-p . ) 12II 1 1 J
(same as before)(3.5)
( 3 • 6 )
To further validate the above results a few more runs
were made 0 In one case we consider a tandem of 8 nodes
Yith different link capacities. The specifications of the
path and obtained ~esults are given in table 3.5. The
comparison of 'complete', 'partial' and 'non independence'
clearly shows the superiority of 'partial independence'
over 'complete independence'.
65
Table 3.5: Comparison of 'partial', 'complete' and'non' independence cases for a tandem of 8links (external traffic 3000bps, mean mes-sage SOObps) ·
I Ilnk IDelay
Var. at I delayN 1
I capacityl individual I square II (bits) I (ms ) nodes (ms*ms) I II I I (ms *rns) II I y N Y I N I II ! I'
6766:I
I 9600 82 82 6724 I 6724 II 4800 385 390 148225 I 1411651 152100 II 6400 172 181 29584 I 31016, 32761 II 9600 82 88 6724 I 6948 1 7744 II 4800 385 394 148225 I 1303311 155236 1I 6400 172 193 29584 I 333461 37249 II 9600 82 94 6724 I 7621
1
8836 II 4800 385 403 148225 I 118243 162409 II II End-to-end delay: II Y = 1745 II N = 1824 II End-to-end variance (varianceE): II y = 524015 1
I p = 861912 ,I N = 927116 I
I 1
*Y=Cornplete Independence; P=Partial Independnce;N=Non-independence.
66
The following table (table 3.6) shovs the same
compa~ison for calculation of third moment of end-to-end
delay, in a tandem of 2-nodes. Each node has an external
traffic of 3000 bps and link capacities are 9600 bps. The
curve for the same data is given in fig. 3.20
67
Table 3.6 Comcarison of 'partial' , 'complete' and 'non'independence cases as applied to compute thirdmoment of end-to-end delay(external traffic 3000bps, mean message SOObps)
Third moment oftandem I 1trafficl end-to-end 1(bits) 1 delay I
1 I1 y I p I N II 1 t I125.0 0.0110511 0.0152601 0.0150821
250.0 O.Ol1717r 0.0159751 0.0172441375.0 0.0124371 0.0167471 0.0173821500.0 0.0132171 0.0175811 0.018996,625.0 0.014: 064 1 0.018-*83: 0.0201911750.0 0.014985
10.019463
10.0214471
875.0 0.0159881
0.0205261
0.02464511000.0 0.017083
10.021685
10.0244571
r1125.0 0.0182801 0.0229481 0.02442511250.0 0.0195911 0.0243291 0.03020811375.0 0.0210311 0.0258421 0.03052711500.0 0.0226161 0.0275031 0.0314791
I !
1625.0 0.0243641
0.029331i 0.03005311750.0 0.026296
10.031348
10.0412711
1875.0 0.0284391
0.0335791
0.03346812000.0 0.030821
10.036055
10.0411641
I2125.0 0.0334771 0.0388091 0.04621112250.0 0.0364461 0.0418831 0.044499
12375.0 0.0397781 0.0453261 0.05143612500.0 0.0435281 0.0491931 0.05981°1
I
2625.0 0.047765, 0.053555, 0.07203312750.0 0.052570
10.058493
10.0749131
2875.0 0.0580421
0.0641071
0.06598113000.0 0.064300
10.070517
10.0791851
I
3125.0 0.0714921 0.0778711 0.116923,3250.0 0.0797971 0.0863511 0.094326\3375.0 0.0894401 0.0961821 0.117847,3500.0 0.1007021 0.1076471 0.110200,
*Y=Complete Independence: P=Partial Independnce;N=Non-independence.
68
o
.....
>D =... ~IJ 0
"La
~Q
c CcEc&t ~
-00L. Q.-
No.o
complece i~dependence
non-independencepa~tial independence
o ....----~----,.----......-----.,.._---...,a 0.1 D.J 0.3 o.~
tandcn utilizationo.~
fig 3.20: A comparacive look ac the three approaches when used tocompute the chird moment of end-co-end delay.
69
3.2. Conclusion:
In this chapter we studied the validity of
'independence assumption', especially when applied to
calculate the higher moments of end-to-end delay.
'Independence assumption' was found not to provide good
approximation for higher end-to-end moments even when the
individual nodes behaved as if they were having no
dependence between incoming traffic stream and message
length. In this context the concept of 'partial
independence' was introduced and was found to give better
results than the 'complete independence'. Wong's [14]
results were accordingly modified.
Finally, in the light of above results, in all our
analytical treatment to follow, we will use 'partial
independence' rather than 'complete independence'.
70
CHAPTER 4
Modeling the Network
In this chapter we will develop a suitable model to
represent the network interconnection as given in chapter
1. We start with rather an oversimplified model to get
the first feel of what is going on. Then ye go on to a
more realistic model for our representation of the network
interconnection base on Wong's [14] model and the analysis
carried out in the last chapter. This model is used to
find delay and its variance under different conditions of
internetwork traffic, intranetwork traffic, pathlength
through the networks, gateway processing times and message
lengths. Later on, the delay and variance in case of
different alternatives (as described in chapter 1) a~e
calculated based on this model. All this gives us a
suitable tool to measure the supe~iority of one
alternative over the other.
71
4.1. Some Assumptions:
For recapitulation fig. 1.5 is reproduced overleaf.
As described in chapter 1, the two alternatives which
gateway G1
has, to deal with the packets originating in
network A and destined for network A', are the following:
i) The first one (hereafter referred as pad-and
pass) is to embed the incoming packets (from A) in
header and trailer as required by network B, and
transmit.
ii) The second approach (hereafter referred as
strip-and-pad) is to first strip the header and
trailer from a packet which were attached to it
for its journey in network A, and then attach new
header and trailer as required by network B.
Net"orkA
NetworkB
fig 4.1: Interconnection of networks A, A' <identicalarchitecture} and network 8 (different architecture).
.......f\..)
73
Assumptions:
For all the models to be described the~e is common
set assumptions which will be applicable to all of them.
This set is given below:
i) Partial independence assumption is invoked
invariably (with the possible exception of
simulations where we do not have to face the
problem of analytical tractability).
ii) In calculating end-to-end delay error
probability is not taken into account. Although
any real network will have some error probability,
it still is a good approximation to assume error
free channels.
iii) Arrival rate associated with any traffic
stream is Poisson and is stationary.
iv) All queues have FCFS discipline.
v) Any message that enters the network has a
single destination, and will make to it sooner or
later without defection. This necessitates that
all nodes be having infinite waiting room. In a
real network buffer space is sufficiently large to
minimize loss of messages.
vi) A message is fully received before its
74
transmission can begin.
vii) Propagation delay and nodal processing time
are taken to be zero unless other~ise specified.
vii) The maximum packet length supported by
netvo~k B is a~ least as much as supported by
networks A and At. This assumption implies that
we will never have to face the problem
fragmentation and reassembly of packets9
In what follows, whenever we need any assumptions
other than those described above, they ~ill be given unde~
the heading 'Additional Assumptions'.
4.2. A Tentative Model:
As a first approximation (a
interconnection of fig. 4.1
following network of queues.
75
rather crude one) the
can be represented by the
- 6<
~I,I
_J -, II I .-- C
N IN
0N::..
n~....
~J
wU
N 1 a.JCC'.j
I e0
I u\II
1_, IQ.I...J,- "..-
I I .:I.~
r 0r , :I...
'N ~
=-1 J=- ~
c
=:r] J ~J:,
I 4t.J
I,....
I 0, I e0
_I J ..,-, a:Ie.... I N •...I 0
:L .. s..~
n I nQ.c.~
to:)I ..,
a'tI w
I :&.
N,IN .
~I I. '4'~ w ~"I r.c. Q. .....
-< nt
76
77
Here path 1 corresponds to pad-and-pass and path 2
corresponds to strip-and-pad.
In this model every delay element (gateways, networks
A/A', network B etc.) has been modeled as a single server
queue with exponential service time distribution. In fig.
4.2 either gateway has been shown as two queues. In fact
there will be only one queue at a gateway. The two queues
represent the fact that there are two alternate paths for
a message. Once a particular path (shown as dotted lines)
is chosen, all messages will travel through the same path.
In addition, complete independence is assumed among
different queues. Hence all our queues can be treated as
independent M/M/l queues.
Some notations:
A :
DP. :1
Mean number of messages originated per second
in network A and destined for network A'. This
process has Poisson distribution.
Service rate at gateTJay 'i' (i=1,2) if path j
(j=1,2) is selected.
End-to-end delay if path 'i' (i=1,2) is chosen.
Processing time at gateway 'it
path j (j-l,2) is taken.
(i=1,2) when
u . :1
Service rate in network S, if path i (i=1,2) is
chosen. It may be noted that to optimize delay
78
by choosing any of the two paths, delay or
message length through networks A and At is not
affected at all. So in subsequent discussion
we will concentrate only on delay encountered
at gateways and network B.
The delay on path 1 and path 2 can now be represented
as:
1 1 1(4: .1)DP l = A
+ A + - A.&0'11 - ~l - .u21
DP2 = 1 + 1 + 1 ( 4 .2)"'12 - A ""2 - A .u22 - A
A:so ~e know, processing time for pad-and-pass is
less than for strip-and-pad at both the gateways. i.e.
Also let,
( 4 .3)
The message length for strip-and-pad is less than for
pad-and-pass. Hence average service time at network 8
~ill be more for pad-and-pass. Let the excess mean
service time at network B for pad-and-pass be given by
'y'. Then ~e have,
79
1 = 1 + PI~12 ~ll
1 1= + P2
~22 ~21
1 = 1 - Y)J.2 ~l
( 4,. 5)
( 4 • 6 )
( 4 .7)
Special case:
Assume that gateways G1
and G2
have the same traffic
characteristics. Also it is reasonable to assume that
1.u11
= 1~21
( 4 .8)
1 1= (4.9)
}L12 . ,u22
Then from eqn. we get the delay on path 2 is given by:
DP2 = 2 + 1~12 - A ~2 - A
2 1= 1 + 1
1 - A 1 - A (4.10)-- + p - yJ.L 11 }Ll
Now if p=O, meaning, gateway delay is same
irrespective of the path chosen, then
2 1< DP 1DP2 = A + 1}Lll - (4.11)1 - A
,ul- y
The implication of the above result is rather
trivial. It simply means that if delay at the gateways is
80
not a function of the path chosen, then we must do strip
and-pad, so that shorter messages travel through net~ork B
and encounter less delay.
Without going anv. further in this oversimplified
model, we turn to a little mo~e realistic model for our
network representation.
81
4.3. Network-modeling based ~ Wong'~ Analysis:
In this section we model our network based on Wong's
model given in chapter 2 and the analysis carried out in
the last chapter.
Additional Assumptions:
i) For the time being we assume that the only
traffic to be carried through the gateways, is the
one going between networks A and A'. i.e.
gateways are fully devoted to the traffic under
consideration.
ii) The processing times at gateways are
exponentially distributed random variables and do
not depend on the length of the message to be
processed. This assumption will be removed in the
successive refinements.
iii) The traffic internal to the network S, is
assumed to have exponential distribution with the
same mean as tandem traffic. In practice, the
mean of internal traffic of network E does not
depend on the mean of tandem traffic. Thus while
traversing network B, We may encounter two
different message classes with different mean
message-lengths. In the
Wong [14], there is no such
original model due to
distinction and all
82
messages have the same mean. In the present case
we assume that there is only one message class and
will remove this assumption later on.
iv) The message lengths on two paths differ by a
constant length, implying if on st~ip-and-pad we
assume exponential distribution, then on pad-and
pass it must be a combination of constant plus an
exponential component with the same mean as on
strip-and-pad. Eut we assume that message length
on pad-and-pass too has exponential dist~ibution
but with higher mean.
Under above assump~ions the network can be modeled as
the following tandem of queues:
G1
:=J o(
fig 4.]:
Network 8
0 0 \( ,,;~
Representation of a path in the network.
G2
~Q-.-
oow
84
External traffic to all nodes has been represented as a
single traffic-stream. We have not taken into account, in
the above model, the external traffic to the gateways.
End-to-end delay calculations are carried out using
?artial independence and mean and variance are given by
(4.12)1Tr = . 1:( ) j.l.C. ( l-p . )
ll:a r 1 1
and
=(I 1
11 I
}r,uc.1 21l 1 J J
(4.13)
Having devised the above model, the delay
=alculations were done by varying the different t:affic
pa~ameters like gateway processing times and their
difference on t~o paths, mean message length of tandem
traffic and its difference on tvo paths, arrival ~ate of
tandem messages and the number of hops traversed through
network 8. ~he exte~nal t~affic on all the links was
assumed to be 3000 bps. All links have a capacity of 9600
bps.
The folloying notation has been used to represent the
various parameters:
pathl+:
path2-:
delay:
variance:
A :tandem
85
Indicates the curve corresponding to path 1.
Plus (+) sign indicates the fact that longer
messages will traverse the network if path 1
is chosen.
Same as for path 1. Minus (-) sign denotes
the presence of shorter messages on path 2.
Mean end-to-end delay on path 1 or path 2.
variance of end-to-end delay on path 1 or
path 2.
Internetwork t~affic (messages/sec).
differencex: Difference in message lengths on the two
paths.
m len.:1
n-hop:
Mean message length on path i (i=1,2).
Gateway processing time at gateway on path i
( i =1,2) .
Number of hops traversed through network B.
The results obtained as a comparison of path 1 and
path 2 are given in fig. 4.4. through fig. 4.15. These
curves clearly depict how the variation in different
86
parameters affect the mean and variance of end-to-end
delay. One thing clearly stands out by a mere look at
these curves: path 2 is to be preferred over most of the
range of controlling parameters. Path 2 has less delay
and variance for most part of 'difference in bits' for two
paths. Only close to a difference of 32 bits in message
lengths, path 1 seems to perform bette~. The same way
path 2 appears superior over most of the range of number
of hops to be traversed through the netvork a, the mean
message length to be taken care of and the input rate of
tandem traffic. In fact as we go from left to right on
these curves (i.e. as we increase the value of tnese
parameters) path 1 gets worse and ~orse.
The only cross-over points whe~e pa~hl gets better
are ~hen gateway processing time becomes excessive for
path 2 as compared to path 1.
Finally, based on the above model we can say that
path2 is to be preferred 'most of the time' (the phrase/
'most of the time' not to be confused with 'always') and
we postpone the furthe~ interpretation of t~e curves
obtained, till we go a step ahead to refine our model in
succeecing chapters.
87
In'-,....
Legend:~ -:> A = 3
tandem0 -> A = 1
tande..rn
;path2-
-z... pathl+
// --.r- pathl+
/ path2-
fCD ...--....--....- .......Q.
o
........~
IS
o-0 ~1
a-.0
't16128 12-.
di+f~r'ence
&It
Q ....------.----..,.------,-----,~J
Variation of delay with difference in message lengthson two pachs (P1=O.1, p =O.ll,n hoo=3,m len = 5002 _. - 2bics).
88
eN.o
.."o......._---..----.....---..,...-----,o ,:2 12B J2'1 320 'tl~
diffe~ence {bits)
fig 4.5: Variation of variance ~ith difference in messagelengths on two paths (P1=O.1, p =O.ll,n hop=3,m ten =
2 - - 2500 bits).
L'1,...,...
...
Legend:L1 -> ,.\
tandemo -> ,.\
tandem
= 5
= 1
89
L.,....>-
".....LJ /'1J ~,
r~
/...
...
U1
o1 3 't
no. of hops5 6
fig 4.6: Variation of delay with number of hops traversedthrough network B (Pl=O.l, P2=O.11,difference Inlengths=160 bits, m_1en2=SOO bies).
o
90
Legend:d -:> Atandem :2 5
a -> Atandem = 1
./
~ pachl+
,..o~ ....---.....----.....----,....----..---~
1 J It
no. of hopes
fig 4.7: Variation of variance with number of hops traversedthrough net~ork B (Pl=O.l, P2=O.11,dif:erence Inlengths=160 bi~s, m_len2=SOO bics)o
91
Legend:d -> Atandem = 3
o -> Atandem = 1
0 •.1.6
path2-
1~»>:~
~ pathl+
~
4~' O.1~ 0.15
98t~,-,sf pr'ocess.
Variation of delay wich gateway processing cime forpath 2. Gateway processing time is held constant at0.1 sec (difference in message lengths =160 bits,m_1en2=SOO bies, n_hop=l).
LI')e-
Q ....-----~=-""...-=~---....------------..
fig 4.8:
94
L.agend:o -> differenceX = 224 bits
N.
L)
ce rII •.... 0
i,g:>
C -> di::erenceX = 160 b:'cs
o-L----.......---~----_r_---..,..---__,10 SS loa J.~~ 1 ~fJ
mea8sge 1ensih on psth2 {bi~a)
fig 4.11: Variacion of variance wiCh mean messag~ lengch on pach2 (Atanaem=l,Pl=a.l, P2=O.11).
95
'l)
c-:0
Legend:o -> Pl = 0.13 sec \c:~ -> Pl = 0.10 sec
tQ
path2-
,,) JPco.Q
~ /~\
c pathl+>-G /0-o~
~c
0.48a.~~ d.09 O.D~ a.DSproce~8in9 time di~f.
~co ......----...----...-----....,...---~----..----...a.~ D.D'
fig 4.12: Varia~ion of delay ~ich gaceway processing cimediffe~ence on t~o paths (Atandem=l, n_hop=l,differencein message lengchs=160 bits, m_ Len2=500 bits).
96
.....o
c....o
n...00o&:II...t. ...G~
,.~
Legend:o -> Pl = 0.13 sec
d -> Pl 2 0.10 sec
pach2-
"pachl+
f
D.ar~.D' 0.0' 0.03 O.4aproce~sin9 time dj~~.
fig 4.13: Variation of variance with gateway processing cimedifference on two paths (Acandem=l, n_hop~l.diffe~encein message lengths=160 bits, m len~=500 bits).- ..
97
... Legend:~ -:> n_hop =3
LIt. o -:> n_hop =1...
55
pach2-
~ ,......-e~athl+
pathl+~~/
3 't
tand~m tra.f1=icJ1
fig 4.14: Variation of delayP2=O.11,difference Inm_1en2=SOO bits).
with tandemmessage
traffic (Pl=O.l,lengths=160 bits,
98
6
pathl+
5
pach2-
pachl+
3 '+
tandem ts-a-f.f:ic
Legend:d -> n_hop =3
1
N~.
...N
Q
1)
fJ ,..C m15 •.... 0LI,.
fig 4.15: Variacion of varianceP2=O.11,difference inm_1en2=500 bics)o
with tandem t~affic (Pl=O.l,message lengchs=160 bits,
99
CHAPTER 5
Towards a Refined Model
In this chapter we try to make a few refinements in
our model by removing some of the unrealistic assumptions
made earlier. Tvo of the assumptions made in the last
chapter can now be removed. The first one that
processing time at the gateways is an exponentially
dist~ibuted random variable independent of the length of
the messages to be handled, is not so reasonable. In
practice, only the latter part of the above statement i.e.
processing time at the gateways is independent of the
length of the messages to be handled, is true. But so far
as the distribution is concerned, it tends to be constant
rather than exponential. It depends only on the path
chosen.
The second assumption to be removed is that of single
class messages in the network B. Previously, we assumed
that the mean message length of the traffic internal to
the net~ork B is the same as that of tandem traffic. In
real life traffic internal to the network B and tandem
traffic may have different means.
5.1. Verification of Partial Indeoendence for ./2/1
100
tv~e~
gatewavs and ~ classes of messages:
We noted previously that the backbone of Wong's model
is Kleinrock's independence assumption coupled with the
fact that output of an MIMic system is identical to input
process. So it does not allow any cons~ant-time service
stations or multiple classes of customers. We, therefore,
first verify the effect of constant service time sta~ions
(in our case gateways) and multiple classes of custome~s
under pa:tial independence assumption.
To investigate the behavior of the network -i~h
constant service time servers we consider the falloYing
tandem of queues. The two extreme servers are our
gateways with constant service time. The remaining th~ee
se~vers represent the intermediate nodes .ith
exponentially distributed service time (message lengths
being assumed exponentially distributed). The external
traffic to the gateways is assumed to be 3 messages/sec.
The service time at the gateways is 0.1 sec. Exte~nal
t=affic to the other nodes is 3000 bps. Mean message
length for external as well as tandem traffic is 500 bits.
Capacity of all the links is 9600 bps.
101
~
.........Q
..........~
~
t123~....(If
0.0
QJe
en
~...
~
~
~
~
~
~
(J0.
0-
>
....
I.CIJ
--.....
02
%.........
..,
.e
nG3
*-J~
C0U
~"-
~.u....:teQ,)
~
e
n~
~
......
<;
.'"00.....
c.....
nt'--
102
The analytical model used differs a little from
Wong's model because of the presence of constant service
time stations. Mean delay at these stations was
calculated using M/G/l results. End-to-end variance was
calculated under partial independence assumptions. The
following M/G/l recurrence for~ula due to Takacs [21] was
used to find mean and variance of delay at individual
queues (not including the service time).
kw = 1 - p
where ;K is the k-th moment of delay in the queue.
"0w = 1. is the i-th moment of service time
distribution. From the above equations we get,
2 Ab 3var of queueing delay = (w) + 3(1-p)
( 5 • 3 )
End-to-end delay and its variance for the non-
independence case were found using simulations and were
compared with analytical results computed unde~ partial
independence assumption. The tandem traffic rate yas
va~ied from 0.25 messages/sec to 6.0 messages/sec. The
results obtained are given in table 5.1 and figs. 5.2
through 5.9. End-to-end delay calculations using
analytica~ methods are pretty close to the simulations
103
results with non independence. As gateway utilization is
increased these two curves start drifting apart. But even
in the worst case error in delay calculation using
analytical methods does not exceed -5%.
At low gateway utilization, there is very little or
no queueing at the first gateway. So any time a tandem
customer comes it will join the service with high
probability without waiting. Thus under low utilization
conditions the first gateway can be approximated as an
M/G/~ system whose output process we know 'is identical to
the input process[22,23]. Hence we get almost overlapping
curves in figs. 5.2 and 5.4 at low gateway utilization.
As we increase the gateway utilization the output process
of first gateway starts drifting towards one of a constant
interdeparture interval. But the decomposition of the
output in two streams, which are, tandem traffic and
external traffic, and the fact that this forms only a
small portion of the total traffic to the succeeding nodes
helps to keep the delay calculations within tolerable
limits of error. In fig. 5.6 through 5.9 the curves
showing the delay using the above mentioned two approaches
has been plotted against utilization of the links due to
tandem traffic alone. These curves clearly depict that
our constant service time gateways do not affect the
analytical results in any appreciable manner.
104
~able 5 . 1 : Compa~:son of pa~tial-independence and non-independence ~ith constant se rv : c e timegateyays(exter-nal traf:ic 3000bps)
I i End-to-endtandem DelayI gateway I Var.trafficl utiliza-l (ms) (ms*:ns)
I I(bits) I tion I ? N
fp I N I
125 0.325 1 ~ao 4791 38534: 390321250 0.350 I 490 4921 39916
1408351
375 0.375 I 501 5021 414311
423151500 0.400 I 513 5201 43099
1467491, I ,
625 0.425 525 527 1 44943' 46172i750 0.450 538 ~~~I 469931 50910,875 0.475 553 ~b 492841 54365,
1000 0.500 568 574 518591 541621
1125 0.525 Se4! SS9 ~~~~~l 6354811250 0.550 603 1 614 6791911375 0.575 622 1 632 61920 7062811500 0.600 644 1 657 66362 776511
IaS90S:1625 0.625 6681 687 71579
1750 0.650 6951 706 77787 90317,1875 0.675 7251 747 85285 966-i 4 t2000 0.700 7601 782 94499 1133~4!
2125 0.725 799: 836 106057 13677012250 0.750 845 1 878 120913 13962012375 0.775 SOOI 927 1~aS78 16280912500 0.800 966
1
975 167563, 18763il
1040:,
2625 0.825 10491 2062991 ~33"''''·it. :0'1
2750 0.850 11561 116°1 2652171 27ia6~12875 0.875 13031 1321
13620091 3724921
3000 0.900 15171
1429, 5391761 502102,
"?:Par~ al IndependenceN=Non- ndepencence.
105
o... rot0'"....•~ Q
• ce ...E~
~II~ 0
LJ =:I-C
:'egend:a -> non-independence~ -> ?ar~~al ~nQe?endence
D.'i a.! 0.5 0.7
sate~sy utilization
fig 5.2: Comoa~isvn
i.ndependence'gateways.
~f 'non-independence' and ?ar: al~n a canaem navlng constant ser~~ce : me
~
~
DD lSI ~~ ..a 4Il ~
OS..t... . .A ~ A .. A
~ 0 ~ .. ~ ~
-6 •
..IJ
•L.0C g...., =.
106
a.~ O.S O.~
s.te~ay utilizatian0.8
fig 5.3: Delay computed under partialnormalized wich ~espect coindependence.
independence hasdelay found by
bee:'\non-
107
0."o.~D.'1
Legend:o -> non-independencej -> ?arcial independence
D.~ O.S 0.7
S8te~ey utilization
. f" d d' fComparlson 0 non-in epen e~ce and partialindependence' on a tandem having constant service cimegaceways.
~ ..ll~!:::=-~-----r-----,-------,r-----....,..----.,D.~
0=:r
"'='..., 0cr-• rtt
"~E
• Q
J\ '=',..,e
'-"
1)
IJc: '='D --1
N
'-J:>
<:)r-4--t
fig 5.4:
r,.,...
,..N...
108
.--..g:IJ •! 0L-ae....,
..
0.8
fig 5.5:
a.5 0.5 0.1
9~t:~sy utili%s~:an
Variance ~omputea unde~ parcial independence has be~n
normaLi%ed ~ich ~especc ~o vari.nce found by ncnindepenaence.
.. 109
o~...
_ .... Q
• C1(....
E....,
i..eg~~(J:
o -> ~on-:nde?endence
~ -> parcial i~de?endence
oJ- ...- -,.. ..,... ,.- -,~
a D.1 a,J 0.3 o.~
tanden utilization
fig 5.6: Com~arlsc~
:':lae?enc:e~c~
saceways.
~t '~on-inde?endence' ana ?art~a.~~ a tandem havin~ conscanc ~e~vice :i~e
110
~n·
..·,..
"1)
1 =~~--'-0I:EleC 1:1~ ~·>.0r
0.1 a.J 0.3 o.~
t3nden utilization
&II J..----....----,....-----,r-------.,r-------,o O.S
fig 5.7: Delay compuced ~nder ~artial
nor~ali%ed wi~~ ~espect ~o
independenc:eo
independ~~ce ~as
delay touna :;y :'10n-
111
oPut
J:'=r
"Q.... 0c:r• ",
"f'E
• 0Lege~ci:1.\ '='M
E 0 -:> non- nc:ependence,"'
.l -> part al inde?e~cie!":ce0oc: '='~~,.~...
L,;:'-
0,...--1
~ .....----.....----~---.....,~-----,----....,a 0.1 a.J 0.3 o.~
t3nden utilization0.'5
fig 5.8: Comcarisonindep~ndence'gat evays .
ot 'non-lndependence' and parclal~n a ~andem having constanc service cime
....
...·"
.."g
D...c::I-'" ~
D · .~.. .Ite 0 ~
1- '" ~
0 .a. ..c...., ~
UI .a ... ...G
U · ~... .. • ••1) <:)
e 4E' It.,.L. ..J! ~> r:
c::a
112
'"oa D.l a.J 0.3 o.~
tanden utilizationo.s
fig 5.9: Va~iance computed unde~ par:ial independence ~as bee~
normalized wich respect to variance found by nonindependence.
113
Now we try to find out how the presence of multi
class customers in the network affect our results. For
this purpose we used the same model as described before.
We don't have any M/M/l queues in our tandem any more. So
results under 'partial independence' were found using
M/G/l analysis. Two different runs were made with mean
tandem message length 1000 bits and 2000 bits. Mean
message length of the exte~nal traffic was maintained at
500 bits. Tandem traffic was varied up to 3000 bps. The
results obtained with s:mulations and analytical model a~e
compared in fig. 5.10 to 5.17. For the case of 1000 bits
mean tandem message length, delay obtained for simulation
and analytical models match pretty closelya But for
variance the situation deteriorates as we increase the
tandem traffic. For the case of 2000 bits mean message
length, the identical observations can be made though the
drift between 'partial independence' and 'non-
independence' is somewhat apparent, especially for
'variance' calculations. But for low tandem utilization,
in both cases we get tolerable errors in mean delay and
variance using analytical models.
In brief, we can say that presence of multi-class
customers as well as constant service time stations is not
going to hamper our analytical results as long as the
tandem traffic is not very high. In practice ~e can
114
expect tandem traffic to be about 300 bps (with link
capacity 9600 bps). We have seen that in this range we
indeed get good results analytically. So, Wong's results
can now be extended to include our tandem consisting of
constant service time gateways and multiple classes of
messages, though they must be applied with utmost caution.
Let us denote by 'x' the random variable representing the
message length of the 'overall arrival process' (i.e.
tandem as well as external traffic) and by 'x ' the randomt
variable representing the message length of the tandem
t:-affic alone.
encountered in
*The Laplace transform, T (5), for delayrq
queues only can be ~ritten as (following
M/G/l results and the assumption that waiting times in the
individual queues are completely independent):
1 - Pi= n
i e a t r ) 1 _ r1 - a*(s)l,p.C.I-----1 1 L sx J
( 5 .4)
Here i is the mean message length for the overall arrival
process to link i (i.e. the mean taken over tandem and
*external traffic). B (s) is the Laplace transform for
service time distribution of the overall a~rival process.
The multiplicands on the right hand side are just the
expressions for the Laplace transform of queueing delay on
individual links represented as M/G/l systems.
115
*Let T (5) denotes the Laplace transform for thers
service times at all the nodes on path a(r). Then T *(s)rs
is given by:
( x t 1I -s 1: -}* . () c.T (5) = E[e l i e a r lJ]
rs
*Now we can write T (s) asr
( 5 • 5 )
T *(s) = T *(5) X T *(s) (5.6)r rq ~s
End-to-end delay and its variance can be calculated,
for class 'r' messages by inverting the above transform or
by using Takacs results and partial independence:
;2 x tP.) + . L( )c:-
1 i e a r 1
( 5 • 7 )
and
a 2 =r
( 5 .8)
116
:'e~e!'1C::
o 0 -> ~on-~ndepende~ce
: ~ -> ?ar:iai inde?enaence
0.. Q
Q ...~•'"• C
D rB~
>-D-- 0D =~
o,..
'/P
/:i-+-----.....---------".-----.....-----..a D.l a.J 0.3 o.~
tanden utilization0.5
fig 5.l0: A com~ara:~~e .JO~ It non-:nde?endence anc ~a~::aL
inde~encen~2 :~ ~ao:~13~~ ~elay ~n a :Ance~ ~l:~
c~ns:a~c ~e~v~:e ::~e za~e~ays a~Q 2 ~~3ssas j:message. l)ne :~aS3 ~as 500 bi:s mean ~eng:~ a~a ~~e
oc~e~ a mean or :000 oi~s.
117
NN...
...·~....-.LJ
1:1aJ ED .A..... ~ ~· A-. C) • .AD .A
~.a-
e ~ ~
LDr: La....,. =·Q~.,~
~
-C .,.r-:Q
0.1 a.J 0.3 o.~
tanden utilizationa
"'-L-----,.-----.-----,-----..,------.,o 0.5
fig 5.11: Same as fig 5.:0. Delay compuced using ?a~:ialindependence has been normalized wi:h ~espect to delayunde~ non-inde?endence.
118
oC).. Legend:
o -> non-independe~ce
~ -> partial independence
"o.... 0.~
eoc '='D 2...1JJ>0
<:>II!...
a
o~ ....----.....---~----...,..----.,...-----.,
0.05~.l a.~ 0.3 o.~
tanden utilizationfig 5.12: A comparative :ook ~c non-lnde?en~ence an~ partial
inde~encence :0 calculata va~iance on a tandem withconstant 5e~vice cime gat~~ays and 2 ciasses ~f
message. One :lass ~as 500 bits mean length and ~he
othe~ a mean oi :000 bics.
119
...·"""
..."IJI)........ CDrr
D · ..E 0
i,
e ~
c:"" aD
L)I: ~· ~
1j 0t: .A.,
·rtL.
.6IS ~ ..:> r: •Q
0.1 a.J 0.3 o.~
tanden utilizationa
III
0 ....----...-----....----.-,------.,,------.0.5
fig 5.13: Same as f:g 5.:2. Variance :ompuced using parcial:ndependence has been normaliz~ci ~ich ~espect co~ariance unae~ ~on-independence.
120
I
Legend:o -> non-independenced -> partial independence
'".NQ...
~ ....----~----,.----...-----,.---- .....a D.1 a.J 0.3 o.~ o.~
t3nden utilizationfig 5.14: A com~arative look at non-independence and par:ial
independence to calculate delay on a tandem wicheonstant service time gateways and 2 classes ofmessage. One class has 500 bits mean leng:h and theother a mean of 2000 bits.
121
NN....
......~
-00 ....I) ell.... ~
~· .A~ 0 ... ..D .a .A
EL0c LD
",' =·0>-D.....L)
" ~
r:Q
N'II·C
0.5D.l D.J 0.3 o.~
ta~de~ utilization
5.15: Same as fig 5.14. Delay computed using partialindependence has been normalized wich respect co delayunder non-independence.
a~J..._---..,~---~----r------r----,C>
fig
122
eo~
er
.,.Q 0~ Q• a:I
'"DII
• c'-' ...Ii
'-'
Ll0 Legend:e
0D uz o -> non-independenceL. 4 -> parcial independenceQ)-
caIII
a 0.1 a.J 0.3 o.~
tandcn utilizationfig 5.16: A comparacive look at non-independence and partial
independence to caleulace variance on a tandem withconstant service time gateways and 2 classes ofmessage. One class has 500 bies mean length and t~e
other a mean of 2000 bics.
123
....
....~
...."Lj
D......... a:!~
" .! 0
L.~0
c" ~
La
0as.
D C> ~~
..cIJ~
L-a ~
" r:Q
NLa.o
&II
~~----...------....----....,~-----r----....,a D.1 a.2 0 •3 0 • 't 0 .5
t3nden utilizationfig 5.17: Same as fig 5.16. Variance computed using partial
independence has been normalized with respect tovariance unde~ non-independence.
124
5.2. Comparison of the two Paths:
We devoted the last section to establish that even
with constant service time gateways and multi-class
customers we can apply analytical techniques in the range
of interest. In this sec~ion we investigate our refined
model for network-interconnection representation. In this
model both gateways spend constant time to p~ocess any
message. The processing time depends only on the path
chosen. End-to-end delay and variance were calculated by
varying the different traffic parameters like gateway
processing times and their difference on two paths, mean
message length of tandem traffic and its difference on two
paths, arrival rate of tandem messages and number 0: hops
traversed through network 8. The external traffic on all
the links was assumed to be 3000 bps. All links have a
capacity of 9600 bps. So the network characteristics are
essentially same as for the model in the last chapter. In
figs. 5.18 through 5.29, is shown the comparison of the
two paths for different traffic conditions. Most of the
curves have been drawn for mean message length on path 2
as sao bits and tandem message arrival rate as 1
message/sec, thus giving tandem traffic of 500 bps. Also
most of the time we consider only one hop through the
network B. In the following we take up discussion of all
these curves one by one.
Leg~:1d:
.l ->.~ tandem = 3
a -:> Atandem = 1
125
=Q
.,£J~ pathl+
~~-"30---~--~_~ pa th2-
5~
lit
o32 1211 22'1
di1=fcrence'tlPS
fig 5.18: Var:ac~on oi 1eiay -len diffe~ence in ~essage ~eng:hs
on :wo ?acns (p~=O.i, ?~=O.11,n_hop=3.~_Le~~= 500bi:3).
126
rN
o
~....-.z- pathl+
Leg~nd:
.l ->~~anaem = 3
a -> \:anaem = !.
pathl+
4-z- pach2-
NN·
r...I) ·QL) -:c:D
/---z......4
L /lit
/II ...> ·c
~--~--'i!i---o£l~~ pach2-·
'"oo~----.....----,.----.,...-----,
32 1.21 :2'1 :0 '115
dl+fe~ence (bits)
fig 5.19: Variacion ~i ~ariance ~i~h difference in messa~~
Len~t~s on :~o pachs (Pt=O.:. ~2=O.11,n_hop=3.m_~en:=500 bi:sJ.
127
In figs. 5.18 and 5.19,we have plotted delay and its
variance against the difference in message lengths on two
paths. The two sets of curves are for tandem traffic 3
messages/sec and 1 message/sec. The message length on
path 2 is 500 bits and gateway processing times for path 1
and path 2 are as shown in the fig. As expected if we
increase the difference in lengths of path 1 and path 2,
the delay and variance on path 1 start getting worse.
After a while path 1 moves beyond delay and variance for
path 2, and keeps getting worse as we further increase the
difference in lengths. The cross-over point moves towards
right as we increase the tandem traffic. This implies
that at higher tandem traffic, to offset the excess
processing times at gateways on path 2, the difference in
lengths on two paths has to be higher.
Next, we look at how the number of hops to be
traversed through network B affect our choice of a
particular path. As the number of hops is increased delay
and variance for both paths increase but with unequal
rates because with every additional hop, path 1 because of
longer message lengths, suffers more additional delay than
path 2. Thus we see that although initially delay for
path 2 , for tandem traffic equal to 5 messages/sec, is
higher than on path 1, v i th increase in number of hops,
1ftr:·....
&IIo
I) III-eN·
...
\It,.·
fig 5.20:
128
:"eg~nd:
.1 -> A:andem = S
0 -:>~:andem = 1
pachl+
3 't
no. o~ hops
Va~iation of delay wich number of hops erave~sed
:hrou~h network 8 (p~=O.l, ~2=O.11,diffe~ence ~n
leng~hs=:60 bies, m_1en2=30o bits).
129
m&It·'='
-Z- pathl+
,.,~·<:)
l) / Legend:1)
'"~
E: m ~ -:>~:a:ldem = 5
IS ·... e\. IJ -> \:ancem = ,g:>
'"~Q
55
pach2-
1
o
m....
,.,oo
3 't
no. of hOp3fig 5.21: Variation at ~arlance ~lth ~umber of hops :~ave~sed
cnrough ~e:~orK 3 (o~=O.l. D~=O.ll.diffe~ence In. ~
len~~ns=:bO OlC3. ~_len~=SOO bics).
130
delay increases faster on path 1 and we see a cross-over
at n hop = 3 • Hence if we concentrate on delay and
variance for Atandem
=-5 messages/sec, path 1 appears
super'ior for n_hop<=3. But this high traffic is very rare
to come across in p~actice. For A d =1, path 2 is atan em
better candidate both on the basis of delay and variance.
Another point to be noted is that cross-over point moves
towards the right as tandem traffic increases.
Fig. 5.22 and 5.23 show variation of delay and
variance ~i~h gate~ay processing times for the parameters
shown i~ the respective figures. Quite naturally the
delay anc variance on path 2 inc~ease as gate~ay
processing time is increased for path 2 (for path 1 it
being held constant). Hence after a certain value of
gateway processing time, path 1 becomes p~eferable. Eut
as a~gued earlier sao bps is a more likely situation in
practice and for this path 2 performs better up to around
0.125 sec of gateway processing time yhen processing time
for path 1 is 0.1 sec.
Figs. 5.24 and 5.25 show how the delay and variance
vary with mean message length. For the given set of
parameters path 2 is better throughout.
The next set of curves is for difference in gate~ay
processing times. Initially path 2 seems to perform
131
~agend:
In .l -.:>.4.:anciem = 3c0-....
0 -:> A.. d = 1~an em
0 •.113
pathl+
f
d.~' O.l~ O.l~
sate",., p,...occse.D•.1:2
fig 5.12: Va~iaclon or deLay ~i~~ gaceway ?~OCesslng cime Eorpac~ 2. Gac~~ay ?~ocessing cime is heid conscant ac0.1 sec (dif:ere~ce In message le~gths =160 bies,m_len2=500 bi~s. n_hop=l).
132
Leg~nd:
.l -> A = J~andem
o -> \ = 1:andem
o0 ...c N~ .
..... 0C-o>
r.....Q
pachl+:QJ.-~'-':::::::'_-----_---.l~-----------"'"
~ path2-
4.1' O.l~ 0.15 o.!e9ate".r pr-ocea8.
fig 5.23: Variacion of variance ~ith ~ateway ?rOCesslng ~ime :orpath 2. Caceway p~ocessing ~lme is neld c~nstanc at0.1 sec (diffe~ence in messag~ Lengchs =160 bi~s,m Len =500 bics, ~ hoc=l).
- 2 - ·
133
N
~egenci:
o -> di£ferencel = ,., I,--- :>l:SU'1r-....
&n.
U'!....
C -> di::erenceX = :50 bi:s
.} pathl+
1ft
~ ....----....-----.,.----~----.,...---~l~ 55 loa l~~ L~O
meassge length on pa~h2 {bi~s)
fig 5.24: ~ariacion ot Je~ay ~ltn mean ~essag~ Lengch on path 2, l. _. p -.1' ~ -0 . 1 )'"':andem-·· L-v ..... ~2-·J. •
134
~egend:
0 .. > cii:ferencel .:: 214 oics....C -> di:fe:-~:lc:el = 160 bi~s
~.o
cue UIII •... 0
L.
";
O+-----...----......----r-----.....,.---~
pathl+
10 SS loa
mea.age length onJ..-'S 1.'0
peth2 (bi cs)
:i& 5.25: Variation of yariance ~i:h ~ean messag~ ~eng~h on ?ach2 (A~andem=l,?l=O.l, P2=a.ll).
135
r..2ge~ci:
o -> ~, = O.~J 5ec- .~ -> ?l = 0.10 sec
1.1)e-.Q
pathl+
!«).Q
....
0.1e~-L------.----....,----r----.....-----r-----,'='0.0.1 D.al'a.o~ 4.0' O.D~ o.as
proccasing time diff.
fig 5.26: Variacion ot delay ~ith gateway processing ~lme
diffe~ence on cwo pachs (A:andem=l, n_hop=l.diffe~~ncein message Lengchs=:60 bies, ~_len2=SOO bits).
It)-e.Q
ro0.0oeG....to __
a~
"0
N<>.Q
:ig 5.27:
136
pathl+
~
pachl+
path2-
:'e~enc::
a -> ~, = 0.13 sec. -
a.o~ 4.D' 0.0' o.asproce•• lns time dLFf.
Va~ia:lon ai ?arlanCe ~i~~ ga:e~ay ?r~cess:n~ c~me
ciiference ~n t~c ~achs (A~ande~=l, n_ho?:l.dii:e~~nc~~n messa~e Lengths=L60 bies, m ien~=500 bi:sJ.- '"
137
Leg~nd:
~ -> n_hop =3
o -:> n_hop =1
L>U
IJ',
o1 3 't
tandem tra-f.J=ic
path2-
6
fig 5•28 : Va r i a c ion 'J t" ,:e Lay
P2=O.ll,diif~~e~ce ~n
m_Len2=500 btts).
~ich ~andem :raffic (0,=0.1,:ne5 5 age l eng t hs= ~ 6a '. b i cs ,
138
N'&1
:1~e~enc::
.l -> ~ nop =3
0 -> n hop ="!-I:' 1-,. path2-
N
I~·0
I)
0N
paChl+c mII ·.. 0
LaPJ>
r-4
~~
N...·
N
0J.. ~---...,..----~----~---_..,o1 3 't
~andc:m ira.f~ic
....,fig 5.29: Variation of varian~e
o sO.ll,diffe~ence in~2~en =500 bics).
- 2
wi:h tandem :ra:fic (? =0.1,message lengc~s=i5a lbi~s,
139
better. But as we increase the difference path 2 starts
getting closer to path 1, ultimately crosses over and
performs worse than path 1.
Finally, mean delay and variance have been plotted
against tandem traffic (messages/sec). With increasing
traffic the delay and variance on path 2 increase faster
than on path 1. The cross-over point moves towards right
as number of hops through network B are increased. Path 2
outperforms path 1 in the region ~hich is most likely to
occur in p~actice.
In brief we can say that path 2 is more likely to be
a better choice than path 1, although ultimate choice of
any path depends on the overall traffic environment
(gateway processing time, message lengths to be taken care
of on tvo paths, tandem traffic, external traffic etc.)
specific to any ne~work.
It will be a nice idea to compare our previous model
with the one discussed in this chapter. For both models
~e concluded that path 2 is a better choice than path 1.
But model described in the last chapter is biased more
towards path 2 and pLedicts it to be better even while it
is just the reverse if we use our present model.
140
Finally it may be commented that there is still some
room for further ,efinements and we make an effort to do
so in the next chapter.
141
CHAPTER 6
Final Model
In this chapter one last attempt has been made to
refine our model for network interconnection. We know
that message length distribution in a network is not
truly exponential. In fact any message will consist of a
fixed header and a randomly chosen length. Moreover, the
difference in lengths over two paths will be only in the
constant part of the message. In what follows we make an
attempt to incorporate these features in our model. We
start with the validation of independence assumption for
the new model and then find the trade-offs between the two
paths.
142
6.1. Verification of Partial rndeoendence fo~ D+M tyoe of
message distribution:
We assume that randomly chosen part of a message is
exponentially distributed. To investigate the behavior of
the network ~ith constant plus exponential distribution ~e
consider the tandem of queues shown in fig. 6.1. The two
extreme servers are constant service time gateways. The
remaining three se~vers represent the intermediate links.
The external traffic to the gateways is assumed to be 3
messages/sec. The service time at the gateways is 0.1
sec. External traffic to the nodes is 3000 bps. The
capacity of all the links is 9600 bps.
143
~
........Q
.........
(J....~
""'"'=
en
w
QJ
u
:sQJ
~
~
Q3
c-
c
nw
...QJ
.........
J.J
Q
>C
+
~
:z:.........
.&:
.~"-
~....3
eQI
~C
n~
i-
.......<;
-.&J
eo....\.w
nt'-
144
The analytical model is again based on M/G/l results
due to Tackcs (21] and partial independence assumption.
Tackcs recurrence :ormula is reproduced below:
kw
A k k! bi+l ~:I 1 - P .Eli!(k-i)!·(l+l)·~11
1=
( 6.1)
kwhere w is the k-th moment of delay in the queue.
-0·11 = 1. b·1 is the i-th moment of service time
districutlon. From the above equations we get,
w = ( 6 • 2 )
var of queueing delay = (w)2( 6 • 3 )
End-to-end delay and its variance for the non-
independence case were found using simulations and were
compared with analytical results computed under pa~tial
independence assumption. The tandem traffic ~ate ~as
varied from 0.25 messages/sec to 6.0 messages/sec. TMO
different cases for message length distribution Ye~e
considered. In one case exponentially distributed part of
messages had a mean of 500 bits and cons~ant part
consisted of 96 bits. In second case exponentially
distributed part of messages had a mean of 500 bits and
constant part consisted of 224 bits, thus fOLming almost
50% of the exponential part of the messages. A comparison
of non-independence and pa~tial independence assumption
145
has been given in figs. 6.2 through 6.9.
The results obtained are really g~atifying and we get
a close match bet~een analytical and simulation results,
especially at low tandem utilization (defined as
utilization of the links due to tandem traffic alone). As
we increase the tandem utilization the error using
analytical results start increasing. But in any real
network we can expect a tandem traffic of about 500 bps
(tandem u~ilization =.05) and the analytical results are
quite close to simulation results in that low range of
tandem traffic. So the Laplace transforms presented in
the last chapter and equations 5.7 and 5.8 can directly be
applied to our new model.
and
x2+p . )
1
X t1: -
. () C .1 t: a r 1
( 6 . 4 )
a 2r
(r a , ~ 12
P. j 1I 1 X I + 1. X \
= i£;(rd Ixc. '2(1 - Pi)J1 xc? 3tl-Pi)1l l 1 1 J
2( 11
+ i L c-} var(x t )li£a(r) iJ
( 6 • 5 )
146
o......
Leg~nd:
o -> nOn-lnae?enaenCed -> par~iai inJepenaence
~Q...o ...•....,
~';"':~---""----"""------:"-----r------,0.5a.l 0.2 0.3 O.'t
t8nde~ ucili:stion
fig 6.2: Compar1.son of 'non-i~depe:"1de~cev and 'parcialindependence' used to compuca end-eo-end delay.Message len~th consists of a neade~ of 96 bi:s and anex?one~tially distributed ~arc ha~:n~ mean sao bics.~e gace~ays have constanc se~vice c~me.
147
..,~
""-0 JItC ~ A0 qz.... rr ....... 0G&C-OC ~
'" ~.>.0(7.....•-0 ~
r-:Q
hasnon-
0.50.'1'a a.l 0.2 0.3
tande~ ueilizationSame as fig 6.2. Delay for partial-independencebeen normalized with respect to delay forindependence.
~J-----.,------.,~-----r-----r------,o
fig 6.3:
148
&It
'II.
.,.Q ~~ at• ~
'"6• &II
• .0
('lit
1ft•""G
o '"ea::--a:-t)-
III
G'I..
Legend:a -> non-independenced -> partial independence
a a.l 0.2 O~ O.'t
tandCB u~ili%ation
fig 6.4: Compa:-ison of 'non-independence' dr:d ~ar:iaL
indepencence' ~sed ~o compu~e ~~~-:~-~~c ':a~iance.
Messag~ ~e~~:h ~onsists of a heade~ ~t 96 O~~S i~d a~
expcnenc i a l l y distributed ;:a:-t ~a'/ln~ me an SOO 01.':5.
The ga t eva y s have ccns eanc sere/lee ci me,
...·'"
..-0
C)
"... rzs.,.G ·2- 0C-OC~
GI
0lSI·0 e
CG...c,G ~
)- r-:Q
D D.I 0.2 0.3 o.~
tende~ u~ilization
149
fig 6.3: Same as fig 6.4.been normalizedindependence.
Variance for partial-independenc~ haswich respect to va~iance for non-
150
ooC't
'"....c
... '"0"--'•
Legend:a -> non-independence~ -> partial inde?endence
et ..... ...... .-.,. ..,.- p-. -,
WID.l G.2 0.3 OG~
tandeR ucili%stiono.s
fig 6.6: Comparison of 'non-independence' and 'partialindependence' used :0 compute end-~o-end deLay.Message length consists of a heade~ Ot 224 bits ana anex?onencially dist~ibuted part havi~g mean sao bics.!he gace~ays have conscant se~vice cime.
151
...."-0
(,
o ~.... .".
~Q
GI:~oC -=...,. cz;
>-cG
0.50.1 Q.~ 0.3 O.~
tande~ utilizationa
Ln
o~-----r-----""'-----""'---"""-----
fig 6.7: Same as fig 6.6.been normalizedinde?endence.
Delay~ich
for partial-independence hasrespect co delay for non-
152
Legend:a -> non-inde~e~de~ce
~ -> part:ai :nce?e~dence
&It~ ..s::::::;;:.._..... ,... ,...- ..., ...,
o.~
1ft.•III
,.0 ~.-1 ..• :r
"6S III• .0 fSI~
Sv
0Q 1ftC 0
a &ItC4...
"a>'"~..
6.3: Comuarison of '~on-i:lc:e?ende!"lce' and 'parcialindep~ndence' used co c~m?uce ~~d-~~-en~ variance.Message length conS1SCS of a header vi :2~ bics and anexponentially distr:ouC!d ?ar: havin~ mean 500 bics.Thega CetJa y 5 havee 0 ns : ar, C s ~ :"' "/lee : irne •
153
...·,..... ...-0 ..00........ ID~e ·! 0 .. .641
..C-o ~ ..C ... ...~
.A
CZI
CaD .. ~ .....·0 0 ..
CG
"r1
C.
" ~
)- ~Q
o
o.sa.l 0.: 0.3 o.~
tende~ utilizationa
III~ .. .....----.,...----,.----~r-------'
fig 6.9: Same as fig 6.8.been normalizedindependence.
Variance for parcial-independence has~ith respect to variance for non-
154
6.2. Comoarison of path 1 and path ~:
Having investigated the effect on our analytical
model, of the messages with fixed length header we move on
to find out the t~ade-offs between the two paths (pad
and-pass and strip-and-pad). The network configuration is
essentially the same as considered previously. External
traffic to the gateways has a mean interarrival time of
0.3 sec. Service time for external traffic is constant
(0.1 sec). External traffic on the links is assumed to be
3000 bps with mean message length of 500 bits, of which 96
bits form a fixed length header and the rest is
exponent~ally distributed with mean of 404 bits. All
li~ks have a capacity of 9600 bps. Tandem traffic passing
through network B has to have a header of 96 bits to
comply ~ith the protocols in this network. So for pa~h 2
(strip-and-pad) all tandem messages will have a fixed
length heade~ of 96 bits and the rest of the leng~h will
be chosen randomly with exponential distribution. On path
1 (pad-and-pass) fixed part of the tandem message consists
of the header for net~ork 3 plus the header with which it
started its journey in network A. To find the trade-of:s
between the two paths their behavior was investigated for
varying traffic environments as in the last chapte~. The
curves obtained have been sho~n in fig. 6.10 to 6.21.
155
...
en Legend:N... d -:> A = 3tandem
0 -:> A = 1tandem--....
>--rL) ~ pathl+-C~.
0
..e-::.-- paCh2-
~pathl+
mQ
III ~.......- ......--~--~~path2IQ.
U'I
oSJ US J21
di1=,cerence't16
fig 6.10: Variation of delay ~ith difference:n message lengchson t~o paths (Pl=O.l, P2=O.11.n_hop=3,m_1en2= saobits).
156
rN
o
r..\) <;)1)
I:D...L ,.~IS ...
> ~
Legena:.1 -> Atandem = 3
o -> Ata.ndem =a 1
-z. path2-
pathl+
i ~~
12! 22'1
di1=fer-enee
:
~re:) ~i----""'-'-----r-----r------,~ .,1'5
=:IQ
e
N...
fig 6.11: VariationLeng~hs onsao bies).
of ~ariance ~i:h diffe~ence in messa~e
t~o paths (Pl=O.l, P2=Ooll.n_ho~=3,m_ten2=
157
...~.
"!.z.. ~...... ...,... ,..... -,. ---,o
U'I.Legend:~
>- d -:> ,.\ = 5IS tandemo pathl+
0 -:>"tandem :: 1."
653 't
no. of hope
Variation of delay wich number of hops traversedthrough network a (Pl=O.l, P2=O.11,difference Inlengths=160 bits, m_len2=SOO bits).
1
fig 6.12:
158
Legend:~ -> \:andem = 5o -> " = 1:a~de:n
1ftUI.cr
'"~.o
oo '"emI ·
o..e CLJ:!:>
'"....Q
~ pat~l+
1ft...,
'"oo 4.t:=:=:--...,..----r------r----.,.------,
3 't
no. ot hope
fig 6.13: Variation oi variance wieh number ~f hops :~ave~sed
through nec~orK 3 (Pl=O.l, P2=O.11,dif:erence inlengths=160 bies, m_Len2='OO bits).
159
In Legend:e-
~ -> A = 3.... candem0 -:> A = 1
candem
~pat:h2-
...
pathl+
1
O.J.S
~path2-
a•.1~
It)e-c ...~....... ..... ...
~10-+------.-----..-----..,-.-----,.----,..-.---..,a.~ D.~~d.~' 0.1' 0.1'
98te~8T process.
fig 6.14: Variation of delay wich gateway processing time forpath 2. Gateway p~ocessing cime is held constant ac0.1 sec (difference in messa~e ~engths =160 bies,m len =500 bics, n hop=l).
- 2 -
160
/path2--.../
//
//
pachl+
I:
~~andem :: 1o ->
Lagend:~ -> ~t:andem = 3
".
-.Q
......
:t ..J.' 0 •1'1 0 .15 e•.1.es-to'W.y p....oc:c:s ••
fig 6.15: Variacion of va~iance ~i~h ga~e~ay processin~ cime forpath 2. Cateway processing time is held ccns~anc ac0.1 sec (difference In message tengc~s =too bits,m len~=500 bies, n hop=l).- - -
162.
N
Legend:0 -> differenceX = 224 bits
U1r- C -> differenceX. = 160 bit:s...
} pathl+LIt....
'"o~--- ......----...,...-----r----..,-----,10 S~ lOa ~~5 ~,o 235
~ea8ege length on peth2 {bi~8) -ta!fig 6.16: Variation of delay with mean message length on path 2
(Atandem=l,Pl=O.lt P2=O.11).
162
Legend:o -> dif:erenceX = 224 bics
C -> differencaX = 160 bics
} pathl+
O~----..----..----......----....----~10 55 loa l~S l~O 235
me •• _ge length on pe~h2 (bits) -ta!fig 6.17: Variacion of variance with mean message length on pach
2 (Atandem=l,PlzO.l, P2=O.11).
163
\
-.path2-
1
""C!0
Legend:0 -:> PI = 0.13 secfJ:!
Q
;,l -:> PI = 0.10 sec
II')e-.Q
C-011'1~
C> ~ -.,.~ --.:,. - ......-----------.:J....
/,/
pathl+
f
0.46a.o~ d.O' O.D' O.DS
procc88ing time diff.
~..L..._--.....---...,...---..,....---...,-----r---.,Q D.enQ.Q1
fig 6.18: Variation of delay ~ich ~ac~~ay p~oce~St~g :i~e
difference on ~~o pachs (A~anciem=l, ~_~o~=i.aiffe~encein message lengchs=160 bies, ~_L~n2=500 ~l~S).
164
Le~end:
a -> 0 = 0.13 sec• T-
.1 -:> P. = o. :0 secJ.
~oOcoCG
D.OTa.~~ t.a, O.D' o.~s
procea.ing time dLf~.
O-t,----....---~--- ......~--- ..,.1----......----...a~
fig 6.19: Variacion of ~arlance wi:h ~acayay ~~ocessing timedifference ~n t~o pa:hs (A:andem=l, ~_~oc~i.differ~nce
in message len~:~s=:~a bic5. ~_~en1=SOO OlCS).
N
:.Itr:.
Legend;~ -.> n_hop =3
o -> n_hop =1
3 't ~
tandem trs-ffic
165
path2-
fig 6.20: Variation of delayP2=O.11,difference inm_1en2=500 bits).
~ir:h tandemmessage
c~aific (Pl=O.l,lengchs=160 bits,
166
J 3 ~ S
'tandem tr-9.f~ic
Va~iaciQn of variance wich candem cra:fic (p,=O.l,~2=O.11,difference in message lengchs=160 ·bi:s,m_1en2=SOO bits).
1
Noo
N...
NLDQ
Legend:d -> n_hop =3
NIII
C' a -> n_hop =1
~,I} ~ath2-
~. /'10
IJIJ NI: m., ..... Q
L.g:"
N
~Q
fig 6.21:
167
Fig. 6.10 and 6.11 show the variation of delay and
variance as the difference in message lengths on the two
paths is increased. The two sets of curves are for tandem
traffic 3 message/sec and 1 message/sec. The other
traffic parameters (gateway processing time etc.) are
sho~n in the figures. For A = 1 message/sec, path 2tandem
seems better both with regards to delay and variance
though the difference in variance for the two paths is not
1pathmessage/sec,3=appreciable. For Atandem
outperforms path 2 with respect to delay for difference in
length less than 64 bits. with regards to variance path 2
is better only when the difference in length is more than
160 bits.
Similar interpretations can be sought for other
curves. On the whole path 2 again appears to be superior
than path 1 for the traffic conditions often met in
p~actice. But here we would like to address another
important issue. Starting with a very modest model for
our network interconnection, we have brought it into the
current form. But do we really gain anything by adding
more and more complications? Does it really affect our
choice of a particular path? To answer these questions it
will be sufficient to compare the current model with the
one described in the last chapter.
168
In the previous model all message lengths were
assumed to be exponentially distributed. The difference
in lengths for the two parts was considered as the
difference in the mean of this exponential distribution.
But in our present model, this difference lies only in the
constant part of the message length which really is the
case in practice. So the coefficient of variation for
path 1 message lengths is less in the present model than
in the previous one. This facto~ has an important impact
on the relat:ve performance of path 1, especially on the
variance, due to decreased randomness. Path 1 is no
longer that muc~ of a villain now though we still get
better perfo~ance on path 2 most of the time. The slope
of the variation for delay and variance with cifferent
parameters has decreased because of fixed length header.
Eu~ as expected this dec~ease in slope is more for ~ath 1
~han for path 2. If we compare the variation of variance
with difference in lengths on the two pa~ts, the previous
model shows clear superiority of path 2 over path 1, but
with the present model path 1 appears better for A =3tandem
and almost as good as path 2 for A d =1.tan em
The similar observations can be made in fig. 6.12 and
6.13 where ~e have plotted delay and variance against
number of hops t~aversed through network B. The margin
between path 1 and path 2 is less than in the previous
169
model. The same type of behavior can be seen in the rest
of the curves. In all these we see that the cross-over
point has moved in favor of path 1. This movement is
rather substantial for variance curves (compare curves
6.17,5.25; 6.19,5.27; 6.21,5.29). All these factors
indicate that the present model gives better insight into
the network behavior than any of the previous model.
This concludes our discussion of the present model.
It may again be stressed that no fixed boundary lines can
be drawn between the relative performance of path 1 and
path 2. The network parameters must be studied carefully
before any decision can be taken.
170
CHAPTER 7
Conclusion
As noted p~eviously, this thesis is devoted to a
somewhat specific configuration in interconnection of the
communication networks. But some of the tools developed
and the conclusions ~eached are quite general in nature
and may find wide spread application in the area of
computer communicationo In what follows ~e present an
overview of the work done and the conclusions reached.
7.10 Overview:
In chapter 1, the issues related to interconnection
of networks (homogeneous as well as heterogeneous) were
discussed and a detailed discussion of the thesis problem
was presented. Chapter 2 ~as devoted to an introduction
of Kleinrock's 'independence assumption' (10] and J.W.
Wong's analysis of computer networks (14]. It was
followed by verification of the same in chapter 3 • We
:ound in chapter 3 that Kleinrock's 'independence
assumption' was quite a good approximation when invoked to
find mean delay. The results computed analytically match
very closely v i th those obtained using simUlations. It
vas discovered that if the same assumption is applied to
find the variance of delay, it works fine if ye a~e
171
concerned with variance at the individual nodes, but in
computing variance of end-to-end delay we may make an
error as much as 50%. An explanation to this behavior ~as
sought and the concept of 'partial independence' was
introduced. By 'partial independence' we mean that the
delay encountered in queues alone (excluding service time)
is calculated the same way as under 'independence
assumption', but the total service time (transmission time
on the links) for the messages is calculated with the same
message length traversing across the path. A comparison
of 'complete independence' and 'partial independence' with
simulation results was convincing enough to adopt 'partial
independence' in our subsequent analytical treatment.
,In chapter 4 a comparison of the delay on two paths
described in chapter 1 (viz. pad-and-pass and strip-and-
pas) was carried over. Strip-and-pad was found to be
better than pad-and-pass for most of the traffic
conditions. But for the modeling purposes we had made
some assumptions which are not so reasonable. In chapter
5 we introduced constant service time gateways in place of
the gateways with exponential distribution. Moreover we
allowed the situation where internetwork traffic and
intranetwork traffic for net~ork B could have different
mean message lengths. Thus two classes of messages had to
be treated in the new model. For these new conditions
172
independence assumption was analyzed and gave good results
as long as 'tandem' traffic is low as compared to the
'external' traffic at the individual nodes. Again a
comparative study of pad-and-pass and strip-and-pad vas
made based on the refined model.
Chapter 6 ~as the final step towards netwo~k
modeling. Here we considered messages to be consisting of
a fixed length header plus a randomly chosen length having
exponential distribution. Independence assumption ~as
again put to verification fo~ this type of message length
distribution, and was once again found to be sat:sfacto~y
as long as the 'tandem' traffic is low compared to
'external' traffic. For this last model a comparative
study of the two paths showed that pad-and-pass is not
that bad as shown by the models in chapter 4 and 5. In
fact a careful analysis of t~affic environment is required
to be done before one can settle on either of the t*o
paths.
173
7.2. Future Work:
In all the analytical treatment done so far a number
of assumptions were made - some of these for the. sake of
mathematical tractability and others so that we don't
expand our problem too much (still taking care that our
model does not misrepresent the real system). Some of the
aspects where further investigation can be carried out are
suggested below.
We concentrated all the time on the case ~here
network B supports at least as large packets as supported
by networks A and At. This saves us the trouble of
dealing with packet fragmentation and reassembly.
Investigation of a situation where we may have to break
messages into smaller packets before feeding in network S,
will be highly desirable.
It was assumed that all channels are noiseless. This
is not true. The probability of error is also a function
of message length. In pad-and-pass longer messages travel
across the network and as such will be more prone to
errors than the relatively short messages on strip-and
pad. The probability of error has a direct effect on the
delay (caused by retransmissions). A comparison which
takes this factor into account should be looked into.
174
Finally, the buffer occupancies on the two paths is
likely to be different. It is affected by the tyO
factors:
i)On pad-and-pass service time ~ill be more than
on strip-and-pad, thus resulting in higher link
utilization on path 1. This in turn implies that
on the average there will be more messages
waiting in queues at the store-and-for~a~d nodes
on path 1.
ii)Not only the queue lengths ~ill be more O~
path 1, but also the buffer space occupied by
individual messages ~ill be larger because of
increased message-length on path 1.
Hence we will have relatively less buffe~ occupancy
on path 2 (strip-and-pad). It will be interes~ing to
study the network behavior taking into account t~e fini:e
buffer size and the possible buffer overflow. ~he
probability for blocking for path 1 and path 2 may then be
compared. A nice account of net~orks with blocking may be
found in [24,25].
Lastly, different parameters of performance (i.e mean
delay, variance, probability of blocking, error
probability etc.) can be assigned certain weights and a
ney measure of performance which takes all the above
parameters in:o account can be defined. Then this measure
175
can be ~sed to find which of the two paths performs
better.
176
List Of References
[1] Carl A. Sunshine, "Interconnection Of ComputerNetworks," Computer Netvorks 1 (1977) pp. 175-195.
[2] Vinton G. Cerf, Peter T. Kirstein, "Issues InPacket-Network Interconnection,ft Proc. IEEE, Vol. 66,No. 11, Nov. 1978, pp. 1386-1408.
[3] Andrew S.Tanenbaum, "Computer Netwo~ks,ft PrenticeHall Inc. Englewood Cliffs, New Jersey 07632.
(4] John F. Shock, "Packet Fragmentation In Inter-networkProtocols," Computer Netwoks 3 (1979) pp. 3-8.
[5] Jonathan S. Postel, "Internetwork P~otocol
Approaches,ft IEEE Trans. Commun., Vol. COM-28, No.4,April 1980, pp. 604-611.
(6] A.M. Rybczynski, M.S. Unsoy, "X.7S Internet~orking
Flow Control Considerations," ICC 1981, pp. 50.1.150.1.6
(7] Ross Callon, "Inte~network Protocol," Proc.Vol. 71, No. 12, Dec. 1983, pp. 1388-1393.
IEEE,
(8] A. Rybczynski, "X.25 Interface And End-to-end VirtualCircuit Service Characteristics," IEEE Trans. Commun.Vol. COM-28, No.4, April 1980, pp. 500-510.
(9] D.R. Boggs, J.F. Shock, E.A. Taft, R.M. Metcalfe,·PUP: An Internetwork Architecture," IEEE Trans.Commun., Vol. COM-28, No.4, April 1980, pp. 612-624.
[10] L. Kleinrock, "Communicatlon-Het~~,asticMessageFlow And Delay," New Yo~k, McGraw Hill;-!364.
~
[11] R.R.P. Jackson, ~Queuing Systems With Phase TypeService," Opere Res. Quart., Vol. 5, 1954, pp. 109120.
(12] E. Reich ,"Waiting Times When Queues Are In Tandem,"Ann. Math. Stat., 1957, pp. 768-773.
[13] P.J. Burke ,"Output Processes And TandemProceedings of the Symposium onCommunication Networks and Teletraffic,N.Y., April 4-6., 1972, pp. 419-428.
177
Queues,"Computer
Ney York,
[14] J.W. Wong, "Distribution ofMessage Switched Networks,"(1978) pp. 44-49.
End-to-end DelayComputer Networks,
In2
[15] F. Baskett, K.M. Chandy, R.R. Muntz, F.G. Palacios,"Open, Closed And Mixed Networks Of Queues WithDifferent Classes Of Customers," J.A.C.M. 1975, pp.248-260.
[16] F. Baskett, F.G. Palacios, "Processor Sharing In ACentral Server Queuing Model Of Multiprogramming WithApplications," Proc. 6th Ann. Princeton Conf. onInformation Science And Systems, Princeton, Ne~
Jersey, 1972.
[17] K.M. Chandy, "The Analysis And Solutions For GeneralQueuing Networks," Proc. 6th Ann. Princeton Conf. onInformation Science And Systems, Princeton, NewJersey, 1972.
[18] B. Avi-Itzhak, "A Sequence Of Service Stations WithArbitrary Input And Regular Service Times,"Management Science, Vol. 11, Ne. 5, March 1965, pp.565-571.
[19] Izhak Rubin, "Message Path Delays In Packet SwitchingCommunication networks," IEEE Trans. Commun., Vol.COM-23, no. 2, Feb. 1975, pp. 186-192.
[20] M.C. Pennotti, M. Schwartz, "Congestion Control inStore-and-forward Tandem Links," IEEE Trans. Commun.,COM-23, No. 12, Dec. 1975, 1434-1443.
[21] L. Takacs, "A Single Server Queue With PoissonInput." Opere Res., 10 (1962), pp. 388-394.
(22] N.H. Mirasel,poisson," opere
"Theres.,
Output of A M/G/~
1963, pp. 282-284.Queue Is
(23] G. F. Ne~ell, "The M/G/- Queue," SIAM J. Appl. Math.,Vol. 14, No.1, Jan 1966, pp. 86-88.
178
(24] A.G. Konheim, M. Reiser, ftFinite Capacity QueuingSystems With Applications In Computer Modeling," SIAMJ. Computing, Vol. 7, no. 2, May 1978, pp. 210-229.
[25] F.G. Foster, E.G. Perros, ·On The Blocking P~ocess InQueue Networks," European Journal of OperationalResearch 5 (1980), pp. 276-283.
92
Legend:~ -> ACandem = 3
a -> Atanciem :: 1
/par:hl+
a ..1~ -J •.1, 0.1" 0 .1S O.le D•.1.1'
sete1'eT pr-oceaa.
Variation of variance with gateway ?rocessing ci~e forpath 2. Cateway processing time is held constant at0.1 sec (diffe~ence in message lengchs =160 bits,m_1en2=SOO bits, n_hop=l).
....... ...--~ ........_-------.....----------~~--- ..CJ
~
~-=+----..",..----....----~----.----....---....
D•.u
fig 4.9:
93
224 bits
160 bitsC -> differenceX =
Legend:o -> di£ferenceX =
In.....
&II~
I
o
anC;; .....---....,..---~,.....---..,...----....----..l~ S5 10D ~~5 ~'O
mea8~9c length on p~th2 {bits)
fig 4.10: Variation of delay with mean message length on path 2(Atandem=l,Pl=O.l, P2=O.11).
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