8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
1/21
Mo d eling and Perform ance A nalysis
of
Mu lt ih o p Packet Radio Netw ork s
FOUAD A. TOBAGI, FELLOW,
IEEE
Invited
Paper
The design of packet radio networks involves aarge number of
issues whic h interact in a very complex ashion. Many of these
pertain to the RF channel and its use, others pertain to th eopera-
tional protocols. Clearly, no single mod el an be form ulated whic h
incorpora tes all the necessary parameters and leads to the opti -
mum solution. T he one ssential e lement w hich complicatesmat-
ters is that, contrary to point -to-p oint networks in wh ich each
channel is uti l ized by a s ingle pair ofodes, the radio channel in
packet radio networks
i s
a
multiarcess
broadcast
resource: i) in a
given ocality determine d by adio connec tivity, he channe l is
shared by man y conte nding sers, hence the need or channel ac-
cess protocols; ii) radios a broadcast medium and thus the action
taken by a nodeas an effect on theactions taken by neighboring
nodes and the ir outcom e.
Despite the complexity of the pro blem, there has been signif i-
cant progress worth reportin g on. The work accomplishe d
so
far
has been either the nalysis of spec ific xamples of networks or an
attempt to create models that would be useful n the design of
general networks. The purpose of this paper is to survey the var-
ious modeling echnique s hat have been used for the perfor-
mance analysis of packet radio networks, and to discuss the as-
sump tions underlying these models, their scope of applic ability ,
and someof the esults obtained.
I. INTRODUCTION
Packet Radio s
a
communications technologywhich
ap-
plies packet switching to the domain
of
broadcast radio.
It
is an attractive concept because
it
brings together the ad-
vantages of both broadcast communications and packet
switching. The broadcast radio medium is
a
readily avail-
able resource, easily accessible, and particularly suitable
for communication among mobilesers; packet switching
offers the air and efficient sharing of communicationse-
sources by many ontending users with unpredictable nd
bursty demands. A packet radio network s typically a col-
lection of packet switching odes that communicate with
each other via broadcast radio. Each node consists of a
broadcast radio and a digital controller.The broadcast ra-
dio acts like the modem andine in a ire-based terrestrial
Manuscript received March I O , 1986; revised August 14, 1986.
This
work
was supportedby th e Defense AdvancedResearch Proj-
ects Agency (DAR PA) unde r C ontr act MDA-903-84-K-0249.
The author
i s
wi th th e Departm ent of Electrical Engineering,
Comp uter Systems Laboratory, Stanford University, Stanford, C A
94305, USA.
network, providing the connectivityetween neighboring
radios. The igital controller provides theacket switching
functions, receiving packetsfrom neighboringadios, mak-
ing rout ing decisions, and forwarding packets to the next
radio
[26]-[28], [76].
The design of a acket radio ystem involves aarge num-
ber of issues which interact in a very complex fashion
43].
Many of hese pertain to the
RF
channel and
i t s
use, others
pertain to he operationalprotocols. Clearly, no single
model can be formulated which incorporates all the nec-
essary parameters and leads to the opt imumsolution. It is
often the case that certain design choices are dictated by
physical or technological constraints; others re driven by
past experience or he desire for simplicity n system man-
agement. For example, for rapid development and forase
of communication among mobile users, it may be advan-
tageous to have
all
users employ omnidirectional ntennas
and share a single high-speed channel.In fact, by provid-
ing he available communication bandwidth as a single
channel to be dynamically accessed by the users, the ad-
ditional benefit of tatistical load averaging
i s
gained, lead-
ing o higher channel efficiency.)ith such considerations,
the design space i s somewhat reduced. Additionally, one
may select many of the remaining design variables on in-
tuitive grounds,with the hopehat the designwill work;
however, with a system as complex as this, the end result
is very likely to be uncertain
at
best.
More important thanever before, performance analysis
via mathematical modeling and simulation should be an
integral step of the design process. It i s a sensible and in-
expensive way by hich one i s able to compare various e-
sign alternatives, gain insight into the behavior of packet
radio systems, predict heirperformance,
etc.
Unfortu-
nately, even when the system functionality has been sim-
plified nto
i t s
most essential elements, the modeling of
packet radio networkshas proven to be a formidable task.
What makes packet radio networkso different from point-
to-point networks which we seem to handle airly ade-
quately? Why re the existing techniquesevised for po int-
to-point networks ot readi lyappl icabletopacket radiosys-
tems?Theoneessential difference that complicates matter
is that, contrary to point-to-point networks in which each
PROCEEDINGS
OF THE
IEEE, VOL.
75,
NO.
1,
JANUARY
1987
135
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
2/21
channel is utilized by a single pair of nodes, the channel
in packet radio networks i s a multi-access broadcast re-
source: i) in a given locality determined by radio connec-
tivity,theradiochannel issharedby manycontending users,
hence the need for a channelccess protocol;
ii)
he chan-
nel i s a broadcast medium, and thus the action taken y a
device has an effect on the actions taken by neighboring
devices and their outcome.
Despite the complexityof the problem, there has been
significant progress worth reporting upon. The work ac-
complished so far has been either an attempt to create
models that would be useful in the design of general net-
works, or theanalysis
of
specific examples of networks. In
some cases, the effort has permitted
a
comparison of al-
ternatives pertaining o some specific aspect of thedesign
problem, such as, for example, channel access protocols.
But overall, the effort as not yet led o a true network e-
sign methodology, and a reat deal of work remains to be
done. The purpose of this paper i s to survey the various
modeling techniques that have been used for the perfor-
mance analysisof packet radio networks.We shall discuss
the modeling assumptions underlyinghese models, their
scope of applicability, and the results obtained. Through-
out the paper, particular attention wi ll be given o the ap-
propriateness of the modeling assumptionsmade in each
case in view of he intended objective. his
is
important in
order to clearly identify the sefulness of the modeling f-
fort accomplished. Indeed, we recall that': A theory has
only the alternativef being right or rong. A model has
a
thirdpossibility: i t might be rightbut irrelevant." (No dis-
cussion, however, will be provided of network design and
optimization; in[39]he reader will find aurvey of studies
related o spatial reuse of channel bandwidth n packet ra-
dio networks and to topological design.)
The content of the papers
as
follows. In Section I I , we
provide some background to assist the reader in under-
standing the problems underlying thenalysis of multihop
packet-switched networks. We first review the work ac-
complished in point-to-point wi reetworks, and then dis-
cuss briefly the characteristics of packet radio networks
which render hese different and more complex. n Section
I l l we address i n more detail two main aspects of packet
radio operation which are particularly relevant to perfor-
mance modeling. The first pertains to channel signaling
(narrow-band and spread-spectrum)nd the resulting con-
ditions for the correct reception ofackets. The second re-
fers to channel access. We escribe
a
number of ccess pro-
tocols uitableornarrow-bandand pread-spectrum
systems. Sections IV-VI deal with performance modeling
and analysis. The performance measures of interest are i)
networkcapacity(defined asthe maximum throughputthat
the network an support for given traffic requirement pat-
tern) and
ii)
the network's throughput-delay characteris-
tics. As the models used to derive these two performance
measures tend to be of different ypes (and more complex
for the atter than the former), e discuss thesewo issues
separately. We address he networkcapacity in Section IV.
We begin there by examining thepecial case of single-hop
fully connected networks;hese are impler to analyze and
have thus been the subject f many studies; their models
From
A Select ion of Scient i f ic Quotat ions, col lected by A .
L.
Mackay
M.
Ebison,
Ed.).
are important as they constitute the groundwork for the
analysis of multiple-access protocols, and multihop packe
radio networkswith general topologies.We then conside
the problem of deriving networkapacity for networkswith
general topologies. We present a general Markovian mod
which can accommodate he many different combination
of channel signaling characteristics and channel access
protocols, and which has been used to compare various
such alternatives. In Section
V,
we consider the problem
of
deriving the throughput-delay performance ofpacke
radio network.We review various methodssed in solving
this problem. We presentome approacheswhere model
are analyzed exactly and areypically useful o investigat
small or specially constructed networks.We then discuss
some approximate analyses suitable for more general to
pologies. In Section VI, we discuss the role of computer
simulation in relaxing many of the simplify ing assumptio
introduced in the various mathematical models, and dis
cuss their own limitations. Section V l l l concludes the pa
per.
I . BACKGROUND
To best understand the task of modeling he perfor-
mance of amult ihop packet-switching network n general
and to appreciate he complexity underlying that f packe
radio networks n particular, we begin by reviewing brie
the work done on point-to-point packet switching net-
works, such as ARPANET, and then discuss those features
of packet radio networkswhich are responsible for he di
ficulty in analyzing them.
A. Modeling Point-to-PointPacket Switching Networks
A packet switching network i s a complex network of
queues towhich there i s no tractable solution. According
many simplifications are introduced, reducing the view
the network o merely a simple abstraction.n the ase of
ARPANET, for example, the network was stripped of all
i t s
operational protocolslink level, routing, flow control,
tc.),
so that, with theassumption ofa known fixed routing, th
only network aspect that
is
modeled
is
that of queueing
packets in ntermediate nodesfor the appropriateinks, and
then transmitting themon these links when the atter be-
come free. (Note that n point-to-point networks theres no
contention on the links n the network, as only one node
pair may transmit and receive on any given directed ink;
furthermore, each node s equipped withs many modems
transceivers
as it
has outgoing links
so
that transmissions
on separate links are totally independent). With the addi-
tion of other simplifying ssumptions, suchas inf ini te buf
ers, exponential packet length,Poisson packet generatio
at
the sources,and independencebetween nterarrival
times and service times (Kleinrock's celebrated ndepen-
dence assumption [31], [36]), the model i s a network of
queues
of
the Jacksonian type.
The network i s considered to be consisting ofN nodes
and M channels numbered 1 , 2 , .
. .
,M. Let yjk denote the
average number ofpackets per unit time to e transmitte
from source j to destination k
N N
y
e
yjyir;
j = l
k = l
136
PROCEEDINGS OF
THE
IEEE, VOL.
75,
NO. 1, JANUARY 1987
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
3/21
and
f f / k = * f j k / Y .
A
The matrix
[ f f j k ]
represents thetraffic pattern, and
y,
the net-
work throughput. Let
Ci
denote the capacity of channel i
in bits per second and let X; denote the mean traffic rate
in packets per second that flows on channel i.Given [ y j k ]
and the rout ing scheme, the vector { A,}y=, i s easily com-
puted. Let
I /p
denote the mean packet length, and
nj,
the
number of packetswaitingfor or usingchannel
.
or a Jack-
sonian network, the vector (nl, . * ,nM)s a complete state
description; moreover, the state probability s given by Rn,)
P(n2)
. . .
P(nM),where P(nj)
s
the probability of finding nj
in anM/M/l queue with arrival rate X i and service rate /pCi.
From this resuIt,theexpected end-to-end packetdelayaver-
aged over all user traffic is easily derived as [36 ]
M
T =
1 = 1
x[-].pcj 1
hj
Equation
1)
i s important in that
it
provides the network
throughput-delay tradeoff for the traffic patternf f j k ] .
t
also
provides the network capacity for the pattern
[ f f / k ] ; his
is
simply the value of y at which the delay becomes infinite;
or thevalue of y for which the hannel with themaximum
X i / p C i ,
say channel
io,
ecomes saturated (i.e., when X;
=
pCi,). Furthermore, (1)proved o be extremely useful
n
net-
workoptimization. Indeed,
it
represents anobjective func-
tion which has the important property of being eparable
inthedesignvariables
{Ci,hj)E1.Thiswastakenadvantage
of immensely
n
the design of point-to-point networks
36 ] ,
1241.
B . Packet Radio Network Features
A major designssue in packet radio networkss the man-
agment of the
RF
bandwidth. As pointed out in the Intro-
duction, to satisfy communication among mobile users, and
to gain higher bandwidth efficiency,
it
i s advantageous to
consider the entire network toe operating n a single ra-
dio channel to be shared by all nodes. The sharing canbe
done either n the timeomain, or in the ode domain; or
a combination of both. For simplicity i n presentation and
discussion, we limit urselves in this paper only to th eask
of modeling such networks.
Even whenmodelingsimplifications similar to those
made for point-to-point networks re made, he analysis of
packet radio networks emains complex. This i s due to the
nature of wave propagation in radio channels, and to the
multi-access/broadcastnatureof hese networks. First,con-
nectivity between wo nodes i s not a Boolean function. De-
pending on many factors, such as the radio frequency in
use, power of the ransmitters, terrain, antenna type and
orientation, and he distance separating the two nodes,
there are cases when two nodes are, for all practical pur-
poses, considered disconnected; but when radio connec-
tivity exists, the quality of the radio linksamong nodes i s
not thesame for all links andat all times, and depends on
the factors listed above as well as on thedynamic activity
in the network.Thus, the topology of the networks not
as clearlydefined as with point-to-point wireetworks, and
this
fact is rendered even more complex when nodes are
mobile.
Secondly, in a given locality (as determined by radio con-
nectivity), the radio channel s a server which is contended
upon and shared by the many nodes in that locality. The
success of a transmission undertaken by a node depends
not only on the quality of theadio link between the node
and i ts intended neighbor, but also on the particular to-
pology of he network in theocality, transmission activity
of other nodes, state of the receiving node, the signaling
method used, etc. Consequently, essentialo theperation
of a radio networks a policy according o which odes de-
termine their right o access the channel. The objective of
such a policy
i s
to improve the likelihood of success of a
transmission to be undertaken, and thus maximize the
overall network hroughput. Thus in packet radionet-
works, the decision as to whether a node should transmit
or not may be a function of the state of the network in i t s
neighborhood (as dictated by the access protocol); and once
the transmission has been initiated, the outcome
i s
de-
pendent on the state of the neighboring nodes and their
evolution during the ransmission t ime in question. Thus
we are in presence of a network of queues
in
which a pack-
ets service time
s
dependent on the tate of the entireys-
tem and
i t s
evolution in time. This i s indeed a complex
queueing problem to which no imple solution exists.
The conditions underwhich atransmittedacket may be
correctly received and the channel access protocols used
(also called link activation protocols)are so essential to the
performance of packet radio networks and so central to
every modeling effort reported upon, that e devote Sec-
tion Ill to a clear description
of
these issues.
In the equel, unless otherwise stated, we shall consider
a packet radio network to be consisting of N nodes dis-
tributed over some geographical area. The onnectivity of
the network s considered to be a known Boolean function
and specified by an N
X
N hearing matrixH
= [h , , ] ,
where
h , = 1 if j can hear
i,
nd 0 otherwise. Thus each nonzero
entry
h,
in the earing matrix correspondso a directed ra-
dio link in the network from nodeo node, and viceversa.
We denote such a link by
(i,
,
and call node
i
t s source
and node i t s destination. Alternatively, a network can be
represented graphically by a directed graph in which ver-
tices represent nodes in the network, and directed edges
represent one-way connectivity. A link i s said to be active
whenever a transmission i s taking place over that link; i.e.,
whenever the source node i s transmitting a message des-
tined to thedestination node on that link.
Ill. PACKET RADIO NETWORK
PERATION
The operat ion and hence performance of a packet radio
network differ accordingo the hannel signaling method
used: narrow-band or spread-spectrum. Indeed, he two
methods result
in
different capture effects, whereby cap-
ture refers to the ability ofa receiver to receive correctly
a packet despite the presence of other time-overlapping
packets, and thus different conditions for the correct re-
ception of a transmitted packet. Another aspect by which
the operation and performance of packet radio network
differ
i s
the choice of channelaccess protocol. In his sec-
tion, we first define the two above mentioned signaling
methods and he corresponding capture ffects, and then
describe the various channel access protocols that can be
used with each method.
TOBACI: MULTIHOP PACKET RADIO NETWORKS
137
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
4/21
A.
Channelignaling
and
Capture preambleode assignment methods
as
space-homoge
-
-
Two types of channel signaling exist: narrow-band
sig-
nalingand spread-spectrumsignaling. In narrow-band sig-
naling, databits are modulated directly n the carrier.hus
the overlap at some receiver of two packets with roughly
the same signal power typically results in theoss of both
packets. If the two overlappingackets are f distinctly dif-
ferent signal power, thensome form of powercapturemay
come into effect, whereby the strongerpacket
is
received
correctly, and the weaker one is lost. (This, however, some-
times depends on timing as a weak signal may tie up the
synchronization circuit at the receiver.) Considering
a
net-
w o r k w i t h a f i x e d t o p o l o g y i n w h i c h a l l l i n k s a r e o f t h e s a m e
quality, (thus removing the possibility of power capture),
we then say that the network operates in a zero-capture
mode: the overlap ofwo ormore packets at some receiver
results in the destruction of all. (Note that, on the other
hand, in narrow-band systems, it s easy for a nodeo detect
activitydue to other nodes within radio connectivity simply
by sensing carrier.)
Spread-spectrum refers to signaling schemes which are
based on some form of codingnd which se a much wider
bandwidth than do narrow-band schemesfortheame data
rate
[63].
One way of achieving spread-spectrum operation
consists of using directsequence pseudo-noise (PN) mod-
ulation. Another i s frequency hopping (FH). Spread-spec-
trum techniques have been introduced to combat multi-
path effectsand jamming; but they also offerspecific
advantages in multiuser interference which center around
two properties: code-division and ime-capture. Codedi -
vision refers to the fact that transmissionswith orthogonal
codes may overlap in time with little or noffect on each
other. Time-capture referso the ability of n idle receiver
to successfully receive a packet with agiven code despite
the presence of other time-overlapping transmissionsith
the same code (or othercodes) [21], [28],
49].
Packet radio
networks may take advantage of these two properties in
different ways as described in the following.
Since the transmission of ackets is asynchronous, each
packet transmission must e preceded by the transmission
of a preamblewhich the receivers use to acquire bit syn-
chronization. Often, the preambleonsists of a nown code
with strongautocorrelationpropertieswhich i s used
throughout he network, and which dle receivers con-
stantly search for. Furthermore, this code
s
chosen to be
orthogonal to those used in encoding the ackets. In order
to be able to receive a packet, an idle receiver must first
process successfully the preamble. Failures are pr imarily
due to overlaps in time with another packets preamble;
they may also be due to the presence of other ongoing
transmissions, which act as background noise and thuse-
duce the effective signal-to-noiseatio. Errors due to over-
lapping preambles can be reduced by the use of receiver-
directed codes. In this mode of operation, ach receiver s
assigned a distinct preamble code waveform, and trans-
mitters must use the waveform ssigned to the receiver they
are trying to ommunicate with in theransmission of the
preambles. The result i s that a receiver will process the
preamble of only those packets that are specifically des-
tined to it,
all
other transmissions by neighboring trans-
mitters, including heir preambles, appearing as back-
ground noise. We shall refer to the two above-mentioned
neous and receiver-directed, respectively. Havingsuccess
fully processed the preamble of an incoming packet, the
receiver at a node ill thus lockonto theata portion of th
packet until i t s transmission i s completed.
Once a packets locked onto bya eceiver, it s importan
that future overlapping packets do not interfere with th
completion of t s successful reception. The conditions fo
such a success are, here too, dependent on the ways the
PN waveform patterns are selected and used, in addit ion
to theset of active linksand their signal strengths. Severa
cases can be identified. Consider first thease in which
fixed PN spread-spectrum pattern dentical or each bit
transmitted throughout the network is used. An overlap
ping packet would not interfere with packet lockedonto
earlier as long as i t s autocorrelat ion peaks do not coincid
with those of the earlier packetsee Fig.
1).
This means hat
Des i redSigna l
I n te fe r l n g S l g n al
Sum
Fig. 1.
Effect
of interference with no multipath
in
a PN
spread-spectrum
network.
(T i s the bit duration; n is a mea-
sure
of
the degree
of
spread and i s given
by
2TW, where
W is
the RF bandwidth
[20].)
there is
a
vulnerable period of a few chip times foreach bit
during which the rrival of a new packetwould cause de
struction. (Note, however, that the cumulative vulnerabl
period for a packet remains small compared to i t s entire
transmission time, hence the capture effect. Clearly this
represents the ideal picture when there i s no multipath;
multipath has the effect of smearing the correlationeaks
spreading them over time, and causing intersymbol inte
ferences. Such effects are ignored in this paper.) We shal
refer to this case as the space-and-bit-homogeneous code
assignment. To reduce the effect of overlapping packets
one could assign distinct orthogonal code waveforms to
different receivers, and have the transmitters se the wave-
form assigned to the intended eceiver in coding thei rits
This mode reduces the amount ofraffic that interferes ith
a packet transmitted to a receiver o only that raffic which
is
destined forthat particular receiver. We shallefer to this
case as the receiver-directed bit-homogeneous code
ssign
ment.Theelimination of interference fromny overlapping
packets can e approached by using a patternhich varies
on a bit-by-bitasis, and equipping the eceiver with a pr
grammable matched filter which follows the pattern as i
varies from bit to bit. If the attern is long enoughso that
it
does not repeat itself during theransmission ofhe long
est packet, then anyoverlapping packetwould not inte
with thepacket locked ontoearlier, since
it
would notpro
duce any autocorrelation peak during the ent ireeception
138
PROCEEDINGS OF
THE IEEE,
VOL. 75, NO. 1, JANUARY
1987
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
5/21
time of the earlier packet. (The problem of packet acqui-
sition incurred with arrival times too close to each other
would persist, since one would require the PN sequence
to start from thesame point, known to theeceivers. Note
that the startingpoint can be thesame for
all
receivers, or
different for each receiver.) We refer to this scheme as the
bit-by-bit code changing cheme or bit-nonhomogeneous
code assignment. Assuming that backgroundnoise is neg-
ligible andall codes are ruly orthogonal, this mode of p
eration achieves perfect capture, whereby correct recep
tion i s guaranteed once he packet is locked onto. Perfect
capture can also be achieved by assigning distinct orthog-
onal codes to nodes which the latter use to encode their
packets when transmitting. n his case, the preamble con-
tains information on the spreading waveform used, thus
allowing the receiver to programts matched filter accord-
ingly. Note that, in this case, perfect capture does not ne-
cessitate bit-by-bit code changing as long as transmitters
are assignedorthogonal codes. We refer to this scheme as
the transmitter-directed code ssignment.
As true orthogonality ofcodes i s not always guaranteed
on a bit-wise basis, and the background noise cannot al-
ways be negligible, some form of encoding (e.g., convo-
lutional coding) is performed on he nformation to be
transmitted prior tospread-spectrum coding,
o
that, with
the use of a decoder (e.g., sequential decoder), the cor-
rection of bit or symbol errorsan bedone at the receiving
end. Consequently, the correct receptionf a acket s con-
sidered to be achieved if the decoder does not make any
error in decoding the entire acket. Perfect capture refers
then to the ideal case whereby, once locked on, a packet
is always successfully received.
B.
Channel Access Protocols
A channelaccess protocol defines the conditions under
which
a
node may access he channel.For some prootcols,
thesecondi t ionsareexpressed intermsof notonlythestate
of the node n question, but also the state of other nodes
in the network.For practically realizable protocols, the rules
embodied in theaccess protocol are constrained to be de-
fined only in terms of information thatan be acquired lo-
cally at the node, such as the state of the receiver at that
node, and the state of the transmitters in some neighbor-
hood oft.Thus the feasibility of a protocol dependsn the
ability of a node to dynamically acquire knowledge re-
garding the tate of othernodes, which in turnepends on
the particular channel signaling scheme and code-assign-
ment scheme in use. Furthermore, the choice f an access
protocol i s heavily influenced by the capture propertiesn
effect, since certain actions may be desirable under one
capture mode but undesirable under others.2
For simplicity in presentation and iscussion, we assume
a t th i ss ta g e th a t th e h e a r i n g ma t r i xH i ssymme t r i c ,a n d th u s
all edges in the raph representationare bidirectional. For
When di f feren t codes are assigned to nodes (e.g., rec eiverd i-
rected, t ransm it terdir ected, etc.), th e termCode-Division Mul t ip le
Access (CDM A ) is generally empl oyed. How ever, the term C D M A
in i tsel f is not suff ic ient to descr ibe co mp letely the operat io n of
a netwo rk as i t does not sp eci fy any part icular chann el access pro-
tocol .
To
be precise, on e must indic ate, n addit ion to the code-
assignment scheme in use, a chann el access prot oco l wh ich odes
fo l low
i n t ransmi t t ing the i r packets .
any node i let X i )denote the set of
all
nodes connected
to it including itself. Let
X* / )
X ;) -
{ i } .
The elements
of X
* i )
re called neighbors of
.
For a collection of odes
A,
we
let
X ( A )
2 U,,,X( i )
and X 2 ( A ) P
X(X(A)).A
trans-
mission over link i ,
)
s denoted by
3 ( ( i , ) ) .
Given 3 ( ( i ,
j ,
a node
is
said to be hidden with espect to the trans-
mitting node i if
k is
a neighbor of the destination node
but not of the source node
i ;
i.e., if
k
E
X / )
-
X ;).
For all the protocols described below, it is considered
that nodes attempt transmission of theirackets
at
random
points in time defined by ome point process. In the event
that a packet scheduled for transmissions inhibited by the
operation of the protocol,r in the vent that
a
transmitted
packet i s not received correctly, then that packet is again
considered for transmission at some future point in time
determined by the scheduling point process.
1) PureALOHA: In this scheme,
a
node is allowed to
transmit only f
it
i s not already transmitting, regardless of
any other activity n the network. t was first introducedby
Abramson for the single-hop narrow-band LOHA System
which uses separate channels for inbound and outbound
traffic [I]. When applied o multihop acket radio networks
with
a
single broadcast channel, thedefinition implies that
the reception of
a
packet by
a
node is aborted if a packet
transmission i s scheduled at that node during the timeof
reception, i.e., transmission has priori ty over reception. In
systems where nodes receive packets indiscriminantly,
(e.g., narrow-band systems or spread-spectrumsystems in
which the code ssignment used allows an idle receiver to
lock onto any incoming packet), then aborting the recep-
tion of a packet may be beneficial since the packet may be
destined to another node. If, on the other hand, the only
packets that can be received are destined to the node, (as
would possibly be thecase n spread-spectrum systemswith
some code assignments), then abortingan ongoing recep-
tion results in thrashing. The following scheme overcomes
this problem.
2) Disciplined ALOHA: A node may transmit as long as
it
is not already transmitt ing nor locked onto any packet.
This scheme s certainly ideal for spread-spectrumystems
in whicheceivers lock onlyonto ackets destined to them.
For other systems, this scheme achieves someimited form
of activity sensing which might lead to improved perfor-
mance; indeed,
i f
a node i s locked onto a packet not des-
tined to
it,
then it i s likely that there exists a neighboring
node to which the packet i s destined, and it may be ad-
visable to inhibit transmission, thus increasing the prob-
ability of correct reception of the earlieracket.
3) Slotted ALOHA
[2], [3],
[32], [Sl] ,
1521:
The time axis is
considered to be universal for all nodes and divided into
equally sized slots. A node with a packet scheduled for
transmission in
a
particular
slot
transmits thatpacket, syn-
chronizing the start of transmission to coinc ide with the
beginning of the slot.3
It
i s assumed that the packet trans-
mission fits entirely within the slot. Since the slot size i s
Note that a synchronous scheme such as s lot ted ALOHA makes
good sense in a satel l ite network wh ere synchron ization to a uni-
versal t im e axis s easily accom plish ed; it does, how ever, present
imp lementa t ion d i f f i cu l t ies in a mu l t i h o p packet radio network,
especia l ly i f nodes are mobi le. t
i s
nevertheless an interesting
scheme to consider because i t of ten lends i tsel f to t ractable per-
formance models.
TOBAGI:
M U L T I H O P P A C K E T RADIO N ET WOR KS
139
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
6/21
fixed, slotted ALOHA is best suited for fixed-sizepackets.
Clearly, in narrow-band systems, thearrival of everal pack-
ets in the ame slot at a receiver results in the destruct ion
of all. The same effect results n spread-spectrum systems
with certain code-assignment policies. This is the case, for
example,when all preambles use the same code, since hen
slot synchronization would always cause overlap among
preambles in a given slot. Some improvement can be
ob-
tained with receiver-directed preamble code assignment,
bu t transmissions destined to the same receiver still inter-
fere with each other. Additional improvement can be ob-
tained by considering a slot size slightly arger than the
packet transmission ime, and randomizing he start of
transmissions within a lot so as to achieve some degree f
time-capture [14]. In such a case,
it
i s conceivable that the
f i r s t packe t t oa r r i ve t oa rece ive r inag ivens lo t w i I I be locked
onto f later packets happened to be delayed by a suffi-
ciently long periods of time. Beyond that, the conditions
for correctly receivinghe entire packetill depend on the
code assignment used in the ransmission of he packet.
4)
Carrier Sense Mult iple Access (CSMA) [33], 1351: In this
protocol,
it
s required that a node beble to sense the pres-
ence of transmissions by ts neighbors. (Recall that in nar-
row-band systems this i s simply accomplished by sensing
carrier.) A packet wi ll be transmitted by a node onlyf that
node
is
not 'already transmitting and no ongoing trans-
missions are sensed. Consider first i t s use in narrow-band
systems. In spite of carrier sensing, two factors remain
w h i c h c o n t r i b u t e to c o l 1 i s i o n s . Th e f i r s t is t h e n o n z er o p r o p
agation delay etween neighbors: given that atransmission
3 ( ( i , j ) )has been initiated, all nodes in he set
3t
*(i)are not
blocked from transmittingntil theransmission rom node
i has been sensed by them all; thus the propagation time
from node i o
i t s
neighboring nodes constitutes a vulner-
able per iod or 3 i, j ) ) as far as transmissions rom
3t *(i)
n 3t( ) are concerned. The second actor i s hidden nodes:
given that a transmission 3 ( ( i , j ) ) has been undertaken,
nodes in 3t(
- ( i)
cannot sense the presence of 3 ( ( i , j ) ) ,
andarethusneverblocked
bythat t ransmission; inthiscase,
the entire packet transmission time constitutes a vulner-
able period for ( ( i , j ) ) as far as transmissions rom hidden
nodes areconcerned. In spread-spectrum systems, the ca-
pability of sensing the activity of neighboring transmitters
depends on the mode of operation of the network[q.n
systems using space-and-bit homogeneous codes, activity
sensing
i s
possible, and
s
done by observinghe output of
the matched ilters corresponding to the desired wave-
forms. In systems which employ receiver-directed bit-ho-
mogeneousor t ransmit te r -d i rected bit-homogeneouscode
assignment, activity sensing requires morehardware: if di-
rect sequencepseudonoise modulations used, a nodewill
have to possess a bank of filters matched to all possible
codes usedby the neighbors;f frequency hopping s used,
sensing is done by maintaining bank of filters for ll fre-
quencies used. In systems which employ bit-by-bit code
changing, activity sensing s dif ficult to chieve, since here
i s no way to know which ode to try to etect at any given
t im e . It i sno t de f i n i t e l y c1ea r t ha tt heuseo f CSM A insp read -
spectrum systems improves overall network performance
sincespread spectrum already exhibitsstrongcapture
properties, and inhibi ting transmissions may actually de-
crease network throughput. It may, however, be useful if
increased probability of correct receptions desirable, es-
pecially for systems with nonperfect capture, or if lower
channel traffic i s desirable in view of improving anti-jam
capabilities
[28].
5) Busy Tone Mult iple Access (BTMA) [62],
[67l,VI:
h
problem of collisions caused by hidden nodes can be a
leviated by he use of a busy tone on a separate channe
which i s emitted by a node o indicate that it is currentl
receiving a acket. The usytone s then used to inhib it
receiving node's neighbors from transmitting and there
interfering with
t.
Consider first ts use in narrow-band sys
tems. Several variants of BTMA exist dependingon whic
set of nodes transmit abusy tone in any given situation.
Conservative BTMA (C-BTMA),henever a nodeenses ca
rier (due to the presence of a transmission),
t
emits abus
tone. Then any node that wishes to transmit
is
allowed t
do so only if
t
i s not already transmitting and no busy
is sensed. Accordingly, given that a transmission 3((i, j )
has been undertaken, a one-hop propagation delay con
stitutes its vulnerable period as far as transmissions from
nodes in X*(;) f l X / ) re concerned, and two-hop p ro
agation delay as far as transmissions from nodes in
X j
- X ;)
are concerned. Basically in C-BTMA, for a short-p
riod of time following thetart of a transmission,ll node
w i t h i n a t w o - h o p r ad i u s a r o u n d t h e t r an s m i t t er g e t b l o c k e
(Note that if the propagation delay between nodes wer
zero then C-BTMA would be a collision-free protocol.)h
remaining variants are similar to C-BTMA except hat on
the destinations f active links transmit the busytone.he
form the class referred to as destination-based BTMA
D
BTMA). In dealisticdestination-basedBTMA (ID-BTMA),he
destination of every active link emits the busy tone (eve
if that node could not receive the packet destined to
i
hencethe name idealistic). It s introduced for performan
comparison purposes. In receiving destinationBTMA (RD
BTMA), the destination node emits he busy
tone
only if
i s able to synchronize on and start receiving thepacket. I
narrow-band systems, this implies that thechannel aro
the receiving node i s idle prior to the arrival time of the
packet. In his protocol, given that transmission( ( i , j ) ) ha
been undertaken, then after a vulnerable period equal o
atwo-hop propagationdelay, all nodes in the et X * ( j )ar
blocked from tran~rni tting .~
Since a node does not generally knowf a particular tran
mission s destined to it r not, receiving-destinationTMA
i s also idealistic. It too s considered for comparison pur
poses. In practice, this nformation is obtained from the
packet header; assuming that he packet header i s pro
cessed as soon as it s received and before the entire pac
i s received, the time at which a node can first determin
whether ornot
it
i s the intended immediate destination
a particular packet reception
s
at the end f the processin
of the packet header. In Hybr id destination-based BTMA
(HD-BTMA) a nodeoperates as in C-BTMA unti l theheade
i s processed, upon which time it operates as in receivin
destination BTMA.' For spread-spectrumsystems, the pa
Alternat ively, one may addit ional ly makeuse
of
carr ier sensing
This way, a l l nodes n
X*(i)
which nc ludes
3t*(i)l
t ( j ) ) ar
blocked af ter a vulnerable per iod
of only
a one-hop p ro pagat io
t ime, whi le nodes in th e set X )
- (i)
re b lo cked af t er a vu
nerable per iod equal to a two-hop pr opagat ion t ime. This is pre
cisely the RD-BTMA scheme cons idered n the s im ulat ion stud
repor ted u pon in Sec t ion
IV
b e lo w [VI.
Al te rna t ive ly , one may conce ive
of
a scheme in wh ich he nod
op erates as in
CSMA
u n t i l t h eheader is processed pr ior t o sw itch
ing to RD-BTMA.
140
PROCEEDINGS OF THE
IEEE, VOL.
75,
NO.
1, JANUARY 198
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
7/21
ticular variant of BTMA which makes most sense imple-
mentation-wise i s receiving-destination BTMAand is re-
ferred to in th is ontext as locked-onto Destination BTMA
(LD-BTMAI [8]: the destination node of a link emits a busy
tone whenever the link s active and the destination node
i s locked onto the packet (receiver-directed code assign-
ment
i s
assumed in thiscase).
The performance of any of the access protocols cannot
be easily predicted in a multihop packet radio network, as
many conf lict ing effects actually take place; furthermore,
the performance ofn accessprotocol s highly dependent
on the topology of the networknd the imposed traffic e-
quirements. consider, for example, ALOHA and CSMA in
narrow-band systems. If the network is fully connected,
then CSMAwould
ign i f icant lyoutper formALOHA[35] .
But
in a general multihop topology, the presence of hidden
nodes may have such a degrading effect that CSMA may
only be slightly better thanLOHA. (See numerical results
below.) The situation becomes more complex wi th BTMA.
In conservative BTMA,a ransmission may e blocked, while
in destination-based BTMA t may be allowed o take place
and may result in a successful transmission. On theother
hand, in destination-based BTMA (and CSMAor that mat-
ter) it i s possible to allow a transmission to take place that
wi ll be unsuccessful, and whosepresencemay block a
number of other potentiallyuccessful transmissions,while
in conservative BTMA, the former transmission would be
inhibitedbytheprotocol,allowingthe later potentiallysuc-
cessful ransmissions to then take place.6
It
is not clear
whichof theef fectswould redominate in agiven situation,
and hence one must resort to analysis and simulation to
compare the different schemes.
IV . NETWORK
APACITY
As with point-to-pointpacket-switched networks, we de-
fine the network nput traffic requirement by the matrix
r = [?,&I, where y i k , j k, i s the average number ofpackets
per unit timeoffered at nod ej and ultimately destined for
k. (We let y i i =
0,
vi.)
We let y again denote the total ser
traffic, i.e.,
N N
and definea, y i k l y : The matrix [alk] represents he traffic
pattern. We also deftne network capacity to be the maxi-
mum value
y
that the network can support, keeping the
matrix {aik] fixed. In this section, we focus o n analytical
models appropriate for the determinationof network ca-
pacity.
As wi th point-to-point networks, the analysis of packet
radio networks s simplified by assuming that the routing
algorithm is of the ixed type (i.e., all packets from a given
source to a given destination flow through the ame set of
nodes), all nodes have nfinite buffers, and no link nor flow
control protocols are in effect. While under such assump-
tions the determination of network capacity for point-to-
point networks ecomes a trivial task, for packet radio net-
works i t r em a inscom p lexdue t o t hem anyopera t i ona lcha r -
acteristics hat may be n effect, as discussed
in
the previous
section. Models different rom those used for point-to-point
networks are needed. Beforewe proceed with a discussion
A
6For mo re de tai ls on such a discussion and an example, see [77].
of themodels introduced andused for multihoppacket ra-
dio networks, weshall first devotehe followingubsection
to special cases of single-hop networks ando a brief review
of relevant models used in heiranalysis; these models con-
stitute a precursor to multihop network odeling, andwill
thus assist the reader in the understanding of the latter.
A. Single-Hop Networks
Extensive analysis of single-hop networks operating un-
der a variety of channel access methods have appeared n
the literature. Most analyses related to narrow-band sys-
tems with zero capture, although some did address the case
of spread-spectrum and time-capture. In [I] ,Abramson in-
troduced and analyzed pure ALOHA. Slotted ALOHA was
then introduced byRoberts [51], [52] and later analyzed by
Abramson [2], [3] and Kleinrock andam [32], [34].Kleinrock
and Tobagi introduced and analyzed CSMA [33], [35] and
BTMA[67]. Raychaudhuri nd analyzed slotted ALOHA with
codedivision [50]; Cronemeyer andDavis considered
spread-spectrum slotted ALOHA with capture due to time
of arrival [14]. Musser and Daigle derived the throughput
o f pu reALOHAwi t hcoded iv i s ion [ 46 ] . F ina l l y , Pu rs leys t ud -
ied the throughput ofrequency-hopped spread-spectrum
communications [MI. While these represent the key ref-
erences, additional publications on the subject have ap-
peared in he literature. Basically,two models haveemerged
from this work: an infinite-population model and a finite-
population model. Weiscuss thesehere briefly in pecific
representative cases.
In the nalysis of single-hop networks, the environment
i s assumed to consist of a populat ion of nodes commu-
nicating wi th a single station. For the ALOHA schemes, ra-
dio connectivity need not exist among the nodes them-
selves. For CSMA,t s first assumed that all odes are adio-
connected, and thus every terminal can sense the activity
due toany other terminal.This assumption is then relaxed
allowing hidden nodes to be present. Note that, i f the sys-
tem under consideration
s
fullyconnected and of the ero-
capture type, hen all packets need not be estined to a sin-
gle node for the esults to apply.
I
An Infinite-Population Model or the Determination f
ChannelCapacity Under Various Channel AccessProto-
cols: In the infinite-populationodel, the numberofusers
i s
assumed to be very large (infin ite) .Users collectivelygen-
erate packets at some aggregate rate,ayy packets per sec-
ond. Assuming fixed-size packets wi th transmission timeT
seconds, and normalizing timeo T the packet generation
rate becomesS
=
yTpackets perpacket transmission ime.
The assumption of a large population allows
us
to consider
thateach nodewill
beabletosuccessful ly transmit
its packet
in a period of time small compared to the time
t
takes
it
to generate a new one. Thus no queueing of packets ever
takes place, and ll outstanding packets (i.e., generated ut
not yet successfully transmitted) belong o different nodes.
From the point of view of multiuser interference, this is
clearly the worst case. Due to packet collisions and block-
ing by the access protocol, the rate at which packets get
submitted to or ransmitted on thechannel i s larger than
S, and is here denoted by
C.
n this model, it
is
furthermore
assumed that the mean rescheduling delay X incurred by
a packet that was blocked orunsuccessfully transmitted is
very large compared to thepacket transmission time (the-
oretically infinite). Recall that this models aimed towards
TOBACI: MULTIHOP PACKET RADIO NETWORKS
141
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
8/21
deriving he capacity of henetworkunder anaccess
method, and not packet delay.) Accordingly, l it tle corre-
lation exists between newpacket arrivals and their re-
scheduling. Assuming the process of new packet genera-
tion to be Poisson with rate S, the channel traffic process
can also be onsidered to be Poisson with rate G. (We note
that this has been confirmed for slotted ALOHA by con-
ducting
a
more exact analysis which takes in to account X,
and then letting
X
00 [32].) The analysis then reduces to
considering
a
channel traffic which
i s
Poisson, rate
G,
and
deriving the corresponding ate of successful packets
S
as
S = GP,, where P, is the probability that n arbitrary sched-
uled packet is successful.
P,
varies with theaccess scheme
and capture assumptions used, and may involve various
relevant system parameters. Sincehe packet ize has been
assumed fixed, S represents also the throughput. In sys-
tems with zerocaptureor pure LOHA we have
=
Ge-2C;
for slotted ALOHA we get S = Ge-; for nonpersistent
CSMA we get S
=
Ge-aC/[G(l+ 2a)
+
e-a], where a i s the
ratio of propagation time 7 (assumed pessimistically o be
the same for all pairs ofusers) to packet transmission ime
T.
Similar expressions can be derived for other protocols
and other captureassumptions. The network capacity Cis
simply obtained by maximizing wi th respect to G. It was
shown, for example, that
C =
0.18,0.36, and 0.85 for pure
ALOHA, slotted ALOHA, and CSMA
a
=
0.01),
respectively.
2) A
Finite-Population Model for
the
Determination o f
Channel Capacity Under Slotted ALOHA: Abramson also
considered a finite-population model for the analysis of
slotted ALOHA [2], [3], [36]. There are N nodes numbered
1,2, . . ,N. Node
i, =
1, * ,N, follows a Bernouilli pro-
cess rate
G;
to determine the slots during which toransmit
packets. Assuming that at each such slot node i oes ac-
tually have
a
packet to transmit,
Gi
i s then the ate of chan-
nel traffic due to node
i.
he total channel traffic is
N
G
=
E
G;.
,=1
Let Si denote the probability that noderansmits a packet
and
is
successful.Si also representshe throughput for node
i.The total throughput i s
N
s
=
c s;.
i = l
Under the assumption that nodes always have packets to
be transmitted, the following set of equations is derived:
N
Si
=
G;
I I
1 -
C,),
i
= 1, 2,
.
* , N. (2)
j i
/ = 1
This set of equations
s
used to determine the boundaryo
theegionor This i s done by computinghe Ja-
cobian
1
= )(Sl, z, * * , SN; G1,Gz, * * , G N ) ,and setting
it to zero. This leads o the condi tion Ysl
Ci =
1, which, in
turn, leads to the determination of a urface in the S1,
S2,
* ,
SN)plane representingthe boundary. or agiven traffic
pattern {ai}Y=l {Si/S}Y=l,he network capacity i s given
by the point on the surface which corresponds to
i t s
in-
tersection with the line
Let {C };= denote that point. Let
N
c
= x
;.
r = l
We note that C
=
maxcc,);-,(S) such that Si
=
ais,
v i ,
an
i s achieved by unique set ofransmission rate
{G:}Y=l.
Note that he set of equations (2) is valid only
s
long as each node hasalways packetso betransmitted, an
thus i s valid at the boundary. We therefore deduce th
approachisvalidforthedeterminationof networkcapacity
3)
The Hidden Terminal Problem in CSMA:As pointe
out in Section
I l l
CSMA suffers performance degradati
in he presenceof hidden terminals.his problem has bee
studied for single-hop networks by Tobagi and Kleinroc
in [67]. Anenvironment consisting f a large number of er
minals communicatingwith asingle station s considered
Packets are of fixed size. All terminals are in line-of-sigh
and with in range of the station ut not withespect to eac
other. All assumptions ntroduced above for the nfinite
population model are considered to hold true. From th
hearing matrix describing connectivity or he environ-
ment, the population can then be partit ioned into say N
groups, such hat
all
nodes within a group ear eachothe
and hear the same subset of nodes in the rest of the pop
ulation. Aswith the infinite populationmodel,
it
s assume
that each group consists of a large number of nodes who
collectivelyform an independent Poisson sourcewith mea
ratesi packets per unit time. Heretoo, given atraffic p
where
N
s =
csi
i = l
the problem is to find {Ci}Y=lsuch that
N
c = C C;
= max {s)
i = l
and Ci
= ajC.
Note that forhe ALOHA schemes, the resu
i s
the same whether or notall nodes are radioconnected
For CSMA, major degradation s incurred, rendering such
an analysis essential.ConsideringGi to be the rate of chan
nel traffic offered by group i, set of relationships of th
form
S; =
C;
fi(G1,
Gz
. , G N , i = 1, 2,
9
,
N (3
is derived [67]. This setf equationss similar to(2) obtained
with the finite-population modelhat Abramson devised
slotted ALOHA; the difference s simply the complexityo
(3) as compared to (2) requiring numerical procedures fo
the determinat ion of network capacity. This was done b
writing (3) in the form
and by solving iteratively the set of equations for
{Gi}, =
given
{ S i } : = ,
starting with the initialvalues
If the iterative procedure results in a finite traffic vecto
{G,},
hen the inputvector
i s
feasible; otherwise
it
i s not
Thus the networkcapacity can be etermined numerical
by gradually increasing
Si }
until the procedure diverges
142
PROCEEDINGS
OF
THEEEE,
VOL.
75, NO. 1, JANUARY 198
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
9/21
B. Multihop Networks observed that it s applicable to twother schemes;amely,
While there is an abundance of papers in the literature
on single-hop networks, ntil 1980, very fewdealtwith mul-
tihop systems; and even then, those which did examined
specific topologies. he first paper of this typeas perhaps
Gitmans [25] in which network capacitywas derived for a
two-hop centralized network consisting of user terminals
communicating with a single station via repeaters located
around the station, and operating under slotted ALOHA.
The model used is acombination of the infinite-population
model representing the terminals,and Abramsons finite-
population model representing theepeaters. Lateron,
To-
bagi considered the same topology in 70]-[73]; he derived
the network capacitys well as the throughput-delaychar-
acteristics or both slotted ALOHA and CSMA, and dis-
cussed their dependence on such key parameters as re-
peaters connectivity, transmission protocols, and repeat-
ers storage capacity
(see
Section V). Yemini analyzed the
capacity of one-waytandem slottedLOHA networks n 79];
the slotted nature of the system and the consideration of
unidirectional networks in which one or moreueues feed
a
third resulted in
a
problem that could be analyzed (see
also 70]). Network capacitycanalso bededuced from
throughput-delay analyses by considering the asymptotic
value of throughput as delay goes o infinity. Discussionof
such analysis is deferred to the next section.
In this sectionwe are concerned with the determination
of network capacity for systems with a general topology,
and operating underny possible combination of channel
access protocol and captureode. We describe thevarious
models that have been devised and discuss the major re-
sults obtained. To that end, we begin with the description
of a network abstraction adequate for modeling ultihop
networks and present some basic defin itions. Nextwe ad-
dress slotted ALOHA and show that the finite-population
model which Abramson introduced for slotted ALOHA in
single-hop networks can be easily extended to study net-
works with general topologies.
In order to study networks operating under other (un-
slotted) access protocolsandvariouscapture modes,
models based on continuous-time processes are needed.
The first work in that direction was by Boorstyn and Ker-
shenbaum [4]. They formulated a Markovian model for the
analysis of
a
network with general but symmetric hearing
matrix, and ero-propagationdelay, operating under SMA,
with the assumption that the first packet o arrive to a re-
ceiver after the channel as been idle is captured perfectly
(regardless of later events), and all other packets are lost.
The zero propagation delay assumption suggests that the
ratio a is very small, and hat interference
s
mostly caused
by hidden nodes. Under these conditions, it was observed
that
a
sufficient state description for the network
s
the set
of transmitting nodes (irrespective of their intended neigh-
bors.) Analysis then led to
a
steady-state dist ribution with
a product form, and relatively simple throughput equa-
tions, allowing the design of efficient algorithms for the
analysis of large networks, of the order of 100 nodes [30].
Tobagi and Brazio considered the same model in [75], and
ALOHA with zero and perfect capture, and C-BTMA wiih
perfect capture. (C-BTMA with perfect capture was also
treated independently by. C.-N. Chen [lO].)Theyalso ob-
served that not all schemes could be modeled simply by
tracking the set of transmitting nodes, nor could they
al l
lead to a product form solution. n their subsequentpaper
161, they extended the model by considering thestate de-
scription to consist of theset of active links (i.e., the set of
transmitting nodes along with their respective ntended
neighbors).This newmodel could thusccommodate other
access schemes such as ID-BTMA), and etworks wi th non-
symmetric hearingmatrices. They also rovided necessary
and sufficient conditions on the network (topology,raffic
requirements, andaccess protocol) for the nalysis to lead
to
a
steady-state distribution with
a
product form. In the
same paper, they also solved for the throughput in net-
works operating under the zero-capture mode. Withhese
results, the network apacity for narrow-band ystems with
arbitrary topologies under pureALOHA, CSMA, C-BTMA,
and ID-BTMA could be determined.
The aforementioned models are not appropriate or
spread-spectrum systems. Indeed, they accommodateys-
tems in which the nodes decision to transmit, as well
as
thecaptureof
a
packet
at i ts
ntended receiver, depend only
on the et of active inks just before the transmissiontarts.
A
spread-spectrum system exhibits
a
different type of be-
havior. Whenever the transmitter
at
a
given node
s
not ac-
tive, the receiver wi ll alternate between being lockednto
a
packet and being idle (searching for new ackets); and,
in order to determine the uccess or failure ofpacket, one
must not only knowthe
et
of active inks, but also the state
of he receivers (locked or free). Brazio and Tobagi ad-
dressed this problem by augmenting thetate description
so
as to include thetate of receivers. n [7], they restricted
themselves to access protocols for which the decisionade
on whether ornot toransmit can be considered function
of only the set of active links, independent of the state of
the receivers (this i s precisely the case for pure ALOHA,
CSMA, and C-BTMA). For theseprotocols, the modeleads
to an analysis which takes advantage of results obtained
previously. For the remainingaccess protocols (disciplined
ALOHAand locked-onto destinationTMA) the model oes
not enjoy any special structure, and the solution i s thus
more tedious o reach [8]. In all cases, errors due to noise
and interfering users can be properly incorporated
so
that
most pread-spectrum code assignments and apture
modescanbe effectively addressed. Note hat or he
models referred o in his paragraph, the analysis tends to
be exact n nature. Unfortunately, theize of networks that
can be effectively handled
is
rather small.
In
[Ill,
[12], M.-S.Chen and Boorstyn considered theis-
ciplined ALOHA receiverdirectedCDMAprotocol, and
provided an approximate analysis which can accommodate
networks
of
general topology with everal hundred nodes.
A noise model
i s
introducedand analyzed wherebya
threshold
i s set
on the number of transmitting neighbors
that would cause packet loss. Based on this approximate
analysis, the performance of several protocols was corn-
Note that i t maypro ved i f f i cu l t to haveas igna l ingsch emewhich
Note that
fo r
C-BTMA, given the zero-propagat ion delay as-
a l lows both CSMA and thepecial cap tu re effect assum ed (see Sec-
sumpt ion, the assumpt ion o f per fec t capture
s
a natura l one,ven
t i on 111 .
in narrow-band systems.
TOBACI: MULTIHOP PACKET RDlO NETWORKS
143
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
10/21
paredand the effects ofnodeconnectivitystudied in
[15].
we have
In [55],Sen considered a more detailed model ofnode and
link and derived better approximations.
s-
=
G n 1
- Gk).
l k = 1 , 2 ;..,
N
5
A completelyifferent approach to modeling packeta-
ht, = 1
dio networks has been taken by Yemini80]. e used tech-
niques of statistical mechanics which,or some topologies,
led o similar results as those obtained with he above
models. (For more detail, consult also [39].)
7
Network Abstraction and GeneralModel: Given a
static routing function, the throughput requirement for
each link
i,
) , denoted bySl j ,can be computed in terms
of the nd-to-end traffic requirements
.
For agiven source-
destination raffic pattern, here is a corresponding link
traffic pattern. It may happen that for some links the re-
quiredthroughputiszero.Werefertothese1inksasunused
links, and all other inks as used links.
Since the enti re packet radio network operates using a
single radio channel, each node in the network has one
transmitter, but can in general have more than one out-
going link. We consider that each outgoing linkat a node
has a separate queue for thepackets to be transmitted on
it, and that the transmitter i s shared among all queues at
that node. To avoid repeated interference between trans-
missions in the network, ransmission requests or the ar-
ious queues ata node are scheduledaccording to random-
point processes, one for each queue. Consider a point in
time defined by the poin t process for some link i. f the
queue is empty, this scheduling point
is
ignored. If the
queue
is
nonempty then a packet in thequeue is consid-
ered fortransmission.Thetransmission mayor may not take
place depending on the status of the transmitter at the
source node (busy or idle), the priority structure (if any)
among the queues at the source node, the channel access
protocol in use,
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
11/21
Let
X i C,X,.
Due to thessumptions of exponential mes-
X ( t ) s
Q(D)
denote the
ady-state probability of state
D,
the following balance
r 1
+
C
X,Q(D
-
{ i } ) .
i c D
Let Gi = A i l p i . It
i s
easy to check that the expression
(7)
isfy the balance equations and therefore constitute the
Q 4) s obtained by normalizing the
(8), (6 )
can then be written as
I
D n X { r . j ) ) = 6
Here again, in general, an iterative numerical procedure s
used to derive the network capacity for a given traffic pat-
i ,
)
with nonzero desired throughput,
(9)
requires the computationof the sum of products forall
states
D
containing nodes only in agivenubsetA of nodes,
namely, the complement of {;,
} ) .
Let SP(A) denote such
sum of products corresponding to subset A. The compu-
tation ofSP(A) i s aided by recognizing the existence of the
following recursion:
SP(A)
=
SP(A
-
{k } )
+
GkSP(A
- Wk)),
for k E A. IO)
for large networks with a general topology this task re-
mains formidable. KershenbaumandBoorstyn [4],301
nd iterative relations for these sums of products, and
developed a set of algorithms to efficiently generate these
expressions and perform the iterations of
4).
Other rela-
tionsallow hedecompositionof larger networks into
smaller, more tractable, segments. The complexity of their
algorithm, althoughexponential in general,growsqua-
dratically or cubicallywith the number of links forost net-
works on the order f 100 nodes. Running time s minutes
on an IBM-PC and ten times faster on a VAX
780.
Maglaris eta/.
[44]
eneralized the above analysis o allow
nonexponential packet length distributions. The method
of stages developed by Cox to approximate any given
dis-
tributionfunction byan rlang-branchingconfigurationwas
used to model packet length distributions.
It
was shown
that the above analysis i s
st i l l
valid and he form of thee-
suItsarepreserved.Inaddition,itwasshownthatthemodel
could accommodate different packet length distributions
for each link emanating from a node. In [5], this modelwas
also extended to include theeffects of acknowledgments.
b)
Pure ALOHA and C-BTMA in networks with sym-
metric hearingmatrix: Tobagi and Brazio
[75]
showed that
the state description required for the analysis of ALOHA
and C-BTMA in symmetric topologies is, as above, the set
of nodesbusy transmitting. ndependently, C.-N. Chen
treated C-BTMA the same way
in [IO].
To write thebalance
equations for these systems, or each state
DES
we define
U(D)
o be the set of nodes that are not blocked y any node
in the set
D,
given the access protocol. The balance equa-
tions then become:
r
+
x , Q O
-
{ i } )
11)
for which the solutioniven
in 8)
also holds. (Note that for
CSMA
U(D) = { i ; i
@ Z(D)}wh ile for C-BTMA
U(D) = { i ;
i
3Z2(D)}.) The throughput equation for link
( i , j )
in C-
BTMA i s simply
r eD
h i
Si,
=
-
Pr
{X2 ;)
dle}.
(12)
Given the result in
(8), 12)
can be expressed in terms of a
sum of products similar to
(9).
Consider now pureALOHA. Here
S
= 2 { 1 v 2 c - . , N ) ,nd all
nodes act ndependently. Moreexpli cit expressions maybe
written. for every node
i,
we simply have
Pi
1
Pr { i idle} =
+ Cj
Thus
Q(D) i s
given by the following closed form expres-
sions:
In writing the throughputquations, the two cases of per-
fect capture and zerocapture are distinguished. In either
case, a necessary condition for a scheduling point corre-
sponding to link i , j ) o result i n a successful transmission
i s
that 32 be idle at that time. Furthermore, the average
length of a successful transmission s no onger
1Ip;
Indeed,
thesuccessorfailureinthereceptionofapacketisnotsolely
dependent on the tate of the networkt the beginn ing of
reception, as eventsmay occur during its reception af-
fecting
i t s
successful completion. fo r the perfect-capture
case,for example, receiving nodejmayabort the reception,
giving prior ityto
t s
transmitter (see Section
l l .
In he zero-
capture case, in addition to the ossibility of node abort-
ing
t s
reception, transmission by any neighbor of
j
causes
a collision. Let
T i , ) )
denote theaverage period of time
that a transmission over
( i , )
contributes to throughput,
given that
X ) s
idle at the beginn ing f transmission. The
general throughput equation for ALOHA
i s
given by
which results in the following closed-form expressions:
perfectcapture
zerocapture
I t
4)
LinkActivity Model[6]: In order to accommodate net-
workswith nonsymmetric hearingmatrices,aswelI asother
access schemes such as ID-BTMA, Brazio and Tobagi in-
TO BAG I :
MULTIHOP
PACKET RADIO NETWO RKS
145
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
12/21
troduced the linkactivity model
[6].
The focus here
s
still
on access protocols in which the ecision to transmit can
only be basedon thestate of the ransmitters, and not the
receivers (namely, pure ALOHA, CSMA, and BTMA).
Recall that a used link i s said to be active whenever a
transmission s taking place over that link. Clearly, the ac-
tivityofthelinksofthenetworkisconditioned
bytheaccess
protocol in use. For the access protocols considered here,
an access protocol is a set of rules which, givenhe current
set of active links in the network, determines whether or
not a given inactive link can become active. Consider all
used links to be numberedg1 , 2,
. . . , L ,
and let 6:
=
{I ,
2, * , }. The point process for link
i , i
E
C,
s again con-
sidered to be an independent Poisson process with rate
X I
X I
>
O).Thetransmission imeof themessagestransmitted
over link
i
i s assumed to be exponentially distributedwith
mean I/pi ( p i
>
0), and to be redrawn independently from
thisdistribut ion each time themessage s transmitted. (Note
that different links emanating from a nodemay have dif-
ferent means.) Let
X ( t )
denote the set of all active links at
time t .
X ( t )
i s a continuous time Markov chain.
Given an access protocol, link i
E
C s said to block link
j if, whenever link i i s active, the protocol used does not
a l l o w a s c h e d u l i n g p o i n t f o r I i n k j t o r e s u It i n a n a c t u a lt r a n s -
mission. (Note that, in general, if link blocks link j, it does
not necessarily follow that link
j
blocks link
i.)
Let
D
be a
set of links in. blocks linkj E C - D f there exists some
link
;E
Dwhich blocks . (The definition of blockingcan be
extended to include the case where links
i
and k do not
blockj individually, but doogether.) Define
U D)
o be the
set of all links inC
-
D which are not blocked by D. t was
shown in
[6]
that the Markov chain ( t ) is irreducible, pos-
itive recurrent, and ergodic. Denote by { Q D ) ,D E
S }
its
stationary probability distribution.The equilibrium equa-
tions are shown to take the form
A
+
x
Q(D
U
{ ; } ) p i ,
D E S
18)
where M(D) s the set of all l inks i D such that D U
{ i }
E
S,andI(D)isthesetofallIinksj~Dsuchthatjisnotblocked
In general, the solution to
18)
does not have
a
product
form. (Note the difference between
18)
and
II).)
n
[6]
it
was shown that the existence of
a
product form i s equiv-
alent to
X ( t )
being reversible
29],
and that X ( t ) is reversible
i f and only ifD
=
/ D)or all D E
,
or equivalently,
U D)
M(D) for
all D
E S. (Note that under these conditions,
18)
and
(11)
become similar.) This result, n turn, was shown to
lead to the following imple characterization:A necessary
and
sufficient condition for a channel access protocol, to-
gether with a given network topology and traffic require-
ments, to have a product orm solution s that, for a l l pairs
of used links i and j, link j blocks link i whenever link i
blocks link j
As an application of he above characterization, one an
easily prove that, with asymmetric hearing matrix,CSMA
always eads to a product form solution. If, however, the
IC
M(D)
by
D -
{j}.
'Note that in the link activity modelrefers to
a
link,
as
opposed
to a node.
hearing matrix
s
not symmetric, one oes not get a produ
form solution, unless all pairs of nodesn, and n, for which
h =
1
and h = 0 are such that at least one elemen
of the pair
i s
not the source of any used links. As anothe
application, it iseasyto prove that, ineneral, ID-BTMAwi
not lead to
a
product orm solution, since symmetry in
blocking does not usually prevail, even when the hearin
matrix s symmetric. For somespecifictopologiesand raffic
patterns, however, ID-BTMA will have
a
product form so
lution. Examples of these are
a
star network with arms o
length 1 and arbitrary traffic pattern, or
a
four-node chai
in which the outer odes generate no traffic.
When he condition stated above is satisfied, the sta
tionary distribution is simply given by
When it does not hold, onehas to resort to a numerical o
lution of
18).
The th roughpu to f l i n k i ,S i , i s t he long - run f rac t iono f t ime
that link
i
is engaged in successful transmissions. Using re
sults from the theory f regenerativeprocesses, it was for
mally shown in
[6]
that SI
is
given by
20
whereUs ;)s the collection oftatesD
E
S that do ot bloc
link i and do not contain an active link which causes link
i
to be unsuccessful from i t s start, and T D ,
) ,
D E U5( i )
the average time that a transmissionver link icontribute
to throughput given that the tate just prior to the tart o
the transmission i s
D.
For CSMA and BTMA with perfec
capture, we haveT D,) =
l/pI.
Equation 20) hen takes on
a simple form; appliedo CSMA, it i s similar to 9); he same
i s true forC-BTMA. Under zero-capture, however, similarly
to what was encountered above i n the analysis of pure
ALOHA, the average transmission time of auccessful me
sage is not
l /pi,
due to the dependency thatxists between
the message length and ts success. In order to derive T D
i ) ,
an auxiliary Markov chains created as follows. Let
a,
be the collect ion of states in which
i
i s active and no in
terfering link
i s
active, let
&()
be the collection of state
in which i i s active and some interfering link s also active
and let
9 ( i )
be the set of states obtained from
a, ;)
y deac
tivating link . The start of a ransmission over lin k iwh ic
does not suffer a collision at i t s very start corresponds to
a transition of
X ( t )
from astate D E u , ( i ) into stateD U { i }
E e&).Link
i
remains free of collisions as long as
X ( t )
re
mains in
a, /]
while i i s active. The successful completion
of link i's transmission corresponds to a transition from
some state in
a, ;)
nto astate in 9 ( i ) (without having eve
visited any state in @, i)).The derivation of T D,
)
s then
simply obtained by considering the auxiliary Markov
consisting of the states in a, ;), nd two absorbing states
representinga, i)and 9 i).This approach s used to derive
the capacity of any narrow-band systems operating unde
pure ALOHA, CSMA, C-BTMA, and D-BTMA. When a p
plied to pure ALOHA, it provides results similaro 16)and
17).It s also used o model errors dueo noise and captu
effects other than zero and perfectapture, as encountered
in spread-spectrum systems [ 8 ] ,
[MI.
Numerical example
illustrating the se of these analyses areiven below n th
following subsection.
5) Models forSpread SpectrumSystems: The models ex
146
PROCEEDINGS
OF
THEEEE, VO L.
75,
NO. 1, JANUARY
1987
8/10/2019 Modeling and Performance Analysis of Multihop Packet Radio Networks
13/21
amined so far accommodate systems in which theaccess
protocol and the outcome of transmissions are indepen-
dent ofhe receivers state at each node.hile these models
are adequate for some narrow-band systems described i n
his paper, for the analysis of other narrow-band systems
(such as receiving destinationBTMA) and spread-spectrum
systems, models which incorporate the operation of the
receivers must be considered. At any given poin t in ime,
each node may be in one of threebasic states: idle, trans-
mitting, or receiving (i.e., locked onto a packet).
a)
A
general Markovian model17,
[ti]:
Recall that for
anode t oco r rec t l y rece iveapacke t , i ) t henodem us t be id le ,
ii)
t must process successfully the preamble so as to lock
onto thepacket, and
iii) it
must complete the receptionf
the packet free of error. Given that the node is idle, the
probability that
t
locks onto the incomingacket depends
on the et of links whichre activeat the start of the ream-
ble, the evolution of theystem dur ing theeception of he
preamble (which,n urn, depends on theet of parameters
{X , } and { p i } defined previously), the topology, the prop-
agation characteristics, the power levels of neighboring
transmitters, the receivers threshold, etc. An accurate rep-
resentation of most of these aspects i s possible. The main
difficulty arises from having to track the evolution of the
system during theeception of he preamble, which would
lead o non-Markovian models. Modelsof probabilisticcap-
ture have first been considered by phremides [161, [ Iq. In
[ 7 l ,
Brazio and Tobagi gave an approximate model of the
operation of receivers so as to maintain the tractable Mar-
kovian nature of the model. It is assumed that, given the
set of links Dwhich re active just before thestart of trans-
mission of some link , the receiver at node n, if idle, suc-
cessfully locks onto s transmission with probability
6 J D ;
j ,
independently from trial torial. In essence, the model
assumes that preambles are of zero ength, and that6 J D ;
j ) equals the average probability of success for thesystem
with preambles of nonzero length, given and , and aver-
aged over all possible evolutions of the system during the
preambles transmission time.As long as the lengthof the
preamble i s much smaller than min {l/Xi , l/pi} , one can
consider the use of 6,(D; which i s independent of
{ X i }
and {pi} o be a good approximation of the real behavior
of the receiver. By appropriately selecting { 6 , ( D ;
i ) }
one
can model the preamble capture behavior of both space-
homogeneous and eceiver-directedpreamblecode as-
signments as well as the effect of background noise. If the
preamble i s not successfully received, the receiver wil l re-
main idle, waiting for a new packet to lock onto, and the
whole process is repeated. If the preamble
is
successfully
received, the receiver remains locked onto that acket unt il
the end of
t s
transmission (unless, in the meanwhile, the
node switches to transmit mode, as for example
in
pure
ALOHA). The successfuleception of he entire acket then
depends on the activity of he transmitters in the network
during the acket reception time and the capture mode n
effect.
With
this
receiver model, the state of a node at anypoint
in time
,
z, t), can be defined o be (o
i f
the nodes idle,
+k
if linkk (with source node n) s active, and
- P
if node n is
locked onto a packet transmitted over link 0. The state of
the system s then Z t)
= z, t),
*
,
zN(t)).Furthermore, Z t)
i s a Markov process. Writing the alance equations for
Z t)
and solving for the steady-state dis tribut ion numerically,
the throughput ver each ink may be derivedn away sim-
ilar to
(20).
This i s the approach used
in
[8] to analyze
dis-
ciplined ALOHA and D-BTMA.
For systems in which theccess protocol
is
such that the
decisiontotransmit isfunction onlyofinkactivity,Brazio
and Tobagi[ilhave shown that a simplification in the nal-
ysis manifests itself in the form of some degree of sepa-
ration in the epresentation of transmitters activity nd re-
ceiversactivity,whichallowstheresuItsobtainedintheIink-
Top Related