Modeling and Performance Analysis of Multihop Packet Radio Networks

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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

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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

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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

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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

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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:

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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.

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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

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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

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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


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,


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


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


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-

Modeling and Performance Analysis of Multihop Packet Radio Networks

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