The Throughput of Packet Broadcasting Channels. NORMAN ABRAMSON, FELLOW, IEEE. 117. Abstract-Packet broadcasting is a form of data communications.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-25, NO. 1 , JANUARY 1977
117
The Throughput of Packet Broadcasting Channels NORMAN ABRAMSON, FELLOW,
Abstract-Packetbroadcasting is a form of data communications architecture which can combine the features of packet switching with those of broadcast channels for data communication networks. Much of the basic theoryof packet broadcastinghasbeenpresented as a byproduct in a sequence of papers with a distinctly practical emphasis. In this paper we provide a unified presentation of packet broadcasting theory. In Section I1 we introduce the theory of packet broadcasting data networks. In Section I11 we provide some theoreticalresults dealing with the performance of a packet broadcasting network when the users of the network have a variety of data rates. In Section IV we deal with packet broadcastingnetworks distributed in space, and in Section V we derive some properties of power-limited packet broadcasting channels,showing that the throughput of such channels can approach that of equivalent point-to-point channels.
IEEE
channels in theirnaturalbroadcastmode can lkad to significant system performance advantages [ 1 13 , [ 121 .
B. Outline of Results
Packetbroadcasting is a form of datacommunications architecture which can combine the features of packet switching with those of broadcast channels for data communication networks. Much of the basic theory of packet broadcastinghasbeen presented as a byproduct in asequence of papers with adistinctlypracticalemphasis. In thispaper we provide a unified presentation of packet broadcasting theory. In Section I1 we introducethetheory of packetbroadcasting as implemented in the ALOHA System at the University of Hawaii; also in Section I1 we explain a modification of the basic ALOHA method, called slotting. In Section Ill we I. INTRODUCTION provide some theoretical results dealing with the performance of a packet broadcastingchannelwhen the users of the A. Packet Switching and Packet Broadcasting channel have avariety ofdata rates. In Section IV we deal HE transition of packet-switched computer networks from with packet broadcasting networks distributed inspace, and experimental [ l ] t o operational[2]status during 1975 present some incomplete results on the theoretical properties provides convincing evidence of the value of this form of com- of such networks. Finally, in Section V we derive some propmunications architecture. Packet switching, or statistical multi- erties of power limited packet broadcasting channels showing plexing [ 3 ] , can provideapowerfulmeans of sharing com- that the throughput of such channelscan approach that of munications resources among large number of d a t a c o m m u n i equivalent point-to-point channels. This result is of importance cations users when those users can be characterized by a high in satellite systems using small earth stationssince it’implies that ratio of peak t o average data rates. Under such circumstances, the multiple access capabilityand the complete connectivity data from each user are buffered, address and control informa- (in the topological sense) of packet broadcasting channels can tion is added in a “header,” and the resulting bit sequence, or be obtained at no price in average throughput. “packet,” is routed through a shared communications resource by a sequence of node switches [4], [5]. 11. PACKET BROADCASTING CHANNELS Packet-switched networks, however, still employ point-topoint communication channels and large multiplexing switches A . Operation of a Packet Broadcasting Channel for routing andflow control inafashion similar to convenConsider a number of widely separated users, each wanting tional circuit switched networks. In some situations [6] - [ 101 it is desirable tocombinethe efficiencies achievable by a to transmit short packets over a common high-speed channel. packet communications architecture with other advantages ob- Assume that the rate at which users generate packets is such tained by use of broadcast communication channels. Among that the average timebetween packetsfrom a single user is much greater than the time needed to transmit a single packet. these advantages areelimination of routingandnetwork switches, system modularity, and overall system simplicity. In In Fig. 1 we indicate a sequence of packets transmitted by a typical user. addition, certain kinds of channels available to the communiConventional time orfrequency multiplexing methods cations systems designer, notably satellite channels, are basically broadcastin their structure. In such cases use of these (TDMA or FDMA) or some kind of polling scheme could be employed to share the channel among the users. Some of the disadvantages of these methods for users with high peak-toManuscript received January 19, 1976; revised June 11, 1976. This average data rates are discussed by Carleial and Hellman [ 131 . work was supported by The ALOHA System, a research project at the In addition, under certain conditions polling may require unUniversity of Hawaii which is supported by the Advanced Research Projects Agency of the Department of Defense and monitored by NASA acceptable system complexity and extra delay. Ames Research Center under Contract NAS2-8590. The views and conIn a packet broadcasting system the simplest possible soluclusions contained in this paper are those of the author and should not tion to this multiplexingproblem is employed. Each user be interpreted as necessarily representing the official policies, either expressed or implied, of the Advanced Research Projects Agency of the transmits its packets over the common broadcast channel in a United States Government. completely unsynchronized (from one user to another) Theauthor is with The ALOHA System, University of Hawaii, Honolulu, HI 96822. manner. If each individual user of a packet broadcasting chan-
T
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IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1977
n n
n time
Fig. 1. Packets from a typical user.
-
overlap
Fig. 2.
ne1 is required t o have a low duty cycle, the probability of a packet from one user interfering with a packet from another user is small as long as thetotalnumber of users onthe common channel is not too large. As the number of users increases, however, the number of packet overlaps increases and theprobabilitythat a packet will be lost due t o anoverlap also increases. The question of how many users can share such a channel and the analysis of various methods of dealing with packets lost due t o overlap are the primary concerns of this paper. In Fig. 2 we show a packet broadcasting channel with two overlapping packets. Since the first packet broadcasting channel was put into operation in the ALOHA System radiolinkedcomputernetworkatthe University of Hawaii [ 6 ] , they have been referred t o as ALOHA channels.
B. ALOHA Capacizy A transmittedpacket can be received incorrectly or lost completely becauseof two different types of errors: 1) random noise errors and 2 ) errors causedby packet overlap. In this paper we assume thatthefirsttypeoferror can be ignored, and we shall be concerned only with errors caused by packet overlap. In Section 11-D we describe several methods of dealing with the problem of packets lost due to overlap, but first we derive the basic resultswhichtell ushowmany packets can be transmitted with nooverlap. Assume that the start times of packets in the channel comprise a Poisson point process with parameter h packets/second. If each packet lasts r seconds, we can define the normalized channel trafficG where
G = AT.
(1)
If we assume that only those packets which do not overlap with any other packet are received correctly, we may define A' < h as the rate of occurrence of those packets which are received correctly.Then we definethe normalized channel thruput S by
s = h'r.
(2)
The probability that a packet will not overlap a given packet is just the probability that no packet starts T seconds before or T seconds after the start time of the given packet. Then, since the point process formed from the start times of all packets in the channel was assumed Poisson, the probability that a packet will not overlap anyotherpacket is e-2h*, or e-2G. Therefore
and we may plot the channel throughput versus channel traffic for an ALOHA channel (Fig. 3). From Fig. 3 we see that as the channel traffic increases, the throughput also increases until it reaches its maximum at S = 1/2e = 0.184. This value of throughput is known as the capac-
I
time
-
Packlsts from several users on an ALOHA channel.
I ;2e S, channel thruput
Fig. 3.
Channel throughput versus channel traffic for an ALOHA channel.
ity of an ALOHA channel, and it occurs fora value of channel traffic equal t o 0.5. If we increase the channel traffic above 0.5, the throughput of the channel will decrease.
C. Application ofan ALOHAChannel In order to indicate the capabilities of such a channel for use in an interactive network of alphanumeric computer terminals, consider the 9600 bits/s packet broadcasting channel used in the ALOHA System. From the results of Section 11-B we see that the maximum average throughput of this channel is 9600 bits/s times 1/2e, or about 1600 bits/s. If we assume the conservative [ 141 figure of 5 bits/s as the average data rate (includingoverhead) from each active1 terminal in thenetwork, this channel can handle the traffic of over 300 active terminals and each terminal will operate at a peak data rate of 9600 bits/s. Of course, the total number of terminals in such a network can be muchlarger than 300 since only a fraction of all terminals will be active and a terminal consumes no channel resources when it is not active.
D. Recovely of LostPackets Since the packet broadcasting technique we have described will result in some packets being lost due to packet overlaps, it is necessary to introduce some technique to compensate for this loss. We may list four different packet recovery techniques for dealing with the problem of lost packets. The -first three make use of a feedback channel to the packet transmitter and the repetition of lost packets, while the fourth is based on coding. 1 ) PositiveAcknowledgments(POSACKS): Perhaps the mostdirect way tohandlelostpackets is to require the receiver of the packet to acknowledge correct receipt of the packet. Each packet is transmitted and thenstored in the transmitter'sbufferuntil a POSACK is received fromthe receiver. If a POSACK is not received ina given amount of time, the transmitter can repeat the transmission and continue t o repeat until a POSACKis received oruntilsomeother criterion is met. The POSACK can be transmitted on a sepal A terminal is defined as active from the time a user transmits an attempt to log on until he transmitsa log off message.
ABRAMSON: THROUGHPUT BROADCASTING O F PACKET
119
CHANNELS
rate channel (as in the ALOHANET [6] ) or transmitted on the same channel as the original packets (as in the ARPA packet radiosystem [ 151). An errordetectioncodeand a packet numbering system can be used to increase the reliability of this technique. 2 ) Transponder Packet Broadcasting: Certain communication channels-notably communication satellite channelstransmit packets on one frequency to a transponder which retransmits the packets on a second frequency. In such cases all units in a packet broadcasting network can receive their own packet retransmissions, determinewhether apacket overlap has occurred, andrepeat the packet if necessary. This technique has been employed in ATS-1 satellite experiments in the Pacific Educational ComputerNetwork (PACNET) [16] and in the ARPAAtlanticINTELSAT IV packetbroadcasting experiments [ 171 . 3) Carrier SensePacket Broadcasting: For ground-based packetbroadcasting networks where the signal propagation time over the furthest transmission path is much less than the packet duration, it is feasible to provide each transmission unit with a device to inhibit packet transmission while another unit is detected transmitting. A carrier sense capability can increase the channel throughput, even if these conditions are not met, when used in conjunctionwithother packet recovery methods. Carrier sense systems have been analyzed by Tobagi [ 181 and by Kleinrock.andTobagi [ 191 . A comprehensive yet compact analysis of such systems is provided in [42] . 4 ) Packet Recovery Codes: When a user employs a packet broadcasting channel to transmit long files by breaking them into large numbers of packets, it is possible to encode the files so that packetslost due to broadcasting overlap can be recovered. It is clear that some of the existing classes of multiple bursterror-correctingcodes [20] and cyclic product codes [21] can be used for packet recovery in transmissions of long files. It is also clear that thesecodesare not as efficient as possible for packet recovery and that considerable work remains to be done in this area.
users, and that whether or not a user transmits a packet in a given slot does not depend upon the state of any previous slot. Ifwe have n users, we candefine the normalizedchannel traffic for the slottedchannel G where n
G = C Gi.
Note that G may be greater than 1 . As before, we can also consider the rate at whicha user sends packets which do not experience an overlap with other user packets. Define S i < Gi as theprobabilitythat a user sends a packet and that this packet is the only packet in its slot. If we have n users, then we define the normalized channel throughput for the slotted channel S where
s=
n si. i=l
Note that S is less than or equal to 1 and S < G. For the slotted ALOHA channel with n independent users, theprobabilitythat a packetfromtheith user will not experience an interference from one of the other users is n
n
(1 - Gj).
j= 1 j# i
Therefore we maywrite the followingrelationshipbetween the message rate and the traffic rate of the ith user: n
si
= Gi
n (1 - Gj). j= 1 j# i
If all users are identical, we have
s. = -S n
E. Slotted Channels It is possible to modify the completely unsynchronized use of the ALOHA channel described above in order to increase the maximum throughput of the channel. In the pure ALOHA channel each user simply transmits a packet when ready without any attempt t o coordinate his transmission with those of other users. While this strategy has a certain elegance, it does lead to somewhat inefficient channel utilization. Ifwe establish a time base and require each user to start his packet only at certain fixed instants, it is possible to increase the maximum value of the channel thruput. In this kind of channel, called a slotted ALOHA channel, a central clock establishes a time base for asequence of “slots” of the same duration as a packet transmission [41]. Then when a user has a packet to transmit, he synchronizes the start of his transmission to the start of a slot. In this fashion, if two messages conflict they will overlap completely, rather than partially. To analyze the slotted ALOHA channel,define G, as the probability that the ith user will transmit a packet insome slot. Assume that each user operates independently of all other
(4)
i= 1
and G G. = n
-:r-’
so that (6) can be written S = G (1
and in the limit as rz S = GeFG.
-+ 00,
we have
(1 0)
Equation (10) is plotted in Fig. 4 (curve labeled “slotted ALOHA”). Note that the message rate of the slotted ALOHA channel reaches a maximum value of l / e = 0.368, twice the capacity of the pure ALOHA channel. This result for slotted ALOHA channels was first derived by Roberts [41] using a different method.
I
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IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1977
(13) becomes
exp [-2G1 - G2, - G,]
.
Therefore 6
1
\
\
S1 = G1 exp [--2G, - G21 - G,]
(164
and, by a similar argument, the throughput of long packets is
S i = G2 exp .184=1/2e
.368=l/e
S, ihrupui
Fig. 4.
Traffic versus throughput for an ALOHA channel and a slotted ALOHA channel.
111. PACKET BROADCASTING WITH MIXED DATA RATES
A . Unslotted Case: Variable Packet Lengths
Again assume that only those packets which do not overlap with any other packet are received correctly and define hi!< Xi as the rate of occurrence of those packets of duration ri which are received correctly. Qefine the normalized throughput of packets ofdurationri as
i = 1, 2.
- G,
- 2G2].
(1 6b)
For any given values of hl and h, we maycalculate G1, G,, G , ,, and Gzl ; substitution of these values into (16a) and (16b) will allow calculation of the throughputs' S 1 and S, . Therefore (16a) and (16b) may be used to define an allowable set of throughput p?&s (S1,S2)in the (S,,S,) plane. To determine the boundary of thisregion we define 72
In Section I1 we were concernedwiththe analysisof ALOHA channels carrying a homogeneous mix of packets. If some channel users have a higher average data rate than others, however,the high rate'usersmusteithertransmitpackets more frequently or transmit longer .packets. In this section we shall analyze theunslotted ALOHA channel whencarrying packets of different lengths, and we shall analyze the slotted ALOHA channelwhentheprobability of transmitting ina given slot varies from user t o user. Let us assume an unslotted ALOHA channel with two different possible packet durations, 7, and 7 , . Assume 7, 2 r l , and therefore we refer to the two different length packets as long packets and short packets, respectively. Assume also the start times of thelongpacketsandshortpacketsformtwo Poisson point processes withparameters h2 and h, packets/ second, and that the two Poisson point processes are mutually independent.Then we candefine,the normalized channel traffic for those packetso f duration ri:
S i = Xi'ri,
[-.G,,
(12)
Since we assumed two independent,Poisson point processes, the probability that a short packet will be received correctly is
aL-. 71
Note that a 2 1. We may rewrite (16a) and (16b) in terms of a, the ratio of long packet duration to short packet duration: S , = G1 exp[-2Gl
S 2 = G2 exp
[-(1
-
( + :) 1
G,]
+ a)G, - 2G2] .
(1 8b)
The boundary of the set of allowable (Sl,S2) pairs in the (S1,Sz) plane is defined by setting the Jacobian
equal t o zero. A simple calculation shows that the Jacobian is zero when
Note that this checks forG1 = 0 and for a = 1. We need only :substitute this expression for G2 into (1 sa) and (18b) t o obtain two equations for S, the short packet throughput, and S 2 , the long packet throughput, in terms of the single parameter G,; and as G, varies from 0 (all long packets) to1/2 (all short packets), we will traceoutthe boundary of the achievable values of throughput in the (SI, S,) plane.These achievable throughput regions are indicated for several values of a in Fig. 5. The basic conclusionof this analysis is thatthetotal channel throughput can undergo asignificantdecrease if all packets are not of the same length. Thus if the two different
ABRAMSON: THROUGHPUT OF PACKET BROADCASTING CHANNELS
121 ~n
(25) ,i=l 1fJ.k
after some algebra we may write the Jacobian as
,104 S I long packel c h a m / thruput
2
Fig. 5.
Achievable throughput regions inan unslotted ALOHA channel.
packet lengths differ by a large factor, it is often preferable to break up long packets into many shorter packetsas long as the overhead necessary to transmit the text in each packet is small. Ferguson [23] has generalized theseresults to show that channel throughput is maximized over all possible packet length distributions withfixed length packets. In view of this discouraging result, we might conclude that an inhomogeneous mix of users inevitably leads to a decrease in the maximum value of channel throughput. Surprisingly, this conclusion is not warranted, and we shall show in Section 111-Bthat a mix of users of varied data rates can lead to an increase in the maximum values of channel throughput.
B. Slotted Case: Variable Packet Rates In the section we shall consider a slotted ALOHA channel used by n users, possibly with different values of channel traffic Gi. From (6) we have a set of n nonlinear equations relating the channel traffics and the channel throughputs for these n users:
Thus the condition for maximum channel throughputs is p i = l i
This condition can then be used to define a boundary to the n-dimensional region of allowable throughputs S1,S 2 , -., S,. Consider the special case of two classes of users with nl users in class 1 and n2 users in class 2:
SI and GI be the throughputs and traffic rates for users in class 1, and let S2 and G2 be the throughputs and traffic rates for users in class 2. Then the n equations (21) can be written as the two equations
Let
s1
= Gl(l - Gl)nl-l(l
s, = G,(1 Define n
n (1 -Gj);
CY=
j=
1
then (21) can be written
Forany set of n acceptable traffic rates Gl, G2, ..., G,, these n equations define a set of channel throughputs S1,S 2 , -, S, or a region in an n-dimensional space whose coordinates are the Si. In order to find the boundary of this region, we calculate the Jacobian:
Since
- G2)"2-l(l
-G2y2
(294
-G 1 p .
(29b)
For any pair ob acceptable traffic rates G, and C,, these two equations define a pair of channel throughputs S1 and S 2 or a region in the (S, , S 2 ) plane. From (27) we know that the boundary of this region is defined by the condition
nlGl
+ n,G2
= 1.
(30)
We can use (30) to substitute for Gl in (29a) and (29b) and obtain two equations for S1 and S 2 in terms of a single parameter G,. Then as G 2 varies from 0 to 1, the resulting (S1,S,) pairs define the boundary of the region we seek. These achievable regions are indicated for various values of nl and n 2 in Figs. 6 and 7. The important point to notice from Figs. 6 and 7 is that in a lightly loaded slotted ALOHA channel, a single large user can transmit data at a significant percentage of the total channel data rate, thus allowing use of the channel at rates well above the limit of l / e or 37 percent obtained when ali users have the same message rate. A throughput data rate above the l / e limit
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IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1977
nl users at rate S, n2 users at rate S, (nl'n,)
bl
SI = e
I/e
Fig. 6.
"/
3
Allowable channel throughputs.
I .o
nl users a t rate SI n, users at rate S,
in,,n,)
Additional deta.ils dealing with excess capacityand the delay experienced with this kind of use of a slotted ALOHA channel may be found in [l I ] and [25] . A different view of the use of a slotted packet broadcasting for different sources may be found in [4.3]. IV. SPATIAL PROPERTIES OF PACKET BROADCASTING NETWORKS
A . Packet Repeaters
I/e
"I
Fig. 7.
3
Allowable channel throughputs.
has been referred to as "excess capacity" [24] . Excess capacity is important for a lightly loaded packet broadcasting network consistingof manyinteractiveterminal users and a small number of users who send large but infrequent files over the channel. Operation of the channel in a lightly loaded condition, of course, may not be desirable in a bandwidth-limited channel. For acommunications satellitewhere the average power in the satellite transponder limits the channel, however, operation in a lightly loadedpacket-switchedmode is an attractivealternative. Since the satellite will transmit power only when it is rela9ing a packet, the duty cycle in the transponder will be smail and the average power used will be low (See Section VI. Finally, we note, that it is possible to deal withcertain limiting cases in moredetail, toobtainequationsforthe boundary of the allowable (S1,S2) region. I ) For n1 = n 2 = I : Upon using (30) in (is),we obtain
In thissection we deal withcertain spatial properties of packet broadcasting networks. Not long after the initial units of the ALOHA System went into operation, it was realized that the range of the network could be extended beyond the range of a single radio link in the network (about 200 km) by the use of packet repeaters. A packetrepeateroperates in much the same manner as a conventional radio repeater with onemajor exception. Since radio transmissionin a packet broadcasting network is intermittent, a packet repeater can receive apacketandretransmitthatpacket in the same frequency band by turning off its receiver during a retransmission burst. Thus a packet repeater can sidestep many of the frequency allocation and spatial cell problems [26] of conventional land-based repeater networks. The use of packet repeatersleads to the consideration of packet broadcasti:ng networks withmore thanonecentral stationdistributed over very large areas. Users transmita packet, and if the packet cannot be received directlyby its destination, it is forwarded t o its destination by one or more packet repeatersaccording t o some routingalgorithm[27] . The study of such networks has led to the analysis of two communicationtheory issues related to theperformance of thenetworks: 1) captureeffectand 2) thedistribution of packet traffic and packet throughput in space.
B. Capture Effect Up tothispoint we have analyzed packet broadcasting channels under the pessimistic assumption that if two packets overlap atthe receiver, bothpackets are lost. In fact,this assumption provides a lower boundto the performance of
123
ABRAMSON: THROUGHPUT O F PACKET BROADCASTING CHANNELS
real packet broadcasting channels, since in many receivers the stronger of two overlapping packets may capture the receiver and may be received withouterror. Metzner [ 4 0 ] has used this fact to derive an interestingresult, showing that by dividing users into two groups-one transmitting at high power and the other at low power-the maximum throughput can be increased by about 50 percent. Thisresult is of importance for packet broadcasting networks with a mixture of data and packetized speech traffic. In order to include the effect of capture in a packet broadcasting network, we consider a distribution of packet generators over a two-dimensional plane and a single packet broadcasting receiver which receives packets from these generators [ 4 1 ]. The receiver then may be viewed as a “packet sink” and the packet generators as a distribution of “packet sources” in the plane. We assume that the rate of generation of packets in a given area depends only on r, the distance from the packet sink, andis independent of direction 0 . Then we may define a trafficdensity andathroughput density analogous to the normalized traffic G and normalized thoughput S defined in Section 11-B.
C(r) = normalized packet traffic per unit area at a distance r. S(r) = normalized packet thruput per unit area at a distance r. The traffic due to all packet generators in a differential ring of width dr at a radius r is
G(r) 27rr dr.
(34)
We assume that packets from different users are generated so that the packet starting times of all packets generated in the differential ring constitute a Poisson point process. Then since the sum of twoindependent Poisson processes is a Poisson point process, if users in different rings are independent, the start times of all packets generated in a circle of radius r also constitutea Poisson point process, andthetotal traffic generated by all users within a distance r of the center is
dx.
G(x)2nx
(3 5 )
If we assume that a packet from auser at a distance r from the center will be received correctly unless it is overlapped by a packet sent from a user at a distance ar or less (a > l), then using the results of Section 11-B the probability that such a packet will be received correctly is exp
[
- 4 a l r dx] C(x)x
.
(36)
Any packet generated from a packet source in the circle of radius ar shown in Fig. 8 will interfere with packets generated from a source in the circle of radius r. A packet generated outside the circle of radius ar will notinterferewith packets generated from a source in the circle of radius r. We canrelate the normalized packetthroughputtothe normalized packet traffic in the usual way:
Fig. 8.
Regions of interferingpackets.
- [ lr
2nrS(r)dr = 27rrC(r) exp -4n
C(x)xdxdr
1
or
[
S(r) = G(r) exp - 4 7 r l r G(x)x d x ] .
(37)
If we take a derivative of (37) with respect to r and use ( 3 7 ) to substitute for the exponential,we get
S’(r)G(r)= G‘(r)S(r)- 4nra2S(r)G(r)G(ar). (3
8)
We have not found a general solution of ( 3 8 ) for relating S(r) to G(r) in the presence of capture. We have been able to analyze two special cases, however.
C. Two Solutions In the first of these special cases we assume a constant traffic density G(r). We can then show that the throughput density S(r) has a Gaussian form, due to the fact that those packets generated further from the receiver will be received correctly less frequently than thosepacketsgenerated close to the receiver. In the second special case analyzed we assume a constant packet throughput density S(r) and perfect capture (a = 1 ) . Underthese assumptions,thepacket trafficdensity will increase as the distance fromthe receiver increases. We show that there exists a radius ro such that the packet trafficdensity is finite within a circle of radius ro around the receiver, while the packet traffic density becomes unbounded on the circle of radius y o . For the important case of a packet broadcastingchannel distributed over some geographical area and using apacket retransmission policy(Section 11-D), this result has an interesting interpretation. In such a situation any packet transmitted from a terminal located within the circle of radius ro will be received correctly with probability one (after a finite number of retransmissions), while theexpectednumber of retransmissions required for a packet transmitted from a terminal furtherfromthecenter than ro willbe unbounded. Thusthereexists a circle of radius ro such that terminals transmitting from within this circle can get their packets into the central receiver, while terminals transmitting from outside this circle spend all their time retransmitting their packets in vain. We call ro the Sisyphus distance of the ALOHA channel. I ) Constant Packet Traffic Density: Assume the density of normalized packet traffic is constant over the plane
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IEEECOMMUNICATIONS, TRANSACTIONS ON
C(r) = Go
JANUARY 1917
(39)
and define the distance r1 as the radius of a circle within which the total packet traffic is unity:
rrl 2Go n 1. Then (38) reduces to
Y ( r )=-
4ra
-’
Fig. 9.
S(r)
Region of constant packet throughput So for a single packet sink.
for
r1
with the boundary condition
so = Go
(41b)
where
so that the packet throughput density is and ro is the Sisyphusdistance mentionedinSection IV-C. Note that the Sisyphus distance also has the property that and the total normalized packet thruput from a circle of radius r is
S
=
[
S(r‘)2rr‘ dr‘
=L 2a2 (1 -exp
[-2(
:)‘I)
(43)
As in the previous case, the total packet throughput which can be supportedbya single packet sink operatingwith perfect capture is one half. V. PACKET BR.OADCASTING WITHAVERAGE POWER LIMITATIONS
and
1 lim S = ,.+2a2 Notethatatotalthroughputwhichcan
1 nro2So = 2
(44) be supported by a
single packet sink with “perfect capture” (a = 1) is equal t o one half. 2 ) Constant Packet Throughput Density: Another case of interest where we have found a solution for(38) is that of constant packet throughput density in the plane. Assume
S(r) = So
(45)
over the region in the plane where S(r) and C(r) are bounded. Then ( 3 8 ) becomes
G’(r) = 4nra2G(r)G(ar).
(46)
For the case of a = 1 (perfect capture), (46) becomes
(47) G’(r) = 4nrG2(r) with the boundary condition G(0) = s o so that
(49) Note that the normalized packet traffic per unit area is finite
A . Satellite Packet Broadcasting
In previous sections we have analyzed the performance of packet broadcasting channels and compared the performance of these channels to that of conventional point-to-point channels operating at the same peak data rate. Such a comparison is of interest in the case of channels limited by multiple access interference rather than noise, since an increase in the transmitted power of such channels will not lead to improved performance. But juut as the average data rate of a packet broadcasting channel can be well below its peak data rate when it is operated at a low duty cycle, the average transmitted power of a packet broadcasting channel can be well below its peak transmitted power. In thissection we analyze thethroughput of apacket broadcasting channel when compared to that of a conventional point-to-point channel of the same average power. This analysisis of interest in the case of satellite information systems employing thousands of small earthstations.Fora satellite system thefundamentallimitation in thedownlink is the average power available in the satellite transponder rather than the peakpower. Our results show that in the limit of large numbers of small earthstations,the.packetthroughput approaches 100 percent of the point-to-point capacity. Thus the multiple access capability and the complete connectivity (in the topological sense) of an ALOHA channel can be obtained at no pric.e in average throughput. Furthermore, since our results suggest the use of higher peak power in the satellite
125
ABRAMSON: THROUGHPUT O F PACKET BROADCASTING CHANNELS
‘4
transponder (while the average power is kept constant), the small earthstations may use smaller antennasand simpler receivers and modems than would be necessary in a conventional system. In existing satellite systems the TWT output power in each transponder cannot be varied dynamically. In such systems the advantages impliedby our analysis may be realized byfrequency-divisionsharing a single transponderamong several voice users and a single channel, operating in an ALOHA mode or some other burst mode, and occupying a frequency band equivalent to one or more voice users. The type of operation implied by our analysis also suggests investigation of high peak power satellite burst transponders (perhaps employing power devices similar to those used in radar systems) for use in information systemscomposed of large numbersof ultra-small earth stations.
.9
“hsianal-to-noise
G .2
A
.6
B. Burst Power and Average Power The capacity of a satellite channel can be calculated by the classical Shannon equation
c = Wlog
( +-3 1
where C is the capacity in bits (if the log is a base two logarithm), W is the channel bandwidth, P is the average received signal power at the earth station, and N is the average noise power attheearthstation.Equation (53) expresses the capacity of the satellite channel under the assumption that the
ratio (db)
1.0 1.2 channel trafflc
.8
D = G,
(544
while for a hard-limiting transponder
D = 1 -e-G.
2.0
Fig. 10. Linear transponder;unslottedchannel.
1) We replace W by SW to account for the fact that the channel is only used intermittently. 2 ) We replace P in (53) by P/D to keep the average power of the channel fixed at P. We should note that when we make these changes, we are assuming that the packet length of the system is long enough so that the asymptotic assumptions which are used to derive (53) still apply. In practice, this is not a problem. With these two changes then, we have four different cases. I ) Unslotted channel, linear transponder:
transponder transmits continuously.
If the channel is used in burst mode the transponder will emit power only when a data burst occurs, and the average power out of thetransponder willbe less thantheburst power. Let D be the ratio of the average power transmitted tothe power transmitted during adataburst.Fora linear transponder D will equal the channel traffic G, and for a hardlimiting transponder D will equal the duty cycle of the channel. For both the unslotted and slotted ALOHA channel the duty cycle is 1 - eCG. Thus for a linear transponder2
1.41.8 1.6
C, = Ge-2G Wlog (I
+
&)
.
2) Unslotted channel, limiting transponder:
3 ) Slotted channel, linear transponder: C,
= Ge-GWlog
(1
+
2)
4 ) Slotted channel, limiting transponder: (54b)
C, = Ge-GWlog Note that in the case of a hard-limiting transponder with small values of channel traffic, the duty cycle approaches that of a linear transponder. If we retain P as the notation for the average signal power received at the earth station, the power received during a data burst will be P/D. Thus (53) should be modified in two ways. *Our analysis is of significance only for G < 1 . The analysis is formally correct, however, for all G, even though the designation of the powertransmitted during bursts as “peak power”becomes inappropriate for the linear transponder case when G > 1 . (In such a situation the “peak power” is less than the average power.)
We have calculated the normalized capacities Ci/C for i = 1, 2, 3, 4 for different values of P/N, the signal-to-noise ratio of the earth station when the transponder operates continuously. The normalizedcapacities are plotted in Figs. 10, 11, 12, and 13 for PIN equal to -20, -10, 0, 10, and 20 dB. Of particular interest in these curves is the fact that the highest values of Ci/C occur just where we wouldwant themto occur-for small values of channel traffic (C) andfor small earthstations (low PIN). In the limit we have (fora fixed value of G)
126
IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1977
""1
O
signal-to-noise ratio (db)
.2
L
.A
ciwnnel traffic
Fig. 13.
Fig. 11. Limitingtransponder;unslottedchannel.
':I
~
I
.6
~
I
~
I
.8 1.0 1.2 channel traffic
'
I
1.4
~
1.6
I
'
1.8
G' I 2.0
l
'
l
Limitingtransponder;slottedchannel.
and in all cases r\
stgnpl-to-noise ratio
(db)
lim G-0
Ci
lim - = 1. C
(58)
N+O
Thusthis multiplexing technique allows a network of small inexpensive earth stations to achieve the maximum value of channel capacity,atthe same time providing completeconnectivity and multiple access capability. VI. BACKGROUND AND ACKNOWLEDGMENT I
' .Z '
Fig. 12.
s
ci= lim C. D ' E'O
.A .6 '
'
I b 112 ' 1'4 channel traffic .E
'
'
1'6 ' 118 ' 2!OG
Lineartransponder;slotted channel.
i = 1,2,3,4
so that 1) unslotted channels,linear transponder
Cl = e--2G lim P Z' O C 2) unslotted channels, limiting transponder
3) slotted channel, linear transponder
4) slotted channel,limiting transponder C, Ge c G lirn - = $0 C (1 -ecG)
The term "packet broadcasting" was first coined by Robert Metcalfe in his Ph.D. dissertation [28]. As is often the case with simple ideas, the concept of combining burst transmission and Poisson user statistics to provide random access to a channel has occurred independently to a number of investigators. (56) The first attempt at an analysis of such a system of which I am aware is contained in an internal Bell Laboratories memorandumbySchroeder [29], suggested by anearlierpaperby Pierce and Hopper [30]. Two other early related papers were written by Costas [31] and Fulton [32]. Of course, a theoretical analysis is not necessary in order to build such a system, and anyone who has sat in a taxi listening to the staccatovoice (57a) bursts of a radio dispatcher and a setof taxi drivers sharing a single voice channel will recognize theoperationofa voice packet broadcasting channel using a carrier sense protocol. And even after an analysis is available, the concept of packet broadcasting may be suggested without reference t o the (57b) theory [331. The first papers analyzing packet broadcasting in the form implemented in the ALOHA System [6] assumed fixed packet throughput and a retransmission ,protocol as described in Section 11-D-1). This approach leads toanumber of questions involving optimum retransmission policy [28], the behavior of the channel with a finite numberof users [39.] , stability of the channel [ 131 , and transmission of long files by means of various reservationschemes [34] , [44] . A comprehensive treatment of these as well as other interesting packet broadcasting questions may be found in Kleinrock 1421 . In this paper we (57d) have takenadifferentapproachby assuming a given packet traffic rather than throughput. With such a starting point, the
ABRAMSON: THROUGHPUT OF PACKET BROADCASTING CHANNELS
1
questions mentioned above donot assume key importance in thetheory,although theirpractical importance is not diminished. Much of the theory of packet broadcasting was developed in two working groups sponsored by the Advanced Research Projects Agency of the Department of Defense. These groups circulated a private series of working papers-the ARPANET Satellite Systemnotes (ASS notes) and the Packet Radio Temporary notes (PRT notes)-where many of the theoretical results described or referenced in this paper appeared for the first time. Unfortunately, the several references to ASS notes in papers subsequently published in the open literature may have produced some confusion in the minds of those trying t o trace the references. Among the most significant of the ASS note rind PRTnote results was the firstderivation of the capacity of a slotted ALOHA channel and the first analysis of the use of the capture effect in packet broadcasting, both by Larry Roberts.Thatnote has since been republishedin the open literature [41]. The results ofSection 111-A dealing withtwodifferent packet lengths were suggested by an ASS note written by Tom Gaarder, and the results of Section 111-B dealing with the excess capacity of a slotted channel were suggested by an ASS note written by Randy Rettberg. Other problems which were first analyzed in ASS notes or PRT notes but not emphasized in thispaperinclude various packetbroadcasting reservation systems [22] [35], [36], carrier sense packetbroadcasting [ 181 , [ 191 , and questions dealing with packet routing and protocol issues in anetworkof repeaters [ 3 7 ] .Thereader interested in theoreticalnetworkprotocolquestionsshould also see Gallagher [38] , although this work did not originate in an ASS note’or PRT note. The first system to employ packet broadcasting techniques was the ALOHA System computer network at the University of Hawaii in 1970. Subsequently, packet repeaters were added to thenetwork andpacketbroadcasting by satellite was demonstrated in the system. Some of the people involved in theimplementation anddevelopmentof the system were RichardBinder, Chris Harrison, Alan Okinaka,and David Wax. The historical relevance of [29] and [ 3 2 ] was pointed out to me by Joe Aein, to whom I am indebted, in spite of my embarassment at having forgotten I was thesis supervisor on the second of these papers. REFERENCES L. G. Roberts and B. D.Wessler, “Computer network development to achieve resource sharing,” in 1970 Spring Joint Comput. C o n f , AFIPS Conf Proc., vol. 36. Montvale, NJ: AFIPS Press, 1970, pp. 543-549. L. G. Roberts, “Data by the packet,” IEEE Spectrum, vol. 11, pp. 46-51, Feb. 1974. W. W. Chu, “A study of asynchronous time division multiplexing for time-sharing computer systems,” in 1969 Spring Joint Comput.Conf:,AFIPS Conf Proc., vol. 35. Montvale, NJ: AFIPS Press, 1969, pp. 669478. F. Heart, R. Kahn, S. Ornstein, W. Crowther, and D. Walden, “The interface message processor for the ARPA computer network,” in 1970 Spring JointComput.ConJ, AFIPS Conf Proc., vol. 36. Montvale, NJ: AFIPS Press, 1970,pp.551567. H. F h k , I. Frisch, and W. Chou, “Topological considerations in
127 the design of the ARPA computer network,” in 1970 Spring Joint Cornput. Con$, AFIPS Con$ Proc., vol. 36. Montvale, NJ: AFIPS Press, 1970, pp. 581-587. N. Abramson, “The ALOHA system-Anotheralternative for computer communications,” in I 9 7 0 Fall Joint Comput; C o n f , AFIPS Con$ Proc., vol. 37. Montvale, NJ:AFIPS Press, 1970, pp. 281-285. R.E. Kahn,“Theorganization of computer resources into a packetradio network,” in Nat.Comput.Confi, AFIPS Con& Proc., vol. 44,May 1975, pp. 177-186. N. Abramson and E. R. Cacciamani, Jr., “Satellites: Not just a big cable in the sky,” IEEE Spectrum, vol. 12, pp. 36-40, SePt. 1975. I. T. Frisch, “Technical problems in nationwide networking and interconnection,” IEEE Trans. Commun., vol. COM-23, pp. 7888, Jan. 1975. R. M. Metcalfe and D. R. Boggs, “ETHERNET:Distributed packet switching for local computer networks,” Commun. Ass. Comput. Mach., to be published. N. Abramson, “Pakket switching with satellites,” in Nat. Comput. C o n f , AFIPS Confi Proc., vol. 42, 1973, pp. 695-702. R. D. Rosner, “Optimization of the number of ground stations in a domestic satellite system,” in EASCON’75 Rec., Sept. 29Oct. 1, 1975, pp. 64A-645. A. B. Carleial and M. E. Hellman, “Bistable behavior of ALOHAtypesystems,” IEEE Trans. Commun., vol. COM-23, pp. 401410, Apr. 1975. P. E. Jackson and C.D. Stubbs, “A study of multiaccess computer communications,” in 1969 Spring Joint Conzptrt. Con&, AFIPS ConJ Proc., vol. 34. Montvale, NJ: AFIPS Press, 1969, pp. 491-504. T. J. Klein, “A tacticalpacketradiosystem,” in Proc. Nat. Telecommun. Con&,New Orleans, LA, Dec. 1975. K. Ah Mai, “Organizational alternatives for a Pacific educational computer-communicationnetwork,” ALOHA Syst. Tech. Rep. CN74-27, Univ. Hawaii, Honolulu, May 1974. R. Binder, R. Rettberg,and D. Walden, TheAtlantic Satellite Packet Broadcast and Gateway Experiments. Cambridge, MA: Bolt Beranek and Newman, 1975. L. Kleinrock and F. A. Tobagi, “Packet switching in radio channels: Part I-Carrier sense multiple-access modes .and their throughput-delaycharacteristics,” IEEE Trans. Commun., vol. COM-23, pp. 1400-1416, Dec. 1975. F. A. Tobagi and L. Kleinrock, “Packet switching in radio channels: Part 11-The hiddenterminal problem in carrier sense multiple-access and the busy-tone solution,” IEEE Trans. Commun., vol. COM-23, pp. 1417-1433, Dec. 1975. R. T. Chien, L. R. Bahl, and D. T. Tang, “Correction of two erasure bursts,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 186187, Jan. 1969. N. Abramson, “Cyclic code groups,” Problems oflnform. Transmission, Acad. Sci. USSR, Moscow, vol. 6, no. 2, 1970. L. G. Roberts, “Dynamic allocation of satellite capacity through packet reservation,” in Nat. Comput. Con5 AFIPS Con$ Proc., vol. 42, June 1973, pp. 711-716. M. J. Ferguson, “A study of hnslotted ALOHA witharbitrary message lengths,” in Proc. 4th Data Commun. Symp.. Quebec, Canada, Oct. 7-9, 1975, pp. 5-20-5-25. R. Rettberg,“Random ALOHA with slots-excess capacity,” ARPANET Satellite Syst. Note 18, NIC Document 11865, Stanford Res. Inst., Menlo Park, CA, Oct. 11, 1972. N. Abramson, “Excess capacity of a slotted ALOHA channel (continued),” ARPANET Satellite Syst. Note30, NIC Document 13044, StanfordRes. Inst., Menlo Park, CA, Dec. 6, 1972. L. Schiff,“Random-access digital communicationfor mobile radio in a cellular environment,” IEEE Trans. Commun., vol. $OM-22, pp. 688-692, May 1974. H. Frank, R. M. Van Slyke,and I. Gitman, “Packetradio network design-System considerations,” in Nat. Comput. Conf, AFIPS Con& Proc., vol. 44, May 1975, pp. 217-232. R. M. Metcalfe, “Packet communication,”Rep. MAC TR-114, Project MAC, Massachussetts Inst.Technol., Cambridge, July 1973. M. R. Schroeder,“Nonsynchronoustimemultiplexsystemfor speech transmission,” Bell Lab. Memo., Jan. 19, 1959. J. R. Pierce and A.L. Hopper, “Nonsynchronous time division with holding and random sampling,” Proc. IRE, vol. 40,pp. 1079-1088, Sept. 1952.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-25, NO. 1 , JANUARY 1977 J. P. Costas, “Poisson, Shannon, and the radio amateur,” Proc. IRE, vol. 47, p. 2058, Dec. 1959. F. F: Fulton, Jr., “Channelutilizationby intermittent transmitters,’’ Tech. Rep.2004-2, Stanford Electron. Lab., Stanford Univ., Stanford, CA, May 12, 1961. K. D. Levin, “The overlapping problem and performance degradation of mobile digital communication systems,” IEEE Trans. Commun.. vol. COM-23, pp. 1342-1347, Nov. 1975. R. Binder,. “A dynamic packet-switching system for satellite broadcast channels,” in Con$ Rec., Int. Con$ Commun., vol. 111, June 1975, pp. 41-1-41-5. S. S . Lam and L. Kleinrock, “Packet switching in a multiaccess broadcastchannel:Dynamic control procedures,” IEEE Trans. Commun., vol. COM-23;pp. 891-904, Sept. 1975. L. Kleinrock and S.S. Lam, “Packet switching in a multiaccess broadcast channel: Performance. evaluation,” IEEE Trans. Commun., vol. COM-23, pp. 410-423, Apr. 1975. I. Gitman, “On the capacity of slotted ALOHA networksand some design problems,” IEEE Trans. Commun., vol. COM-23, pp. 305-317, Mar. 1975. R. G. Gallager, “Basic limits on protocol information in data communicationnetworks,” IEEE Trans. Inform.Theory, vol. IT-22, pp. 385-398, July 1976. M. J. Ferguson, “On the control, stability, and waiting time in a slotted ALOHA random-access system,” IEEE Trans. Commun., vol. COM-23,pp. 1306-1311,Nov. 1975. J . J . Metzner, “On improving utilization in ALOHA networks,” IEEE Trans. Commun., vol. COM-24, pp. 447-448, Apr. 1976. L. G. Roberts, “ALOHA packet system with and without slots and capture,” Comput. Commun. Rev., vol. 5, pp. 28-42, Apr. 1975. L. Kleinrock, Queueing Systems, Volume 2: Computer Applications. New York: Wiley, 1976,pp.,360-407. I. Gitman, R. M. Van Slyke, and H. Frank, “On splitting random accessed broadcast communication channels,” in Proc. 7th Hawaii Int. Con& Syst. Sei.-Suppl. on Comput.Nets, Western Periodicals, Jan. 1974, pp. 81-85.
‘
[44] W. Crowther, R. Rettberg, D. Walden, S. Ornstein, and F. Heart, “A system fo:r broadcast communication: Reservation ALOHA,” in Proc. 6th Hawaii Int. Con$ Syst. Sci., Western Periodicals, Jan. 1973, pp. 371-374.
*
A Multiaccess Model for Packet Switching with a Satellite Having Some Processing Capability
Abstract-A multiaccess model for packet switching with a satellite having the capability of interrogating the uplink header and creating the downlink header is proposed. The satellite broadcasts slot assignments, based on the users’ reported queue status, to the users for transmission in the next frame. With the protocols being done at both the earthstationsand atthe satellite, the proposedmultiaccessmodel avoids collisions that areprevalent in schemes of the ALOHA type. Manuscript received February 27, 1976; revised July 13, 1976. This paper has .beenpresented attheThirdInternational Conference on Computer Communication, Toronto, Ont., Canada, August 3-6,1976. This work was supported in part by the National Research Council of Canada under Grant A7779 and in part by a Graduate Scholarship. F. W. Ng is with the Department of Electrical Engineering and the Computer Communications Networks Group, University of Waterloo, Waterloo, Ont., Canada. J . W. Mark is with the Department of Electrical Engineering and the Computer Communications Networks Group, University of Waterloo, Waterloo, Ont., Canada. He is currently on leave at the IBM Thomas J . Watson Research Center, Yorktown Heights, NY 10598.
The actual model is too complex to handle analytically. We derive analytical equations for a two-gioup model. Calculated and simulated buffer overflowprobabilitiesasa function of trafficintensityand buffer size are compared. We alsoevaluate theperformance of the actual model in terms of average system delay as a function of.traffic intensity by means of computer simulation.
I. INTRODUCTION
S data traffic grows, the demand on computer communication using satellites,whichoffer wide transmission bandwidths over long distances, will continue to increase. The trend has been t o findmore efficientschemes forchannel sharing. Synchronous time-division multiplexing (STDM) represents an attractive scheme which permits many users to share the same channel. One of the serious drawbacks associated with a snychronous transmission system is that it assigns
A
