BRAM: The Broadcast Recognizing Access Method

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I E E E TRANSACTIONS O N COMMUNICATIONS, V O L . COM-27, NO. 8 , AUGUST 1979

BRAM: The Broadcast Recognizing Access MethodIMRICH CHLAM TAC, WILLIAM R. F RANTA, A N D K. DAN LEVJN

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Absrru ct-In his paper, we first present he broadcas t recognizingaccess meth od (BRAM ), an access protocol suitable for regulating inter-node communicat ion in e i ther a radio or (coaxial or fiber) cable basedcommunicat ion system.The methodavo ids collisions, impos es negli-gible compu tat ional requirements on the nodes a t tempting to t ransmit ,and is fair in the sense that no n ode will be indefinitely prevented fromtransmit ting. Next we introduce parametr ic BRAM which a t tempts tobalance he engthof nsert ed hann el idle eriods, esulting fromschedulin g effects, against the proba bility of allo wed message collisions.We show that param etric BRAM can be used to realize a method whichbalances nserted channel dle ime against the probability of messagecollision to yield enhanced performance. Fo r high message loads, para-metric BRAM converges to BRAM, while for low and medium loadingsityields hroughputs in excessof BRAM, and othermethods.Bo thBRAM andparametric BRAM are discussed u nder he assumption ofhom ogeneo us message arrival rates at the nodes. We conclude by show-ing how the parametric BRAM can be applied when the nodes operatewith heterogeneous or mixed message arrival rates.

M I . INTRODUCTIONANY protocols havebeendevised to handle he use byseveral nodes (computers, erminals, etc.) of a co m m o ncommunication medium or channel , e.g . , radio, coaxial cable,etc. for comm unication. As pointed out in 141, hese proto-cols can be categorized as

a) fixed assignmentb) controlled assignment, orc) no assignment (random access).

an d [4 , 51 cont ains a discussion of the relative meri ts of p roto-cols in each category.Rando m access schemes are characterized by not requiringa centraliz ed control mechanism to regulate he ransmissionof messages. Instead each node data source regulates its ownuse of the comm unication medium by use o f an access proto-col which requires only information available to the nodes bysensing thechannel .The andom accessschemes range fromsimple ALOHA [41 (which does ot use any informationwhich may be available on the channel), to a variety of carriersense multiple access (CSMA) variants [4 ], to th emini-slottedaccess protocol (MSAP) [7]. The performa nce of these proto-cols is directly elated tohow well node ransmissions aresynchronized to avoid message collisions,with heobviousconseq uence that transmissions destroye d by collision m ust berepeated. ALOH A includes no provisions to limit the prospectofcollisions,but provides random izationof etransmissions

Paper app roved by the Editor for Computer Communicat ion of th eIEEE Communicat ionsSociety for publicationwithout oral presenta-tion. Manuscript received March 13 , 197 8; revised Decem ber 28, 1978.I . Chlamtac and W. R. Franta are with the Department of Compute rScience, University of Minnesota, Minneapolis, M N 5 5 4 5 5 .K. D. Lcvin is with the Faculty of Management, Tel-Aviv University,Tel-Aviv, Israel.

(after collision) while the CSMA variants provide node atten-tion o deference(deferenceoccurswhenanodewishing totransmit senses the channel before doing so , and defers to anongoing transmission, if any). Finally, certain protocols, e.g.,MSAP [7] areable to avoidcollisions altogether.(Time divi-sionmultipleaccessTDMA [8] alsoavoidscollisions, bu t isnot a randomaccess protoc ol.)

In th is not e, we discuss BRAM-The Broadcast RecognizingAccessMethod-whichallocates thechannel to nodes via adecen tralized protocol. BRAM (in part) regulates channe l ac-cess via a scheduling function and four variants, namely:

fair BRAM (FB)prioritized BRAM (PB)parametric fair BRAM (PFB)parametric prioritized BRAM (PPB)

are discussed in subsequ ent sections. The first two avoid colli-sions, the latter two limit their probability of occurrence. Aswe shall find, for small values of the product (number of nodesin networkchannel ,end-to-endpropagation delay) FBandPFBoffer fairallocationproto col s (fair in thesense that anupperboundexists on ransmissiondelays or all message soriginatingatan yan d all nodes) which exhibitperformancemeasures at least as good as those of other published methods(which do notgua rante e fairness). As theproductdefinedabove increases, PFB provides good performance measures forsmall channel throughput values, while PPB exhibits perform-ance measures better than for the other methods, independentof the value of the products defined above.

11. THE BROA DCAS E RE COGN IZING ACCESSMETHOD-BRAM

The BRAM is a random access protocol which exh ibits de-centralize d control (channe l access is not regulated by a singleagent) and is applicable to n etw ork s which emp loy a coax ial orfiber optics busor radio channel as the comm unication medium .

The BRAM protocol is designed:d . 1) To allow collision-free transmissionsd.2) To be air to all nodes in hesense hat heepoch

when a node desires to secure the channel and the epoch whenit successfully secures the channel is bounded,d.3)oxhibithannelhroughputnd transmissiondelay measures as good as or better han hose exhibited byother published methods (even though hey may not be fairto the nodes) .It requires that nodes:

r.1) defer to ongoing ransmissions,r.2) are ble to discernhe identity of the odero m

0090-6778/79/0800-1183$00.75 0 1979 IEEE


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1 1 8 4 IEEE TRANSAC TIONS ON COMMUNICATIONS, VO L. COM-27, NO . 8 , AUGUST 1979

T ,n2 = index of nodewhich lasttransmitted.. . .. . . b h JV V

t r a n s m i s s m n periodidleeriod or schedulingeriod Transmission period Ofduration T for node with

iridex n

Figure 1 . Channel periods fo r th e BRAM protocol.

whicha transmissionoriginates (accomplished orexample,by assigning a node an in dex, and having the node encode theindex as the beginning portionof its transmission),

r.3) stagger their attem pted transmission starting times bya, the commun ication medium end-to-end propagation delay(so that objective d.1 can result).Additionally, to simplify our presentation,e also assume hat:

r.4) the chann el is noiseless;r.5) thereexists an independent acknowledgmentchannel

an dnodeacknowled gment processing time equirem ents arenegligible;r .6) there is n o capture effect;r.7) each of the K nodes in the netw ork is able to transmit

and receive, b ut not simultane ously, and turn around timefrom ransmittin g tate o receiving state (o r vice versa) isnegligible;

r .8) the nodes can detect the busy/idle status of the chan-nel in a negligible time;

r.9) all nod es are with in range and in ine of sight (LOS)of each other .

T o achieve d.1-d.3 BRAM acts as follows. One consid ers thechannel state (and hence the time axis structure) to consist ofa sequen ce of cycles compos edof idle (and/or), sched uling andtransmission periods. An idle period occurs when no node at-tempts to gain co ntro l of the channel following its deferenceto a ransmission. A scheduling period (of random length as weshall see) occurs when one or more nodes a t tempts to obta incontrol of the channel for a transmission; and a transmissionperiod T secon ds ong results wh en a collision free ransmis-sion occurs. These periods are shown n figure 1. It is duringthe scheduling period that one of the nodes desiring to trans-mit gains exclusive control of the channel for the ( i ts) trans-mission pehod which follows.

Fbr ord ering acquisition of the chan nel BRAM requires th atall nodes have a com mon understan ding of the t ime ep och,,, ,(see figure l ), whic h marks the beginnin gof the nth, say, che-duling p eriod. This common understanding is easily had by as-sociating T,, with the end of the (n - 1)st transmission period(if the re are nod es whic h wish to trans mit and are deferring)or wi th the uni t boundar iesf clocks (one in each node) whichincrement in K. a units and are reset to zero at the terminationof each ransmission(for hecasewherenodesapproach anidle channel for scheduling).

During the scheduling period, channel control is granted toone of he ready nodes as a resultof node use of he sche-duling function H (which specif ies when to transmit) and node

a t tent ion to r.1. The sch eduling fu nction has the form

n, = n 2

the index of any node j , with j E {rj } , he set of readynodes (i.e. , the set of nodes wishing to transmit), andthe index of the n ode which transmitted last (obtainedas a result of r.2, and maintained by all nodes, idle orready), andthe number of nodes in the network.

I t is o bvious that H is integer-valued , with 1 < H packet transmission time, then between T,, an d T,, +H( j , n2 ) . a a transmission can occ ur so tha t j may still find thechannel idle at T,, +H (j , n2)- .a , .a lthough it may be so due toth e (n + 1)st scheduling period rather than the nth. To avoidthis difficulty in step p.1, we insist th at if the channel becomesactive betw een T , an d T,, + H ( j , a2)-a he node proceeds tostep p.2.From the preceding discussion, we have that the length of ascheduling period is thus given by m in H ( j , n2)a . In [ l ] , wewere able to prove tha t each no de i E { r j }which does not cap-ture hechann el in a given schedulingperiod, say th en th ,improves its chances for capturing the channel in the ( n + 1)sttransmissionperiod, in the sense that oreachsuchnode iwhichdoesnot apture he hannel in thenth cheduling


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CHLAMTAC e t a l . : BROADCAST R E C O G N I Z I N G ACCESS METHODperiod we have H ( j , n z n + l ) .01 simulation esultsmust beused.Theequation (6) does serve to provide a lo wer bo und on ex pecte ddelays, and we haveobserved tha t for a > .01 he error be-tween (6) and imulation esults is proport ional o he dif-ference between (2) and ( 6 ) ,as we might expect.

111. PARAMETRIC BRAMFrom the curves labeled m = K , figure 4, we see that fair

BRAM performance s degraded as a increases, especially forlow and mid-rang e G values. More generally, we can observethat fair BRAM performance, as given by S/C , degrades asth eproduct K - a increases (thisobservation is also true orMSAP [ 7 ] ) . The reason for he degradation rests on the factthat as K - a increases,so does the u pper bound n the expec ted


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CHLAMTAC e ta l . : BROADCAST RECOGNIZING ACCESS METHOD 1187

0.2

.01 0.1 1 10 100

Offered Traific c

Figure 4. Comparisonof hroughput n m = K BRAM an doptimalm-value param etric B R A M for various a-values. (Simulation results).

allowable m values.2 The simula tion results n Figure 4 com-pare throughp ut for various a values, for fair BRAM (using Kin (1)) with S optimized fair parametric BRAM using m* in(7). The results for fair BRAh4 are labeled m = K while thosefor fair parametric BRAM are labeled m*. As can be seen fromFigure 4, use of m* with R increases th roughput for low andmidrange values of G (as desired ), with the improv eme nt be-coming more appreciable as a increases. Additio nally , we seefrom Figure 4, that m* +K (and, of course, the length of thescheduling periods approach a ) as G becomes large.In Figure 3 : we show, or various a values, K = 10 an dhomogeneous raffic curves ofD versus S for MSAP, prior-itized BRAM (PB), parametric BRAM (PFB)an dparametricprioritized BRAM (PPB), the atter woboth using the m*value which yields he owest value of D . Figure 3 can beconside red representative of he families of curveswhich wehave for a variety of K , a configu rations. We have includedMSAP delay curves for com paris on, as, see [ 7 ] , MSAP displaysthe best overall performanc e of all schemes proposed to da te(including CSMA, TDMA, etc). Specifically as shown n [ 7 ] ,MSAP exhibits delay urves lightlyworse than CSMA forsmall K , S values, but exhibits superior delay performanc e toall other pub lishe d rand om access schemes for all K - a valuesunder heavy trafficoads.Additionallyhe omparison issimple, since b oth prioritized BRAM and MSAP performanceis given by (2).From Figure 3 (and curves ike it) we can conclud e hat:

I . ( a - K small). For small values of a.K ( K = 10, a = .01,a = .I are shown in Figure 3 ) prioritizedparametric BRAMexhibitshe est erformance (of all method s) ut sinceK w is small, the delay curves it exh ibi ts do not deviate sub-stantially from those for MSAP, fair BRAM, prioritized BRAMfair parametric BRAM, or results predicted by MIDI1 queueing

2 The allowable etof m values being, of course n the range1 < m Q K. h e m * values used in the performance curves were foundb y a limited search procedure and simulation, based on he fact h atm* = 1 fo r S - 0, that m * increases as G does, and hat m* K fo rlarge G.

theory. For low a - K values, then, he primary tem of notecomes rom heobservation hat or fair BRAM (andpara-metric fair BRAM with fairness applied to he groups) thevariance of he delay distributionmust be maller than orother met hods, since as we haveshown the delay for nodesis bounded while for MSAP no such guaranteed bound on de-lay exists. Said differently,FB is fair to henodes,otherprotocols, e.g;, MSAP, are not .

11. ( a - K not small) . For a*K values which are not small(say > . l ) , h e performance curves for he metho ds diverge,and orcomparison Figure 3 show s curves for heextremecase a = 1 ( K = lo). From Figure 3 (andothe r curves forvarious a , K configurations) we can conc lude:

1. The expected normalized delay exhibited by param etr icprioritized BRAM (PPB) is superior tohat or all othermethods (including MSAP) for all S values (as G +m an d S +1 , PPB approaches PB and MSAP since m + K . This fact isdepicted in Figure 3 fo r a = 1. ) Note also tha t the delay curvesfor PPB and PFB begin at 1 and 2 respectively.These valuesare explained by observing that for PPB and S - 0, m = 1 ,SO that H(ni , 6,) = (1 ~ 1 + 1) mod 1 = 0 and every eadynode transmits immediately: so that its transmission delay be-com es he packe t transmission time alone. For PFB , we alsohave m = 1 fo r S - 0, but in this case H(ni,2 ) = 1 (see (7))so tha t transmission delays must include on e scheduling period(of magnitude one fora = 1) plus the packe t transmission time .Thus, for a = 1 the delay curve begins with D = 2. Fur ther ,PFBapproaches FB and PPB approaches PB in performanceas S increases ince forbothprotocols m* + K as S in-creases.

2. The exp ecte d delay for param etric fair BRAM (PFB) isalso (as it is for PPB) lower han MSAP for low S values.Additionally, the second moment of the delay distribution isalso lower fo r PFB ince, as statedbefore, hereexists anupper bound on delay for FB.

3 . The capac ity (i.e., the largest acheivable throu ghpu t be-comes l for prioritized variants and l/ (] + a ) for fair variants,see Figure 3 .


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1 1 8 8 IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-27, NO. 8 , AUGUST 1979Mixed Data Rates

The results reported in Figures 2 , 3 and 4 reflect perform-ance under assumptions r.11 and r.12. If r.11 is relaxed, thatis if we cannot assume that hi = h / K , i = 1, .-, K then caremust be taken in the process of associating nodes with groups(for homogeneous traffic the group simply co ntains as nearlythe same numb er of n odes as possible). Specifically whe n thehi are node depende nt we can easily obtain groups for whichthe collisionprobabilities re so imbalance d hat collisionswill occur very frequently in somegroups and infrequentlyin others. Such mb alances can adversely affect performa nce.It is of nterest, herefore, o ask how henodesshould begrouped so as to obtain he best perform ance possible whenusing parametric BRAM, with mixed traffic rates. The answeris given by the following theorem.

Theorem: The opt imal g func tion in the sense of reducingto a minimum the probability of intragroup collisions, is theone which divides thenetworknodes ntogroupsofequalaggregated arrival rates.

Proof: Let h = Z E l hi. Let us divide the K nodes intom disjoint groups, Is,s = 1 , -.,m . Then the probability that atransmission is originatedbynode i , given that toriginatesin Is, s

hiC hji E I sThis transmission will be destr oyed by collision if othe r nodesbelonging to Is became ready during the previous transmission.

: The probability of collision s, n group Is given that t origi-nates a transmission, therefore,.given by:

with T the packe t transmission tim e. Assuming the absence ofa dominant hi, (9) becomes (approximately)t 2 h j T +O ( h j T ) ,

j E I sj # i

which since T E becomes

- Cj E I sj # iThe probability of a collision within group Is is therefore3

3 Recall that by construction there are only ingroup coll isions, andinter-group coll isionsdo not occur .

(at least approximately)

which equals

~i E s

or

Then, since theprobabilityofa ransmissionoriginating ngroup Is is

whereKx = 2 h j ,

j = l

the probability,P, f a collision on any transmissio n is

which reduces to

To minimize the probability P of collision, we must minimizethe right-hand side of (1 2). This is done by minimizing


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C H L A M T A C e t a l . : BRO ADC AST RECO G NI ZING ACCESS M ETH O D

(with X(s) th e aggregate rrival rate orgroup S) which sachieved by setting

xKX ( S ) =- s = 1, 2, ..,K

thus establishing the theorem. The theorem remains true evenif heassump tion used to simplify (9) is not used, but heproof becomes more difficult.

Ofcourse, here is no guarantee hatgroupsca nbe con-structed which satisfy this criterion (even approx imately). Th einability to con struc t groups satisfying he heorem ncreasesas cer tain of the hi values become extreme, Le., muc h largerthan m ost. This case is not of particular concern, however, forwhen extreme values occur, performanc e is essentially de ter-mined by the nodes possessing these extre me values, which wenaturally associate with single node groups.

SUMMARYBRAM is adecentralized access pro toc olbasedonnode

deference to ongoing ransmissions, an ability to iden tify thenode (or group) ndex associated with asuccessful ransmis-sion, and use of a scheduling function which staggers the po-tential ransmission ime or henodesby at east) a , thechannel end to end propagation delay. By appropriate settingofhechedulingunctionan dts arameters)heou rvariantsFB ,PB ,PFB,and PPB (see Figure 3 ) ar epossible.PPB exhibits performance measures (S as a fu nction of G orD as a f u n c t i o n of S) b e t t e r h a n fo r all o t h e r methods ( seeFigures 3 and 4), for all K - a values. D is expectedpacketdelay, S is throughput or channel ut i l izat ion, and is the totalnumber of packets awaiting transmission. F or small K - a valuesthe performance characteristics of the four variants andMSAPbecome close. Morespecifically, when K - a . is small, FB andPFB xhibitperformancemeas ures t least as good s orMSAP and CSMA while providing a protocol which is fair inthat a bound on consecutive packet transmission opportunitiescan be calculated; a prope r ty not exhibited by other methods(e.g., CSMA or MSAP). Th is is significant, since especially forlocal compu ter networks, it is likely tha t a < .005 and K < 50,se e [2] or [ 7 ] , giving K - a < .2 so that FB andPFBallowfair protocols which exhibit attractive performance measures.Fur ther , the use of the scheduling function allows two variants(PFB,PPB) in which m* can be set o yield either he bestS / G (Figure 4) or D /S (Figure 3) perfo rma nce allowable overthe range ( 1 < m < K ) o f m values, and for which he per-formance in mos t cases is bet ter han orothermethods.(Since o u r discussion has bee n on a theoretical level, we havenot been concerned with cases wherein it is desirable to haveth enodesadjust m* in esponse to changes n he oading,G. t is notdif f icult ,howe ver, o imagine chemeswhereinthenodesmonitor , orexample , S , and via nod e onodepacketransmissionsynchronizehemigrationsf m.*)Additionally, the protocol allows the node s to be groupe d soas to minimize the possibile degradations resulting (for PFB,PPB) from mixed arrival rates, Le., heterogeneo us node trans-mission requirem ents.

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1189REFERENCES

Chlamtac, Imrich, Franta , W. R., Levin, D., BRAM: Th e BroadcastRecognizingAccessMethod.TR 8-4,Dept.ComputerScience.University o f Minnesota, March, 197 8. (Certain of h e results pre-sented n his report are also available n th e Masters thesis of I.Chlamtac,entitledRadioPacketBroadcastedComputerNetwork,TheBroadcastRecognizingAccessMet hod, Tel-Aviv University,December 1976) .Fr ant a, W. R., andBilodeau,Mark B., Analysis of aPrioritizedCSMA Protoco l Based o n Staggered Delays, TR 77-18, Dep t. Com -puterScience, University of Minnesota. To appe ar in ActaInfor -matica.Kleinrock, L. And S. S. Lam. Packet-Switching n a Slotted Satel-liteChannel , NationalComputer Conference. 1973 , Vol. 42, pp.Kleinrock, L., QueueingSystems. Vol. 2, ComputerAppl ica t ions ,Wiley-Interscience, 197 6.Kleinrock, L.. Performanceof Distributed Multi-access Co mpu ter-Communications Systems, Proc. IFIP-77, pp. 547-552.Konheim. A. G., Meister, B., Waiting linesand imes nasystemwith polling,JACM, Vol. 21, July 1974, pp. 470-490.Scholl, M., MultiplexingTechniques orDataTransmission OverPacket-SwitchedRadioSystems,Ph.D.dissertation , University ofCalifornia, Compu ter Science Dept., 1976: or Kleinrock, Leonard,and Scholl. Michel,Packetswitching n adiochannels: New con-flict-freemultiple ccess chemes fo r a small numberof users,Proc. ICC, June 1977, Chicago, Paper No. 22.1.Tobagi , Fouad, A . , an dKleinrock,L. ,PacketSwitching in RadioChannels:Part 111-Polling and Dynam ic)Split-Channel Reserva-tionMultiple Access, IEEE Trans. onommunications, Vol.

703-710.

COM-24, NO. 8 , August 1 976. *Imrich Chlamtac was born in Czechoslovakia nMarch 21, 949. He received the B.Sc. andM.Sc. (Distinction)degrees rom he Tel AvivUniversity, srael, n May 1975 andFebruary1977 espectively. He is currentlycompletingrequirements for the Ph.D. degree n the Com-puter Science Department, University of Minne-sota .From 1975 hrough1976he served as aconsultant ndprojectmanager in computer-aided ystemanalysiswith th e Jsraeli Govern-ment . During 1977 he was an instructor a t theDepartment of Mathematics at Tel Aviv University.SinceSeptember1977 he has been aTeaching Associate a t the Com puter Science Depart-men t at the University of Minnesota and a mem ber of he Special In-teract ive Computat ion Laboratory a t the Universi ty Computer Center .In September 1979 he is jo ining he facul ty of he Computer ScienceDe par tm ent at the University of Minnesota. His research nterests arein multiple access communications, simulation and computer networkmodeling and analysis. *William R. Franta was born inMinneapolis,MN, o n May 21, 1942. He received the Bache-lor of Mathematics degree from the Institute ofTechnology, University of Minnesota, n 196 4,the M.S. degree in mathem atics in 196 6, andthe Ph.D. degree n com puter science n 1970,also from he University of Minnesota, Min-neapolis, MN. He is currently Assoc iate Profes-sor of Computer Science and Associate Directorof the University Comp uter Cente r, UniversityofMinnesota. His researchnterestsncludequeueingheory,imulat ionmethodology,omputererformanceevaluationand ocalnetworking. He s theauthor of t he book , TheProcessV iew o f S imula t ion , published by Elsevier-NorthHollandpublishers.


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K. Dan Levin was born in Jerusalem, Israel o n cente r. From 1971 to 1973 he was a consultant at he Auerbach Corp.J u n e9 ,19 44 . He received the B.S. and M.S. inPhiladelphia nd n1974-1975 after eceiving hePh.D.)hew as(CumLaude)degrees in Economics and Busi- visitingassistantprofessor at he Dep artme nt of DecisionSciences, ThenessAdministrat ion rom heHebrewUniver-WhartonSchool,UniversityofPennsylva nia. n1975he eturned osity of Jerusalem, Israel in 1967 and 1970 and Israel and joined the Faculty of Management, Tel-Aviv University whereth ePh.D.degree rom heWhartonSchool,he is lecturing on nforma tionSys temsandComputerCommunica t ionUniversity fPennsylvania in 197 4.From1964Netw orks. His esearchnterests re in DistributedProcessing ndt o 1 9 7 1 h eworked at hecomputer center of ComputerCommunication Networks an dhe is also consultant in theseth eMinistry fFinance, sraelwhere ewas reasoheChiefScientist-TheMinistry of Communica t ion , ndresponsible for plan ning nddevelopment ndvariousotherorganizations.served as the deputy d i rec tor of the computer

A Diffusion Approximation Model for a CommunicationSystem Allowing Message InterferenceDONALD P. GAVER A N D JOHN P. LEHOCZKY

Abstrocf-Probabil ist icmodels re resented nd nvestigatedodescribe he ervice urnished to messages ent via communicatio nschannels on which messages in progress may be destroyed by a newmessagedemand.Retriesbydestroyedmessages remodeled.Thegeneral approach utilizes diffusion approximations. The quality of thenumerical results is evaluated by comparison with simulations, and isfound to be sa t i s fac tory .

1 . INTRODUCTIONW STUDY the operating characteristics of an element ofa commu nication system; the element consists of a largenumber , c, of channels whichservice an arriving stream of mes-sages i n the following manner. When a message a r r i v e s i t f i r s tselectsachannel andomlyand nitiatesa ransmissio n (ser-vice) time of randomdura tion. If the hannel selected isalready occupied, i.e. is being used for transmission, both mes-sages may be destroyedor erminatedbeforecompletion,and he channel reverts to an empty or open condit ion. Thetransmitters of the messages are assumed capable of detectingthe event of such destructions by means of a (short) reply,acknowledgment ,or directive, from he ecipient.Followingdestruction or interrup tion message enters etryor re-transmission population, from which it later attempts to findan empty channel and eventually complete message ransmis-sion. Such an array of channels, with described message inter-

Paper approved by the Editor for Com puter communic ation of th eIEEE Comm unications Society for publication without oral presenta-t ion. Manuscript received March 15, 1977; revised March 13, 1979. Thiswork was supported by the Office of Naval Research.D. P. Gaver is with the Depa rtmen t of Operations Research, NavalPostgmduate School, Monterey, CA 93940.J . P. Lehoczky is withheDepartment fStatist ics,Carnegie-Mellon University, Pittsburgh, PA 1 5 2 1 3 .

action, is characteristicof ertainmilitary ommunicationsystems. The possibility for message interr uption and destruc-tion also occurs in other communication systems.

Approx imate probability models for such systems are writ-ten down directly in the form of stoch astic differential equa-tions; see Gaver andLehoczky 1976).TheApproximationstend obecomeexact when c, th enumbe r of channelsbe -comes large (c + w), as ollows rom he esults of Kurtz(1971)an dofBarbour (1974),particularlyTheorem K , p.23; or etails seeGaver andLehoczky 1977)l .Samplenumerical (Monte Carlo) studies indicate that the approxima-tion technique may be quite adequate for c as small as ten.The r e s u l t s o b t a i n e d m a y be s u m m a r i z e d b r i e f l y . F i r s t , f iv edifferent models are introduced; these differ in the manner inwhich hedemandsmanifest hemselves,bothdirectly ndfrom the retry population. Model 1 permits messages to arrivein a (nonhom ogeneous) Poisson process, and sends interruptedmessages to the retry pop ulation, from which re trials occur atrandom. It is shown that if Poisson arrival rate is X (a constant)and serv ice rate is p and no customers defe ct from retry, thenthe expected number of channels occupied is simply 4 = p =X / p provided 0 < p < 1/2. Long-run waiting time is shown tobe p-l + v-l2p(l - 2p)-l , where v is retry rate; see (2.16);a different expression holds if a fraction (11of messages defec t,see (2.19). The ong-run number of channels occupied is ap-proximatelyGaussian,with simplecovariancematrix.Simu-lated results for the multi-dim ensional birth-death process arecompare d o diffusion approxima tions; he numerical resultsin Tables 1 and 2 are in close agreem ent. In Model 2 carriersensing of busy channels by memb ers of the retry populationis modeled . Here oc cupancy of channels is increased to 4 = p ,0 < p < 1 and hewaiting imebecomes horter: p- l +v-l2p(l - p)- l, see (3.9). Thus system service is improve d.

0090-6778/79/0800-1190$00.750 1979 IEEE

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BRAM: The Broadcast Recognizing Access Method