In this paper we discuss the benefit of par allel computing in propagating orbits of ob jects.   Several analytic methods are now in
use operationally .  We will discuss three such schemes.   We demonstrate the benefit of par allelism by using an INTEL iPSC   hyper 
cube and by using a cluster of Unix based
workstations running Parallel Virtual Machine  PVM.    The software PVM allows a
heterogeneous set of networked workstations
to appear as a multicomputer.   We will show that one can achieve near 100% efficiency on the hypercube 

Brewer

Dongara J.

Figure 0

Hilton C.G.

Hujsak R.S.

Lane M.H.

Phipps W.E.

Quinn M.J.

Stone L.C.

Neta, Beny

Danielson, D. A.

Ostrom, Sara

Brewer, Susan K.

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveFaculty and Researcher Publications Faculty and Researcher Publications1994Performance of Analytic OrbitPropagators on a Hypercube and aWorkstation ClusterNeta, BenyNaval Postgraduate School, Monterey, CA.http://hdl.handle.net/10945/34063PERFORMANCE OF ANALYTIC ORBITPROPAGATORS ON A HYPERCUBE AND AWORKSTATION CLUSTERBeny Neta D  A  DanielsonSara OstromandSusan K  BrewerNaval Postgraduate SchoolDepartment of MathematicsCode MANdMonterey CA email bnetamoon math nps navy milAbstractIn this paper we discuss the bene t of parallel computing in propagating orbits of objects Several analytic methods are now inuse operationally We will discuss three suchschemes We demonstrate the bene t of parallelism by using an INTEL iPSC hypercube and by using a cluster of Unixbasedworkstations running Parallel Virtual Machine PVM The software PVM allows aheterogeneous set of networked workstationsto appear as a multicomputerWe will show that one can achieve near	 eciency on the hypercube Author to whom all correspondence should beaddressed   IntroductionThe Naval Space Command NAVSPACECOM and the Air Force Space CommandAFSPACECOM currently track daily over objects in elliptical orbits around theEarth To assist in identi cation and tracking of these objects in orbit they both use ananalytic satellite motion model The Navy isusing the subroutine PPT based on variation of elements model of arti cial satellitemotion around the Earth The theory is dueto Brouwer and Lyddane  Given a set ofsatellites mean orbital elements at a givenepoch the model predicts the state positionand velocity vector at a future time Themodel considers perturbing accelerations dueto atmospheric drag oblateness of the Earthand asymmetry of the Earths mass aboutthe equatorial plane The Air Force is usingSGPSDP Simpli ed General Perturbations based on the theory of Lane and Cranford  The Deep space capabilities are dueto Hujsaks  work They replaced the oldversion SGP which was based on the work ofKozai  and Brouwer  and made operational by Hilton and Kuhlman  The oldversion had no capabilities to track objectsin deep space ie period greater than minutesWith the current increase in space operations the number of objects necessary tobe tracked is expected to increase substantially Additionally if there exists a desireto increase the accuracy of prediction the resulting model would require even more computing resources and make achieving resultseven more time consumingParallel computing oers one option todecrease the computation time and achievemore realtime results Use of parallel computers has already proven to be bene cial inreducing computation time in many other applied areasTwo common measures of eectiveness accounting for both the hardware and the algorithm are speedup and eciency Thespeedup Sp of an algorithm is de ned asSpTsTporT Tpwhere Tsis the time on a serial computer andTiis the time on a parallel computer havingi processors The eciency Ep is de ned byEpSppand it accounts for the relative cost of achieving a speci c speedup many factors couldpossibly limit the eciency of a parallel program These factors include the number ofsequential operations that cannot be parallelized the communication time between processors and the time each processor is idledue to synchronization requirements see egQuinn Two decomposition strategies can be usedin parallelization of any algorithm ie control decomposition and domain or data decomposition It was shown by Phipps etal  that control decomposition is inecient for orbit computation using the analyticmethods mentioned aboveIn this paper we will summarize the resultsof parallelization of the analytic orbit propagators using domain decomposition strategy The INTEL iPSC hypercube is usedWe will also discuss the use of a cluster ofUnixbased workstations networked and allrunning the Parallel Virtual Machine PVMsoftware PVM was developed by Oak RidgeNational Laboratory It is a software system that enables a collection of heterogeneouscomputers to be used as a coherent and exible concurrent computational system Geistet al  In the next section we discuss theresults of parallelization when using the INTEL hypercube We give a brief introductionto PVM software in section  The results ofparallelizing PPT on a cluster of workstations will be detailed in section  In section we discuss PVM use in parallelizing the AirForce models We give our conclusions in section  Parallel Versions of PPT SGPSGPSDPIn this section we discuss the parallelizationof PPT as well as SGP SGPSDP Theidea id to let one processor read and distribute the data to the other p    processors which propagate the orbit and sendtheir results to another processor the collector which writes to the disk see  gure The results for n satellites    n   are given in Table  for a hypercubeconsisting of  processorsIt is clear that PT is more ecient Thisshould not be of a surprise since PPT requires more computation time  msecthan the others Note also that the eciencyis improving with the number of processorsThe question is now how to  nd the optimal number of processors to use Phipps etal   have developed a model for the execution time to propagate n objects using pprocessor The time tp is given bytp  tw p  twp  tcp where tw p is the time the last node mustwait to receive its  rst data set twp is thetotal time the last node must wait for all itssubsequent data and tcp is the time for eachnode to propagate its share of the n objectsIt was shown there thattw p  p   tm twp   if tw   t np   tw p   t  if tw  t tcp nt p  where tm is the time to send a single message between the distributing and workingnode and t is the time to propagate one object These were found to betm   msec and t   msecTherefore the speedup and eciency for n  objects can be plotted as a function ofthe number of processors It can be seen inthe next  gure that for PT the maximumeciency is 	 and is achieved when using  processors For PSGP the maximumeciency is over 	 using  processorsFor PSGP and PSDP the maximum eciency over 	 can be achieved when using processors Figures  show the plots ofthe eciency of each code as a function of pNote that the number of objects propagatedby each code is dierent When using SGPone handles all orbits the same but when using SDP only the deep space orbits areconsidered The rest are handled by SGPAs a result of discussion with AFSPACECOM we realized that the propagator in usually called several times for each object Eachcall corresponds to a speci ed time beyondepoch SGP propagates data for low earthobjects which requires more frequent trackingthan deep space satellites Thus a relativelylarge number of observations are received perday by the AFSPACECOM for each low earthsatellite The estimated number of calls toSGP for each object is  and to SDP is To analyze the speedup and eciency wenote that each time a new set of satellite datais received by SGP an initialization subroutine is called before the SGP main subroutine is called For every other incrementedtime speci ed for the same satellite the initialization program is not called Thus theexecution time can be modeled by Ostromt  tf m  tswhere tfis the time to propagate the satellite including initialization and tsis the propagation time without initialization The values of tf tsas measured on the iPSC hypercube aretf msec ts msecthust  msecFigure  depicts the speedup and eciencyversus hypercube dimension when propagating  satellites to  times each Clearlymuch higher speedups are obtainable in thiscase The maximumeciency is nearly 	when using a hypercube having  nodesA similar analysis for SDP Ostrom shows that t   msec Using now satellites 	 of a total of  objects one nds near 	 eciency using a nodehypercube see Figure  This analysis canbe extended to PPT Parallel Virtual MachineParallel Virtual Machine PVM is a small Mbytes of C source code software package that allows a heterogeneous network ofUnixbased computers to appear as a singlelarge distributedmemory parallel computerThe PVM package is good for largegrain parallelism that is as least  kbytesnodeThe term virtual machine is used to designate a logical distributedmemory computerand host is used to designate one of the member computersThe PVM software developed at OakRidge National Laboratory see Dongara etal  and Sunderam et al  suppliedthe functions to automatically start up tasksto communicate and synchronize with eachother A problem can be solved in parallel bysending and receiving messages to accomplishmultiple tasks similar to send and receive onthe hypercubePVM handles all message conversion thatmay be required if two computers use dierent data representations PVM also ensuresthat error messages generated on a remotecomputer are displayed on the users localscreenThe PVM system is actually composed oftwo parts the daemon and a library of PVMinterface routines The daemon pvmd orpvmd resides on all the computers makingup the virtual machine When a user desiresto run a PVM application heshe executespvmd on one of the computers which in turnstarts up pvmd on all the others The libraryof PVM interface contains routines for message passing spawning processes coordinating tasks and modifying the virtual machine Parallelization ofPPT using PVMStone  has tried four possibilities of domain data decomposition The master sends one satellite to eachworking node then sends one satellite ata time upon request ds The master sends one satellite to eachworking processor then continues inroundrobin fashion ds The entire data set is divided to p number of working nodes blocks The master sends a block to each working nodeds The entire data set is divided to pblocks The master sends one block toeach and then the other block to eachdsIn the second option we save on communications In the third case we save even more oncommunication because we reduced the number of times required to send data On theother hand sending such large blocks forcesthe others to wait Thus the last case is anattempt to compromise between the previoustwoFor these experiments PVM was startedon eighteen dierent workstations so measurements could be taken for one to sixteenworking nodes The workstations are SUNSparc II and Sparc IPX having  MHz processors and con gured with  Mbytes ofsystem memory The workstations are connected by a  Mbytes Ethernet based network Stone experimented with  and objects in the data set We give here theresult for   gure  It is clear thatfour working processors suce to minimizethe computing time and that the fourth possibility is the best Stone  has shown thata speedup of almost  was achieved when using  SUN workstations Parallelization of SGPusing PVMBrewer  has tried three possibilities for domain data decomposition Answer Back Method ABMThe master sends one block of m satellites to each working node Upon request a working processor receives another block of m satellites until the dataset is processed Successive Deal I SDIThe master sends one block of m satellites to each working node and continuesto deal such blocks in roundrobin fashion Successive Deal II SDIIThe master sends one block of m satellites to each working node The rest ofthe data set is divided by p twice thenumber of working nodes Blocks of thissize are given to each working nodes inroundrobin fashion  blocks eachThe second method will eliminate the communication time by the workers requestingmore data The third method will cut thecommunication overhead This is dierentfrom SDI with a larger m because in SDIIlarge blocks are sent while the workers arebusy propagating the  rst m satellitesWe have experimented with various values of m and chosen  and  processorsie  working nodes respectively Thenumber of satellites taken to be  	of which were considered deepspace Fora deepspace satellite  calls were made toSDP For the other satellites  calls weremade to SGPThe  rst measure is the endtoend timeThis is the most important since it is a reection of the total performance of each algorithm The Answer Back Method was superior when using  or  processors Whenusing  processors ABM was faster in mostcases See Figures We can look at this from another point ofview In the next three  gures we plot theendtoend time for each method It is clearfrom  gure  that a choice of  or  processors is the best shortest time for ABMFor SDI and SDII a choice of  processors isbestThe second measure is the percent of timea working processor spent on communicationFrom the next three  gures  is clear thatSDII requires less communication time whichshouldnt be surprising It is also clear thatthe more working nodes we have the higherthe percentageThe third measure is eciency In all threecases the ABM was more ecient The nextthree  gures  show that for each methodit is more ecient to use  or  processorsrather than In closing we should note that with the useof an open network there are great uctuations in the amount of time taken to performa given task The execution time depends onthe number of current users and the percentage of the CPU allocated to each user Topartially compensate for that we averaged run times to arrive at our results ConclusionsIn this paper we have shown the bene t ofMIMD parallel computers in predicting theorbit of objects Analytic orbit propagatorscurrently in use by the Navy and Air Forcewere implemented on an INTEL iPSC hypercube and on a cluster of networked Unixbased workstations running PVM The eciency of the algorithms nears 	 when using the optimal number of processors Thisoptimal number depends on the number ofsatellites the orbit propagator used and thenumber of calls to the propagator per satellite For a cluster of workstations we haveused the software PVM and have shown thatit is more ecient to use  or  workstationsthan  The speedup is almost  when using workstationsREFERENCES SK Brewer Air Force Space CommandSatellite Orbit Predictor Using Parallel Virtual Machines M Sc ThesisNaval Postgraduate School Departmentof Mathematics Monterey CA  Brouwer D Solution of the Problem ofArti cial Satellite Theory without DragAstronomical Journal     Dongara JGeist G A Mancheck R SunderamV S Integrated PVM Framework Supports Heterogeneous Network Computing Computers in Physics pp  Geist G A Beguelin A Dongara JJiang W Mancheck R and SunderamV S PVM  Users Guide and Reference Manual Oak Ridge National Laboratory Technical Report ORNLTM Oak Ridge TN  Hilton CG and Kuhlman JR Mathematical Models for the Space DefenseCenter PhilcoFord Publication NumberU pp  November  Hujsak RS A Restricted Four BodySolution for Resonating Satellites withan Oblate Earth AIAA Paper Number June  Kozai Y The Motion of a CloseEarth Satellite Astronomical Journal    Lane MH and Cranford KH An Improved Analytical Drag Theory for theArti cial Satellite Problem AIAAPaperNumber  August  Lyddane RH Small Eccentricities orInclinations in the Brouwer Theory ofthe Arti cial Satellite AstronomicalJournal    Ostrom SR Parallelization of the AirForce Space Command Satellite MotionModels MS thesis Naval PostgraduateSchool  Phipps WE Neta B and DanielsonDA Parallelization of the Naval SpaceSurveillance Satellite Motion ModelJournal Astronautical Sciences   Phipps WE Neta B and Danielson DA Parallelization of the NavelSpace Surveillance Center Satellite Motion Model Proceedings of the Space Surveillance Workshop LincolnLaboratory Massachusetts Institute ofTechnology March April  Vol II pp Quinn MJ Designing Ecient Algorithms for Parallel Computers McGrawHill New York  Stone LC Parallel Processing of NavySpeci c Applications Using a Workstation Cluster MS Thesis Naval Postgraduate School Department of Electricaland Computer Engineering MontereyCA December  Sunderam VS Geist GA MancheckR The PVM Concurrent Computing System Evolution Experience andTrends Parallel Computing  submittedSerial Parallel maxnEmaxnEtime msecCode Code to propagatean objectSGP PSGP   SGP PSGP   SDP PSDP   PPT PT   Table  Maximum eciency of various propagatorsList of FiguresFigure  Distribution of Satellite DataFigure  Estimated Speedup and Eciencyof PT for Various Hypercube SizesFigure  Estimated Speedup and Eciencyof PSGP for Various Hypercube SizesFigure  Estimated Speedup and Eciencyof PSGP for Various Hypercube SizesFigure  Estimated Speedup and Eciencyof PSDP for Various Hypercube SizesFigure  Estimated Speedup and Eciencyof PSGP m   for Various HypercubeSizesFigure  Estimated Speedup and Eciencyof PSDP m   for Various HypercubeSizesFigure  PVM Applied to PPTFigure  Comparison of Models Four NodesFigure  Comparison of Models EightNodesFigure  Comparison of Models SixteenNodesFigure  Answer Back Model Node ComparisonFigure  PercentWorker Communication forEach Model Using  NodesFigure  PercentWorker Communication forEach Model Using  NodesFigure  PercentWorker Communication forEach Model Using  NodesFigure  Answer Back Model EciencyFigure  Successive Deal I EciencyFigure  Successive Deal II Eciency

Performance of Analytic Orbit Propagators on a Hypercube and a Workstation Cluster

Beny Neta

D. A. Danielson

Sara Ostrom

Susan K. Brewer

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D. Danielson

Susan Brewer

Performance of analytic orbit propagators on a hypercube and a workstation cluster

http://calhoun.nps.edu/bitstream/handle/10945/34063/danielson_performance_1994.pdf?sequence=4&isAllowed=y

Performance of Analytic Orbit Propagators on a Hypercube and a Workstation Cluster

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