Scheduling High-Level Tasks Among Cooperative Agents

Transcription

1 Schedulng Hgh-Level Tasks Among Cooperatve Agents Bradley J. Clement nversty of Mchgan Artfcal Intellgence Laboratory Ann Arbor, MI 89 Edmund H. Durfee nversty of Mchgan Artfcal Intellgence Laboratory Ann Arbor, MI 89 Abstract Schedulng tasks among cooperatve agents requres tradeoffs between varous factors ncludng task prortes and contet-dependent eecuton tmes. We have specfcally been nvestgatng the space of functons for evaluatng alternatve dstrbuted task schedules for mult-operator applcatons. In ths paper, we descrbe some canddate functons and converge on ntutvely appealng functons, whch we show to lead to equvalent preferences over dstrbuted schedules. We then look at the computatonal complety of fndng schedules that (appromately) optmze ths functon. When contetswtchng costs are thrown nto the m, moreover, the complety becomes even more dauntng. To address these problems, ths paper summarzes our work on forgng correspondences between our problems and those studed n operatons research. Moreover, we have developed a new hll-clmbng strategy for solvng these problems, and n ths paper we show that these strateges yeld very good performance wthn the range of parameter settngs that are representatve of our applcaton doman.. Introducton The problems of allocatng and schedulng tasks among dstrbuted agents has been a central concern n the mult-agent systems lterature for many years. Often, makng good allocaton and schedulng decsons requres a careful analyss of task needs and the relatonshps between tasks [,]. Moreover, n many domans the prortes of tasks factor nto the decsons, as do the costs of swtchng between dssmlar tasks by an agent. In partcular, some tasks may have dfferent degrees of urgency and/or mportance and are to be accomplshed by human operators, such as customer phone servce operators, operators of nuclear reactors, or personnel n a submarne control room. Specfcally, we are lookng at the latter applcaton doman; TAIPE (Tactcal Assstants for Interacton Plannng and Eecuton) s a system desgned for the Navy to help automate shp operaton [6]. In TAIPE nteracton plans determne the functonal components that must be used to manage a dsplay dalogue for operators to process tasks. A crew of operators needs to apply ther abltes to process the most mportant tasks quckly whle consderng the nherent dffculty of umpng between dssmlar tasks. In ths paper, we focus on strateges for automatcally managng the task assgnments and schedules among the human operators. Operator task management n TAIPE has, as one of ts responsbltes, the need to balance the task processng loads among the operators such that urgent tasks are processed quckly, operators are all utlzed, each operator's tme s used effcently, and operator stress s mnmzed. These dfferent demands can at tmes be contradctory. For eample, usng operators effcently can mean that each operator specalzes n a partcular subclass of tasks so as to mnmze contet swtchng overhead. But ths can lead to underutlzaton (some operators mght be dle when there are no tasks of ther subclasses to do), poor response to urgent tasks (whch mght be concentrated at a few operators), and hgh stress for the operators that have the urgent tasks. Systems lke TAIPE have an unusual combnaton of propertes that set them apart from the knds of dstrbuted and real-tme systems more often studed n computer scence. The tasks are of coarse granularty; the message passng costs are neglgble relatve to the task processng tme; the processng elements are comple (humans); the contet swtchng between tasks can requre substantal effort; the tasks can have dfferent levels of mportance; the tasks should not be preempted (due to contetswtchng costs); and hard deadlne nformaton s not avalable snce tasks are best effort. The goal of ths paper s to begn to lay out the desgn space for alternatve load balancng strateges; specfcally, we are concerned wth the desgn of metrcs for evaluatng the utlty of a partcular dstrbuted task schedule for operators, gven the prortes and eecuton tmes of the tasks. Wth such metrcs n hand, we then

2 need to develop algorthms for formng (appromately) optmal dstrbuted schedules, where we can characterze the complety of such algorthms. The contrbutons of ths paper, therefore, are n descrbng a process for artculatng the task management obectve functon n a decentralzed applcaton nvolvng sophstcated agents and for desgnng algorthms that are cost-effectve n ths comple doman. In Secton, we relate how ths work fts n wth prevous research by provdng background on related work n dstrbuted computng, operatons research, and artfcal ntellgence. Secton proposes a strategy of ncorporatng prortes and tme needs nto utlty functons and sngles out one that s ntutvely appealng and flebly meets the needs of systems lke TAIPE. After revewng the complety of dfferent varatons and smplfcatons of the schedulng problem assocated wth the performance evaluaton functon, Secton presents smulaton results revealng nterestng aspects of the performance based on the search space topology. Secton talks about the complety of addng contet-swtchng nto the problem, ntroduces a heurstc-based hllclmbng algorthm for tractably searchng for good solutons n the more comple doman usng the proposed performance functons, and descrbes how t performs n varous contets.. Related work In dstrbuted and real-tme systems lterature, prortes are assgned to tasks to resolve system resource contenton. These resources could be processors, memory, nput, etc. Prortes n TAIPE are dfferent n that the eecuton tme of tasks also plays a part n the schedulng of tasks (whch resolves processor contenton). TAIPE s task prortes represent urgency, but there s no hard deadlne nformaton to consder. These prorty values are not assgned based on orderng relatonshps among the tasks but are ntrnsc values possbly representng the mportance or urgency of tasks, the rewards for processng them, or the costs of not processng them. Secton eamnes the selecton of a functon to evaluate the performance of task schedules based on ths noton of prorty. Recently, the operatons research communty has produced varous appromaton algorthms for schedulng that gve performance guarantees for dfferent performance evaluaton functons. Some of these algorthms guarantee to produce a schedule that has a performance no worse than a constant factor of the optmal schedule s performance [,,9]. These knds of guarantees are helpful n buldng real-tme systems that have hard deadlnes. For systems lke TAIPE, knowng the epected performance would also be a valuable, so we gve results of smulatons amed at fndng that nformaton n Secton.. Although these methods for performance evaluaton shape strateges for load balancng, there are also ssues concernng the actual load balancng algorthms and the archtecture of the dstrbuted system. Many studes of load balancng algorthms have centered on partcular hardware archtectures [,,,,6]. They usually seek greater performance by reducng communcaton overhead, whch s greatly nfluenced by the way the processors are connected. These archtectures and assocated algorthms offer nsght nto desgnng scaleable systems of dstrbuted agents for task management for applcatons lke TAIPE. On a software level, load-balancng algorthms generally adopt one of two approaches. One approach s to optmally balance the load gven the current task set; ths approach works well f the optmzaton problem can be solved quckly relatve to the evoluton of the task set. The other approach s to use suboptmal mechansms that locally adapt schedules to rapdly-changng task sets [9,]. Our work tres to move the frst approach toward the second. The characterstcs of TAIPE, where the arrval rate of tasks s relatvely low and the processng tme of tasks s relatvely large, motvate makng (near) optmal dstrbuted schedules, yet there s enough potental task turnover to suggest the use of faster, anytme algorthms. Contet swtches between tasks can also be vewed as a hnderng relatonshp between tasks. Task relatonshps, such as hnders, enables, facltates, and precedence are computatonally descrbed n the TÆMS (Task Analyss, Envronment Modelng, and Smulaton) framework []. TÆMS provdes a method for modelng comple computatonal task envronments. A TÆMS model ams to analyze, eplan, or predct the performance of a system or some part of t. TÆMS may be useful for representng comple agent and task nterrelatonshps for TAIPE when more nformaton about possble task relatonshps are avalable.. Load balancng strateges To desgn algorthms for formng dstrbuted schedules, we frst must ask what defnes a good schedule. A good schedule mamzes the performance of the dstrbuted system, but we then must be clear about what performance means. The performance of schedules have conventonally been measured n many ways: average wat throughput

3 operator utlzaton turnaround tme average response tme schedule completon tme (makespan) deadlnes met avalablty relablty In a system lke TAIPE, the response tme of tasks s the most mportant crteron because at crtcal tmes the system must react quckly to ts envronment. In other applcatons, such as a telephone customer servce, average wat may be more mportant than response tme. We epect that ssues dealng wth performance evaluaton n terms of average wat to be smlar to those nvolvng response tme. Because some tasks are more urgent than others, we would rather mnmze the response tme of hgher prorty tasks. Dealng wth both of these varables n characterzng the performance of the system s the subect of ths secton... Intal strategy; prorty = load Currently n TAIPE, not only does the prorty of a task determne ts order n an operator s queue, but t s also used to represent the task s load. If we assume that the prorty represents the mportance of eecutng a task, then load balancng actually tres to balance cumulatve responsblty or stress nstead of actual workload n terms of process tme requrements. That s, t s possble for prorty to be balanced among operators and yet one operator could have substantally more work to do because of the eecuton tmes of hs/her tasks. Even gnorng ths workload mbalance for the moment, balancng based on prortes does not ensure that the hghest prorty tasks are addressed frst snce balancng the cumulatve prortes does not ensure that the hghest prorty tasks are at dfferent operators. If we balance load as the sum of prortes of tasks for each operator (Fgure a), we see that not only are the hghest prorty tasks not necessarly processed soonest, but also workload (n terms of processng tme) may not be dstrbuted evenly. In Fgure a, there are two operators, and the boes are tasks labeled wth prortes and havng unform process tme requrements. Tasks at the bottom of the queue are eecuted frst, and we assume that prorty values are represented as non-negatve real numbers. The prortes of such tasks can be eaggerated to mprove the dstrbuton of hgh prorty tasks; ths s what happens n the current TAIPE, whch actually balances the sum of the squares of the prortes. Even so, cases lke the one n Fgure b llustrate that ths strategy can also fal both to ensure that the hghest prorty tasks are addressed soonest as well as to dstrbute workload. Op load Op 6 6 Op Op 6 6 prorty load prorty C Op Op 6 6 (a) (b) (c) g prorty ( ) h( response_ tme ) g( prorty ) h( response_ tme ) Fgure. Strateges nvolvng eecuton tme.. Strateges nvolvng eecuton tme If we wsh to mnmze response tme or average wat on hgher prorty tasks, we should mamze a utlty functon that gves greater value to hgher prorty tasks wth smaller response tmes. The response tme for a task s the tme t takes to complete ts eecuton from the tme t enters the system; t s the sum of ts eecuton tme and the eecuton tmes of all tasks before t n the queue when t enters the system. For ncreasng functons g and h, the class of general utlty and cost functons gven n Fgure c balance both the stress and work load. Some smple eamples we wll look at n some detal follow. prorty response tme _ () prorty response tme C prorty response_ tme () _ () sng any of these utlty or cost functons acheves the same purpose of mnmzng response tme for hgher prorty tasks, but the meanng of relatve prorty values s dfferent for each equaton and varatons of them. Ths s llustrated by comparng the utlty and cost functons descrbed above for the followng alternatve orderngs of tasks n a queue n Fgure. The frst number n each bo

4 s the task s prorty, the second s the task s eecuton tme. =,, =.8 =, =.8 C =, C = 9 Fgure. Performance functon dfferences We see that the left schedule s better for the utlty functon, Functon, and the rght s better for the other utlty functon, Functon, and the cost functon, Functon. Ths shows that prorty assgnments have dfferent meanngs based on the nature of the functon measurng performance. Moreover, f we were to square the prortes, Functon wll prefer a dfferent schedule. Ths llustrates the need to codesgn the functon for schedule evaluaton and the functon for computng prortes... The cumulatve prorty strategy << What s the space of perf. eval. fcns.? Can t be traversed? >> What TAIPE needs s to process hgher prorty tasks as soon as possble. We can thnk of ths as tryng to accumulate as much task prorty value as possble as soon as possble. If we order tasks accordng to prorty / eecuton_tme ratos, operators wll frst perform tasks that get the most bang (prorty value) for the buck (tme nvested).,,,, Avg Cum. Prorty /6 /6 tme ³ cum_ prorty( t) dt tme_ perod () Fgure. Average cumulatve prorty cum. prorty In general, we can fnd the optmal orderng based on ths bang for the buck noton by mamzng the tmeaveraged cumulatve prorty. Fgure shows how orderng the tasks for a sngle operator n decreasng prorty / eecuton_tme ratos mamzes the area under the cumulatve prorty curve and, thus, the average cumulatve prorty. Note that ths shows a gradual accumulaton of prorty value over the eecuton of tasks. The curve could have been plotted as a step functon that added the prorty value of tasks at ther completon. From the summaton formula gven n Secton. for ths functon, t s smple to see that the step functon has the same propertes as the functon that gradually accumulates prorty value.,,,,,,,,, Op Op Op Avg. Cum. Prorty = 7.8 cum. prorty tme 6 tme 8 6 cum. prorty,,,,,,,,, Op Avg. Cum. Prorty =. Fgure. Cumulatve prorty n a multoperator schedule We can also use ths measure to optmze a multoperator schedule. The cumulatve prorty functon now represents the total accumulated prorty of the system whch s the sum of the cumulatve prorty functons of all of the operators, and the utlty functon s smply the tme-averaged cumulatve prorty. Fgure shows how ths utlty functon gves preference to dstrbutng load evenly among operators. The left sde of Fgure shows the stuaton where nne tasks are spread evenly among three operators. We want to mamze the area under the cumulatve prorty curve whle mnmzng the total eecuton tme. The chart at the bottom shows how the cumulatve prorty curve s constructed from those of the three operators. On the rght, we have the same tasks all allocated to a sngle operator, and ts cumulatve prorty functon s also shown n the bottom chart. As we should epect, we see that processng tasks n parallel produces a greater cumulatve prorty curve. Other propertes of ths functon are eplored n the net secton... Comparng performance functons We have descrbed dfferent functons whch can be used to evaluate the performance of a schedule and mprove strateges prevously used by TAIPE. The dfferent functons represent dfferent vews of the approprate relatons among prorty, eecuton tme, and the performance of a schedule. However, although the tme-averaged cumulatve prorty, or bang for the buck, functon (Functon ) and the cost functon summng the products of prortes and response tmes (Functon )

5 represent two such perspectves, they are, n fact, equvalent. For any two schedules for a set of tasks, they wll always agree on whch s better. A proof follows: The tme-averaged cumulatve prorty of a schedule, S, s smply computed as follows: ( S) ma e p t p ( t t ) where p s the prorty, t s the eecuton tme, and t e s the end or fnsh tme (response tme for ths statc case) of th task. t ma s some constant greater than or equal to the sum of the eecuton tmes of all of the tasks. We need ths constant as the tme perod over whch the cumulatve prorty s averaged so that schedules can be compared farly. So, f we have two schedules, S and S, and (S ) > (S ), then t ma p t p ( t t ) ( ) p t p t t ma e ma e! t ma t ma p t p t e e But these terms n the above nequalty are ust sums of products of prortes and response tmes of the two schedules, whch s the cost functon, Functon. So, we have C(S ) < C(S ). Thus, f the bang for the buck functon prefers one schedule to another, the cost functon wll prefer t, too--the functons are equvalent. Q.. Schedule evaluaton functon summary In summary, by consderng the relevant factors for schedule evaluaton n TAIPE, we have been able to eplore the space of alternatve evaluaton functons. Clearly, the TAIPE startng pont s napproprate. Moreover, we can enumerate some ntutvely-appealng functons, and show that the choce between them s moot, n that they represent consstent preferences over schedules.. Complety of schedule optmzaton Now that we have dentfed a good measure for comparng alternatve dstrbuted schedules, we have to determne whether we can use ths measure frutfully wthn our applcaton. In partcular, we have to assess the computatonal complety of fndng schedules that satsfy the preferences. nfortunately, n general computng an optmal dstrbuted schedule based on ths measure s very costly. For eample, the cost functon s known n operatons research as weghted average completon tme or total weghted completon tme. The correspondng decson problem, Schedulng to Mnmze Weghted Completon Tme, s strongly NP-complete for an arbtrary number of machnes and s pseudopolynomal for a fed number of machnes [7]. What ths means s that we need to consder technques for smplfyng or appromatng the calculatons... Smplfyng strateges Several smplfcatons of the problem gnore factors that ntroduce complety. These factors nclude the dstrbuton across multple operators, the non-unform prortes, and the non-unform eecuton tmes. If we can rela one of these, we can generate smpler strateges for generatng optmal schedules. If we assume there s no dstrbuton, the problem becomes very easy. For the sngle machne case, schedules are optmzed by applyng Smth s Rule [], whch orders tasks accordng to prorty / eecuton_tme ratos. If we assume unform prortes, computng a dstrbuted schedule s smlarly easy. Snce the utlty functon and the cost functon are equvalent, when two schedules are compared, the prorty varable, p drops out, and only the sums of response tmes are compared. Thus, the best schedule mnmzes the average response tme. The shortest-ob-frst (SJF) strategy s known to optmally mnmze average response tme, so the bang for the buck strategy becomes SJF wth ths smplfcaton. The optmal polcy s to order all tasks accordng to ascendng processng tmes and then to deal them out to the operators such that the net task s assgned to the operator assgned the shortest total processng tme. []. If we assume unform eecuton tmes, the best schedule never has a hgher prorty task begn after a lower prorty task. Ths means that tasks can smply be sorted by ther prorty and dealt out to the net avalable operator to acheve the best performance; we wll refer to ths as the greatest prorty frst (GPF) rule. To prove ths, f we assume that ths property does not hold, then there must be an optmal schedule wth tasks and k that have prortes p and p k and response tmes t e and t ek such that p > p k and t e > t ek. The contrbuton of these two tasks to the cost functon would be p t e + p k t ek. If we swap the postons of the tasks n the schedule, the contrbutons of any other tasks n the schedule to the cost wll not the change because the tme slots are all the same, and the contrbuton of tasks and k would be p t ek + p k t e. Snce the former schedule s optmal, ts cost must be smaller than the latter s, whch means p t e + p k t ek < p t ek + p k t e

6 p (t e - t ek ) < p k (t e - t ek ) p < p k But ths contradcts the assumpton, so t s true that f the eecuton tmes are fed, the tasks wll eecute n order of prorty across all queues. Q.. Appromaton technques Clearly, one appromaton would be to try to use smple algorthms lke TAIPE currently uses, where we can ust order based on a value that can be measured for each task. As opposed to orgnal TAIPE s use of prorty, we want to nclude response tme. nfortunately, ths wll not work for the prorty / response_tme and prorty / response_tme utlty functons (Functons and ); Smth s rule does not apply to them. In Fgure, we see an ecepton for Functon, and a smple eample for Functon can be seen n Fgure below. Although the queue on the rght s ordered n decreasng prorty / eecuton_tme ratos, the left orderng s optmal for Functon. The rght sde gves an allocaton that ensures that the task wth the hghest prorty / eecuton_tme rato s always beng processed; however, the optmal allocaton starts a task wth a rato of / before another wth a rato of. Notce that the optmal confguraton ncreases operator utlzaton whle the other puts a hgher prorty task behnd a lower prorty task. Fortunately, recent work n the lterature has been devoted to developng appromaton strateges for varatons of ths problem [,,9]. An D-appromaton algorthm guarantees that the performance wll be a factor D of the optmal soluton. In Table, we summarze the space of algorthms for solvng varatons of the dstrbuted schedulng problem. The D E J representatons of the schedulng problems come from operatons research. D descrbes the number of machnes, and P corresponds to m dentcal machnes. E s a set of constrants and/or relaatons such as precedence, preempton, or release dates. J s the type of optmzaton, whch n ths case s mnmzng weghted completon tme.,,,6,,,6 = 7. = 7. Fgure. Functon and Smth's rule Alternatvely, we can try to apply Smth s Rule across operator task queues for the tme-averaged cumulatve prorty functon. That s, we could try to construct optmal schedules by orderng all of the tasks n decreasng prorty / eecuton_tme ratos and deal them out to the most underloaded operator. However, t s easy to fnd cases where ths appromaton fals to match an optmal allocaton; an ecepton s shown n Fgure 6: Performance Evaluaton Functon prorty response_ tme prorty response_ tme C pr resp_ tme ³ cum_ prorty( t) dt tme_ perod P w C Performance Evaluaton Functon prorty response_ tme prorty response_ tme m-machs. Sngle Machne No Prortes?? SJF []?? SJF [] NPC [7] D =. [] No Eec. Tmes GPF GPF w C Smth s Rule [] P C SJF [] Precedence Contet Constrs. Swtch? NP-hard []? NP-hard [] Op,, Op, Op,, Op, C pr resp_ tme ³ cum_ prorty( t) dt tme_ perod P w C P p w C GPF P prec w C NPC [9] D =.8 [] NP-hard [] C = C = Table. Complety of schedulng problems Fgure 6. Smth's rule n the mult-operator case << Check to see f table and eplanaton s clear >> 6

7 In addton, recent results for appromaton algorthms are gven. An D-appromaton algorthm guarantees that the performance wll be a factor D of the optmal soluton. In recent years, many constant factor appromatons for schedule optmzaton (such as those n the table) have been dscovered [,,9]. We also wsh to assert that ntroducng contet swtchng adds another level of ntractablty to the problem. In Secton, we dscuss the added complety of contet swtchng. The cumulatve prorty strategy appears to be much better studed than the other canddates we have nvestgated. However, the correspondng problem s yet stll NP-complete, so suboptmal algorthms are epected be more cost-effectve for real-tme systems. The appromaton algorthm referenced n Table ensures a soluton wthn a factor of ( ) or. of the optmal cost by orderng the entre set of tasks accordng to Smth s Rule and then applyng a lst schedulng algorthm. However, gettng a better sense of the topology of the search space could tell us f better performance can be epected from nformed search methods such as hll-clmbng. We net present smulaton results that gve nsght on the epected performance best-frst search... A hll-clmbng strategy << Is ths secton clear? >> So, we know we can effcently mnmze the weghted completon tme cost functon (Functon ) to wthn twenty percent of the optmal usng the appromaton algorthm of the precedng secton. Can we epect to do much better than ths? Best-frst search methods, such as hll-clmbng, offer smple mechansms to quckly mprove on solutons and can be flebly used n conuncton wth other algorthms. The epermental results gven here reveal the epected performance of usng such methods. The smulaton results gven below were collected over a randomly generated sets of tasks to whch hll-clmbng was performed on all equally lkely m n- assgnments to operators to completely eplore the terran of the search space. The hll-clmbng operatons, par-wse task swaps and sngle-task mgraton, were performed on a random allocaton of tasks to operators, and the tasks were ordered wthn operators queues accordng to Smth s Rule. Several parameters were vared (wthn bounds characterstc of TAIPE and computatonally feasble) to see ther effects on the performance: number of operators.. number of tasks.. range of prorty values -; -; - range of processng tmes -; -; - For each permutaton of these parameter values, ten schedule nstances were generated and smulated. Number of Operator Operators Avg. Appromaton by Local Optma Tasks Rato to Optmum Avg. Steps Average Steps to Local Optma Tasks (a) (b) Avg. Local Optma Optma Tasks Operators Probablty of Reachng Global Optma (c) (d) 6 8 Operators. Probablty Tasks Fgure 7. Performance based on number of operators and tasks Avg. Appromaton by Local Optma Prorty Range Proc. Tme Range Rato to optmum Average Steps to Local Optma Prorty Range (a) (b) Average Local Optma Prorty Range Proc. Tme Range Number Optma Probablty of Reachng Global Optma Operators (c) (d).9.8 Steps Proc. Tme Range.9 Probablty Tasks Fgure 8. Performance based on the range of prorty values and processng tmes << Can we make graphs more understandable by changng angle of vew? >> Fgure 7 shows how the performance vares wth the number of operators and tasks wthn the system, and Fgure 8 vares the range of prorty values and processng tmes. The average appromaton by the local optma n Fgures 7a and 8a reveal that hll-clmbng performs 7

8 wthn one percent of the optmal for our test set; however, the appromaton seems to slowly dverge from the optmal for greater parameter values. It s nterestng to note that performance s better for fewer operators and larger ranges of prortes and processng tmes. Fgures 7b and 8b show how the number of hll-clmbng steps ncrease wth respect to the parameters. We noted that the worst case number of steps was. Fgures 7c and 8c show the number of wells (or peaks) n the search space. In correlaton wth Fgures 7a and 8a, the complety s greater for larger numbers of tasks and operators and smaller ranges of prortes and processng tmes. Fgure 7d shows how more operators makes load balancng easer, and Fgure 8d seems to ndcate that there s a better chance of fndng a global optmum when prorty and processng tme ranges are small even though the epected performance (n Fgure 8a) s worse. Ths eperment suggests that other suboptmal search methods, such as smulated annealng or tabu search, can be epected to perform very well wthn bounded ranges of the parameters gven here. We also know that we can bound ths dvergence f we use these search methods n conuncton wth the.-appromaton algorthm. We can epect that good local optma can be found wthn a few hll-clmbng steps, and hll-clmbng can easly be adapted to use dynamc nformaton, such as new task arrvals or changng prorty and processng tme values. Ths bodes well for systems lke TAIPE that can use the bang for the buck performance measure where urgent stuatons requre quck adaptaton.. sng clusterng algorthms for mnmzng contet swtchng In Sectons and, we dentfed the tme-averaged cumulatve prorty functon as a fttng utlty functon for evaluatng the performance of a schedule. Smth s Rule allows the search algorthm to concentrate on the allocaton of tasks and gnore the orderng of tasks wthn a queue when contet swtchng and other task dependences are nsgnfcant. Here we rentroduce contet swtchng nto the schedulng problem. We wll gve a bref analyss of the complety ncurred by contet swtchng and relate how search algorthms can take advantage of dvdng the search space to deal wth the complety added by contet swtches between tasks. We follow wth results from a smulaton to show how well one search algorthm can perform wthn the more comple system and comment on strateges mult-agent systems can apply to adapt to dfferent envronment condtons... Reducng contet swtches through clusterng In Secton, we dscussed how contet swtches can affect the performance of a system. Contet swtches between tasks depend on both the order of eecuton of tasks wthn a queue and the allocaton of tasks among the queues of multple operators. An optmal algorthm must fnd the best allocaton based on the best orderng for each queue. A sub-optmal algorthm, however, can deal wth these two operatons separately. There may be much smpler, faster algorthms for handlng statc stuatons where the arrval of new tasks s gnored. It s thought that n short unusual bursts of task arrvals, the queues wll be unusually long, and statc algorthms could acheve better effcency. In addton, the coarse granularty of the tasks allow more tme for the algorthm to produce a soluton before system changes make the nformaton obsolete. For these reasons, we feel that statc algorthms are approprate for ths system. We assume that the contet swtch tme between two tasks can be estmated by comparng ther attrbutes. Tasks that have smlar or matchng attrbutes wll have a shorter contet swtch tme than tasks wth very dfferent propertes. Thus, f smlar tasks are allocated together, we should epect smaller average contet swtch tmes. Ths heurstc can take advantage of clusterng algorthms whose purpose s to group smlar thngs. A clusterng algorthm may have a measurement that ether computes the smlarty or dssmlarty of a group of tasks. For two tasks, ths may be represented as a weghted sum of matches (or msmatches) of the values of the attrbutes. For a cluster of tasks, we can sum these matches over pars of tasks and normalze them by dvdng by the number of pars, n(n-)/. We assert that the contet swtch tme for two tasks s drectly proportonal to ther dssmlarty, or ncoherence. The ncoherence of a cluster s equvalent to the average ncoherence of all pars of tasks, whch s calculated as a weghted average percentage of msmatched attrbute values. Because we only look at whether attrbute values match or msmatch n the above computaton, we assume that any par of dfferent values for an attrbute wll requre the same amount of contet swtchng. Mnmzng the cost functon (Functon ) for the cumulatve prorty strategy for the case where contet swtches are computed n proporton to the number of msmatched attrbute values s related to the NP-hard mnmum latency problem (MLT), also referred to as the delvery man problem and/or the travelng reparman problem []. Ths s the problem of fndng a tour that mnmzes the sum of the latences (dstances from the startng pont) of all ponts. There s a constant factor 9- appromaton for MLT n the case of a dstance matr 8

9 adherng to the trangle nequalty[]. The contet swtchng problem s the same n the relaed case of a sngle operator and equal processng tmes for tasks. Contet swtches between tasks are dstances between ponts that adhere to the trangle nequalty no two tasks can have more msmatches than the sum of the number of msmatches they each have wth any thrd task. In consderng one very smplfed aspect of ths schedulng problem, we see ntractablty, so we can easly magne that the schedulng problem wthout these relaatons must be at least as hard. Now we look at a clusterng strategy that must deal wth ths antcpated complety... Heurstc-based hll clmbng Ths algorthm has been mplemented n TAIPE to support two dfferent load balancng schemes. One of these s for sngle operators to locally search for a good cluster of tasks that meet a load requrement and are to be shed to another operator. Ths method allows task management to be handled decentrally. The other scheme s centralzed, and our heurstc-based hll-clmbng mechansm searches through the space of task reallocatons among two or more operators n order to mprove the coherence of clusters at each operator and to balance load. The algorthm reles on a dstance functon, D f below, for mprovng clusters and addressng load. For the mult-operator scheme, we could substtute the cost functon (Functon ) to mplement a cumulatve prorty strategy. The basc dea s to restart hll-clmbng at places n the search space that could lead to the optmal soluton. For ths algorthm, these places are ntal clusters of tasks constructed for each attrbute as the largest set of tasks sharng the same value. From these ntal clusters, tasks are added, swapped, and removed untl a local optmum s found. The best of the local optma found s chosen as the soluton. Here s a more detaled descrpton of the algorthm: compute the dstrbuton of values for each attrbute for all tasks n lst fnd the value(s) for each attrbute that has the greatest dstrbuton of tasks construct canddate ntal clusters of tasks for each value wth greatest dstrbuton for attrbute remove duplcate canddate clusters BestLocalOptmum = {} for each cluster := canddate_cluster done := false whle not done f D f (modfed_cluster) t D f (cluster) LocalOptFound := true cluster := modfed_cluster best := cluster for each task TaskLst - cluster // Try addng task f D f (cluster + task) > D f (best) best := cluster + task // Try swap swap orgnal canddate cluster tasks ff to asprate at local optmum f LocalOptFound swap_set = canddate_cluster else swap_set = cluster - canddate_cluster for each c-task swap_set f D f (cluster + task - c-task) > D f (best) best := cluster + task - c-task // Try remove remove orgnal canddate cluster tasks ff to asprate at local optmum f LocalOptFound remove_set = canddate_cluster else remove_set = cluster - canddate_cluster for each c-task remove_set f D f (cluster - c-task) > D f (best) best := cluster - c-task modfed_cluster = best f LocalOptFound AND D f (modfed_cluster) t D f (cluster) done = true f D f (BestLocalOptmum) > D f (cluster) BestLocalOptmum := cluster soluton = BestLocalOptmum The algorthm does not allow any changes to the orgnal tasks n the ntal canddate cluster untl a local optmum s found and then asprates a better soluton f one ests by removng or swappng the orgnal tasks. The dea stems from Tabu Search where tabu moves may be taken f they eceed an aspraton level[8].. Performance wth Contet Swtchng The descrbed algorthm was smulated on data havng the same characterstcs as those of the eperment outlned n Secton. but wth the followng addtonal vared parameters: number of task attrbutes,, 9 number of values per attrbute, ma contet swtch.,. (factor of proc. tme) As epected, the performance of the hll-clmbng algorthm grows worse wth greater numbers of operators and tasks when faced wth contet swtchng (Fgure 9). In contrast wth the eperment n Secton, the number of operators does not make the problem smpler. 9

10 Operators Average Appromaton Tasks Rato to Optmum Fgure 9. Performance based on number of operators and tasks Further work s needed to etend the development of the strategy for evaluatng allocaton and schedulng decsons to accommodate tme-dependent prorty functons and the ncorporaton of hard deadlnes. More work should also be drected towards formulatng and evaluatng centralzed and decentralzed archtectures for performng the task management functons, and ncorporatng them nto systems lke TAIPE. The fnal obectve of ths research s to use nformaton about how the dynamcs of the system fluctuate n order to optmze performance by swtchng between centralzed and decentralzed coordnaton technques and dfferent load balancng algorthms to best handle the fluctuaton of system dynamcs. Average Appromaton Average Appromaton References Range of Prortes.... Rato to optmum Range of Proc. Tmes 9 Attrbutes..... Numbe r of Values (a) (b) Fgure. Peformance based on number of attrbutes and attrbute values Rato to optmum Fgure a shows a slght ncrease n performance for larger ranges of prortes and processng tmes. Fgure b reveals that the number of values per attrbute greatly affects the performance whle the number of attrbutes 6. Concluson In ths report, we have looked at the problem of task schedulng and load balancng for a partcular mult-agent system. The man contrbuton here s a functon sutable for evaluatng the performance of a schedule for such a system. In addton, a lnk has been made to the complety analyss of proposed alternatve algorthms n the operatons research communty. Furthermore, suggestons have been made for choosng search algorthms that can effcently fnd solutons that are valued by ths performance evaluaton functon. We have analyzed the computatonal needs of ths problem, and dentfed smplfcatons and appromatons that show promse. In partcular, besdes makng new connectons to the operatons research lterature, we have also eamned the use of heurstc hll-clmbng methods. For the parameter space manfested n TAIPE, our results ndcate that usng these methods provdes a promsng avenue for managng the task loads and schedules n the dstrbuted system. [] Ishfaq Ahmad, Arf Ghafoor. SemDstrbuted Load Balancng for Massvely Parallel Multcomputer Systems. IEEE Transactons on Software Engneerng, 7():987-, October 99. [] Avrm Blum, Prasad Chalasan, Don Coppersmth, Bll Pulleyblank, Prabhakar Raghavan, Madhu Sudan. The Mnmum Latency Problem. Proceedngs of the 6 th Symposum on the Theory of Computng, pp. 6-7, 99. [] Peter Brucker. Schedulng Algorthms. Sprnger-Verlag, Berln, 99. [] C. Chekur, R. Motwan, B. Nataraan, C. Sten. Appromaton Technques for Average Completon Tme Schedulng. Proceedngs of the Annual ACM-SIAM Symposum on Dscrete Algorthms. pp , 997. [] Keth Decker, Vctor Lesser. Quanttatve Modelng of Comple Envronments. Internatonal Journal of Intellgent Systems n Accountng, Fnance, and Management, (), 99. [6] Edmund Durfee, Jaeho Lee, Marcus Huber, Mchael Kurnow. TAIPE: Tactcal Assstants for Interacton Plannng and Eecuton. Proceedngs of the Internatonal Conference on Autonomous Agents, pp.-, 997. [7] Mchael R. Garey, Davd S. Johnson. A Gude to the Theory of NP-Completeness. W. H. Freeman, San Francsco, 979. [8] Fred Glover. Tabu Search Part I, ORSA Journal of Computng, ():9-6, 989. [9] Lesle A. Hall, Andreas S. Schultz, Davd B. Shmoys, Joel Wen. Schedulng to Mnmze Average Completon Tme: Off-lne and On-Lne Appromaton Algorthms, revsed March 997. [] Renhard Hanleden, L. Rdgway Scott. Load Balancng on Message Passng Archtectures. Journal of Parallel and Dstrbuted Computng, :-, 99. [] Tsuyosh Kawaguch, Sek Kyan. Worst Case Bound of an (LRF) Schedule for the Mean-Weghted Flow-Tme Problem. Sam Journal on Computng. ():6-8, 986.

11 [] J. Mohan Kumar, L. M. Patnak, A. Das. Load balancng algorthms for an etended hypercube. IEE Proceedngs Comput. Dgt. Tech., :98-6, September 99. [] Jy-Shane Lu, Kata Sycara. Multagent Coordnaton n Tghtly Coupled Task Schedulng. Proceedngs - Second Internatonal Conference on Mult-Agent Systems, pp. 8-88, 996. [] Ashwan Raman, Pradp Chande, Pramod Sharma. A General Model for Performance Investgatons of Prorty Based Multprocessor System. IEEE Transactons on Computers, (6):77-7, June 99. [] W. Smth. Varous Optmzers for Sngle-Stage Producton. Naval Res. Logst. Quart. :9-66, 96. [6] Marc Wllebeek-LeMar, Anthony Reeves. Strateges for Load Balancng on Hghly Parallel Computers. IEEE Transactons on Parallel and Dstrbuted Systems, (9):979-99, September 99. [7] Hua Wu, Davd Chang, Wllam Oldham. Dynamc Task Allocaton Strateges. IEEE Transactons on Parallel and Dstrbuted Systems, 6():-6, December 99.