Scheduling in polling systems

Transcription

1 Schedulng n pollng systems Adam Werman, Erk M.M. Wnands, and Onno J. Boxma Abstract: We present a smple mean value analyss MVA framework for analyzng the effect of schedulng wthn queues n classcal asymmetrc pollng systems wth gated or exhaustve servce. Schedulng n pollng systems fnds many applcatons n computer and communcaton systems. Our framework leads not only to unfcaton but also to extenson of the lterature studyng schedulng n pollng systems. It llustrates that a large class of schedulng polces behaves smlarly n the exhaustve pollng model and the standard M/GI/1 model, whereas schedulng polces n the gated pollng model behave very dfferently than n an M/GI/1. 1 Introducton Schedulng s a common mechansm for mprovng system performance wthout purchasng addtonal resources. Tradtonally, smple schedulng polces such as Frst-Come-Frst-Served FCFS and Processor- Sharng PS, whch shares the servce capacty equally among all jobs n the system, have been appled most frequently, and thus domnate the queueng lterature. However, recently, system desgns n a varety of applcaton domans have begun to use polces that gve prorty to jobs wth small servce demands n order to reduce the mean response tme sojourn tme and mean queue length. For nstance, varants of Shortest-Remanng-Processng-Tme SRPT and Foreground-Background FB have been appled n many computer applcatons, e.g. web servers [8, 17] and routers [14, 15]. Ths growng focus on prorty-based polces has led to an ncreasng number of theoretcal studes of such polces as well, e.g. [1, 27, 28] and the references theren. However, almost all theoretcal studes of prorty-based polces are performed n smple settngs such as the M/GI/1 and G/GI/1. Our goal s to explore the mpact of prorty-based schedulng n a more complex queueng model: pollng systems. A pollng system conssts of a sngle server that polls a number of queues n a fxed order. Pollng systems were frst ntroduced n the late 195s by Mack et al. [12, 13] to model a patrollng reparman. Snce the 195s, pollng systems have been used to model a wde range of applcatons n computer, communcaton, producton, transportaton, and mantenance systems. The ubqutous nature of pollng systems has meant that they have receved an enormous amount of attenton n the queueng communty. Extensve surveys on pollng systems and ther applcatons may be found n [11, 2, 21, 22, 25]. Wthn a pollng system there are a number of desgn decsons that the system operator needs to make. The system operator must decde the order n whch to serve the queues, how many customers to serve durng each vst to a queue, and the order n whch customers wthn each queue are served. The frst decson can ether be statc,.e. the pollng order remans unchanged n the course of operaton see, Computer Scence Department, Carnege Mellon Unversty, 5 Forbes Avenue Pttsburgh, PA, USA. Much of ths paper was completed durng a vst to the EURANDOM nsttute wth support provded by an STW grant. Department of Technology Management, Technsche Unverstet Endhoven, P.O. Box 513, 56 MB Endhoven, The Netherlands Department of Mathematcs and Computer Scence, Technsche Unverstet Endhoven, P.O. Box 513, 56 MB Endhoven, The Netherlands EURANDOM, P.O. Box 513, 56 MB Endhoven, The Netherlands 1

2 e.g., [3, 4], or dynamc,.e. the pollng order changes over tme see, e.g., [9, 31]. For the second decson a plethora of strateges has been proposed and analyzed, wth most work focusng on the exhaustve and gated dscplnes. Exhaustve servce means that a queue must be empty before the server moves on, whereas n case of gated servce only those customers n the queue at the pollng start are served. The focus of the current paper s on the thrd schedulng decson: how to schedule jobs wthn each queue. As a result, we wll lmt dscusson to the most common confguratons for the frst two decsons: cyclc servce order and exhaustve/gated servce. Although both the number of papers analyzng pollng systems and the number of papers analyzng schedulng polces are mpressve, the combnaton of the two has receved very lttle attenton. There are only a few exceptons where the effect of prorty-based polces s studed n pollng systems, for example, [6, 19, 23, 24]. However, the results attaned for such prorty-based polces are mostly lmted to pseudoconservaton laws and approxmatons that are exact only n specal cases, e.g. symmetrc pollng systems. Despte the lack of work analyzng schedulng n pollng systems, there are many real-world examples where applcatons may beneft from schedulng n a non-fcfs manner. For example, n the computer scence communty pollng models have recently been used to study the Bluetooth and protocols as well as schedulng polces at routers and /o subsystems n web servers. In many of these settngs, the workloads are known to have hgh varablty, and thus usng a polcy other than FCFS wthn the queues s appealng. Further, n these applcatons t s often desrable to gve dfferent requests dfferent prorty levels n order to provde dfferentated servce. Outsde of computer systems, non-fcfs schedulng s used n pollng systems n the area of producton-nventory control. Specfcally, t s used n the stochastc economc lot schedulng problem SELSP, where multple standardzed products have to be produced on a sngle machne wth sgnfcant setup tmes see [29] for a survey on SELSP. In the SELSP, schedulng wthn the queues s necessary due to orders for the same product beng placed by customers wth dfferent prorty levels. Thus, the lack of research on schedulng n pollng systems s not due to a paucty of applcatons. Instead, we beleve t s manly due to the followng factors: 1. It s natural to beleve that the mpact of wthn-queue schedulng n a pollng system s small because t only nfluences the system performance locally, leavng the amount of tme spent outsde the targeted queue unaffected. 2. The analyss of pollng systems s dffcult even n the smple case of FCFS schedulng: for FCFS explct closed-form expressons for the mean delays are n general not known. Instead, the mean delay s expressed as the soluton of a set of lnear equatons. In ths paper, we llustrate that the mpact on mean response tme from schedulng wthn a queue of a pollng system can be dramatc. Further, n order to llustrate the mpact of schedulng, we provde a unfed analytc framework for studyng schedulng n pollng systems. Ths mean value analyss MVA framework allows the analyss of a varety of schedulng polces many for the frst tme n classcal asymmetrc pollng systems wth ether gated or exhaustve servce. Further, our framework llustrates a strkng dchotomy between the mpact of schedulng polces n exhaustve pollng systems and gated pollng systems. Ths reveals tself n the fact that a large class of schedulng polces behaves the same n exhaustve pollng models as they do n the standard M/GI/1, whereas schedulng n gated pollng models has a dfferent effect than n the M/GI/1. The rest of the paper s structured as follows. In Secton 2 we present a detaled model descrpton. Sectons 3 and 4 deal wth the analyss of schedulng polces n pollng systems wth gated and exhaustve servce, respectvely. Fnally, Secton 5 contans some remarks and suggestons for future research. 2

3 FB FCFS LCFS PLCFS PS SJF SRPT Foreground-Background preemptvely shares the server evenly among those jobs that have receved the least amount of servce so far. Frst Come Frst Served serves jobs n the order they arrve. Last Come Frst Served non-preemptvely serves the job that arrved the most recently. Preemptve Last Come Frst Served preemptvely serves the most recent arrval. Processor Sharng serves all customers smultaneously, at the same rate. Shortest Job Frst non-preemptvely serves the job n the system wth the smallest orgnal sze. Shortest Remanng Processng Tme preemptvely serves the job wth the shortest remanng sze. Table 1: A bref descrpton of the schedulng polces dscussed n ths paper. 2 Model descrpton and notaton Throughout, we consder a pollng system wth one sngle server for N 1 queues, n whch there s nfnte buffer capacty for each queue. The server vsts and serves the queues n a fxed cyclc order. We ndex the queues by, = 1, 2,..., N, n the order of the server movement. For compactness of presentaton, all references to queue ndces greater than N or less than 1 are mplctly assumed to be modulo N, e.g., queue N + 1 actually refers to queue 1. The number of jobs served durng a vst to a queue s determned by ether exhaustve or gated servce, as we descrbed n Secton 1. Then, durng the vst to each queue, we allow jobs to be scheduled for servce accordng to a varety of schedulng polces, summarzed n Table 1. However, we lmt the dscusson to work-conservng dscplnes,.e. the server never dles whle at queue f there s work n queue. Customers arrve at all queues accordng to ndependent Posson processes wth rates λ, = 1, 2,..., N. The servce requrements at queue are ndependent, dentcally dstrbuted random varables wth dstrbuton functon F x, densty f x, mean E[X ], and second moment E[X 2 ] <, = 1, 2,..., N. We denote F x = 1 F x. When the server starts servce at queue, a setup tme s ncurred of whch the frst and second moment are denoted by E[S ] and E[S 2 ] < respectvely. These setup tmes are dentcally dstrbuted random varables, ndependent of any other event nvolved. The total setup tme n a cycle s denoted by S wth mean E[S] = N =1 E[S ]. 1 Two other mportant quanttes are the mean resdual servce requrement and the mean resdual setup tme for queue, whch can be expressed as follows, respectvely, E[R X ] = E[X2 ] 2E[X ], E[R S ] = E[S2 ], = 1, 2,..., N. 2E[S ] The occupaton rate utlzaton ρ at queue excludng setups s defned by ρ = λ E[X ] and the total occupaton rate ρ s gven by ρ = N =1 ρ. A necessary and suffcent condton for the stablty of ths pollng system s ρ < 1 see, e.g., [2]. We wll always assume the queues are stable. The cycle length of queue, = 1, 2,..., N, s defned as the tme between two successve arrvals n case of the gated dscplne or departures n case of the exhaustve dscplne of the server at ths queue. It s well-known that the mean cycle length s ndependent of the queue nvolved and the servce dscplne and s gven by see, e.g., [2] E[C] = E[S] 1 ρ. We note that although the frst moments of the cycle lengths are dentcal for all queues, hgher moments generally dffer. 1 Throughout, we focus on the case E[S] >. When the total setup tme s equal to zero, some subtletes appear due to the fact that the number of cycles wth zero length tends to nfnty. However, wth some mnor adjustments MVA can stll be appled see [3]. 3

4 The vst tme θ of queue, = 1, 2,..., N, s composed of the servce perod of queue,.e., the tme the server spends servng customers at queue, plus the precedng setup tme n case of exhaustve servce or plus the succeedng setup tme n case of gated servce. By vrtue of these two dfferent defntons, a queue s empty exactly at the end of ts vst tme n case of exhaustve servce, whle the queue before the gate s empty at the begnnng of a vst tme n case of gated servce all customers watng for servce are then placed behnd the gate. Snce the server s workng a fracton ρ of the tme on queue, the mean of a vst perod of queue reads, for exhaustve servce, E[θ ] = ρ E[C] + E[S ], = 1, 2,..., N, and, for gated servce, E[θ ] = ρ E[C] + E[S +1 ], = 1, 2,..., N. We defne an, j-perod θ,j as the sum of j consecutve vst tmes startng n queue, j = 1, 2,..., N. The correspondng mean s gven by E[θ,j ] = +j 1 n= E[θ n ], = 1, 2,..., N, j = 1, 2,..., N. Notce that n case j = 1 and j = N, E[θ,j ] s equal to the mean vst perod E[θ ] of queue and the mean cycle length E[C], respectvely. The fracton of the tme q,j the system s n an, j-perod equals q,j = E[θ,j] E[C], = 1, 2,..., N, j = 1, 2,..., N, where q,n equals 1 by defnton. Moreover, the mean of a resdual, j-perod s gven by E[R θ,j ] = E[θ2,j ], = 1, 2,..., N, j = 1, 2,..., N, 1 2E[θ,j ] wth the remark that the second moments E[θ,j 2 ] are stll unknown at ths stage. Notce that snce for each fxed, j the successve, j-perods are not ndependent, they do not form a renewal process. Ths means, among others, that Equaton 1 does not drectly follow from the theory of regeneratve processes. For a proof why ths result s nevertheless stll vald see, e.g., [7]. Our man nterest s n the mean response tme sojourn tme E[T ] of a type- customer, = 1, 2,..., N, whch s defned as the tme n steady state from a customer s arrval at queue untl the completon of hs servce. Often, t wll be more convenent to study the mean delay, E[D ], whch s defned as E[T ] E[X ]. By Lttle s Law, under non-preemptve polces, these mean delays can be related to the mean queue lengths excludng the customer possbly n servce E[L ], = 1, 2,..., N. The analyss of the present paper s orented towards the determnaton of E[L,n ], the mean queue length at queue at an arbtrary epoch wthn a vst tme of queue n,, n = 1, 2,..., N. The correspondng uncondtonal mean queue length E[L ] can be expressed n terms of E[L,n ] as follows E[L ] = N q n,1 E[L,n ], = 1, 2,..., N. 2 n=1 3 Gated pollng systems We wll now develop a smple analytc framework for analyzng schedulng polces n gated pollng systems. Ths framework allows smple arguments to be used to obtan results for the mean delay of a wde varety of 4

5 schedulng polces and llustrates that the comparson between schedulng polces n gated pollng systems s remarkably smple t s far less complex than n the M/GI/1 model. Ths s perhaps surprsng due to the complexty of the underlyng pollng system. We frst derve expressons for the mean delay of a varety of schedulng polces n terms of the mean resdual cycle length n Secton 3.1. The polces we consder are FCFS, LCFS, SJF, FB, and PS. In addton, we dscuss m-class prorty queues, whch are of partcular practcal mportance. After dervng the performance of these schedulng polces n terms of the mean resdual cycle length, we analyze ths quantty n detal n Secton 3.2. Fnally, we present numercal experments n Secton The effect of schedulng on mean delay To begn our study of schedulng n pollng systems, we consder the mean delay of a tagged arrval of sze x, j x, to queue. Frst note that because we are consderng a gated pollng system, job j x wll not receve servce durng the cycle nto whch t arrves where we have to recall that a cycle s defned as the tme between two successve arrvals of the server at queue. Further, the length of tme remanng n the cycle s smply the resdual cycle length, R C. Note that the age of the cycle at the arrval of j x s equal n dstrbuton to the resdual of the cycle. The delay of j x s made up of three components: the resdual cycle length, the amount of servce gven to arrvals after j x and durng the same cycle, and the amount of servce gven to arrvals before j x and durng the same cycle. To smplfy the computaton of the latter two components, we notce that all common schedulng polces obey the followng two propertes: Property 1 The contrbutons to the delay of j x from each job that arrves before j x and durng the same cycle as j x, denoted c 1 X, are..d. Property 2 The contrbutons to the delay of j x from each job that arrves after j x and durng the same cycle as j x, denoted c 2 X, are..d. Now, we have the followng smple representaton for the mean delay for a job of sze x, E[D x], under any polcy that obeys Propertes 1 and 2: N A R C N A R C E[D x] = E[R C ] + E c 1 X j + E c 2 X j j=1 j=1 = E[R C ] 1 + λ E[c 1 X ] + λ E[c 2 X ], 3 where N A Y s the number of arrvals durng tme Y, and X j s the job sze of the jth arrval. Usng 3, we can now easly obtan formulas for the mean delay of a handful of common schedulng polces under gated pollng models. Further, 3 mmedately gves bounds on the attanable mean delay under any work conservng polcy n gated pollng systems. For nstance, we see that E[R C ] E[D x] E[R C ]1 + 2ρ. Both of these bounds are n fact tght. Wth a lttle work, t s possble to prove that the lower bound can be attaned by a polcy that preemptvely gves jobs of sze x hghest prorty, and the upper bound s attaned for all job szes under a polcy that gves prorty to the job wth the longest remanng sze. In addton, ths gves us a tght upper bound on the overall mean delay, but a tght lower bound on E[D ] does not follow 5

6 mmedately. Later, we wll derve a tght lower bound by analyzng SJF, whch optmzes E[D ], n Secton The resultng bounds on the attanable mean delay are E[R C ]1 + λe[m ] E[D ] E[R C ]1 + 2ρ, where M s the mnmum of two ndependent job szes. It s already evdent that the formulas we derve n ths secton wll be qute dfferent than the results for schedulng polces n the M/GI/1 settng. Further, they are very dfferent than the results we wll derve for exhaustve pollng systems. The results n the gated pollng settng are much smpler and more elegant. Fnally, before movng to the analyss, t s mportant to note that there s no dstncton between preemptve and non-preemptve polces n ths settng snce new arrvals must wat untl after the next cycle for servce. Thus, for nstance, PLCFS and LCFS are equvalent, as are SJF and SRPT FCFS We start wth the smplest, most common schedulng polcy: FCFS. The mean delay of FCFS n gated pollng systems s well known, but t s useful to note how easly t follows from 3. In the case of FCFS, only arrvals before the tagged job wll contrbute to the delay of the tagged job. Thus, E[c 1 X ] = E[X ] and E[c 2 X ] =, whch gves the well-known result that E[D x] FCFS = E[R C ] 1 + ρ, 4 whch we refer to as the so-called arrval relaton. Note that ths s not an explct formula snce E[R C ] s unknown. However, t s mportant to note that snce we are consderng only work-conservng polces E[R C ] s ndependent of the schedulng polcy used at each queue. In the remander of ths secton, we derve for each ndvdual schedulng dscplne an arrval relaton n terms of E[R C ]. We wll descrbe how to calculate E[R C ] later n Secton LCFS Another smple, common polcy s LCFS. Agan, obtanng the mean delay of LCFS from 3 s smple. Snce only arrvals after the tagged job contrbute to the delay of the tagged job, we have E[c 1 X ] =, and E[c 2 X ] = E[X ], whch gves E[D x] LCFS = E[R C ] 1 + ρ. 5 Notce that ths s the same mean delay we obtaned for FCFS. In fact, the same result can easly be shown to hold for all blnd, lst based polces,.e. all polces that serve ndvdual jobs to completon usng an orderng lst that s determned wthout usng job szes SJF We now move beyond smple polces to sze-based polces. It s easy to see that SJF optmzes the mean delay and queue length under gated pollng systems snce t s equvalent to SRPT n ths settng, whch has long been known to optmze queue length and mean delay [18]. Though SJF s a more complex polcy than FCFS and LCFS, the mean delay of SJF can stll be derved very easly from 3. In partcular, f we consder the contrbuton of an arrval to the delay of a tagged job, the moment when the arrval occurred durng the cycle s rrelevant the arrval wll contrbute to the delay 6

7 of the tagged job only f ts sze s smaller. Thus, we have E[c 1 X ] = E[c 2 X ] = E[X 1 [X<x]]. Defnng ρ x = λ E[X 1 [X<x]] then gves E[D x] SJF = E[R C ] 1 + 2ρ x. 6 Comparng 6 wth 4, we can mmedately see that small job szes perform better under SJF than FCFS, but that larger job szes perform worse. In fact, the largest job szes have E[D x] SJF E[R C ]1 + 2ρ, whch 3 shows s the worst possble mean delay. To obtan the overall mean delay under SJF, we need to ntegrate 6. The result gves us a smple characterzaton of the optmal mean delay: x E[D ] SJF = E[R C ] 1 + 2λ f x tf tdtdx = E[R C ] 1 + λ t2f tf tdt = E[R C ] 1 + λ E[M ], 7 where M s the mnmum of two..d. job szes and the second lne follows from the frst by nterchangng the ntegrals. Comparng 4 for FCFS and 7 for SJF shows: E[D ] SJF E[D ] FCFS E[M ] E[X ]. It s easy to see that under determnstc job szes E[D ] SJF = E[D ] FCFS, but as the servce dstrbuton varablty ncreases SJF provdes more and more mprovement over FCFS. For example, under an exponental dstrbuton E[D ] SJF = E[R C ] 1 + ρ /2, and under dstrbutons that have a decreasng falure rate DFR the mprovement s even greater, snce t s easly shown that under such dstrbutons E[M ] E[X ]/ FB It s often the case that applcatons do not know job szes, and therefore cannot use SJF to attan the optmal mean delay. In such cases, the age attaned servce of a job can often serve as an ndcaton of the remanng sze of the job. For nstance, f job szes have a DFR IFR servce requrement dstrbuton, then jobs wth larger ages are lkely to have larger smaller remanng szes. Thus, FB FCFS s a poor man s SRPT n the case of DFR IFR job szes. In fact, FB and FCFS have been shown to optmze among polces blnd to job szes the queue length dstrbuton and the mean delay n G/GI/1 queues when job szes are DFR and IFR respectvely [16]. Further, snce these optmalty results hold even when busy perods begn wth an arbtrary batch arrval, they also hold n pollng systems. In ths secton, we focus on FB, wth the motvaton that DFR servce dstrbutons are common n computer and telecommuncaton applcatons. The mean delay under FB agan follows easly from 3. Agan, we consder a tagged job of sze x, j x. The key observaton s that any job that arrves durng the same cycle wll contrbute mnx, x work to the mean delay of the tagged job, because the server wll gve an equal servce rate to all jobs n the system throughout the vst perod snce all new arrvals stay behnd the gate. Thus, E[c 1 X ] = E[c 2 X ] = E[mnX, x]. Defnng ˆρ x = λ E[mnX, x] then gves E[D x] FB = E[R C ] ˆρ x. 8 Lke under SJF, FB clearly benefts small job szes compared to FCFS whle hurtng large job szes. In fact, t s agan true that large job szes are treated as badly as possble under any work conservng polcy. 7

8 Comparng 8 and 6 llustrates that FB behaves very smlarly to SJF, though FB clearly pays a prce for not usng job szes snce ρ x ˆρ x. Ths dfference s accentuated when we look at E[D ]: x E[D ] FB = E[R C ] 1 + 2λ f x F tdtdx = E[R C ] 1 + 2λ F t 2 dt = E[R C ] 1 + 2λ E[M ], 9 where M s agan the mnmum of two..d. job szes and the second lne follows from the frst by nterchangng the ntegrals. The comparson between 9 and 7 gves a clear pcture of the prce FB pays for not usng job szes. In addton, 9 gves a very smple comparson between FCFS and FB: E[D ] FB E[D ] FCFS E[M ] E[X ]/2. Notce that equalty holds under the exponental dstrbuton. Further FB wll be better under DFR dstrbutons and worse under IFR dstrbutons, as expected PS PS s a polcy that s wdely used n computer systems due to ts farness propertes and ts smplcty, so t s mportant that we spend a moment on t here. However, n gated pollng systems, PS s actually equvalent to FB, whch we just dscussed. In partcular, because all jobs that arrve durng a vst perod wll not receve servce untl the next vst, FB wll always end up sharng the server evenly among all jobs n the queue, whch s exactly what PS does. Thus, all the results we descrbed for FB also hold for PS, ncludng the fact that FB s optmal among blnd polces for queue length and mean delay when job szes are DFR m-class prorty queues The last polcy we consder s probably the most mportant from a practcal perspectve, so we wll spend the most tme explorng ts behavor. In an m-class prorty queue, arrvals are tagged by ther class, 1,..., m, and then jobs from class are gven preemptve prorty over jobs from classes >. Wthn each class, jobs are served n FCFS order. These polces are often used n practcal settngs nstead of dealzed polces lke SJF, e.g. [8, 17]. The mean delay of a class j job, E[D j ], s agan easly derved from 3. Throughout, we use a superscrpt j to specfy class j. For a tagged job of class j, all arrvals durng the cycle from classes < j and all earler arrvals from class j wll be served before the tagged job. Thus, we have that E[c 1 X k ] = E[X k 1 [k j] ] and E[c 2 X k ] = E[X j 1 [k<j] ]. Defnng ρ j = λ j E[X j ] then gves E[D j ] = E[R C ]1 + 2 ρ k + ρ j. 1 k<j Notce that the mean delay of SJF can be obtaned by takng the approprate lmt of ths formula. From 1 the overall mean delay can be calculated usng E[D ] = j λ j λ E[Dj ]. 8

9 Though ths formula s easy to wrte, t hdes the answer to mportant questons such as how the dstrbutons of the job szes n each prorty class affect the overall mean delay. In the remander of the secton we wll develop a better understandng of ths behavor. We wll start by studyng the case when there are only 2 prorty classes, and then we wll use the deas llustrated by ths smple case to study the general case of m prorty classes. m = 2: Two prorty classes Let us now look at ths n the case of two prorty classes to see how prortzaton affects the overall response tme. Wth two prorty classes, we have that E[D ] E[R C ] = λ1 1 + ρ 1 λ + λ ρ 1 + ρ 2 λ = 1 + ρ λ1 ρ 2 λ + λ2 ρ 1 λ = 1 + ρ λ1 λ 2 E[X 2 ] E[X 1 ] λ, 11 from whch t follows that E[D ] E[D ] FCFS E[X 1 ] E[X 2 ]. So, prortzng small job szes s an effectve heurstc. In fact, we can see from 11 that the optmal mean response tme for a 2-class prorty system wll occur when there s a threshold t such that jobs wth sze t have hgh prorty and jobs wth sze > t have low prorty. The natural queston, then, s what s the optmal such t? We can determne t as follows: E[D ] = 1 + ρ λ F tf t s f s E[R C ] F t ds t t t s f s F t ds = 1 + ρ λ F t sf sds F t sf sds t = 1 + ρ + λ F te[x ] sf sds, t whch s mnmzed when the fnal term s mnmzed. Takng dervatves, we see that f te[x ]+tf t = only when E[X ] = t, assumng that f x > for all x. It s easy to see that ths s a mnmum, so the optmal threshold s t = E[X ]. The fact that the optmal threshold s smply E[X ] n ths settng regardless of the shape of the dstrbuton s qute specal. In the M/GI/1 settng the optmal threshold s far from nsenstve to the shape of the servce requrement dstrbuton. Smlarly, n the case of exhaustve servce the optmal threshold turns out to be not nsenstve. m prorty classes We wll now move beyond the 2-class prorty settng and consder the behavor of an m-class system. In the 9

10 m-class case we have, cf. 1, E[D ] m = 1 + ρ j E[R C ] j=1 λ j λ = 1 + ρ + m j=1 λ j λ = 1 + ρ λ m j j 1 m k=1 j=1 k=j+1 k=1 ρ k λ j λ 2 ρ k λ k m k=j+1 ρ k E[X k ] E[X j ], 12 where the last lne comes from groupng terms havng the same λ j λ k multpler. Equaton 12 s the extenson of 11. It shows that the mprovement of the mean delay of the prorty queue over FCFS s drectly related to the dfferences between the means of the prorty classes. In fact, ths form mples that the m-class polcy whch optmzes E[D ] wll be a threshold based polcy snce the mean delay of any nonthreshold based polcy can be mproved by nterchangng mass between two prorty classes that overlap so as to separate ther means but not change ther arrval rates. Thus, our goal n the remander of the secton s to determne the optmal threshold values for an m-class threshold based polcy. Consder an m-class polcy wth thresholds t s such that = t < t 1 <... < t m = that assgns jobs wth sze x [t s 1, t s prorty s. We wll prove that the optmal thresholds are defned by t j = 1 t j+1 uf F t j 1 F t j+1 udu. 13 t j 1 Notce the ntuton behnd the form of ths relaton: the threshold dvdng classes j and j + 1 s defned as the mean of the total dstrbuton for jobs of classes j plus j + 1,.e. the mean of the dstrbuton of jobs that the threshold s dvdng. To prove that 13 defnes the optmal thresholds, we start by combnng the terms n 13 that have the same E[X s ] n order to obtan E[D ] E[R C ] m λ s s 1 = 1 + ρ λ E[X s ] λ s=1 Lookng at ths term by term, we notce that j=1 λ j λ m j=s+1 λ j. 14 λ s 1 j=1 λ j λ = F t s 1, m j=s+1 λ j λ = F t s and λ s t s EX s = uf udu. λ t s 1 Returnng to 14 we have that E[D ] E[R C ] = 1 + ρ λ m s=1 t s F t s 1 F t s uf udu. 15 t s 1 Next, we observe that F t s 1 F t s = F t s 1 + F t s. Usng ths, we can wrte the odd s terms n 15 as the LHS and the even s terms as the RHS and see that adjacent terms telescope leavng E[D ] E[R C ] m 1 = 1 + ρ + λ s=1 t s+1 1 [s odd] F t s uf udu. 16 t s 1 1

11 To fnd the optmal cutoffs we look at the partal dervatves of 16 wth respect to t s for < s < m: { d E[D ] d = λ dt s 1 E[R C ] dt s [s + 1 odd] + F t t s+2 s+1 uf udu t s t s [s odd] + F t s uf udt t s 1 } t s + 1 [s 1 odd] + F t s 1 uf udt t s 2 { = λ 1 [s + 1 odd] F t s+1 t s f t s + f t s + 1 [s 1 odd] + F t s 1 = λ f t s { t s+1 t s 1 } t s f t s t s+1 t s 1 uf udu } uf udu F t s+1 F t s 1 t s, 17 from whch we can see that when f x > for all x the only crtcal pont occurs when t s = 1 t s+1 uf F t s+1 F t s 1 udu. 18 t s 1 It s easy to see that ths s the global mnmum. Choosng the optmal threshold: Dscusson and examples We have seen that n the case of m = 2, the optmal cutoff s t 1 = E[X ] regardless of the servce requrement dstrbuton. Ths nsenstvty to the shape of the dstrbuton s qute surprsng, and very dfferent from what occurs n the standard M/GI/1 model and from what we wll see n the case of exhaustve pollng systems. However, when m > 2 the shape of the servce requrement dstrbuton plays a role n the choce of the optmal thresholds. We wll llustrate ths wth two examples: the exponental and Pareto dstrbutons. Example 3.1 In the case of the exponental dstrbuton wth rate µ and m = 3, we can solve for t 1 and t 2 very easly. Let F u = e µu. Then, usng 18 we get that the optmal thresholds satsfy t 2 = e µt1 uµe µu du = t 1 + 1/µ and t 1 1 = t 1 1 e µt2 t 2 uµe µu du, 19 whch gves µt 1 = 1 µt 2 e µt2 /1 e µt2, yeldng 1 µt 1 = 2e 1+µt1. Clearly there s only one soluton where both t 1 and t 2 are postve. Further, notce that the soluton wll always have t 1 < E[X ] and t 2 > E[X ]. As an example of what the thresholds wll be, n the case of an exponental wth mean 1, we obtan t 1 =.59 and t 2 = Example 3.2 In the case of the Pareto dstrbuton, we have F u = k/u α wth α > 1. Then, usng 18 we get that the optmal thresholds satsfy t t 2 1 α = k whch gves 1 k t 2 α = u αkα du = α t uα+1 1 α α 1 α 1 t1, and t 1 = 2 k t 2 1 k t 2 α t 2 1 k/t 2 α k u αkα du, uα+1

12 Clearly t 2 = k s always a soluton, but we are only nterested n t 2 > k. As an example of solvng ths, we can look at the case of α = 2. Lettng z = k/t 2 gves 1 z 2 = 4z1 z, whch has roots z = 1/3, 1. Ths gves t 2 = 3k and t 1 = 1.5k. Specalzng further to the case when E[X ] = 1 gves k =.5 from whch we obtan t 2 =.75 and t 2 = 1.5. Notce that these thresholds are more concentrated around the mean than n the case when job szes are exponental. 3.2 Resdual cycles Throughout the last secton, we derved formulas for the mean delay of schedulng polces n terms of the mean resdual cycle length. Thus, we have solated the effects of the setup tmes and the dependences between vst tmes nto one quantty, whch s ndependent of the schedulng dscplne as long as the schedulng dscplne s work-conservng. Ths allowed us to perform very smple comparsons of the mean delays across all the schedulng dscplnes we have consdered. Unfortunately though, no general explct closed-form expresson for the mean resdual cycle lengths s known. However, n order to calculate these mean resdual cycle lengths numercally, we can make use of the recently developed mean value analyss MVA for FCFS pollng systems [3]. Although [3] studes only FCFS pollng systems, t also provdes as a by-product the mean resdual vst tme, and thus the mean resdual cycle lengths, whch are ndependent of the servce dscplne. Before we can start the analyss, we have to ntroduce some addtonal notaton. That s, n case of gated servce, all customers watng n queue at the start of a vst tme of ths queue are placed behnd a gate meanng that they are served n the current cycle. However, customers arrvng durng a vst tme of ther queue are placed before ths gate and are, thus, only served n the next cycle. Wth ths dfference understood, t s clear that, n case = j, L,j s the sum of two auxlary varables, L, = L, + L,, where L, and L, represent the queue length behnd and before the gate, respectvely. Recall that the customer n servce s excluded. In case j, all customers n queue are obvously located before the gate,.e., L,j = L,j, j = 1, 2,..., N. The correspondng uncondtonal queue length L has mean E[L ] = N q n,1 E[ L,n ] + q,1 E[L, ]. 2 n=1 For = 1, 2,..., N and j = 1, 2,..., N, under FCFS schedulng, we have the followng set of equatons N q n,1 E[ L,n ] + q,1 E[L, ] = λ E[R C ] 1 + ρ, 21 n=1 +j 1 n= q n,1 q,j E[ L,n ] = λ E[R θ,j ], 22 E[R θ,1 ] = E[L, ]E[X ] + E[S +1] E[θ,1 ] E[R S +1 ] + ρ E[C] E[θ,1 ] E[R X ] + E[S +1 ], 23 E[R θ,j ] = q j 1 j 1 j 1,1 E[R θ,1 ] 1 + ρ +n + E[S +n+1 ] + E[ L +n, ]E[X +n ] 1 + ρ +m q,j n=1 n=1 m=n+1 +1 q,1 q,j E[R θ+1,j 1 ]. 24 Elmnaton of E[R θ,j ] from 21 and 22 wth the help of 23 and 24 yelds a set of NN + 1 lnear equatons for equally many unknowns E[L, ] and E[ L,n ]. The soluton to these equatons yelds the mean 12

13 5 4 FCFS 2 class SJF 25 2 FCFS 2 class SJF E[D] 3 2 E[D] ρ Fgure 1: Impact of schedulng n symmetrc gated pollng systems for exponental servce tmes wth E[X] = ρ Fgure 2: Impact of schedulng n symmetrc gated pollng systems for Webull servce tmes wth E[X] = 1 and E[X 2 ] = 2. resdual cycle length E[R C ] = E[R θ,n ], from wth we can easly obtan the mean delays for all the schedulng dscplnes under consderaton. 3.3 Numercal evaluaton We now present some smple numercal experments llustratng the performance of schedulng polces n gated pollng systems. Of course, a wde varety of cases can be studed: dfferent number of queues, choce of servce requrement dstrbutons and ther parameters, choce of setup dstrbutons and ther parameters, etcetera. However, the am of the present secton s to provde some smple llustratve cases whch show the potental of schedulng n pollng systems, so we wll only present a few small examples. For the frst case, we consder a symmetrc two-queue pollng system wth gated servce. Suppose that the servce and setup tmes follow exponental dstrbutons wth means equal to 1. In ths system, we compare three schedulng dscplnes,.e., the optmal polcy SJF, the most mportant one from a practcal perspectve m-class prorty and the standard one FCFS, where we take for the m-class prorty systems the number of prorty classes equal to 2. Recall that n Secton 3.1 we have proven that n the case of two prorty classes the optmal threshold s ndependent of the servce requrement dstrbuton and s gven by E[X ] = 1. It goes wthout sayng that the other polces analyzed n the present paper can be evaluated just as easly, but we have omtted them for reasons of presentaton. Fgure 1 shows the mean delay of an arbtrary customer as a functon of the total load ρ for these three schedulng dscplnes under an exponental servce dstrbuton. The order of the polces s not surprsng, but what s surprsng s the fact that the performance of the two-class prorty dscplne s so close to optmal,.e. SJF, even though we have dstngushed only two prorty classes. Furthermore, t s mportant to observe that, on average, the mean delay for SJF polcy s 15% lower than that of FCFS; a gan system operators can acheve wthout the need of purchasng addtonal resources. For the second case, all nput parameters are taken the same as n the frst case, but now the servce requrement dstrbuton follows a hghly varable Webull dstrbuton wth E[X] = 1 and E[X 2 ] = 2. Under ths more varable dstrbuton E[M] = 1/8, thus the mprovement of SJF over FCFS s even more pronounced. 4 Exhaustve servce dscplne We now move to the case of exhaustve pollng systems. We agan develop a framework that allows smple arguments to be used to obtan results for the mean delay of a varety of schedulng polces. The framework 13

14 wll llustrate that the effectveness of schedulng wthn queues n exhaustve pollng systems s comparable to the effectveness of schedulng n the M/GI/1 model. In fact, we wll see many parallels between the M/GI/1 model and exhaustve pollng systems. Ths s n contrast to the results we just explored for gated pollng systems whch llustrate that schedulng performs very dfferently than n the M/GI/1 model. Agan, we frst derve expressons for the mean delay of a varety of schedulng polces n terms of the mean resdual cycle length n Secton 4.1, and then we analyze the mean resdual cycle length n Secton 4.2. The polces we consder n ths secton nclude FCFS, LCFS, PLCFS, SJF, SRPT, and m-class prorty queues. The case of m-class prorty queues wll be partcularly llustratve of the contrast between gated and exhaustve pollng systems. 4.1 The effect of schedulng on mean delay To begn our study of schedulng n exhaustve pollng systems, we consder the mean delay of a tagged arrval of sze x, j x, to queue. Frst note that because we are consderng an exhaustve pollng system, job j x wll complete durng the cycle nto whch t arrves unlke n the gated case. We recall that, for the exhaustve dscplne, a cycle s defned as the tme between two successve departures of the server from queue. When the tagged job arrves, t wll need to wat at least untl the server returns to queue. Wth probablty E[S] E[C] t arrves durng the setup of queue and must wat E[R S ] before the server returns to queue. Further, wth probablty 1 q,1, the tagged job arrves durng an ntervst perod and must wat E[R θ+1,n 1 ] + E[S ] tme before the server returns to queue. Let us defne E[V ] = E[S ] E[C] E[R S ] + 1 q,1 E[R θ+1,n 1 ] + E[S ], as the expected tme untl the server returns to queue. In addton to watng E[V ] tme before recevng servce and the job sze x tself, dependng on the schedulng polcy, the response tme of j x may nclude tme devoted to servng jobs that arrve after j x begns servce, jobs that arrved before j x, jobs that arrved after j x and before j x receves servce. We denote the contrbuton of the frst pece as c 1 x and the second pece as c 2 W, where W represents the statonary work at queue. To smplfy the computaton of the thrd component, we notce that many common schedulng polces obey the followng property: Property 3 The contrbuton to the delay of j x from each job that arrves after j x and before j x receves servce, denoted c 3 X, s..d. Further, once j x receves servce, no servce s gven to any other jobs that arrved before j x. Many common polces obey Property 3, e.g. FCFS, LCFS, PLCFS, SRPT, and SJF. However, Property 3 does not hold under PS. We wll dscuss ths further n Secton Any polcy whch obeys Property 3 wll have the followng representaton for the mean response tme of a job of sze x: N A V E[D x] = E[c 1 x] + E[V ] + E j=1 = E[c 1 x] + E[V ] 1 + λ E[c 3 X ] 1 λ E[c 3 X ] B c3x j c 3X j + E[B c3x c 2 W ] + E[c 2 W ] 1 λ E[c 3 X ] = E[c 1 x] + E[V ] + E[c 2 W ] 1 λ E[c 3 X ], 25 14

15 where N A Y s the number of arrvals durng tme Y, X j s the job sze of the jth arrval, and B X Y s the length of a busy perod started by Y work where servce requrements of arrvals have..d. szes X. Usng 25, we can now easly obtan formulas for the mean delay of a handful of common schedulng polces under exhaustve pollng models FCFS We start wth the smplest polcy, FCFS. The mean delay of FCFS n exhaustve pollng systems s well-known, but t serves as a useful example of applyng 25. In the case of FCFS, only arrvals before the tagged job wll contrbute to the delay of the tagged job. Thus, E[c 1 x] =, E[c 2 W ] = E[W ] and E[c 3 X ] =, whch gves E[D x] FCFS = E[V ] + E[W ]. To calculate E[W ], we use Lttle s Law to wrte E[D x] FCFS n terms of the mean number n queue, E[L ] FCFS : E[D x] FCFS = E[V ] + ρ E[R X ] + E[L ] FCFS E[X ]. Recallng that E[D x] FCFS = E[V ] + E[W ] gves E[W ] = ρ E[V ] + E[R X ] 1 ρ, and E[D x] FCFS = E[V ] + ρ E[R X ] 1 ρ. In order to vew ths n terms of the mean resdual cycle length, we use the well-known result that: E[D x] FCFS = E[R C ]1 ρ. 26 It follows that E[R C ] = E[V ] + E[W ] 1 ρ = E[V ] + ρ E[R X ] 1 ρ Ths wll be useful for other polces as well snce all work conservng polces have the same mean resdual cycle lengths. The calculaton of E[R C ] s delayed untl Secton LCFS Another smple, common polcy s LCFS, for whch E[c 1 x] =, E[c 2 W ] = ρ E[R X ], and E[c 3 X ] = X and, thus: E[D x] LCFS = E[V ] + ρ E[R X ] 1 ρ = E[R C ]1 ρ = E[D x] FCFS. 28 In fact, LCFS s not alone n havng E[D ] the same as FCFS. As n the M/GI/1 queue, t s easy to see that all non-preemptve polces that do not use sze nformaton have the same mean response tme under exhaustve pollng systems PLCFS Movng beyond non-preemptve polces, let us now consder PLCFS. Obtanng the mean response tme of PLCFS from 25 s smple. Snce all arrvals after the tagged job contrbute to the response tme, we have 15

16 E[c 1 x] = ρ x/1 ρ. Further, E[c 2 W ] = and E[c 3 X ] = X, whch gves: E[D x] PLCFS = ρ x + E[V ] 1 ρ = E[R C ]1 ρ + ρ 1 ρ x E[R X ]. 29 Thus, we can see that E[D x] PLCFS E[D x] FCFS x E[R X ], whch s the same relaton as n the M/GI/1 settng Extendng the framework Though we can handle smple polces usng 25, n order to handle prorty-based polces we need to extend the framework because determnng E[c 2 W ] under such polces can be problematc. To handle such polces we wll vew E[c 2 W ] as the work n a transformed FCFS queue, whch wll allow us to mmc the dervaton n Secton In partcular, we wll see that the followng property holds under SJF, SRPT, and many other prorty-based polces. Property 4 The contrbuton c 2 W can be vewed as the work n a transformed FCFS system where jobs arrve accordng to a Posson process wth rate λ havng..d. szes c 2X and a dfferent maybe dependent stream of jobs may arrve whle the server s dle followng a general maybe non-posson process. The resultng statonary amount of remanng work of the job recevng servce s denoted c 2R X. 2 As a smple example of Property 4, note that under FCFS the transformed system s the same as the orgnal system, whch gves E[c 2X ] = E[X ] and E[c 2R X ] = ρ E[R X ]. We wll see other examples of transformed systems n the next sectons. However, let us frst examne the mplcatons of Property 4. Denote the number of jobs n the queue of the transformed system as L and the delay n the transformed FCFS queue as D FCFS. Recall that the mean delay n a FCFS queue s smply the work n the system plus E[V ], thus E[V ] + E[c 2 W ] = E[D FCFS ]. Gven a polcy obeys Property 4, we can wrte E[D ] FCFS = E[V ] + E[c 2R X ] + E[L ]E[c 2X ], whch gves usng Lttle s law E[D ] FCFS = E[V ] + E[c 2R X ] 1 λ E[c 2 X. ] Combnng the above wth 25 gves E[V ] + E[c E[D x] = E[c 1 x] + 2R X ] 1 λ E[c 2 X ]1 λ E[c 3 X ] = E[R C ] + 1 ρ 2 1 λ E[c 2 X ]1 λ E[c 3 X ] E[c 1 x] ρ E[R X ] E[c 2R X ] 1 λ E[c 2 X ]1 λ E[c 3 X ]. 3 The form of 3 s qute llustratve. The frst term captures the growth as a functon of the mean resdual cycle length and the second term captures the tradeoff between gvng prorty to jobs that arrved 2 Note that ths quantty does not assume that there s a job at the server, and thus s a functon of the load as well as the servce dstrbuton. 16

17 earler versus jobs that arrved later. In addton, 3 llustrates an mportant comparson between the M/GI/1 model and exhaustve pollng systems. Recallng that E[D ] FCFS = E[R C ]1 ρ, we have that E[D x] = E[D x] FCFS 1 ρ 1 λ E[c 2 X ]1 λ E[c 3 X ] ρ E[R X ] E[c + E[c 1 x] 2R X ] 1 λ E[c 2 X. 31 ]1 λ E[c 3 X ] The mportant pont about the above s that the contrbuton functons c [ ] are ndependent of the pollng system. So, the only place the pollng system mpacts 31 s through E[D x] FCFS. Thus, the qualtatve relatonshps between the mean delay of polces that satsfy Propertes 3 and 4 are nsenstve of the underlyng structure of the pollng system and only depend on the fact that queues are served exhaustvely. Note that the quanttatve dfferences between polces wll depend on the structure of the pollng systems though, snce the relatve weghts of the two terms n 31 depend on the magntude of E[D x] FCFS SJF Now, let us consder a sze-based polcy n order to llustrate how to apply 3. SJF s an mportant polcy to consder because t optmzes the mean response tme among all non-preemptve polces. To analyze SJF, consder a transformed FCFS queue where jobs of sze x are only allowed to arrve at the moment they begn to receve servce n the standard SJF queue. Thus, jobs of sze < x stll obey a Posson process but jobs wth sze x do not. The mean response tme for the tagged job s the same n both of these queues. Thus, for SJF, we have that E[c 1 x] =, E[c 2X ] = E[X 1 [X<x]], E[c 2R X ] = ρ E[R X ], and E[c 3 X ] = E[X 1 [X<x]]. Applyng 3 gves: E[D x] SJF = E[V ] + ρ E[R X ] 1 ρ x ρ = E[R C ], 32 1 ρ x where ρ x = λ E[X 1 [X<x]]. Thus, we can see that E[D x] SJF E[D x] FCFS ρ x 1 1 ρ, whch also holds n the M/GI/1 settng. To obtan the overall mean delay of SJF, we can smply ntegrate 32 as follows 2 1 E[D ] SJF ρ = E[R C ] f xdx. 1 ρ x Unfortunately though, no closed-form soluton s avalable for ths ntegral. It s easy to see however that 2 1 ρ 1 ρ x f xdx 1 ρ and thus E[D ] SJF E[D ] FCFS as expected SRPT As n the M/GI/1 settng, SRPT optmzes mean response tme n exhaustve pollng systems. However, the mean delay of SRPT has not been derved n ths settng. But, the analyss of SRPT follows easly from what we have just descrbed for SJF because SRPT also satsfes Property 4. In the case of SRPT, the transformed system that we use has jobs wth orgnal sze < x arrve at the same nstants as normal, but has jobs wth orgnal sze x arrve to the server at the moment they obtan remanng sze x. Thus, they always arrve when the transformed system s dle. Thus, we obtan E[c 2X ] = 17

18 E[X 1 [X<x]] and E[c 2R X ] = ˆρ xe[r mnx,x], where ˆρ x = λ E[mnX, x]. Further, notng that new arrvals contrbute to the response tme of the tagged job only when they are smaller than the remanng sze of the tagged job, we have E[c 3 x] = E[X 1 [X<x]] and E[c 1 x] = x 1 1 ρ 1dt = x ρ t t 1 ρ t dt, where dt 1 ρ t should be nterpreted as the length of a busy perod started by dt work ncludng all new arrvals of sze < t. Applyng 3 then gves: x E[D x] SRPT ρ t = 1 ρ t dt + E[V ] + ˆρ xe[r mnx,x] 1 ρ x x ρ ρ t = E[R C ] + 1 ρ x 1 ρ t dt ρ E[R X ] ˆρ xe[r mnx,x] 1 ρ x As wth SJF, we can obtan the overall mean delay of SRPT by ntegratng 33; however, such ntegraton must be done numercally. But, wthout resortng to numercs, t s already evdent that SRPT can provde sgnfcant reductons n mean delay when compared to FCFS and even SJF m-class prorty queues We now move to m-class prorty queues. We wll lmt our dscusson to non-preemptve prorty queues so that the results can be contrasted wth the results from the gated pollng systems n Secton The mean delay of a class j job, E[D j ], s agan easly derved from 3. Forgong the detals snce they parallel the analyss of SJF, we have that E[c 1 X ] =, E[c 2X ] = E[X k 1 [k j] ], E[c 2X ] = ρ E[R X ], and E[c 3 X ] = E[X k 1 [k<j] ]. Thus, 3 gves: E[D j ] = E[R C ] 1 ρ 2 1 k<j ρk 1 k j ρk, 34 where ρ j = λ j E[X j ]. Notce that the mean delay of SJF can be obtaned by takng the approprate lmts. From 34 we can calculate the overall mean delay usng E[D ] = j λ j E[D j ]. λ As wth the gated case, ths formula s easy to wrte but t hdes the behavor of the mean delay as a functon of the job szes of each class. As n the gated case, t s straghtforward to show that the mean delay wll be mnmzed when prorty s gven to the classes that have small servce requrements. Thus, t agan makes sense to consder threshold based polces. However, unlke the gated case, we cannot derve a closed form expresson for the optmal threshold. Ths s not surprsng snce such an expresson does not exst for the M/GI/1 settng ether. However, n the case of 2 prorty classes, we can determne the optmal threshold and contrast t wth our results for gated pollng systems n Secton m = 2: The optmal threshold In the case of two prorty classes, we can smplfy the expresson for the mean delay. In partcular, lettng t be the threshold used by the polcy, we have E[D ] E[D ] FCFS = λ1 λ 1 ρ 1 ρ 1 + λ2 1 λ 1 ρ 1 = 1 ρ F t. 1 ρ 1 18

19 Dfferentatng ths expresson, we fnd d E[D ] dt E[D ] FCFS = ρ f t1 ρ 1 + λ tf t1 ρ F t, 1 ρ 1 2 whch gves that the mean delay s mnmzed when the threshold satsfes t = 1 λ t sf sds. E[X ] 1 ρ F t Though ths s not explct, t can be solved easly n the case of many common servce dstrbutons. For nstance, f job szes are chosen unformly from the range, a, then the optmal threshold s t = 2 λ 1 1 ρ. Further, f job szes are exponental wth mean 1/µ, the optmal threshold satsfes µ λ e µt µ t 1 = 1. Notce the dfference between these results and what we found for gated pollng systems. In the gated case, the optmal threshold for 2 prorty classes was E[X] regardless of the servce dstrbuton. In contrast, here the optmal threshold s E[X ] for all servce dstrbutons note that the optmal threshold s an ncreasng functon of λ and as λ, t E[X ] and depends greatly on the shape of the dstrbuton Polces that do not obey Propertes 3 and 4 Though we have seen that many common polces obey Propertes 3 and/or 4, there are also polces that do not satsfy them. Foremost, PS does not satsfy ether 3 or 4. Smlarly, all PS-type polces such as Dscrmnatory, Weghted, and Mult-level PS also volate these propertes. Thus, our analytc framework does not apply to these polces. In fact, t s easy to see that these polces are fundamentally more dffcult to analyze n exhaustve pollng systems than they are n the M/GI/1 model and of course more dffcult than n gated pollng systems. To see ths, notce that an analyss of the mean delay of PS n exhaustve pollng systems depends on understandng the transent behavor of the queue length dstrbuton under PS n the M/GI/1 model, whch s known to be a very dffcult problem [1]. Thus, we leave the analyss of PS-type polces as an open queston and note that, unlke polces that satsfy Propertes 3 and/or 4, the behavor of PS wll be very dfferent than t s n the statonary M/GI/1 settng. However, not every polcy that volates Propertes 3 and 4 s dffcult to analyze n exhaustve pollng systems. In partcular, FB volates these propertes but can be analyzed drectly. We do not nclude the analyss smply due to lack of space. 4.2 Resdual cycles In Secton 4.1 we have been able to express the mean delay of a varety of schedulng dscplnes n terms of the unknown mean resdual cycle length E[R C ], = 1, 2,..., N, where ths quantty s ndependent of the specfc schedulng dscplne. To compute these unknowns we agan make use of the MVA for FCFS pollng systems [3], whch yelds, as a spn-off, the mean resdual cycle lengths under all work conservng polces. 19