Yun Bae KIM Jinsoo PARK. Dept. of Systems Management Engineering Sungkyunkwan University Cheon-cheon-dong 300, Jang-an-gu Suwon, KOREA

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1 Proceedngs of the 008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. EW PPROCHES FOR IFERECE OF UOBSERVBLE QUEUES Yun Bae KIM Jnsoo PRK Dept. of Systems Management Engneerng Sungkyunkwan Unversty Cheon-cheon-dong 300, Jang-an-gu Suwon, KORE BSTRCT Many nference methods of queueng systems have been developed on the bass of Larson s QIE(queue nference engne) wth the assumpton of homogeneous Posson arrvals. It nferred the queueng systems wth startng and endng tmes of servce. However, the arrval processes are becomng complex lately, so there are some lmts to apply the method. Our study ntroduces new methods of queue nference whch can fnd the nternal behavors of queueng systems wth only external observatons, arrval and departure tme. Ths study deals wth general GI/G/c queueng systems. (a) FCFC (frst come frst served) (b) LCFS (last come frst served) (c) RSS (random selecton for servce) The accurate nferences were obtaned from FCFS and LCFS systems, and the approxmate solutons from RSS systems. ITRODUCTIO Ths paper suggests new methods that nfer the queueng systems. One mght stll be able to observe, from outsde the system, the arrval and departure tmes of customers. Gven each customer s arrval and departure tmes, f one knows the number of servers, then one can exactly compute both tme n queue and tme n servce of every customer rrelevant of servce dscplne, FCFS or LCFS. For the RSS dscplne, one can approxmately calculate them. If these same assumptons hold except that one does not know the number of servers, one can accurately estmate the number of servers n addton to the queueng and servce tmes. Our only assumpton s that the servce tmes are ndependent. Ths paper s n the tradton of queue nference studes. Much of that work derves from R. C. Larson s (990) semnal paper on the Queue Inference Engne, whch assumed that the only avalable data are the tmes at whch servce starts and stops. Our assumptons are complementary, snce Larson assumes only the nternal operaton of the servce faclty s vsble, whle we assume system arrvals and departures are vsble but both the queue and the nternal operatons of the servce faclty are nvsble. Snce Larson s orgnal work, a number of related papers have appeared whch mprove computng tmes and/or generalze assumptons (Bertmas and Serv 99, Daley and Serv 99, Basawa et al. 996, Bngham and Ptts 999, Jang et al. 00). There are also a number of other papers dealng wth some aspects of queue nference (Masuda 995, Dmtrjevc 996, Pckands and Stne 997, Toyozum 997, Mandelbaum and Zeltyn 998, Jones 999, Coolen 003). To our knowledge, the work presented here s the frst to draw exact nferences about queueng and servce tmes from arrval and departure tmes when the number of servers s unknown. There s some prevous work on estmatng the number of servers n queueng systems. The organzaton of ths paper follows. In secton, we develop nference for three servce dscplnes, FCFS, LCFS and RSS. t frst we ntroduce the method for the case of known number of servers. We expand ths method to the case of unknown number of servers. In secton 3, we present smulatons valdatng the nference methods. In secton 4, we summarze our results. METHODOLOGY. Purpose of Inference The fnal goal s to fnd the watng tme and the servce tme of each customer. From ths result we calculate the mean and varance of watng tme and servce tme. We use only arrval tmes and departure tmes of customers. Let us defne some notatons. : number of customers : arrval tme of customer /08/$ IEEE 80

2 Km and Park D B S Q T : departure tme of customer : servce startng tme of customer : servce tme of customer : watng tme of customer : system sojourn tme of customer c : number of servers Observable terms are, s and D s. If we know the number of servers c, we can fnd the exact B s from the process of number of customers n system. We expand the method to the case of unknown number of servers. To dentfy the servce statng tmes, we use the concept of congeston perod below. In the LCFS systems, more complex concept s requred. In the congeston perod, the servce startng tme of a customer who sees n customers s the frst departure tme that remans n customers. Fgure shows the mechancs of LCFS. If one (grayed customer n Fgure ) sees n customers upon arrval, c customers are beng served and n-c customers are n the queue. s one jons the queue, the queue sze becomes n-c+, whch s (a). There wll be two stuatons follow. One s that an arrval s faster than all servce completons; the other s that a servce completon arses frst. (b) and (c) n the Fgure represent the cases, respectvely. The system eventually reaches (c) whether t takes the state of (b) or not. If the system reaches (c), the number of customers becomes n agan and one starts hs servce at that tme. c+k number of customers start of congeston perod +k servce startng ponts ( from B to B +k ) end of congeston perod c tme Fgure : Congeston perod and servce startng tmes Before congeston perod, the servce startng tme of each customer s equal to arrval tme. fter that servce startng tme s determned by departure tme. departure means a servce s over and a new servce starts. ow, we have to allocate these servce startng tmes to customers.. llocatng the Servce Statng Tmes.. FCFS Systems In the FCFS system, we can drectly determne the servce startng tme wth arrval order. If we denote D () be the - th order statstc of D s, the servce startng tmes are determned by equaton () n the congeston perod B =. () D( c) Obvously, n the non-congeston perod, servce startng tmes are same wth arrval tmes... LCFS Systems Fgure : Servce startng tme of LSFS customer To get the mathematcal form of ths method, let us defne the next. : number of customers that sees at hs arrval D : number of customers that sees at hs departure D ( n, t) : departure tme when D = n after tme t ow, we can get equaton () B ( < c) = D (, ) ( o / w). () The above term of equaton () s the case of noncongeston perod and the below s congeston perod...3 RSS Systems In the RSS systems, t s mpossble to allocate the exact servce startng tmes. We can only observe the characterstc of the queueng systems; probably customers leave the system n the sequental order of ther servce begun. From ths fact, we can estmate the order of servce approxmately. The servce order may follow wth departure order. Therefore we can reconstruct the smulaton wth arrval and departure tmes. Fgure 3 shows the smulaton reconstructon algorthm. The acton of Get a customer from the queue who has the smallest departure tme pont s processng n event Departure. From ths algorthm, we can approxmately allocate the servce startng tmes. We can exactly calculate the mean queueng tme and mean servce tme from ths result. Besdes, we can calculate approxmate varance of them. If the number of servers s smaller, the varance s more close to real one. If the nherent varance of servce tme s small, 8

3 Km and Park then we get the same result. Other than two cases of mentoned we need to develop a dfferent method. Start Read arrval and departure ponts, and D Insert all rrval events to event lst wth the customer Event rrval Is the event lst empty? o Get the frst event and the customer from the event lst Is there a server avalable? Yes Set the customer s servce start tme to the smulaton clock Insert a departure event to event lst Insert the customer to the queue Create customers who have and D smulaton clock = tme pont of the event Process the event o End Event Departure Yes Is the queue empty? Yes o Get a customer from the queue who has the smallest departure tme pont Set the customer s servce start tme to the smulaton clock Insert a departure event to event lst Fgure 3: Smulaton reconstructon algorthm..4 Expanded Method for RSS System If the number of servers s large or the real varance of servce tmes s large, the estmated varance of servce tmes has large error. In these cases, we use the ndependence of queueng tme and servce tme to estmate the varance. Theoretcally, f these tmes are ndependent, the covarance of them s zero. We are expectng a successful allocaton of the servce startng tmes that result n zero covarance. s a result, we construct an optmzaton problem wth covarance of them as objectve functon. To make the mathematcal form of optmzaton problem, we denote C (, r) as the optonal r-th departure tme n the congeston perod that contans customer. ow we have the optmzaton model. = mnmze Q ˆ Qˆ s.t. ˆ B = ( < c) Bˆ = C(, r) ( o / w) B ˆ < D Q ˆ = B ˆ Qˆ and Ŝ mply the sample means of Qˆ and Ŝ. that of secton....3 The Cases of Unknown umber of Servers s In the case of unknown number of servers, we cannot allocate the servce startng tmes drectly. The servce startng tmes depend on the number of servers. For any randomly selected value of c, we can get only estmated servce startng tmes. fter we have estmated servce startng tmes, we can calculate the servce tmes. Usng propertes of servce tmes, we can fnd the real number of servers. ctually, f the number of servers s estmated wrong, the estmated servce tmes can be represented as = S + Δ, where Ŝ s the estmated servce tme and S s the real servce tme. If we know the real number of servers c, we can get the real servce tme S ; but f we estmate the number of servers wrong, we get the estmated servce tme as Ŝ. We can show that S and Δ are ndependent. The proof of ndependency s provded n Park (007). For any cases, the optmzaton problem successfully fnds the real number of servers, c, whch mnmzes the sample varance of servce tmes..3. FCFS systems In the FCFS systems, f we unte both case of congeston and non-congeston perod, we get next term. {, D } B = max (3) ( c ) From equaton (3) we can get the next optmzaton form. = S mnmze ( ) s.t. Bˆ =, cˆ Bˆ = max{, D( cˆ) }, > S ˆ > 0 cˆ 8

4 Km and Park Hat notatons mean the estmated values and barred notatons mply ther sample means. Grd search over the ntegers provdes the estmated number of servers..3. LCFS systems We can construct the optmzaton model wth same objectve functon and equaton (). = S mnmze ( ) s.t..3.3 RSS Systems ˆ ( < cˆ) B = D (, ) ( o / w) S ˆ > 0 In the RSS System, we cannot allocate the servce startng tmes exactly. So we use the result from smulaton reconstructon for constrant term. Let us denote SR c ˆ ( ) as servce startng tme of customer from smulaton reconstructon under number of servers s ĉ. ow we get = S mnmze ( ) s.t. Bˆ SR ˆ ( ) = S ˆ > 0. c Ths problem provdes the real number of servers too. We wll confrm ths fact at Secton 3. 3 SIMULTIOS D IFERECES 3. Smulaton Models We smulated general GI/G/c systems whch have general nter-arrval tme dstrbuton, general servce tme dstrbuton and the number of servers s c. We used determnstc, exponental and normal dstrbuton as nter-arrval and servce tme dstrbuton, and tred, 3 and 7 as the number of servers. We also vared the server utlzaton 0.8, 0.9 and.0. Three dfferent servce dscplnes (FCFS, LCFS, and RSS) are tested. Therefore the number of combnatons we smulated s 43(3 arrval types 3 servce types 3 number of servers 3 server utlzatons 3 servce dscplnes). Table shows the mean servce tme as system parameters. We fx the arrval rate to and change the mean servce tme by server utlzaton ρ. Table : Mean servce tmes c mean servce tme ρ=0.8 ρ=0.9 ρ= We collected,000 arrval and departure tmes from each smulaton to nfer the system of nterest. We also collected,000 servce startng tmes from each smulaton for true values. We only used the arrval and departure tmes to fnd servce startng tmes. 3. Inference Results We nferred the servce startng tmes for the stuaton of known and unknown number of servers. We calculated mean queueng tme, mean servce tme, varance of queueng tmes and varance of servce tmes from nferences. 3.. Case of Known umber of Servers In ths case, we get exact soluton for FCFS and LCFS systems. So we omt the results of these servce dscplnes. However, n the RSS systems, we get approxmate solutons. Table shows some results of RSS systems. The optmzaton model n secton..4 provded the results. Var(Q) means true varance of queueng tmes and Var (Q ˆ) s estmated varance of queueng tmes. R.E. means relatve error and the values are calculated down to four decmal places. In the left most column of Table, we use a normal dstrbuton for GI(or G). ll average values are same to actual values. Table : Varance of queueng tmes for the RSS systems System ρ Var(Q) Var (Qˆ ) R.E M/M/ D/M/ GI/M/

5 Km and Park The small gaps between Var(Q) and Var (Qˆ ) are due to the statstcal nose comng from samplng. The last column of the Table (R.E.) can be gnored f we repeat the replcatons n smulaton experments. 3.. Case of Unknown umber of Servers In ths secton, we deal wth the case of unknown number of servers. We want to show the un-modalty of varance of servce tmes. That s, we want to show that the optmzaton models n secton.3 work correctly. Fgure 4 s the results of some FCFS systems. In these systems, the number of servers s 7 and server utlzaton s 0.8. The rest of the systems whch have hgher utlzaton, another number of servers, or dfferent arrval and servce dstrbutons show smlar shape on ths plot. The horzontal axs means the number of servers and the vertcal axs mples the varance of servce tmes accordng to the number of servers. The uncharted area represents nfeasble soluton area. objectve (varance of servce tmes) number of servers M/G D/G GI/G objectve (varance of servce tmes) objectve (varance of servce tmes) number of servers Fgure 5: Example results of LCFS systems number of servers Fgure 6: Example results of RSS systems 4 SUMMRY M/G D/G GI/G M/G D/G GI/G Fgure 4: Example results of FCFS systems See Fgure 5 to obtan the case of LCFS dscplne. ll of the system condton s same to FCFS prevous results (Fgure 4). Wth same condton, Fgure 6 shows that case of RSS works. The fact that we can realze s the convergence of objectve, the varance of servce tmes. If we overestmate the number of servers larger and larger, the nference understands that all the customers start ther servce as they arrve. Therefore, the servce tmes converge to system sojourn tmes. Ths fact provdes the upper lmt of the number of servers. eedless to say, the rest of those systems show smlar results. We nferred the queueng systems usng arrval and departure ponts. The only assumpton of our research s ndependence of servce tmes. We fnd nternal behavors of FCFS, LCFS and RSS systems. t frst, we proposed nference methods when the number of servers s known. nd we expended to the case of unknown number of servers. We nferred the accurate soluton from FCFS and LCFS systems, and approxmate soluton from RSS systems. REFERECES Basawa, I., U. Bhat, and R. Lund Maxmum lkelhood estmaton for sngle server queues from watng tme data. Queueng Systems 4: Bertmas, D. and L. Serv. 99. Deducng queueng from transactonal data: The queue nference engne, revsted. Operatons Research 40:

6 Bngham,. and S. Ptts on-parametrc estmaton for the M/G/nfnty queue. nnals of the Insttute of Statstcal Mathematcs 5: Coolen, F., P. Coolen-Schrjner nonparametrc predctve method for queues. European Journal of Operatonal Research 45, Daley, D. and L. Serv. 99. Explotng Markov-chans to nfer queue length from transactonal data. Journal of ppled Probablty 9, Dmtrjevc, D Inferrng most lkely queue length from transactonal data. Operatons Research Letters 9: Jang, J., J. Suh and C. Lu. 00. new procedure to estmate watng tme n GI/G/ systems by server observaton. Computers and Operatons Research 8: Jones, L Inferrng balkng behavor from transactonal data. Operatons Research 47: Larson, R The queue nference engne: Deducng queue statstcs from transactonal data Management Scence 36: Mandelbaum,. and S. Zeltyn Estmatng characterstcs of queueng networks usng transactonal data. Queueng Systems 9: Masuda, Y Explotng partal nformaton n queueng-systems Operatons Research 43: Park, J Study of Inference Methods for Queueng Systems wth rrval and Departure Tme Data. unpublshed Ph. D. thess. Sungkyunkwan Unversty, Suwon, Korea. Pckands, J. and R. Stne Estmaton for the M/G/nfnte queue wth ncomplete nformaton. Bometrka 84: Toyozum, H Sengupta s nvarant relatonshp and ts applcaton to watng tme nference. Journal of ppled Probablty 34: UTHOR BIOGRPHIES YU BE KIM s a Professor at the Sungkyunkwan unversty, Suwon, Korea. He receved master degree from Unversty of Florda and Ph.D degree from Rensselaer Polytechnc Insttute. Hs current research nterests are smulaton methodology, agent based smulaton and smulaton based acquston. JISOO PRK holds a postdoctoral poston n the Department of Systems Management Engneerng at the Sungkyunkwan unversty. He receved a master s degree n ndustral engneerng and a Ph. D. n the same subject from the Sungkyunkwan unversty, Seoul, Korea. Hs current research nterests are n queue nference, analyss of queueng systems and agent based modelng and smulaton. Km and Park 85