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1 Saisics and Probabiliy Leers 82 (2012) Conens liss available a SciVerse ScienceDirec Saisics and Probabiliy Leers journal hoepage: Fiing birh-and-deah queueing odels o daa Ward Whi Deparen of Indusrial Engineering and Operaions Research, Colubia Universiy, New Yor, NY , USA a r i c l e i n f o a b s r a c Aricle hisory: Received 6 January 2012 Received in revised for 11 February 2012 Acceped 12 February 2012 Available online 17 February 2012 Keywords: Birh-and-deah processes Epirical birh-and-deah processes Fiing birh-and-deah processes o daa Conservaion laws Operaional analysis Given easureens of he nuber of cusoers in a queueing syse over a finie ie inerval, i is naural o ry o fi a saionary birh-and-deah process odel, because i is rearably racable, even when he birh and deah raes depend on he sae in an arbirary way. Naural esiaors of he birh (deah) rae in each sae are he observed nuber of ransiions up (down) fro ha sae divided by he oal ie spen in ha sae. I is eping o validae he odel by coparing he seady-sae disribuion of he odel based on hose esiaed raes o he epirical seady-sae disribuion recording he proporion of ie spen in each sae. However, i is inappropriae o draw srong conclusions fro a close fi o he sae daa, because hese wo disribuions are necessarily iniaely relaed, even if he odel assupions are no nearly saisfied. We elaborae by (i) esablishing sochasic coparisons beween hese wo fied disribuions using lielihood-raio sochasic ordering and (ii) quanifying heir difference Elsevier B.V. All righs reserved. 1. Inroducion This paper is oivaed by effors o fi sochasic queueing odels o daa fro syse easureens in call ceners and hospials, for exaple, as colleced in Arony e al. (2011) and Brown e al. (2005). Since he nuber of cusoers in a queueing syse ypically increases or decreases by one a each ransiion, i is naural o consider fiing a saionary birh-and-deah (BD) process o he observed segen of he saple pah. Clearly, here also are any oher applicaions where a BD odel igh be fied o daa. Since daa over a finie ie inerval will invariably involve only finiely any ransiions, i is naural o fi a finie-sae saionary BD process o he daa. Thus he ypical sae space for he BD process is {0, 1,..., }. Wih hese + 1 saes, here are 2 paraeers, he birh raes λ j in saes 0 j 1, and he deah raes µ j in saes 1 j. If he odel fi is genuinely good, hen he BD odel can be very helpful because i is rearably racable; see Secion 2. There already is quie an exensive saisical heory for esiaing he paraeers of queueing odels and saionary BD processes; see, for exaple, Bha e al. (1997), Billingsley (1961), Israel e al. (2001), Keiding (1975), Ross e al. (2007), and Wolff (1965). We will be considering he naural esiaor of he birh and deah raes considered in Wolff (1965); i.e., he birh (deah) rae in each sae is esiaed by he observed nuber of ransiions up (down) fro ha sae divided by he oal ie spen in ha sae. I is eping o validae he BD odel fi in ha way by coparing he seady-sae disribuion of he odel based on hose esiaed raes o he epirical seady-sae disribuion recording he proporion of ie spen in each sae. However, i is inappropriae o draw srong posiive conclusions fro a close fi o he sae daa, because hese wo disribuions are necessarily iniaely relaed, wihou any odel assupions being ade. For exaple, he acual syse could be highly non-saionary. Neverheless, under inor regulariy condiions, hese wo fied disribuions are asypoically idenical as he saple size increases; his is a special case of ore general epirical global balance equaions E-ail address: ww2040@colubia.edu /$ see fron aer 2012 Elsevier B.V. All righs reserved. doi: /j.spl

2 W. Whi / Saisics and Probabiliy Leers 82 (2012) in he saple-pah analysis of queues in Ch. 4 of El-Taha and Sidha Jr. (1999). In he queueing lieraure, he close relaion beween hese wo epirical fied disribuions sees o go bac o he operaional analysis of Buzen (1976), Buzen (1978), Buzen and Denning (1980), and Denning and Buzen (1978); see Secions 4.6 and 4.7 of El-Taha and Sidha Jr. (1999) for discussion and references. Given ha he wo epirical fied disribuions are necessarily nearly idenical, wih virually no condiions a all, i is eviden ha he relaion has no direc iplicaion abou eiher (i) wha is an appropriae sochasic odel, or (ii) he syse perforance a oher ies. These iporan issues require furher properies ha us be checed epirically. For exaple, in jusifying he applied relevance of operaional analysis, in Secion 2 of Denning and Buzen (1978) he auhors discuss his issue and enion a variey of invariance assupions o jusify predicions a oher ies. Oherwise, his relaion serves only as a (useful) consisency chec on he daa processing. In his paper, afer quicly reviewing basic BD heory in Secion 2, and carefully specifying he fiing procedure in Secion 3, we elaborae on he relaion beween hese wo fied disribuions based on daa fro a finie ie inerval in Secion 4 by (i) esablishing sochasic coparisons beween hese wo fied disribuions using lielihood-raio sochasic ordering and (ii) quanifying heir difference. Aferwards, in Secion 5, we briefly discuss addiional daa analysis seps o fully validae he fied BD process. In Secion 6, we discuss he associaed saisical proble of esiaing confidence inervals for he birh and deah raes. Finally, in Secion 7, we discuss reaining probles. 2. Review of birh-and-deah process heory Le X {X() : 0} be a (finie-sae saionary) BD process on he sae space {0, 1,..., } wih sricly posiive birh raes λ i, 1 i 1, and deah raes µ i, 1 i. Tha eans ha X is a reversible irreducible coninuous ie Marov chain (CTMC) wih all ransiions up one or down one; for exaple, see Keilson (1979), Ch. 5 of Ross (1996), and Ch. 6 of Ross (2010). Thus, he characerizing ( + 1) ( + 1) rae arix (infiniesial generaor) Q of he CTMC has eleens Q i,i+1 λ i, 0 i 1, and Q i,i 1 µ i, 1 i, wih all oher off-diagonal eleens 0 and all row sus 0. Equivalenly, he successive holding ies in sae i are independen and idenically disribued (i.i.d.) exponenial rando variables wih ean 1/(λ i + µ i ), and he probabiliy of an upward ransiion a each ransiion ie fro sae i is λ i /(λ i + µ i ), independen of he holding ie and all prior hisory. An (irreducible finie-sae) BD process X has a unique liiing seady-sae probabiliy disribuion α, i.e., α j li P(X() = j X(0) = i) for all i, which is also he unique saionary disribuion, i.e., α j = α i P(X() = j X(0) = i) for all > 0. (2) i=0 Because of he reversibiliy, he saionary disribuion α can be expressed as he unique soluion o he local balance equaions α i λ i = α i+1 µ i+1, 0 i 1, such ha i=0 α i = 1. The local balance equaions (3) can be solved recursively o give (1) (3) α i = r i, r j j=0 where r 0 1 and (4) r i λ 0 λ i 1 µ 1 µ i, 1 i ; for exaple, see Secion 6.3 of Ross (2010). I is also no difficul o copue ransien perforance easures of finie-sae BD processes, as shown by Keilson (1979). The BD odels considered in os applicaions have special srucure. For exaple, any queueing applicaions involve he classical M/M/s/rqueueing odel, which has a Poisson arrival process wih rae λ (he firs M), exponenial service ies wih ean µ 1 (he second M), s hoogeneous servers woring in parallel and r exra waiing spaces. Thus he M/M/s/r odel has consan birh raes λ i λ, 0 i s + r 1, and siple deah raes, µ i in{i, s}, 1 i s + r. However, i ofen is of ineres o consider ore general BD queueing odels. For exaple, when here is baling (arrivals refusing o join when he line is oo long), see Whi (1999), for exaple, he arrival rae ay be decreasing; when here is cusoer abandonen if hey have no progressed rapidly enough, he deah rae ay be increasing ore rapidly han above; when cusoers have non-exponenial paience disribuions, i can be effecive o approxiae by a BD odel wih a ore general sae-dependen deah rae; see Whi (2005). For coplex real applicaions, i is naural o le he daa dicae wha he relevan birh and deah raes are. For exaple, he nuber of woring servers ay no be a fixed deerinisic quaniy, bu neverheless a BD odel could be useful. (5)

3 1000 W. Whi / Saisics and Probabiliy Leers 82 (2012) Fiing he BD odel o queueing syse daa Consider a queueing syse in which arrivals and deparures occur one a a ie. Le X(s) be he nuber of cusoers in he syse a ie s. We now consider fiing a BD odel o daa colleced over an inerval [0, ]. Le λ i () and µ i () be naural direc esiaes of he birh raes and deah raes based on saple averages over he ie inerval [0, ]. Siilarly, le ᾱ i () be naural direc esiaes of he saionary disribuion based on saple averages over he ie inerval [0, ]. In paricular, le A i () be he nuber of arrivals during he inerval [0, ] when he syse is in sae i; le D i () be he nuber of deparures during he inerval [0, ] when he syse is in sae i; and le T i () be he oal ie during he inerval [0, ] in which he syse is in sae i; i.e., T i () 0 1 {X(s)=i} ds, 0, where 1 A is he indicaor funcion of he se A, equal o 1 on A and equal o 0 oherwise. Then le λ i () A i() T i (), µ i() D i() T i () and ᾱ i () T i(), 0. (7) In general, his esiaion procedure need no produce an irreducible BD process, because here can be iniial and final ransien saes. However, here is a larges subse of saes ha is he sae space of an irreducible BD process, wih all oher saes being ransien. Necessarily, λ i () > 0 for a 1 i a 2 wih λ i () = 0 oherwise, and µ i () > 0 for d 1 i d 2 wih λ i () = 0 oherwise, for soe consans a 1, a 2, d 1, and d 2. There are hree possibiliies for hese inervals of posiive raes : (i) a 1 = d 1 1 and a 2 = d 2 1, (ii) a 1 d 1 1 and a 2 d 2 1, wih a leas one of hese wo inequaliies being sric, or (iii) a 1 d 1 1 and a 2 d 2 1, wih a leas one of hese wo inequaliies being sric. In case (i), he process is irreducible and he sae space is {a 1,..., a 2 +1} = {d 1 1,..., d 2 }; in case (ii), here are ransien saes, so he BD process is reducible, wih he sae space of he irreducible BD process being {d 1 1,..., d 2 }, while all oher saes are ransien, being visied by iniial or final birhs, bu never by deahs; in case (iii), again here are ransien saes, so he BD process is reducible, wih he sae space of he irreducible BD process being {a 1,..., a 2 + 1}, while all oher saes are ransien, being visied by iniial or final deahs, bu never by birhs. In all hree cases, here is a unique saionary disribuion, which places 0 probabiliy on each ransien sae, if here are any. For sipliciy, henceforh we assue ha he irreducible case (i) prevails wih a 1 = 0 < a = d 2 =. Fro Secion 2, we see ha, under he siplifying assupion of irreducibiliy, his esiaed BD process has he unique saionary probabiliy disribuion ᾱ e i () r i(), 0 i, r j () j=1 where r 0 () 1 and r i () λ 0 () λ i 1 (), 0 < i. (9) µ 1 () µ i () Equivalenly, ᾱ e () is he unique probabiliy vecor saisfying he local balance equaion associaed wih he esiaed birh and deah raes; i.e., ᾱ e i () λ i () = ᾱ e i+1 () µ i+1() for all i, 0 i <. (10) (6) (8) 4. The relaion beween he wo fied saionary disribuions We now esablish ore precise connecions beween he disribuion ᾱ e () based on forula (8) using he esiaed birh and deah raes and he direc epirical disribuion ᾱ() in (7) for finie values of. We ephasize ha hese wo probabiliy disribuions will no have he sae suppor, and hus of course hey could no be equal, if he irreducibiliy condiion above is no saisfied. We hus assue irreducibiliy below. In pracice, he irreducibiliy can always be achieved by reoving he iniial and/or final ransien fro he saple pah if eiher (or boh) is (are) here. For he sochasic coparison, we use he noion of lielihood raio sochasic ordering (LR) for probabiliy ass funcions (pfs); for exaple, see Secion 9.4 of Ross (1996). Le X 1 and X 2 be wo rando variables, each aing values in he nonnegaive inegers, wih pfs p i () P(X i = ). We say ha X 1 is sochasically less han or equal o X 2 in he LR ordering, and wrie X 1 LR X 2 or p 1 LR p 2, if p 1 ( + 1) p 1 () p 2( + 1) p 2 () for all inegers, (11)

4 W. Whi / Saisics and Probabiliy Leers 82 (2012) where a leas one is posiive. I is well nown ha LR ordering iplies ordinary sochasic order. We say ha X 1 is sochasically less han or equal o X 2, and wrie X 1 s X 2 or p 1 s p 2, if p 1 (j) p 2 (j) for all. j= j= Equivalenly, X 1 s X 2 if E[f (X 1 )] E[f (X 2 )] for all non-negaive non-decreasing real-valued funcions f on R. Theore 1 (Sochasic Coparison of he Two Saionary Disribuions). Consider a saple pah segen over an inerval [0, ] of a sochasic process wih only finiely any ransiions, all of which are ±1. Suppose ha a BD process is fied o his daa, as in (7), and suppose ha i is irreducible wih sae space {0,..., }. For i 0 X(0) and i X(), here are hree uually exclusive and exhausive alernaives: If i 0 = i, if i 0 < i, if i 0 > i, hen ᾱ() = ᾱ e (); hen ᾱ() LR ᾱ e (); hen ᾱ() LR ᾱ e (). Moreover, he difference, i () ᾱ e i () ᾱ i(), can be quanified by solving he finie recursion (for 0 i ) i () λ i () = i+1 () µ i+1 () + ēi() wih where ē i () 1 {i0 i>i } 1 {i0 i<i }, so ha ē i () = 0 for all bu i i 0 values of i. Proof. By he definiions in (7), Ti () Ai () ᾱ i () λ i () = T i () and Ti+1 () ᾱ i+1 () µ i+1 () = = A i() Di+1 () T i+1 () (12) (13) i () = 0, (14) i=0 (15) = D i+1(). (16) However, since all birhs in sae i ae he syse o sae i + 1, while all deahs in sae i + 1 ae he syse o sae i, A i () D i+1 () 1 for all i. We can say ore if we loo a he iniial sae i 0 and he ending sae i. Firs, if i 0 = i, hen A i () = D i+1 () for all i. Cobining his wih (15) and (16), we see ha ᾱ() saisfies he local balance equaion (10). Hence, in his case, wih i 0 = i, we us have ᾱ() = α e (), as claied in (13). Nex, if i 0 < i, hen A i () = D i+1 () + 1 for i 0 i < i ; else A i () = D i+1 (). (18) As a consequence, insead of he local balance equaions in (10), in his case, he probabiliy vecor ᾱ() saisfies he associaed syse of inequaliies (17) ᾱ i () λ i () ᾱ i+1 () µ i+1 () for all i. (19) However, we can iediaely rewrie (19) as ᾱ i+1 () ᾱ i () λ i () µ i+1 () = r i+1() r i () so ha ᾱ() LR α e (), as claied in (13). By siilar reasoning, if i 0 > i, hen = αe () i+1, α e (20) i () A i () = D i+1 () 1 for i 0 i < i, else A i () = D i+1 (), (21) so ha ᾱ() LR α e. In general, we have (10) and ᾱ i () λ i () = ᾱ i+1 () µ i+1 () + e i / for all i. (22) Subracing hese equaions direcly yields (14).

5 1002 W. Whi / Saisics and Probabiliy Leers 82 (2012) Fro he hree condiions in (13), we see ha he wo disribuions are always idenical wih appropriae iniial and erinal condiions. The inor differences ore generally are only due o edge effecs, jus as in he relaed conservaion law L = λw. If we add addiional regulariy condiions, hen we can also show ha he difference due o hese edge effecs is asypoically negligible as. We can also bound he rae of convergence. For his purpose, we consider he esiaion as a funcion of he inerval endpoin. See Chaper 4 of El-Taha and Sidha Jr. (1999) for relaed asypoic resuls. We avoid probles caused by dividing by sall by resricing he seing o 1. We deduce he following bound on he rae of convergence fro Theore 1; we oi he proof. Corollary 4.1 (Bound on Rae of Convergence). If () < <, 0 < a 1 λ i () a 2 < and 0 < b 1 µ i () b 2 < for all 1, hen i () K/ for all 1, where K is a funcion of, a 1, a 2, b 1, and b 2. (23) 5. Validaing he saionary BD odel 5.1. Tie saionariy In any queueing applicaions, such as call ceners and hospials, here ends o be syseaic variaion in arrival raes and perforance easures over ie. Thus, i is iporan o chec ha a saionary odel is really appropriae for he ie inerval under consideraion. This is a vial sep if any saionary sochasic odel is used. Given daa over a ie inerval [0, ], saionariy can be checed by considering subinervals [ 1, 2 ] wih 0 1 < 2. Firs, le A i ( 1, 2 ) be he nuber of arrivals during he inerval [ 1, 2 ] when he syse is in sae i, and siilarly for he oher processes. Then, as in (7), le λ i ( 1, 2 ) A i ( 1, 2 )/T i ( 1, 2 ). Le f be an arbirary real-valued funcion on he sae space. We can exend he saisics λ i (), µ i (), ᾱ i () and ᾱ e() i o associaed saisics as funcions of he riple (f, 1, 2 ), for exaple, by leing λ f ( 1, 2 ) f (i) λ i ( 1, 2 ). (24) i=0 I is good o chec ha, for various funcions f, he saisics λ f ( 1, 2 ), ᾱ f ( 1, 2 ) and α e f ( 1, 2 ) are approxiaely consan, independen of he subinerval [ 1, 2 ]. For exaple, if f (i) = 1 for all i, hen λ f is he esiaed oal arrival rae; if f (i) = i, hen ᾱ f is he h oen of he esiaed seady-sae disribuion. A siple approach is o choose a few represenaive funcions f, fix 1 = 0, and plo as a funcion of 2, 0 2. Siilarly, if predicion is coneplaed for anoher ie, hen evidence should be sough ha hese esiaed raes are sill relevan. In he spiri of Secion 2 of Denning and Buzen (1978), hose are concree invariance properies o chec The BD odel assupions Given ha he syse is consisen wih a saionary sochasic process for which all ransiions are ±1, i reains o chec he Marov propery. A anageable way o chec ha is o use he fac ha, wihin ha seing, a BD process can be characerized by having he ies spen in each sae and he ransiion a he ransiion epoch be rando variables ha are uually independen and independen of he syse hisory. For a pracical es, le X (i) be he ie spen in sae i afer arriving in sae i fro elsewhere, le J (i) be +1 if he ransiion a he end of ha inerval is up one and le J (i) = 1 oherwise, and le Y (i) be he lengh of ie spen away fro sae i iediaely afer he inerval X (i). The sequence of vecors {(X (i), J(i), Y (i) (i) ) : 1} should be i.i.d. rando vecors wih X being independen of boh J (i) and Y (i) ; for exaple, P(X (i) >, J i (i) = 1) = P(X > )P(J i = 1) = e (λ i+µ i ) λi. (25) λ i + µ i Wih he daa, we should chec ha he epirical disribuion (hisogra) of X (i) is approxiaely exponenial; we should chec ha he covariances beween X (i) and X (i) +1, beween J(i) and J (i) (i) +1, and beween X + Y (i) and X (i) + +1 Y (i) +1 are suiably negligible. We ay also wan o copare esiaes of he asypoic variance for funcions of a BD process o he exac values for a BD process, for which Proposiion 1 of Whi (1992) can be used for coparison. 6. Esiaing confidence inervals Assuing ha he validaion seps have been carried ou, such ha we hin a BD odel is appropriae, i is naural regard he saple averages λ i (), µ i () and ᾱ i () in (7) as finie-saple esiaes of he rue, bu unnown, odel paraeers λ i

6 W. Whi / Saisics and Probabiliy Leers 82 (2012) and µ i of a saionary BD odel and he associaed seady-sae probabiliies α i. Following sandard saisical pracice, i is appropriae o evaluae he effeciveness of hese esiaes by also esiaing he saple variance and confidence inervals. Given ha he daa coe fro a single observed saple pah, i is naural o use he ehod of bach eans as in siulaion oupu analysis in discree-even sochasic siulaion; for exaple, see Secion 3.3 of Braley e al. (1987). Wih ha in ind, we sugges wo alernaive esiaion procedures. The firs esiaion procedure is based on he observaion ha he ehod of bach eans applies ore naurally o siple ie averages, as for ᾱ i () in (7). We can wor direcly wih ie averages if we esiae γ i α i λ i and δ i α i µ i by he ie averages γ i () 1 A i () and δ i () 1 D i (). We hen obain he associaed esiaors λ γ,i () γ i ()/ᾱ i () and µ δ,i () δ i ()/ᾱ i (). These alernaive esiaors are also consisen, bu dividing by he sall unreliable values ᾱ i () can lead o poor efficiency (high variance). The second procedure is based on looing only a he oal ie spen in sae i. Tha can be done by concaenaing he variables X (i) in Secion 5.2. We also eep rac of he ransiions a each inerval end poin. Le U (i) and V (i) be he inervals beween he ( 1)h and h birhs and deahs, respecively. By he Poisson spliing heore, Proposiion of Ross (1996), for a BD process, he associaed wo couning processes are independen Poisson processes. Thus, for a BD process, he sequences {U (i) : 1} and {V (i) : 1} are uually independen sequences of i.i.d. exponenial rando variables wih eans λ 1 i and µ 1 i bach eans. To illusrae, suppose ha we have n = observaions of U (i). We use his represenaion, bu, o avoid assuing ha he BD odel is correc, we wor wih. (Saring fro a fixed inerval [0, ], he acual nuber n is rando, which inroduces bias, bu we shall no consider ha issue.) We hen for he bach eans Ū (i) n, 1 U (i) j, Ū (i,b) 1 j=( 1) and he saple variance σ 2 U (i,b) s 2 U (i,b) ( 1) 1 =1 =1 Ū (i), n 1 j=1 U (i) j. (26) (Ū (i), Ū(i,b) )2. (27) In grea generaliy, even when he process is no a BD process, he bach eans Ū (i), should be approxiaely i.i.d. noral rando variables. If he variance σ 2 U (i,b) were nown, hen he rando variable Ū(i,b), has a noral disribuion wih variances σ 2 U (i,b) /. Since he variance is in fac unnown, we ac as if Ū b has he Suden- disribuion wih 1 degrees of freedo. Thus, for = 20, a wo-sided 95% confidence inerval esiae for E[U (i) ] based on he ehod of bach eans applied o he n = observaions is Ū (i,b) ± 2.09 σ 2 U (i,b) /19 = Ū (i,b) 20 ± 0.48 σ (i,b) U. (28) 20 Given an esiaed wo-sided 95% confidence inerval [a 1, a 2 ] for E[U (i) ] as in (6), he inerval [1/a 2, 1/a 1 ] provides an associaed esiaed wo-sided 95% confidence inerval for λ i = 1/E[U (i) ]. 7. Reaining issues Even hough i is now coon o have large daa ses, here ay no be enough daa o fi a general odel. The available daa ofen us be reduced o obain inervals in which he syse can be regarded as approxiaely saionary. Even for a syse ha is approxiaely saionary over a long inerval, having 2 paraeers for + 1 saes is liely o produce a odel wih oo any paraeers. If he acual sae space is large, hen he daa for any saes will be inadequae o obain reliable rae esiaes. The saisical analysis discussed in Secion 6 should provide guidance. I ofen will be iporan o cobine daa over uliple subinervals and fi a ore resricive odel for which he birh and deah raes have srucure. The references offer guidance; see, for exaple, Keiding (1975); Ross e al. (2007). I is naural o consider piecewise linear funcions, in he spiri of he piecewise-linear esiaes of he rae of a non-hoogeneous Poisson process in Massey e al. (1996). Acnowledgens This research was suppored by NSF gran CMMI The auhor hans a referee for a careful reading of he paper. References Arony, M., Israeli, S., Mandelbau, A., Maror, Y., Tseylin, Y., Yo-Tov, G., Paien flow in hospials: a daa-based queueing-science perspecive. Woring Paper, NYU. Bha, U.N., Miller, G.K., Subba Rao, S., Saisical analysis of queueing syses. In: Dshalalow, J.H. (Ed.), Froniers in Queueing Theory. CRC Press, Boca Raon, FL, pp (Chaper 13). Billingsley, P., Saisical Inferences for Marov processes. Universiy of Chicago Press, Chicago. Braley, P., Fox, B.L., Schrage, L.E., A Guide o Siulaion, second ed. Springer, New Yor.

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