SMOOTH FLEXIBLE MODELS OF NONHOMOGENEOUS POISSON PROCESSES USING ONE OR MORE PROCESS REALIZATIONS. for all t (0, S]

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1 Poceedngs of the 8 Wnte ulaton Confeence J Mason R R Hll L Mönch O Rose T Jeffeson J W Fowle eds MOOTH FLEXIBLE MOEL OF NONHOMOGENEOU POION PROCEE UING ONE OR MORE PROCE REALIZATION Mchael E uhl halaa C eo Industal & ystes Engneeng epatent Rocheste Insttute of Technology Rocheste NY 463 UA Jaes R Wlson Edwad P Ftts epatent of Industal and ystes Engneeng Noth Caolna tate Unvesty Ralegh NC 7695 UA ABTRACT We develop and evaluate a sepaaetc ethod to estate the ean-value functon of a nonhoogeneous Posson pocess (NHPP) usng one o oe pocess ealzatons obseved ove a fxed te nteval To appoxate the ean-value functon the ethod explots a specally foulated polynoal that s constaned n least-squaes estaton to be nondeceasng so the coespondng ate functon s nonnegatve and sooth (contnuously dffeentable) An expeental pefoance evaluaton fo two typcal test pobles deonstates the ethod s ablty to yeld an accuate ft to an NHPP based on a sngle pocess ealzaton A thd test poble shows how the ethod can estate an NHPP based on ultple ealzatons of the pocess INTROUCTION Nonstatonay (te-dependent) pocesses ae coonly encounteed n sulaton studes ncludng applcatons to anufactung health cae and telecouncatons systes Te-dependent pont pocesses ae often odeled as Nonhoogeneous Posson Pocesses (NHPPs) fo exaple the stea of patents avng at a hosptal the avals of custoe-sevce equests at a call cente and the patten of occuences ove te of devce falues n a telecouncatons netwo Let Nt () denote the nube of avals (events) n the te nteval ( t ] fo each t ( ] a fxed obsevaton nteval of nteest If { Nt (): t ( ] } s an NHPP then t s copletely defned by ts ate functon λ(t) o by ts ean-value functon μ() t E[ N()] t λ( u) du = = t fo all t ( ] (Çnla 975) In ths pape we popose a specally foulated polynoal appoxaton to the ean-value functon of an NHPP that can be eadly estated by constaned least squaes fo one o oe ndependent ealzatons of the pocess; and the esultng estato of the ate functon s sooth (contnuously dffeentable) ove the ente obsevaton nteval The est of ths pape s oganzed as follows ecton gves the bacgound and otvaton fo ths eseach ecton 3 detals the NHPP-fttng ethod fo the stuaton n whch we have a sngle ealzaton of the taget aval pocess ecton 4 pesents an expeental pefoance evaluaton of the ethod fo two typcal test pocesses when only a sngle ealzaton of each pocess s avalable ecton 5 extends the NHPP-fttng ethod to handle ultple ealzatons of the ftted pocess and ecton 5 also povdes an llustatve exaple of the extended pocedue ecton 6 contans conclusons and a descpton of futue wo BACGROUN AN RELATE WOR The NHPP lteatue ncludes a nube of ethods to odel the ean-value functon and the assocated ate functon fo a selected pont pocess so as to estate these functons accuately fo saple data and subsequently to geneate ndependent ealzatons of the ftted pocess effcently The cuent wo s otvated by the ultesoluton pocedue of uhl and Wlson () and uhl uant and Wlson (6) fo odelng estatng and sulatng NHPPs that exhbt one o oe nested cyclc effects (fo exaple te-of-day effects and day-of-thewee effects) and that ay also exhbt a long-te tend The ultesoluton pocedue estates the ean-value functon n a sepaaetc anne fo a sngle ealzaton of the pocess Howeve ths ethod cannot handle NHPPs that lac cyclc ate coponents Ths pape addesses ths ssue as well as the case of ultple ealzatons of the taget aval pocess /8/$5 8 IEEE 353

2 uhl eo and Wlson oe closely elated wo on NHPPs ncludes the followng MacLean (974) appoxates the ate functon of an NHPP usng an exponental-polynoal functon Lees (99) estates a pecewse-constant appoxaton to the ate functon of an NHPP fo ultple ealzatons of the pocess Johnson Lee and Wlson (994) use an exponental-polynoal-tgonoetc ate functon fo pocesses havng a sngle cyclc effect o a long-te tend (o both) uhl Wlson and Johnson (997) eploy an exponental-polynoal-tgonoetc ate functon wth ultple peodctes to odel pocesses havng a longte tend o one o oe ate coponents exhbtng cyclc behavo In ths pape we popose a sepaaetc ethod fo odelng the ean-value functon of an NHPP that wll copleent ths set of odelng tools 3 METHOOLOGY In ths secton we see to ft the ean-value functon of an NHPP to a sngle ealzaton of obseved avals ove the obsevaton nteval ( ] We popose a sepaaetc odel of the fo μ() t = μ( ) R() t fo all t ( ] () whee Rt () s a nondeceasng functon epesentng the cuulatve popoton of avals up to te t In pncple a unfoly accuate appoxaton to the functon R(t) can always be acheved usng a polynoal of suffcently hgh degee wth the specal fo t / f = Rt () = () β ( t / ) + β ( t / ) f > = = whee the coeffcent vecto B = ( β β ) s constaned to yeld R () t fo all t ( ] Note that the fo of Equaton () ensues the ntal value s R () = and the fnal value s R ( ) = fo all values of 3 Estaton of the Functon R(t) B Gven the obseved aval tes fo a sngle ealzaton of the taget pocess soted n ascendng ode t () t () t ( N( )) we let W = N( ) denote the coespondng cuulatve popoton of avals up to the te t of the th aval fo N( ) We see to ft () the polynoal functon Rt () to the ponts [ t () W fo = N ( ) va the followng steps: Tansfo the data usng a vaance-stablzng tansfoaton; Fo the tansfoed data estate the degee of the best-fttng polynoal of the fo (4) below (whch s a sutably escaled veson of () above) usng a odfed lelhood ato test; and Gven the polynoal degee estate the vecto B n Equaton () by applyng the ethod of least squaes to the ognal aval data The pocedue fo detenng the degee of the ftted polynoal and then estatng the polynoal coeffcents s based on a fowad-selecton type of egesson analyss Howeve conventonal egesson analyss eques esponses that ae ndependent and noally dstbuted wth a constant vaance The obsevatons { W } fo an NHPP do not satsfy ths equeent Theefoe followng uhl uant and Wlson (6) we eploy a vaancestablzng tansfoaton of the fo Y = sn ( W) fo = N ( ) Fo convenence the aval tes ae scaled to the unt nteval so that we tae Z = t () esultng n the tansfoed ponts [ Z Y fo = N ( ) Equaton (3) yelds esponses { Y } that ae appoxately noal wth a constant vaance σ To fnd the appopate degee of the polynoal Rt () we apply the ethod of constaned least squaes to the tansfoed data set of sze = N( ) We ft a statstcal odel of the fo E[ Y] = f ( Z ; C ) such that π f u = f(; u C) = (4) π Cu C u f + = > = fo u [] whee C s the coeffcent vecto (3) 354

3 uhl eo and Wlson π / f = C Τ C C f > subject to the constant f ( u ; C ) fo all u [] Note that fo the nonnegatvty constant s equvalent to equng that the zeos of the degee- ( ) polynoal f ( u; C ) le outsde the nteval ( ) To detene the appopate degee fo the statstcal odel (4) a odfed lelhood ato test s used Fo successve values of the vecto C s estated va constaned least squaes yeldng C : f ( u; C) ˆ = Y f Z C ˆ ˆ C ag n ( ; ) fo The coespondng eo su of squaes fo the degee- ft s = Y f Z C E ( ; ) fo ; and fo = we tae π E = Y ( ) + The coespondng total su of squaes ( ) T = Y Y whee Y = Y s used n the fst step of the odfed lelhood-ato pocedue to yeld a constant ate functon f such a odel s appopate In tes of the axu lelhood estato of the esponse vaance σ fo each postulated value of E fo σ = = we see that the assocated lelhood functon s gven as Y f( Z ; ) C L( C ; Y) = exp σ π σ / = πσ e ; ( ) and the esultng log-lelhood functon fo degee s Ψ ( C ; Y) = ln( π) + + ln( σ) The degee s detened usng the followng lelhood ato test (uhl uant and Wlson 6) at the level of sgnfcance α (whee < α < ) The fnal estate of s f E / T < o = σ n : ; ln χ α () othewse σ whee χ α () denotes the α quantle of the chsquaed dstbuton wth degee of feedo To ephasze the dependence of the polynoal n Equaton () on ts degee and ts coeffcent vecto B n the est of ths secton we wte ths functon as Rt (; B ) fo t ( ] Gven the estated degee to be used n Equaton () the coeffcent vecto B s estated by applyng constaned least squaes to the ognal (untansfoed) data yeldng ˆ : ( ; ˆ R u ) B B = ag n W Rt ( ; Bˆ ) B () (5) povded ; and f = then we tae B = β = n Equaton () so that the ftted ate functon s constant Thus the fnal estato of the ean-value functon () s μ ( t) = N( ) R( t; ) fo all t ( ] (6) B and the estated ate functon s λ () t = μ () t fo all t ( ] 4 EXPERIMENTAL PERFORMANCE EVALUATION To evaluate the effectveness of the sepaaetc ethod fo odelng the ean-value functon of an NHPP we conduct an expeental pefoance evaluaton n whch we pesent two cases fo evaluaton Fo each case we geneate one ealzaton of the NHPP and use the sepaaetc ethod to ft the ean-value functon Ths pocedue s caed out fo = eplcatons of each case Nuecal and gaphcal esults of the study ae pesented 355

4 uhl eo and Wlson 4 tatstcal Pefoance Measues To evaluate the ablty of the sepaaetc ethod to ft ethe the undelyng NHPP o the obseved data wth suffcent accuacy we use seveal statstcal pefoance easues pevously foulated by othe eseaches specfcally Johnson Lee Wlson (994); uhl Wlson and Johnson (997); uhl and Wlson (); and uhl uant and Wlson (6) Fo copleteness the defntons of these pefoance easues ae ncluded hee In the est of ths secton λ () t denotes the estated ate functon and μ () t denotes the estated ean-value functon fo the th eplcaton of a test poble (case) Two sets of statstcal pefoance easues ae used The fst set of easues allows us to copae the estated ate o ean-value functon wth ts theoetcal countepat whle the second set of statstcs allows us to copae the ftted ate o ean-value functon wth ts epcal countepat defned by the obseved set of avals In estatng the ate functon λ () t on the th eplcaton of a test pocess the aveage absolute eo axu absolute eo δ ae espectvely δ λ () t λ() t dt { } δ λ λ = ax ( t) ( t) : t δ and fo = In estatng the ean-value functon µ(t) sla easues fo the aveage absolute eo Δ and the axu absolute eo Δ ae defned by Δ = μ() t μ() t dt { μ μ } Δ = ax ( ) ( ) : t t t espectvely fo = Aggegate pefoance easues suaze the eo n estatng the ate functon ove all eplcatons of the test pocess The saple ean of the { δ : = } s denoted by δ = δ ; and the saple coeffcent of vaaton ove all eplcatons of the test pocess s / ( ) = Vδ = δ δ δ Maxu-absolute-eo statstcs fo the theoetcal ate functon ae coputed slaly so that the saple ean and coeffcent of vaaton of the { δ : = } ae denoted by δ and V δ espectvely Analogous pefoance easues fo aveage absolute eos n estatng the theoetcal ean-value functon ae denoted by Δ and V Δ Fo axu absolute eos n estatng the theoetcal ean-value functon the coespondng suay statstcs ae denoted by Δ and V Δ espectvely Noalzed statstcs ae also coputed to facltate copason of esults fo dffeent ate and ean-value functons that s fo dffeent test pocesses: δ δ Qδ = Q = δ μ( )/ μ( )/ Q Q Δ Δ =Δ μ() t dt =Δ μ() t dt In addton to the quanttes entoned above statstcs ae developed to easue the ablty of the poposed pocedue to appoxate each obseved aval pocess On the th eplcaton of a gven NHPP ( = ) we let { t () : = N ()} denote the odeed aval epochs obseved n the te nteval ( ] Aveages ae epoted ove all eplcatons of the followng: the su of squaed eos ( E ) and the ean-squaed eo (M E ) along wth the assocated coeffcents of vaaton Moeove n estatng the epcal ean-value functon on the th eplcaton of a test pocess the aveage absolute eo and axu absolute eo ae espectvely N ( ) () N ( ) μ ( t ) { μ () } ax ( t ) : N ( ) fo = To copae the aveages and acoss test cases we use the noalzed statstcs 356

5 uhl eo and Wlson Q Q = ( / ) ( / ) N ( t) dt = = ( / ) ( / ) N ( t) dt = The second type of aggegate pefoance easue s estated by expessng each pefoance easue and obseved on the th eplcaton as a pecentage of the aveage level of the epcal ean-value functon on that eplcaton and then calculatng the aveage acoss eplcatons as follows: H H 4 Expeental Cases = = ( / ) N ( t) dt = = ( / ) N ( t) dt In ths secton we pesent esults fo two test pobles whose ean-value functons have the sepaaetc fo of Equatons () () In Case the undelyng NHPP has = 4 μ ( ) = and degee = 5 In Case the NHPP has = μ ( ) = 5 and degee = 3 Table suazes the confguaton fo each test case In the expeentaton we geneated = ndependent eplcatons of each test pocess; and each of the esultng data sets was suppled to the NHPP-fttng pocedue eplcatons of the fttng pocedue when t s appled to a sngle ealzaton of the pocess To llustate the fttng pocedue fo Case Fgue shows a hstoga fo the nube of avals ove equal subntevals of length te unts Fgue dsplays the step functon fo the obseved cuulatve avals fo the sngle ealzaton Fgue 3 shows the tansfoed data and the estated functon (4) fo the tansfoed esponse Fo ths ealzaton the fttng pocedue yelded a degee-6 polynoal Gven the degee of the polynoal the pocedue ft a degee-6 polynoal to the ognal data The esultng polynoal coeffcents ae shown n Table and the estated ean-value functon s plotted along wth the epcal ean-value functon n Fgue 4 The estated ate functon calculated as the devatve of the ean-value functon s shown n Fgue 5 The nuecal and statstcal esults fo the expeental pefoance evaluaton usng Cases and ae shown n Tables 3 and 4 and Fgues 6 9 espectvely The statstcal pefoance easues ae collected fo = eplcatons of the fttng pocedue fo each case Table 3 contans the pefoance easues fo evaluatng the ft of the estated ate and ean-value functons to the undelyng ate and ean-value functons fo each case Table 4 copaes the fts fo each case wth the epcal ean-value functon Fgues 6 and 8 show 9% toleance ntevals (bands) fo the ftted ean-value functon aound the undelyng ean-value functon n Cases and Fgues 7 and 9 show 9% toleance ntevals (bands) fo the ftted ate functon fo Cases and espectvely Table : Estated polynoal coeffcents fo the eanvalue functon ft to one ealzaton of Case Polynoal Coeffcents β β β 3 β 4 β 5 β E 5 Table : Polynoal coeffcents fo Cases and Polynoal Coeffcents Case β β β 3 β 4 β Results and Analyss The fttng pocedue s used to estate the ean-value functon of the NHPP fo Cases and Fo Case the pocedue s llustated fo fttng an NHPP to a sngle ealzaton of the pocess Gaphs ae plotted at evey step of the estaton pocess Then nuecal and gaphcal pefoance easues ae pesented fo both cases fo = Fgue : Hstoga of aval data fo t [4] n Case 357

6 uhl eo and Wlson Fgue : Cuulatve avals fo t [4] n Case Fgue 5: Estated ate functon fo t [4] n Case Fgue 3: Tansfoed ft vs tansfoed data fo t [4] n Case Fgue 6: 9% toleance ntevals fo µ(t) t [4] n Case Fgue 4: Estated ean-value functon fo t [4] n Case Fgue 7: 9% toleance ntevals fo λ(t) t [4] n Case 358

7 uhl eo and Wlson Fgue 8: 9% toleance ntevals fo µ(t) t [] n Case Table 3: Goodness-of-ft statstcs fo estatng λ(t) and µ(t) Pefoance Measues Case ( = 4) Case ( = ) δ 5 94 V δ Q δ δ V δ Q δ Δ 5886 V Δ Q Δ Δ V Δ Q Δ Fgue 9: 9% toleance ntevals fo λ(t) t [] n Case Table 4: Goodness-of-ft statstcs fo estatng N(t) Measues Case ( = 4) Case ( = ) E V E M E V M E Q Q H 466E-5 859E-5 H 69E-4 63E-4 5 ETIMATING AN NHPP FROM MULTIPLE REALIZATION OF THE PROCE If we wee able to obtan ultple ealzatons of the pont pocess unde study (say P ealzatons) and we wanted to ft an NHPP to the pocess then the sepaaetc pocedue outlned n ecton 3 can be appled wth soe no odfcatons uppose that on obseved ealzaton of the pocess we have a total of N ( ) avals { t : N( ) }; and let P N ( ) = N ( ) P = denote the aveage nube of avals n ( ] taen ove all P ealzatons of the pocess To obtan an estate fo the constaned polynoal () we fo the consoldated set of = P N( ) aval tes { t : N( )and = P} taen ove all P ealzatons; and we sot ths oveall set n ascendng ode to yeld the odeed aval tes t () t () 359

8 t ( ) To adapt the appoach of ecton 3 we let W = / denote the cuulatve facton of the oveall set of avals that occu up to the te t () of the th ealest aval fo = The pas {[ t() W } ae then tansfoed usng the vaance-stablzng tansfoaton detaled n ecton 3 The tansfoed data set {[ Z Y } s used to estate the appopate degee of the ftted polynoal () The least-squaes estate B of the coeffcent vecto s obtaned fo (5) usng the ognal (untansfoed) data {[ t() W } The fnal estato of the ean-value functon then has the fo uhl eo and Wlson μ ( t) = N( ) R( t; ) fo all t ( ] (7) B Fgue : Aval ate ove the nteval ( 45] fo the lunch wagon exaple (Lees and Pa 6) 5 Illustatve Exaple fo an NHPP wth Multple Pocess Realzatons To llustate the fttng pocedue fo fttng the ean value functon gven ultple ealzatons of the NHPP we utlze a coon exaple of avals to a lunch wagon whch appeas n Lees and Pa (6) and elsewhee In ths exaple = 45 Hee we utlzed the pecewse lnea ean-value functon of Lees and Pa (6) to geneate P = 3 ealzatons of the aval pocess The aval ate dung each subnteval s llustated n Fgue The geneated data esulted n ealzatons of N ( ) = 6 N ( ) = 48 and N ( ) 3 = 45 avals fo a total of = 53 avals n the oveall data set Gven 3 ealzatons of the aval pocess the sepaaetc pocedue was eployed esultng n a ftted degee-5 polynoal havng the coeffcents shown n Table 5 Fgue shows the ftted ean-value functon plotted along wth the epcal ean-value functon ceated fo the oveall set of 53 aval events fo the 3 ealzatons Fnally Fgue shows the estated ate functon ove the nteval (45] Fgue : Ftted ean-value functon fo the lunch wagon exaple Table 5: Polynoal coeffcents fo the ftted polynoal n the lunch wagon exaple (ultple ealzatons) Polynoal Coeffcents β β β 3 β 4 β CONCLUION AN FUTURE WOR In ths pape we have pesented a sepaaetc ethod fo fttng the ean-valuefuncton of a nonhoogeneous Fgue : Ftted ate functon fo the lunch wagon exaple 36

9 uhl eo and Wlson Posson pocess when one o oe ealzatons of the pocess ae avalable The expeental pefoance evaluaton deonstates that the poposed ethod s capable of consstently estatng the undelyng ean-value functon of the NHPP n soe cases of nteest Although not dscussed hee the ethod also esults n an effcent ethod fo geneatng ealzatons fo the ftted NHPP n sulaton expeents We beleve that the poposed pocedue fo odelng and sulaton of aval pocesses has dstnct advantages n applcatons fo whch the physcs of the poble eque a sooth ate functon athe than a ate functon that s pecewse constant o exhbts othe types of nonsooth behavo Futue eseach ncludes conductng an extended expeental pefoance evaluaton fo a wde vaety of pont pocesses that ae fequently encounteed n cetan applcaton doans fo exaple odelng and analyss of call centes In addton we plan to stess-test the pocedue to detene the level of coplexty of the undelyng aval pocess that the ethod wll be able to handle elably REFERENCE Çnla E 975 Intoducton to stochastc pocesses Englewood Clffs New Jesey: Pentce Hall Johnson M A Lee and J R Wlson 994 Expeental evaluaton of a pocedue fo estatng nonhoogeneous Posson pocesses havng cyclc behavo ORA Jounal on Coputng 6 (4): uhl ME and JR Wlson Modelng and sulatng Posson pocesses havng tends o nontgonoetc cyclc effects Euopean Jounal of Opeatonal Reseach 33: uhl M E J R Wlson and M A Johnson 997 Estatng and sulatng Posson pocesses havng tends o ultple peodctes IIE Tansactons 9 (3): uhl M E G uant and J R Wlson 6 An autoated ultesoluton pocedue fo odelng coplex aval pocesses INFORM Jounal on Coputng 8 (): 3 8 Lees L M 99 Nonpaaetc estaton of the cuulatve ntensty functon fo a nonhoogeneous Posson pocess Manageent cence 7 (7): Lees L M and Pa 6 scete event sulaton Englewood Clffs New Jesey: Peason Pentnce Hall MacLean C J 974 Estatng and tesng of an exponental polynoal ate functon wthn the nonstatonay Posson pocess Boeta 6 (): 8 85 AUTHOR BIOGRAPHIE MICHAEL E UHL s an Assocate Pofesso n the Industal and ystes Engneeng epatent at Rocheste Insttute of Technology He has a Ph n Industal Engneeng fo Noth Caolna tate Unvesty (997) Hs eseach nteests nclude sulaton odelng ethodologes wth applcaton to cybe secuty healthcae and seconducto anufactung and sulaton analyss pocedues fo nput odelng and output analyss He seved as Poceedngs Edto fo the 5 Wnte ulaton Confeence He s pesdent of the INFORM ulaton ocety and a ebe of IIE and AEE Hs e-al addess s <Mchaeluhl@tedu> and hs Web addess s <peopletedu/eee> HALAA C EO s a student pusung a Maste s degee n Industal and ystes Engneeng at Rocheste Insttute of Technology He eseach nteest s n feld of systes odelng JAME R WILON s a pofesso of the Edwad P Ftts epatent of Industal and ystes Engneeng at Noth Caolna tate Unvesty He s a ebe of AAUW ACM and AEE; and he s a Fellow of IIE and INFORM He has held seveal postons n WC ncludng Poceedngs Edto (986); Poga Cha (99); and ebe of the Boad of ectos (997 4) Hs e- al addess s <jwlson@ncsuedu> and hs Web addess s <wwwsencsuedu/jwlson> 36