A Queuing Model of the N-design Multi-skill Call Center with Impatient Customers

Transcription

1 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o., pp.- hp://dx.doi.org/./ijuess..8.. A Queuig Model of he -desig Muli-skill Call Ceer wih Impaie Cusomers Chuya Li, ad Deua Yue Yasha Uiversiy, chool of Ecoomics ad Maageme, Hebei Qihuagdao Zhijiag College of Zhejiag Uiversiy of Techology, College of ciece, Zhejiag Hagzhou Yasha Uiversiy, College of ciece, Hebei Qihuagdao, [email protected], [email protected] Absrac This paper sudies a ueuig model of he -desig muli-skill call ceer. I his model, here are wo ypes of cusomers ad wo server groups who have differe skills. Group cosiss of specialized servers who ca oly serve oe ype of cusomers, Group cosiss of flexible servers who ca serve boh wo ypes of cusomers. The cusomers waiig for he service i he ueue may leave he sysem due o impaiece. By dividig he sysem s sae space io several ses of saes, we obai he euilibrium euaios ad he seady-sae probabiliies of he sysem. We also obai he compuaioal formula of he service level ad he compuaioal procedure of he saffig problem. Keywords: muli-skill call ceer ueuig sysem impaie cusomer seady-sae probabiliies. Iroducio Call Ceer, also called Cusomer ervice Ceer, is a kid of comprehesive iformaio service sysem, i is based o he Compuer Telephoe Iegraio echology, ad i akes full advaage of commuicaio ework ad compuer ework. Call ceer is widely used i may areas ad idusries, such as commuicaio, fiace, e-busiess, emergecy ceers ad may ohers. Call ceer eables cusomers o obai a fas, exac ad warm respose from he orgaizaios, ad he orgaizaios could provide may services via call ceer, For may compaies, such as hoels, airlies, ad credi card compaies, call ceers provide a primary lik bewee cusomers ad service provider. I is very impora for icreasig he compeiiveess of compaies, improvig cusomer saisfacio ad developig ew cusomers. Rece years, he call ceer idusry has bee rapidly expaded ad i has bee spread all over he worldwide. The oal umber of call ceers is icreasig subsaially, ad call ceers become bigger ad bigger. Wih he rapid developme of he echology, he call ceers also have oher chaels, for example, , fax, iere isa message ad so o. These makes he ages i a call ceer more busier ad he ages have o maser may skills. Bu, i a call ceer, ay oe age could o maser all he skills. owadays, may call ceers are of muli-skill ype raher ha of sigle-skill ype. I sigle-skill call ceers, here is o disicio amog he calls ad all ages are supposed o hadle all calls. However, i muli-skill call ceers, he calls are differe ypes ad he ages ca hadle several ypes of calls, i.e., hey are of muliple skills. O he oher had, muli-skill cal ceers make he call ceer s operaio ad evaluaio more complicaed. Due o he complexiy of muli-skill call ceers, some mehods used o aalyze sigle-skill call ceers are o loger applicable i aalyzig muli-skill call ceers. o, i eeds urgely ew mehods o assess accuraely he service I: - IJUET Copyrigh c ERC

2 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. level of muli-skill call ceers. There are may researches o he performace evaluaio of muli-skill call ceers. Gas e al. [] comprehesively cosidered he muli-skill call ceer for he cases of less skill groups less ha or eual o hree groups, ad provided deailed iformaio o V-desig, -desig, M-desig, W-desig ad so o. Cezik ad Bhulai sudied he cases of more call ypes ad more skill groups of he muli-skill call ceers. Alaso [] preseed a ieraive cuig plae mehod for miimizig saffig coss i a sigle-skill call ceer sysem. Cezik e al. [] geeralized Alaso s mehod o muli-skill call ceers by usig he liear programmig ad simulaio mehod. Bhulai e al. [] preseed a mehod of solvig he model, he mai characerisic of he mehod is o use firsly a heurisic algorihm o calculae he umber of ages i each skill group every period ad he o use liear programmig mehod o solve he shif schedulig problem such ha he cos is miimized. This mehod reduces he umber of imes of simulaio. A he same ime, Gurvich, Aksi ad Jouii sudied he case ha all calls have differe ypes ad he servers ca serve all cusomers. Gurvich e al. [] sudied large-scale service sysems wih muliple classes of cusomers ad fully flexible servers, hey ivesigaed he uesio ha how may servers are eeded ad how o mach hem wih cusomers. They foud ha a sigle-class saffig rule ad a idle-server-based hreshold prioriy corol are asympoically opimal i he may-server heavy-raffic limiig regime. Aksi [] sudied a model ha capured he impac of shared iformaio processig resources o phoe ceer performace, where hey cosidered he reegig behavior. Furhermore, Aksi [] sudided a series of relaed problems. Jouii e al. [8] formulaed ad aalyzed a muli-class call ceer model wih prioriies ad impaie cusomers, where aicipaed delays may be aouced upo arrival. A approximaio based o a ormal disribuio was proposed. Chevalie e al. [9] discussed he saffig problem of a block model, here is o waiig ueue i he ueuig model, whe all ages busy, he call arrivig will be blocked. The evaluaio of he service level is he proporio of he call o be blocked. There are wo ypes ages: he specialized servers oly have oe skill ad he flexible servers ow all of he skills. The coclusio is ha 8% of he specialized servers ad % of he flexible servers is a kid of ideal mach. humsky[] also discussed he mach problem of he specialized servers ad he flexible servers uder wo skills codiio. I addiio, Dai Tao e al. [] discussed a mehod for performace evaluaio of W-desig muli-skill call ceer based o Markov process. This paper sudies -desig muli-skill call ceer wih impaie cusomers, where here are wo ypes of calls ad wo groups of servers who have differe skills. We prese a ew mehod of solvig he model by dividig he sysem s sae space io several ses of saes. The, we obai he euilibrium euaios ad he seady-sae probabiliies of he sysem. We also obai he compuaioal formula of he service level ad he compuaioal procedure of he saffig problem.. ysem Model I his paper, we sudies a ueuig model of he -desig muli-skill call ceer wih impaie cusomers, where here are wo ypes of calls ad wo groups of servers. The model is show i Figure. There are wo ypes of calls or cusomers, Call ad Call. The wo ypes of calls arrive accordig o a oisso process wih raes ad, respecively. There are wo ueues, Queue ad Queue, which cosis of cusomers of Call ad Call, respecively. We assume ha he calls hrough he call ceer s selecio sysem ca be accuraely classified. Copyrigh c ERC

3 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Figure. The -desig Muli-skill Call Ceer Queue Model We assume ha cusomers may leave he ueue due o impaiece. The impaiece ime is expoeially disribued wih he meas There are wo caegories of servers, Group wih servers ad Group wih servers. Group is of specialized servers who ca oly serve cusomers of Call, while Group of flexible servers who ca serve cusomers of boh Call ad Call. The service imes of servers i Group ad are all expoeially disribued wih meas ad, respecively. The rouig policy of he model is based o skills ad he imporace of he wo differe ypes of calls. I is assumed ha Call is impora ha Call. I oher words, Call has o-preemive prioriy ha Call. Whe a server i Group complees his her service, if here are cusomers of Call waied i Queue his server will service a cusomer waied i Queue, ad if here is o cusomers of Call waied i Queue his server will serve a cusomer waied i Queue. Whe a server i Group complees his or her service, if here is a cusomer waied i Queue he she will serves a cusomers of Call, oherwise he she will be free if Queue is empy. There are ifiie waiig spaces for boh ueues. For he same ype calls, hey are served i Firs-come Firs- serviced FCF disciplie. The ueues of Call ad Call are idepede of each oher.. The Calculaio of he eady-sae robabiliy.. The Divisio of he ae pace We divide he sae space by cosiderig he relaioship bewee he umber of calls ad he umber of ages i each group. This divisio is differe from he divisio give by Dai Tao e al. []. There are sae ses i he sysem, le i,i=,,,deoe he specific sae se, defie is he sae se ha he ages i Group are he idle sae <, ad he ages i Group are he idle sae < oo. is he ages i Group are he idle sae <, bu he ages i Group are he jus full sae =. is he ages i Group are he idle sae <, bu he ages i Group are he busy sae >. is he ages i Group are he jus full sae =, bu he ages i Group are he idle sae <. is he ages i Group are he jus full sae =, ad he ages i Group are he jus full sae = oo. is he ages i Group are he jus full sae =, bu he ages i Group are he busy sae >. is he ages i Copyrigh c ERC

4 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Group are he busy sae >, ad he ages i Group are he busy sae > oo. The sae rasiio diagram is show i Figure < < < = < > = < = = = > > >.. The Divisio of he ae pace Figure. The ae Trasiio Diagram I his subsecio, we calculae he rasiio raes amog differe ses of saes by meas of Figure. le i, i=,, deoe he seady-sae probabiliy of each sae se, i j, i, j=,, deoe he sae se rasiio rae. The ueues i he sae ses are idepede of each oher, here are oly wo cases ha he sae will be chaged: arrivals of calls or leaves of calls. The chage of saes due o arrivals of calls - - Criical poi Figure. The Diagram of ae-rasfer of he Ages i Group for e Cosider he sae-rasfer of ages i Group for se. The sae rasfer diagram is show i Figure. From Figure, for -, if a call arrives a he sysem he he se will o be chaged. For = -, if a call arrives a he sysem he se of saes will be chaged from se o se. The rigger of he rasfer from se o se is due o arrivals of Call, ad he arrive rae is. Thus, we ca obai he rasfer rae from se o se as follows: Copyrigh c ERC

5 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Copyrigh c ERC lim fiished o bee has service he prese ad ime Dela i Call of oe arrival here is lim fiished o bee has service he prese ad ime Dela i Call of oe arrival here is lim lim, Where = - is he probabiliy of ha he umber of Call is = - for he ages i Group for se. oe ha < ad <,ad ha he wo ueues are idepede of each oher, he resuls of M/M/c/c loss ueuig sysem ca be used o our aalysis. Thus, we have!! j j j A similar aalysis ca also ge he res of he seve sae rasfer rae caused by call arrivig, as follows:. The chage of saes due o leaves of calls: Figure. The Diagram of ae-rasfer of he Ages i Group for e Oe leaves of calls due o he service compleed, he sae of he ages will from he jus full sae o he idle sae. Cosider he sae-rasfer of ages i Group for se. The sae rasfer diagram is show i Figure. From Figure, i he se ha =, if a call compleed he service he se of saes will be chaged from se o se. The rigger of he rasfer from se o se is due o service compleed of call, ad he service rae is. Thus, we ca obai he rasfer rae from se o se as follows: A similar aalysis ca also ge aoher four sae rasfer rae caused by service compleed, as follows:

6 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Aoher sae chage is from he busy sae here is a ueue o he jus full sae, due o he service compleed or impaiece. Cosider he sae-rasfer of ages i Group for se. The sae rasfer diagram is show i Figure. Figure. The Diagram of ae-rasfer of he Ages i Group for e From Figure, for +, if a call leaves due o he service compleed or impaiece he he se will o be chaged. For = +, if a call leaves from he sysem he se of saes will be chaged from se o se. The rigger of he rasfer from se o se is due o leaves of Call. Thus, we ca obai he rasfer rae from se o se as follows: Where = + is he probabiliy of ha he umber of Call is = + for he ages i Group for se. oe ha <,ad >,ad ha he wo ueues are idepede of each oher, he resuls of M/M/c+M ueuig[] sysem ca be used o our aalysis. Thus, we have where, +! j k j! j! j k j A similar aalysis ca also ge aoher sae rasfer rae, as follows:,.. The Esablishme of he Euilibrium Euaio ad he Calculaio of he eady-sae robabiliy By he former aalysis we ca ge sae rasiio raes, hus we ca obai he euilibrium euaios of sysem, as follows: [ ] [ ]. Copyrigh c ERC

7 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Copyrigh c ERC ] [ ] [ ] [ ] [ ad i i,ha is he sum of he seady-sae probabiliy of all he sae ses is eual o. From he above euaios we ca obai all of he seady-sae probabiliies i, i=,,,. Because he euaios are liear, usig MATLAB sofware o solve, he calculaig speed ca be guaraeed.. The Calculaio of he Call Ceer s ervice Level The service level of mos of he call ceers is defied as: he perceage of he serviced calls i a give fixed waiig ime, deoed by /T. Acually, he 8/ priciple is a geeral rule, ha is o say wihi secods waiig ime, 8% of he calls should be serviced. The service level ca also be expressed as he probabiliy of he calls which cao be serviced i a give fixed waiig ime. We ca ge he service level usig he seady-sae probabiliies above. uch as call, here we assume ha he service level of call is defied as he probabiliy of he calls o be serviced i T ime, call has a ueue oly happe i sae ses. Due o he imporace of call is very high, if a server i group compleed a service, he will choose he call i he ueue o serve immediaely, so i he sae se, he service rae of call is, hus we ca ge, i T ime, he umber of he calls could be served is a ieger o greaer ha T. By he above aalysis we ca ge he probabiliy of call cao be serviced i T ime is: ] [ T oe where, ] [ T i i T,! i j i j, ad,!! j k j k j j j For call, i is more complicaed ha call, bu he aalysis mehod is he same, call have a ueue happes i he sae se, ad, herefore he probabiliy of call cao be served i T ime is:

8 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. oe [ T ] [ T ] The calculaio of he codiioal probabiliies is similar o call.. affig roblem Assumig ha he cos of he servers group is C,ad he cos of servers group is C,i order o miimum he cos, we ry o fid he opimal umber of servers ad, he model is as follows : mi s.. C C oe oe, Z Here he cosrai codiios deoe ha he probabiliies of each ype calls cao be serviced are less ha or eual o,, respecively., are ukow posiive iegers for solvig.. Coclusios This paper sudies a ueuig model of he -desig muli-skill call ceer wih impaie cusomers. The sysem has wo ypes of cusomers ad wo server groups. Oe group ca oly serve oe ype of calls, he oher ca serve boh wo ypes of calls. We divide he sysem s sae space io regios by usig he umber of cusomers waiig before each servers group ad he umber of servers i each group. We obai he euilibrium euaios ad he seady-sae probabiliies of he sysem. We also obai he compuaioal formula of he service level ad he compuaioal procedure of he saffig problem. Ackowledgemes This work is suppored by he aioal aural ciece Foudaio of Chia, ad i par by he aural ciece Foudaio of Hebei rovice, Chia G. Refereces []. Gas ad G. Koole ad A. Madelbaum, Telephoe call ceers: Tuorial, review, ad research prospecs, Maufacurig & ervice Operaio Maageme, vol., o.,. [] M. Cezik ad. L Ecuyer, affig muli-skill call ceers via liear rogrammig ad simulaio, Maageme ciece, vol., o., 8. [] J. Alaso, M. Epelma ad. Hederso, Call ceer saffig wih simulaio ad cuig lae mehods, Aals of Operaios Research. []. Bhulai, G. Koole ad. o, imple mehods for shif schedulig i muli-skill call ceers, Maufacurig ad ervice Operaios Maageme, vol., o., 8. [] I. Gurvich, M. Armoy ad A. Madelbaum, ervice-level Differeiaio i Call Ceers wih Fully Flexible ervers, Maageme ciece, vol., o., 8. [] O. Z. Aksi ad. T. Harker, Modelig a hoe Ceer: Aalysis of a Muli-chael, Muli-resource rocessor hared Loss ysem, Maageme ciece, vol., o.,. [] O. Z. Aksi, E. Karaesme ad E. L. Ormeci, A review of workforce cross-raiig i call ceers from a operaios maageme perspecive, Edied D. embhard, I Workforce Cross Traiig Hadbook, CRC ress, forhcomig,. [8] O. Jouii, Y. Dallery ad Z. Aksi, Queueig models for full-flexible muli-class call ceers wih realime aicipaed delays, roducio Ecoomics, vol., 9. 8 Copyrigh c ERC

9 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. [9]. Chevalier, A. humsky ad. Tabordo, Rouig ad saffig i large call ceers wih specialized ad fully flexible servers, Techical repor, imo Graduae chool of Busiess, Uiversiy of Rochese,. [] R. humsky, Approximaio ad aalysis of a ueueig sysem wih flexible ad specialized servers, OR pekrum, vol.,. [] D. Tao ad J.-z. Huo, A Mehod for erformace Evaluaio of Muli-skill Call Ceer Based o Markov rocess, ysems Egieerig, vol., o., 8. [] M. Armoy ad C. Magiaras, Coac ceers wih a call-back opio ad real ime delay iformaio, Operaios Research, vol., o., a. [] M. Armoy ad C. Magiaras, O cusomer coac ceers wih a call back opio cusomer decisios, rouig rules ad sysem desig, Operaios Research, vol., o., b. [] R. Aar, A diffusio model of schedulig corol i ueuig sysems wih may servers, Aals of Applied robabiliy, vol., o. b, a. [] M. Armoy, Dyamic rouig i large-scale service sysems wih heerogeeous servers, Queueig ysems, vol., os. -,. [] B.-z. Wag ad Y. Liu, The explore of Call ceer echology ad developme, ci-tech Iformaio Developme & Ecoomy, vol., o.,. [] G. Koole ad A. o, A overview of rouig ad saffig algorihms i muli-skill cusomer coac ceers, Deparme of Mahemaics, Vrije Uiversiei Amserdam, eher lads. [8] Y.-h. Tag ad X.-w. Tag, Queuig Theory: Basis ad aalysis echology, ciece ress, ekig, [9] G. Koole ad A. Madelbaum, Queueig models of call ceers: a iroducio, Aals of Operaios Research, vol.,. [] A. A. Borovkov, ochasic processes i ueueig heory, priger-verlag, ew York, 9, pp. -. [] O. Gare, A. Madelbaum ad M. Reima, Desigig a Call Ceer wih Impaie Cusomers, Maufacurig & ervice Operaios Maageme, vol., o.,. [] D. Gross, Fudameals of ueueig heory, Wiley-Blackwell, ew York, 998. Auhors Chuya Li, she received her B.ci. degree i iformaio ad compuig sciece ad he M.ci. degree i operaios research ad cybereics from Yasha Uiversiy, Qihuagdao, Chia. ow she is sudyig he h.d. degree i maageme sciece ad egieerig Yasha Uiversiy, ad she is a uiversiy lecurer a he Deparme of ciece, Zhijiag College, Zhejiag Uiversiy of Techology, Hagzhou, Chia. Her research ieress iclude performace aalysis of ueueig sysems ad reliabiliy aalysis of repairable sysems. Deua Yue, he received his B.ci. degree i applied mahemaics from orheas Heavy Machiery Isiue, Qiihaer, Chia, he M.ci. degree i applied mahemaics from Xi a Elecroic Uiversiy of ciece ad Techology, Xi a, Chia, ad Dr.ci. degree i operaios research ad cybereics from he Isiue of Applied Mahemaics, Chiese Academy of cieces, Beijig, Chia. ow Dr. Yue is a professor a Deparme of aisics, Yasha Uiversiy, Qihuagdao, Chia. His research ieress iclude performace aalysis of ueueig sysems, reliabiliy aalysis of repair sysems, performace evaluaio ad modelig of commuicaio eworks, ad sochasic order ad is applicaios. Dr. Yue has published over papers i jourals icludig aval Research Logisics, Operaios Research Leers, Compuer Commuicaios, Microelecroics ad Reliabiliy, oliear Dyamics ad ysems Theory, Ieraioal Joural of ure ad Applied Mahemaics, Joural of ysems ciece ad Complexiy, ad oher jourals. Copyrigh c ERC 9

10 Ieraioal Joural of u- ad e- ervice, ciece ad Techology Vol.8, o. Copyrigh c ERC