ࡱ> G  bjbjَ &jim:]8@tg|(U6 eeeeeee$hi\keQUe;ee:1q` He Pw Q9eImpatient customers-a factor of impact on counter service organization Marija Marinovi Faculty of Philosophy, Omladinska 14 HR-51000 Rijeka, e-mail: marinm@pefri.hr Zdenka Zenzerovi Faculty of Maritime Studies at Rijeka, Studentska 2 HR-51000 Rijeka, e-mail: zenzerov@pfri.hr Abstract. The subject of research in this paper was the counter service queuing system involving impatient customers since in real life queues with relatively great number of impatient customers can be met quite often, where some of them give up and leave the queue. The objective of this paper is to find out as to how impatient customers impact on the counter service organization. In order to realize the objective set, it was necessary to compare indicators of the queuing system with and without impatient customers. Special attention was paid to the loss probability, occurring in counter service queuing systems involving impatient customers, which shows the probability of leaving a queue by customers. Analysis of the parameters influencing the amount of the loss probability showed that its value can be reduced by changing parameters and thus impact on the performance of the counter service queuing system with impatient customers. The results of analysis of the queuing system with impatient customers are illustrated through the organization of a real life counter service. Key words: queuing theory, queuing system with impatient customers, loss probability, counter service 1. Introduction The paper [5] describes counter service system as a queuing system with an unlimited number of customers in a queue. The starting assumption was that the arrival rate was constant and that it did not depend on system state, i.e., on number of customers already in the system, either with servers or in queues. However, in real life, customers who are faced with a number of other customers already waiting larger than a certain number, leave the system, i. e. give up the service. Customers who cannot or do not want to wait to be served in queue or servers are called impatient customers. So, there are impatient customers who leave either queue or servers. The objective of this paper is to establish as to how impatient customers impact on counter service queuing system. In order to realize the task, counter service is defined as a queuing system involving impatient customers in queue. Queuing system indicators with and without impatient customers are then compared. Since the loss probability shows the quantity of impatient customers that probably will not access the counter, meaning that these are referred to as a "loss" of customers in each queuing system, it is important to explore as to how changes in certain parameters influence its value. At the end a real life example of counter service queuing system and the results used for the assessment of counter service performance are analyzed. 2. Performance indicators of the queuing system with impatient customers Considering the type of leaving from the system by customers, the paper considers the first case when customers in queuing system with impatient customers express impatience only when are in queue. When they approach the server, they patiently wait to be served, meaning that leaving from system after getting into service is not considered. Observed queuing systems is based on the following assumptions: Queueing system has S service places and n customers in the system Arrival rate is a simple one with intensity  Service rate is a simple one with intensity  Average time for which a customer leaves the queue is  EMBED Equation.3  Intensity of giving up waiting is . Taking into consideration the fact that queue leaving influences the probability pn for n>S, the following state probabilities are obtained from the Kolmogorov's equation [8]:  EMBED Equation.3 ,  EMBED Equation.3 , ...,  EMBED Equation.3  (1)  EMBED Equation.3  (2)  EMBED Equation.3  (3)  EMBED Equation.3 , (4) where:  EMBED Equation.3 - traffic intensity  EMBED Equation.3  - leaving coefficient  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  m-queue capacity (max. number of customers) On the basis of formulas (1) and (3) and implementing equation stating that the sum of probabilities in all system states equals to 1, it follows:  EMBED Equation.3  (5) From expression (5), the probability that there is no customers in the system is:  EMBED Equation.3  (6) i.e.,  EMBED Equation.3 . (7) This is the formula for probability p0 for queuing system with impatient customers. In a queuing system without impatient customers of type M/M/S/ EMBED Equation.3  with infinite number of potential customers in a queue (m EMBED Equation.3 ) probability p01 equals to [8]:  EMBED Equation.3 , (8) i.e.:  EMBED Equation.3  (9) where  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3  Formulas (8) and (9) make sense for  EMBED Equation.3 , i.e., when the average number of customers served in a unit of time at all service places is greater than average number of customers arriving to the queuing system in a unit of time. The probability p0 shows the probability of service places being idle and it is an important indicator of any queuing system performance. Considering the subject of the research, it is necessary to compare probabilities p0 and p01 for queuing system both with and without impatient customers. Formula (6) can be transformed in a shorter form as follows [8]:  EMBED Equation.3  (10) where EMBED Equation.3   EMBED Equation.3  for m EMBED Equation.3 . Comparison of probabilities p01 and p0 gives p01 p0 = d where p01 refers to systems without impatient customers, while p0 refers to system with impatient customers. Consequently  EMBED Equation.3 . (11) From (11) follows:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  . Since  EMBED Equation.3 , the following is valid  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  . Since the numerator equals to  EMBED Equation.3  > 0  EMBED Equation.3  i  EMBED Equation.3 , and denominator to  EMBED Equation.3 > 0 for  EMBED Equation.3 <1, it follows  EMBED Equation.3  i  EMBED Equation.3  and d <0, it further follows that  EMBED Equation.3  < p0, i.e., the probability that queuing system with impatient customers is idle is greater than in queuing system without impatient customers. This proves that probabilities p01 and p0 are not equal (for constant parameters S, EMBED Equation.3  and  EMBED Equation.3 ) and that they depend on presence of impatient customers. It is necessary to calculate values for particular indicators in order to perform the analyses of the performance of queuing system with impatient customers. This will obtained by using formulas [8,???]: 1) Expected number of not served customers  EMBED Equation.3  2) The probability of giving up service is proportional to the intensity of giving up waiting  EMBED Equation.3 , and inversely proportional to intensity of arrival rate  EMBED Equation.3 , hence:  EMBED Equation.3  3) Average number of customers waiting for service  EMBED Equation.3  4) Average number of customers being served  EMBED Equation.3  5) Average number of customers in the system  EMBED Equation.3  6) Absolute capacity of the system for service  EMBED Equation.3 , 7) Relative capacity for service  EMBED Equation.3  EMBED Equation.3  8) Average time spent in queue  EMBED Equation.3  9) Average time spent in system  EMBED Equation.3  . Indicators for queuing systems with impatient customers, when implementing formula (6), can be compared with queuing systems without impatient customers in an analogous way. Results show that indicators: average number of customers in a queue, average number of customers in a system, average time spent in a queue and in a system are lower than in queuing system without impatient customers. DAVAC.DOC\PRE8"g/{6QZDENKA\PRIKAZ.DOC8$hArIn queuing system with impatient customers, some users, after having waited for time , leave the system and become lost customers, which adversely impact any queuing system, so the counter service as well. Therefore, it is necessary to measure the possibility of occurring lost customers. 3. Loss probability The performance measure Ploss is defined by: Ploss = the long-run fraction of customers who are lost [3]. The formula for the loss probability was first implemented by Barrer in 1957, while Gnedenko and Kovalenko modified it in 1989. In their work [3], Bootss and Tijms, 1999. presented an alternative formula for the loss probability queuing system including multiple service places, as follows:  EMBED Equation.3  (12) where  EMBED Equation.3 , (13) and  EMBED Equation.3  EMBED Equation.3 . (14) It follows from (12) that the loss probability depends on three parameters: number of serving places S, system utilization coefficient vi\Izbor.2001\Dopis.doc@(}ImRadovi\Stat.prir.doc_@(~zrPrilog.docP6YqRadovi\Naslovna-sadr~aj.docuXDV and average time for which a customer leaves the queue  EMBED Equation.3 . In order to establish the impact of the loss probability on the performance and how to obtain optimal performance of a queuing system with impatient customers, impact of particular parameter to loss probability has been performed. For that purpose, Tables 1, 2 and 3 have been drawn up, containing values for the loss probability of the chosen parameter values: number of service places (S), value of utilization system coefficient ( EMBED Equation.3 ) and average time for which a customer leaves a queue ( EMBED Equation.3 ). Table 1. Loss probability depending on number of service places (S) and average time for which a customer leaves a queue ( EMBED Equation.3 )  EMBED Excel.Sheet.8  Table 2. Loss probability depending on system utilization coefficient () and average time for which a customer leaves a queue ( EMBED Equation.3 )  EMBED Excel.Sheet.8  Table 3. Loss probability depending on number of service places (S) and utilization system coefficient ()  EMBED Excel.Sheet.8  Radi preglednosti the data from Table 1.-3. are presented in Figure 1.-3.  Figure 1.  Figure 2.  Figure 3. From Table 1.-3. and Figure 1.-3. it can be concluded that loss probability decreases with: increase of number of service places (S) increase of time for which a customer leaves a queue ( EMBED Equation.3 ) decrease of utilization system coefficient (). The increase of service place number must not be arbitrary, rather, it must enable efficient performance of a system. Organization cannot considerably influence the increase of average time for which a customer leaves a queue, except for taking some marketing measures, rather, it depends on time available to customer i.e., his or her patience. Decrease of coefficient \REA@(EK%WINWORD2\REPLAND.DOT\REP@(EK%WINWORD2\REPSIDE.DOT\REP@*EK%WINWORD2\REPST can be obtained in two ways: increase of  EMBED Equation.3  which is not cost efficient or increase of service rate EMBED Equation.3 . 4. Organization of counter service U radu /../ promatran je proces opslu~ivanja jedne policijske postaje kao sustava masovnog opslu~ivanja s ekanjem i beskona nim brojem stranaka u redu ekanja. Meutim, budui da je pojava relativno duga kih redova, openito pa tako i u policijskoj postaji esta pojava, dogaa se da korisnici, ukoliko zateknu u redu za ekanje viae korisnika od odreenoga broja, napuataju sustav, tj. odustaju od opslu~ivanja. U ovom radu, koji je nastavak istra~ivanja organizacije aalterske slu~be iz rada / /, proces opslu~ivanja na aalterima bit e razmatran kao sustav opslu~ivanja s nestrpljivim korisnicima, a teorijske postavke iz odjeljka 2. i 3. ovog rada bit e primijenjene na konkretan primjer aalterske slu~be jedne policijske postaje. 4.1. Problem description The analysis of the impatient customer s impact to queuing system efficiency was performed for an existing organization of Police Department counter service the offering following services: Counter 1 - Issuing driving licenses and license plate Counter 2 - Issuing ID and passports Counter 3 - Registration/deregistration of residence. Statistical observation in period from January 2, 2002 to May 2, 2002. produced data on total number of customers requiring one or more of above services or arrived at a counter, as follows: Counter 1 - 6588 customers Counter 2 - 1356 customers Counter 3 - 1019 customers. Considering working hours of six hour a day, the average number of customers per day and hour respectively is as follows: Counter 1 - 77.51 customers/day; 12.91765 customers/hour Counter 2 - 15.95 customers/day; 2.65882 customers/hour Counter 3 - 11.99 customers/day; 1.99804 customers/hour. It was also established that average processing time for any counter was 7.5 minutes.tivehq‚B Ȃ,YȂI"hq‚ Budui da je rije  o opslu~ivanju s nestrpljivim korisnicima, statisti kim promatranjem je utvreno da prosje no vrijeme napuatanja reda ekanja iznosi 6 minuta. The objective is to make business decision, koja e, na temelju svih prethodno navedenih injenica, that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing idle time of clarks to minimum. Usporedbom pokazatelja funkcioniranja obaju sustava, bez i sa nestrpljivim korisnicima, d iObjInfoobit e se ocjena uspjeanosti njihovog funkcioniranja te optimalno rjeaenje organizacije aalterske slu~be. 4.2. Analysis and discussion of the solution Since the counter service is defined as a queuing system, the problem has been solved using the queuing theory. On the basis of adequate formulas, counter operation indicators have been calculated for one system without and for system with impatient customers, and shown in Table 4. Table 4. Comparison of Police Department counter service operation indicators without (A) and with (B) impatient customers No.IndicatorUnitAB1.Arrival rate (()cust./hour17.5745117.574512.Service rate (()cust./hour883.Traffic intensity (()-2.196812.196814.Number of service places (S)counter335.Utilization system coefficient ((/S =) EMBED Equation.3 -0.732270.732276.Average time for which a customer leaves a queue ()hour-0.17.Intenzitet napuatanja nestrpljivih korisnika ()cust./hour-108.Average number of customers in the queue (LQ)customer1.478450.2473479.Average number of customers in the system (L)customer3.675272.1349710.Average time a customer spends in the queue (WQ)hour0.08413 h (5.05min)0.844min11.Average time a customer spends in the system (W)hour0.20913 h (12.55min)0.121481 h (7.2 min)12.Probability that all service places are idle (p0)%8.19%11.34%13.Probability of loss custo- mers (Ploss)%-9.61% Prema rezultatima iz Tablice 4. proizl#azi da je o ekivani broj stranaka u redu ekanja s nestrpljivim korisnicima manji nego kod sustava bez nestrpljivih korisnika. Isti se zaklju ak odnosi i na ostale pokazatelje, tj. prosje an broj stranaka u sustavu, kao i prosje no vrijeme ekanja provedeno u redu i u sustavu. Meutim, vjerojatnost da e aalteri biti slobodni, tj. da stranka nee morati ekati je vea kod sustava s nestrpljivim korisnicima. Takoer se kod sustava s nestrpljivim korisnicima pojavljuje vjerojatnost izgubljenih korisnika Ploss. Na temelju prethodnih rezultata zaklju uje se da na uspjeanost poslovanja aalterske slu~be najviae utje u pokazatelji p0 i Ploss. Vjerojatnost p0 predstavlja vjerojatnost neiskoriatenog kapaciteta aaltera, a Ploss vjerojatnost  izgubljenih korisnika. Prvi pokazatelj djeluje na poveanje troakova poslovanja, a drugi na smanjenje prihoda, ato na kraju rezultira smanjenjem uspjeanosti poslovanja aalterske slu~be. Optimalno rjeaenje problema je broj aaltera koji se odreuje na temelju vrijednosti tih vjerojatnosti koje su u praksi uobi ajene i prihvatljive. 5. CONCLUSION Proces opslu~ivanja na aalterima mo~e se definirati kao queueing system te analizirati njegovo funkcioniranje using the queueing theory. Budui da je u praksi esta pojava da zbog relativno duga kih redova korisnici napuataju red ekanja ukoliko je broj korisnika u redu vei od nekog uobi ajenog, odnosno prihvatljivog broja, potrebno je uzeti u obzir tzv. nestrpljive korisnike. Usporedbom pokazatelja funkcioniranja aalterske slu~be sa i bez nestrpljivih korisnika izlazi da u sustavu s nestrpljivim korisnicima pojedini pokazatelji (broj korisnika i duljina vremena ekanja) imaju manju vrijednost, ato je povoljnije za korisnike. Meutim, poveava se vjerojatnost da e aalter biti slobodan, odnosno neiskoriaten i pojavljuje vjerojatnost  izgubljenih korisnika. Oba pokazatelja utje u na uspjeanost poslovanja sustava opslu~ivanja, o emu bi trebalo voditi ra una pri organizaciji rada aalterske slu~be. Optimalno rjeaenje problema bit e onaj broj aaltera za koji e vjerojatnosti nezauzetosti aaltera i izgubljenih korisnika poprimiti ato manje vrijednosti. REFERENCES [1] N. K. BOOTS, H. TIJMS, A Multiserver Queueing System with Impatient Customers, Managament Science/Vol.45, No. 3, March 1999. 444-448. [2] Z. ZENZEROVI; M: MARINOVI, Impact of service place specialization on the efficiency of queuing system functioning, Operational Research Proceedings KOI 2002, 311-320. Analiza odabranog problema je proairena uzevai u obzir da aalterska slu~ba mo~e biti organizirana s viaenamjenskimaalterima, kao ato je pokazano u tablici 4. ili, pak, da su aalteri specijalizirani s obzirom na vrstu usluga koje pru~a policijska postaja. Vezano za rezultate rada [ ] uzete su u razmatranje dvije varijante organizacije aalterske slu~be: 1. multipurpose (universal) counters, meaning that any customer can be served at any counter, specialized counters, i.e., customers can be served at particular counter depending on service type required. On the basis of adequate formulas, counter operation indicators have been calculated for one system of M/M/3/( type (varijanta A), for one system M/M/2/( type (varijanta B) and two systems M/M/1/( type(varijante C i D), without and with impatient customers, and shown in Table 5., where is: Variant A: multipurpose service places Variant B: specialized service place for issuing driving licenses and license plate Variant C: specialized service place for issuing ID and passports Variant D: specialized service place for registration/deregistration of residence. Vrijednosti u zagradama tablice 5. odnose se na nestrpljive korisnike. Table 5. Comparison of Police Department counter service operation indicators by counter type No.IndicatorUnitABCD1.Arrival rate (()cust./h17.5745112.917652.658821.998042.Service rate (()cust./h88883.Traffic intensity (()-2.196811.614710.332350.249754.Number of service places (S)counter32115.Utilization system-coefficient ((/S =) EMBED Equation.3 -0.732270.807350.332350.249756.Average time for which a customer leaves the queue ()hour0.10.10.10.17.Intenzitet napuatanja nestrpljivih korisnika ()cust./h101010108.Average number of customers in the queue (LQ)customer1.47845 (0.247347)3.02284 (0.284062)0.16544 (0.042894)0.08314 (0.025016)9.Average number of customers in the system (L)customer3.67527 (2.13497)4.63755 (1.543691)0.49780 (0.321629)0.33290 (0.243501)10.Average time a customer spends in the queue (WQ)hour0.08413 h 5.05min (0.844min)0.23400 h 14.04min (1.32min)0.06223 h 3.73min (1.07min)0.04161 h 2.5 min (0.75min)11.Average time a customer spends in the system (W)hour0.20913 h 12.55min (7.2 min)0.35901 h 21.54min (0.119502) (7.17 min)]0.18723 h 11.23min [0.120967]0.16612 h 9.97min [0.013669 (0.82 min)]12.Probability that all service places are idle (P0)%8.19% [11.34%]10.66% [20.48%]66.76% [72.13%]75.02% [78.15%]13.Probability of loss customers (Ploss)%[9.61%][17.84%][13.90%][10.65%] U sustavu s nestrpljivim korisnicima treba uzeti u obzir  izgubljene korisnike, odnosno izra unati vjerojatnost Ploss. Za promatrane varijante proizlazi da je najvea vjerojatnost izgubljenih korisnika 17.84% za varijantu B sustava, za koji je koeficijent iskoriatenja sustava 0.33235, intenzitet napuatanja sustava 10 korisnika/sat, a najmanja za varijantu A 9.61% s koeficijentom iskoritenja sustava 0.73227 i intenzitetom naputanja sustava 10 korisnika/sat. Thus, under given condition, the implementation of specialized counter in Police Department counter service is justified only if the service rate is at least 9.75024 customers/hour; therefore, service time should be reduced to 6.15 minutes, i.e., service rate should be increased by 21.9%. Considering the service type offered at particular counter and proportion of human work required in providing a service, it is not always possible to reduce such service time. However, assuming that in this example is possible to increase service rate to 9.75024 customers/hour, the indicator of expected number of customer in a queue is favorable, but increasing number of counters would severely impact particular counter idle time from 10.66% to 75.02% of total working hours, which is unacceptable in terms of economic considerations. Due to the above, the combinations of multipurpose (universal) and specialized counter service organization have been considered as follows: Specialized counters for driving licenses (S1 and S2) and multipurpose counters for IDs and registration/deregistration of residence (S3 and S4) Specialized counters for driving licenses (S1 and S2) and multipurpose counter for IDs and registration/deregistration of residence (S3) Specialized counter for driving licenses (S1) and multipurpose counter for IDs and registration/deregistration of residence (S3). Counter operation indicators as per above variants are shown in Table 3. Table 3. Police Department counters operation indicators for counter combinations No.IndicatorUnitMultipurpose counter S3 Multipurpose counters S3 i S4 1.(cust./hour4.656864.656862.(cust./hour883.(-0.582110.582114.Scounter125.(/S-0.582110.291056.LQcounter0.810850.053887.Lcounter1.392960.635988.WQhour0.29912 (17.95 min)0.13657 (8.19 min)9.Whour0.17412 (10.45 min)0.01157 (0.69 min)10.P0%41.7954.91 Variants 1 and 2 (due to rather large number of customers) that there will be two counters for driving license (S1 and S2), while other services will be handled either at counters S3 or S4, or only at counter S3, thus reducing the number of counters to three. Variant 3 resulted from experience to date, where driving license were only handled at one counter. It is practicable if, on the basis of past number of customers as noted during observed period, average processing time ranges from 4 to 4.6 minutes, e.g. (=15 customers/hour. Table 4. shows results in comparison of indicators for proposed counter service variants. Table 4. Comparison of indicators for combined counter service variants IndicatorVariant 1.Variant 2.Variant 3. (( = 15)S432LQ2.122.886.15P037.6%27.47%27.8% On the basis of results shown in Table 4. and Table 2., one of the following business decisions should be made: Organize counter service as one system containing three multipurpose counters. Organize counter service with two counters, one specialized for driving licenses and the other multipurpose counter for other services. 1. Loss probability in respect to a constant value of utilization system coefficient () is decreased when the number of service places ( EMBED Equation.3 ) and the average time for which a customer leaves a queue ( EMBED Equation.3 ) are increased. E.g., for  = 0,4 increase of time from  = 0,1 to 0,9, Ploss is decreased for 2,75 times. If loss probability dependence on  and  is considered, the following can be concluded: increase of parameter  increases loss probability, but it drops with increase of the average time for which a customer leaves a queue ( EMBED Equation.3 ). Loss probability for the constant average time for which a customer leaves a queue ( EMBED Equation.3 ) decreased with the increase of service place number (S) and increase with the increase of system utilization coefficient (). It can be concluded that loss probability decreases with: increase of number of service places. increase of time for which a customer leaves a queue ( EMBED Equation.3 ) decrease of utilization system coefficient (). 3. The increase of service place number must not be arbitrary, rather, it must enable efficient performance of a system. Organization cannot considerably influence the increase of average time for which a customer leaves a queue, except for taking some marketing measures, rather, it depends on time available to customer i.e., his or her patience. Decrease of coefficient  !#$%&'()*+,./012345689:;<=>?@BCDEFGHIJLMN can be obtained in two ways: increase of  EMBED Equation.3  which is not cost efficient or increase of service rate EMBED Equation.3 . On the basis of above, it is possible to define the problem. Number of customer requiring a service at any counter is a random variable as well as processing time. Adequate statistical test revealed that distribution of customers arrival number and processing time in observed time unit follow a Poisson distribution, meaning that counter service could be considered as a queuing system. Counter service place system is queuing system of M/M/S/( type, i.e. having an infinite time a customer spent in queue with the queue discipline on the FIFO basis (first in-]_`abcdefhijklfirst out). Uzevai u obzir napuatanje reda ekanja od strane pojedinih korisnika, the counter service become queuing system of M/M/S/( type with impatient customers. Usporedbom pokazatelja funkcioniranja obaju sustava, bez i sa nestrpljivim korisnicima, dobit e se ocjena uspjeanosti njihovog funkcioniranja te optimalno rjeaenje organizacije aalterske slu~be. 4.3. Analysis and discussion of the solution 5. CONCLUSION Proces opslu~ivanja na aalterima mo~e se definirati kao queueing system te analizirati njegovo funkcioniranje using the queueing theory. Budui da je u praksi esta pojava da zbog relativno duga kih redova korisnici napuataju red ekanja ukoliko je broj korisnika u redu vei od nekog uobi ajenog, odnosno prihvatljivog broja, potrebno je uzeti u obzir tzv. nestrpljive korisnike. Usporedbom pokazatelja funkcioniranja aalterske slu~be sa i bez nestrpljivih korisnika izlazi da u sustavu s nestrpljivim korisnicima pojedini pokazatelji (broj korisnika i duljina vremena ekanja) imaju manju vrijednost, ato je povoljnije za korisnike. Meutim, poveava se vjerojatnost da e aalter biti slobodan, odnosno neiskoriaten i pojavljuje vjerojatnost  izgubljenih korisnika. Oba pokazatelja utje u na uspjeanost poslovanja sustava opslu~ivanja, o emu bi trebalo voditi ra una pri organizaciji rada aalterske slu~be. Optimalno rjeaenje problema bit e onaj broj aaltera za koji e vjerojatnosti nezauzetosti aaltera i izgubljenih korisnika poprimiti ato manje vrijednosti. REFERENCES D. Barkovi, Operacijska istra~ivanja, Ekonomski fakultet, Osijek, 2001. R. Bronson, Operations Research, McGraw-Hill, 1982. [1] N. K. BOOTS, H. TIJMS, A Multiserver Queueing System with Impatient Customers, Managament Science/Vol.45, No. 3, March 1999. 444-448. [2] Z. 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