ࡱ>  {{f|h|r|t|||p̀΀ހF&ڃ܃̋.ȍD&ȼȰȰȼȰȰȰȕȰȰȰȼ5mH 0JCJH*OJQJmH  j0J6CJOJQJmH 0J6CJOJQJmH 0J5CJOJQJmH 0JCJOJQJmH  CJH*mH 6CJmHmH 6CJ 6CJmH  jmCJmH CJmH  CJH*mH 8 &P . A!"#$%IMPACT OF SERVICE PLACE SPECIALIZATION ON THE EFFICIENCY OF THE QUEUING SYSTEM FUNCTIONING Zdenka Zenzerovi Faculty of Maritime Studies at Rijeka Studentska 2, 51000 Rijeka, Croatia Telephone: 051/ 338-411;E-mail:  HYPERLINK mailto:zenIMPACT OF SERVICE PLACE SPECIALIZATION ON THE EFFICIENCY OF THE QUEUING SYSTEM FUNCTIONING Zdenka Zenzerovi Faculty of Maritime Studies at Rijeka Studentska 2, 51000 Rijeka, Croatia Telephone: 051/ 338-411;E-mail:  HYPERLINK mailto:zenzerov@pfri.hr zenzerov@pfri.hr Marija Marinovi Faculty of Philosophy Omladinska 14, 51000 Rijeka, Croatia Telephone: 051/345-034; E-mail: marinm@pefri.hr Abstract. This paper analyses the impact of service place specialization on efficiency of the queuing system operation. Using the queuing theory, it was proved that introducing service place specialization is only justified if the service rate of specialized service places is greater than those in multipurpose places. Specialization index has been introduced, showing the increase required of service rate, i.e., it is used to calculate the minimal capacity of specialized service places where the average number of customers in the queue and average time a customer spends in the queue will equal to the multipurpose service places. The method presented is useful in planning and organizing queuing system operation and it represents the basis when making decision as to whether to organize as specialized or multipurpose service place system. The results of analysis of the service place specialization impact are illustrated through example of actual organization of a counter service. Key words: queuing theory, service place specialization, specialization index, counter service INTRODUCTION The efficiency of a queuing system can be determined by the time customer spends in a system and unoccupancy of a service place. The values of these indicators are influenced by the number of service places and service rate of particular service place, considering given or expected number of customers. Depending on the service type offered to customers, service place could be multipurpose (universal) or specialized. Since in real life service places within a queuing system often offer various services, the main issue in organizing such a process is to decide whether service place should be multipurpose or specialized. The decision on service place type results in positive and negative impacts on queuing system functioning. In systems with specialized service places, customer is served at a service place specialized for a particular service; on the other hand, if service is organized with multipurpose service places, each service place handles any customer. If service rates of multipurpose and specialized service places are equal, it can be expected that the average number of customers and time of customers spend in queue would be greater/longer for specialized service places. So called specialization index has been introduced in this research, and it is used to determine limit value of service rate for a specialized service places. This value is used to equalize number of customers and time of customers in a queue for multipurpose system and it justifies implementation of specialized service places. The aim of this research was to present planning method and organization of queuing system considering the type of service place. The presented method was illustrated through a real life counter-service organization. 2. COMPARISON OF INDICATORS OF QUEUING SYSTEM FUNCTIONING FOR MULTIPURPOSE AND SPECIALIZED SERVICE PLACES In accordance with the defined objective, the following task has been defined: to explore whether number of customers waiting in a queue per time unit at all service places will be equal or significantly different, assuming that service places are multipurpose or specialized. By using the queuing theory, it is necessary to compare expected number of customers for a queuing system having a number of specialized service places with expected number of customers for a queuing system having an appropriate number of multipurpose service places. Assuming that two service places, each with respective given values for EMBED Equation  and EMBED Equation , i.e. having a number of customers arriving to the service place and a number of customers that can be served in an observed time unit at a given place, average number of customers in the queue waiting to be served at both places shall be equal to the sum of customers waiting at each service place   (1) If service places are of the same capacity EMBED Equation , and if customer arrival rates at both places are about equal EMBED Equation , average number of customers in the queue can be calculated as follows  (2) By analogy, formula for determining average number of customers in the queue can be derived for S service places operating independently (assuming the same customer arrival rate EMBED Equation  and service rate EMBED Equation )  (3) If service places differ by customer arrival and service rates, formula (3) can be nevertheless used, provided that EMBED Equation  and EMBED Equation  are assigned average values. If service places are not specialized, then each service place is not considered separately, but as an S-channel system of M/M/S/( type, where average number of customers waiting in the queue is calculated using the following formula  (4) where P0 is probability that all service places are unoccupied, i.e. probability that are no customers in a system. According to task defined in this paper it is necessary to compare formulas EMBED Equation  and EMBED Equation   (5) Since in single service place system EMBED Equation , where EMBED Equation  is traffic intensity, the demonstration continues by comparing formulas  (6) If formula (6) is to be greater than 0, the following inequation must be met  (7) i.e.,  (8) EMBED Equation.2 EMBED Equation.2 EMBED Equation.2 . (9) Value of EMBED Equation  in formula (9) is greater than 1, and EMBED Equation  is also greater than 1, since  (10) consequently, the left side of inequation (9) is greater than 1, meaning that EMBED Equation  is greater than EMBED Equation , which was to be proved. Since EMBED Equation , it is concluded that average number of customers and time a customer spends in the queue are considerably lower in one system with S service places, i.e., in multipurpose service place system than in S systems with one service place, i.e., specialized service places, whether customer arrival rates are equal for particular service place or not. SPECIALIZATION INDEX The previous part of this paper assumes that service rates for specialized service places equal to multipurpose places service rates. However, in real life, specialized service places should be more efficient than multipurpose service places since they are better equipped for providing intended services, and in that case the difference in number of customers in the queue will not be so high as mentioned above. In order to equalize specialized and multipurpose service places, considering number of customers in the queue, it is necessary to calculate service rate, using above proof, for specialized service places where number of customers in the queue will be equal (and not greater as shown) to multipurpose service places service rate. Border value of service rate for specialized service places is the solution of relation EMBED Equation , where A = S(LQ(1) and B = 1(LQ(S) Considering above assumptions, service rate ((1) for specialized places is calculated using relation S1(LQ(1) = 1(LQ(S2), (16)  (11) where: EMBED Equation  - multipurpose service place rate ks - specialization index. Specialization index for EMBED Equation  service places is defined as follows: , (12) which leads to  (13) By analogy for  , (14) and for EMBED Unknown EMBED Unknown. (15) Above relations show that specialization index depends on EMBED Equation , EMBED Equation  and number of service places EMBED Equation , where ( represents customer arrival rate to specialized service place. Relations (12)(15) are valid when the number of service places in multipurpose system equals to the number of specialized service places. However, in reality, it is possible that implementation of specialization increases number of service places as compared to the multipurpose system. In such a case, specialization index ks is calculated using relation S1(LQ(1) = 1(LQ(S2), (16) where S1 stands for the number of specialized service places, S2 for the number of service places in one multipurpose system. Such, for S1= 4 and S2= 3, the following relation has been derived for specialization index EMBED Equation.2. (17) Defining value of ks becomes burdensome for higher values of S so, for convenience, it is suggested to enter mentioned values in a table. An example can be found in Table 1 for S = 2, ( = 0.1, 0.9 and ( = 0.2,1.0. Table 1. Specialization index value (ks) for two service places according to values of EMBED Equation.2 and EMBED Equation.2 ctIdIDE DISK TYPE46 ManufacturerGENERIC  DeviceTypeInt13CurrentDriveLetterAssignmentCD @HardwareIDGENERIC  0.10.20.30.40.50.60.70.80.91.00.11.5001.8082.0662.2932.5002.6912.8693.0373.1970.21.3051.5001.6631.8081.9412.0662.1822.2930.31.2261.3751.5001.6111.7131.8081.8980.41.1811.3051.4081.5001.5841.6630.51.1521.2591.3481.4271.5000.61.1311.2261.3051.3750.71.1151.2011.2730.81.1031.1810.91.093 The empty part of the table does not contain calculated values of ks because ratio between EMBED Equation  and EMBED Equation  indicates the instability of a queuing system. Service rate for specialized service places can be calculated by multiplying ks (from Table 1.) (depending of EMBED Equation  and EMBED Equation ) by service rate of the multipurpose service place. If the service rate of a specialized place equals to EMBED Equation , then the number of customers in the queue will equal to that in system having multipurpose service places; if it is greater than EMBED Equation , queue will be smaller and vice versa. Shown method can be used in planning service place types considering the service purpose. Of course, the final decision on specialization of service places should consider equipment required for specialized and multipurpose service places. 4. CASE STUDY - ORGANIZATION OF COUNTER SERVICE 4.1. Problem description The analysis of the service place specialization 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. The objective is to make business decision that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing clerks' idle time to minimum. 4.2. Problem definition 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 number of customers 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-first out). According to the task set, it is possible to make following business decisions: Organize multipurpose (universal) counters, meaning that any customer can be served at any counter Organize specialized counters, i. e., customers can be served at particular counter depending on service type required. Comparison of multipurpose and specialized counter indicators will yield an optimal solution of the observed problem. 4.3. 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 of M/M/3/( type and four systems of M/M/1/( type and shown in Table 2. Table 2. Comparison of Police Department counter service operation indicators by counter type No.IndicatorUnitMultipurpose service placesSpecialized service places1.2.3.1.Arrival rate (()cust./hour17.5745112.917652.658821.998042.Service rate (()cust./hour88883.Traffic intensity (()-2.196811.614710.332350.249754.Number of service places (S)counter32115.Utilization system coefficient ((/S)-0.732270.807350.332350.249756.Average number of customers in the queue (LQ)customer1.478453.022840.165440.083147.Average number of customers in the system (L)customer3.675274.637550.497800.332908.Average time a customer spends in the system (WQ)hour0.08413 (5.05 min)0.23400 (14.04 min)0.06223 (3.73 min)0.04161 (2.5 min)9.Average time a customer spends in the system (W)hour0.20913 (12.55 min)0.35901 (21.54 min)0.18723 (11.23 min)0.16612 (9.97 min)10.Probability that all service places are idles (P0)%8,1910.6666.7675.02 Results in Table 2. show that expected number of customers in the queue is 1.47845 and 3.27142 for multipurpose and specialized counters, respectively. This example supports the proof as per Chapter 2. of the paper, i.e., if the service rates are equal, the queue is greater in specialized than in universal service places. The same conclusion refers to the time customers spend in the queue as well as to busy/idle state of a counter. At last, the fact that cannot be neglected, is that specialization requires increased number of counters, specifically, four counter in this example. However, it is fair to assume that a specialized counter should have higher service rate, and according to relation (17) in Chapter 3. of the paper it comes out that for S = 3/4, (where the first number stands for number of counter in multipurpose system and the second for number of specialized counters), ( = 17.57451/4 = 4.39363, ( = 8, specialization index is ks = 1.21878 which requires specialized counter service rate of (1 = 9.75024 customers/hour. 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, or 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.Wsat0.17412 (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. 5. CONCLUSION In a queuing system, service places can be multipurpose or specialized. If both of service places have the same service rate, than the average number of customers and the time a customer spends in the queue are greater in specialized service places than in multipurpose ones. Using the queuing theory, above conclusion has been proven by comparing the number of customers in a queue for S systems with one specialized service place to one system with S multipurpose service places. However, considering better equipment of specialized service places, their service rate should be higher; specialization index is a number showing the smallest increase of service rate required in order to justify the implementation of specialized service places, i.e., to equal the system efficiency indicators that of multipurpose system. The presented method, recommended for planning of service place types considering its purpose, is illustrated through an actual counter service in a Police Department. REFERENCES D. Barkovi, Operacijska istra~ivanja, Ekonomski fakultet, Osijek, 2001. R. Bronson, Operations Research, McGraw-Hill, 1982. T. Pogny - Z. Zenzerovi, On the Parameter Shift Influence to the Total Waiting Time in M/M/S/( Queueing System, II, Proceeding of the 16th International Conference on Information Technology Interfaces, ITI 94, Pula, University Computing Center, Zagreb,1994,p.p. 397-402. Z.Zenzerovi, Optimizacijski modeli planiranja kapaciteta morskih luka, Ekonomski fakultet, dokt. disertacija, Sveu iliate u Rijeci, Rijeka, 1995. Z. Zenzerovi, Kapaciteti uslu~nih mjesta u funkciji efikasnosti sustava masovnog opslu~ivanja, Zbornik radova Ekonomskog fakulteta Rijeka, Rijeka, 1996. I. }upanovi, M. Gjumbir, Racionalizacija procesa izdavanja prijevoznih karata na autobusnom kolodvoru, Fakultet prometnih znanosti, Zagreb, 1978.    4o p q { | Y׹Ź㴩ݴujCJEHUmH jUCJUVmH jCJEHUmH j48A UVmH jCJUmH 0JCJOJQJmH CJmH 56CJmH 6mH  6CJmH 0J6CJOJQJmH  5CJmH  5CJmH mH B*>*0JB*jU jU5mH -*r"$Fr p q  S$ $ & FnW$$$$$$*r"$Fr p q  SY3?~iop]^de VW^_flm0129:NOPef!"C#D#^#_#`# !  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This paper analyses the impact of service place specialization on efficiency of the q_PQQRRRRRRnSSUWRXxYzY|Y~YYYYǽ$$c  $  $  $  $   $ $$*$+$,$-$$$-ueuing system operation. Using the queuing theory, it was proved that introducing service place specialization is only justified if the service rate of specialized service places is greater than those in multipurpose places. 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The method presented is useful in planning and organizing queuing system operation and it represents the basis when making decision as to whether to organize as specialized or multipurpose service place system. The results of analysis of the service place specialization impact are illustrated through example of actual organization of a counter service. Key words: queuing theory, service place specialization, specialization index, counter service INTRODUCTION The efficiency of a queuing system can be determined by the time customer spends in a system and unoccupancy of a service place. The values of these indicators are influenced by the number of service places and service rate of particular service place, considering given or expected number of customers. Depending on the service type offered to customers, service place could be multipurpose (universal) or specialized. Since in real life service places within a queuing system often offer various services, the main issue in organizing such a process is to decide whether service place should be multipurpose or specialized. The decision on service place type results in positive and negative impacts on queuing system functioning. In systems with specialized service places, customer is served at a service place specialized for a particular service; on the other hand, if service is organized with multipurpose service places, each service place handles any customer. If service rates of multipurpose and specialized service places are equal, it can be expected that the average number of customers and time of customers spend in queue would be greater/longer for specialized service places. So called specialization index has been introduced in this research, and it is used to determine limit value of service rate for a specialized service places. This value is used to equalize number of customers and time of customers in a queue for multipurpose system and it justifies implementation of specialized service places. The aim of this research was to present planning method and organization of queuing system considering the type of service place. The presented method was illustrated through a real life counter-service organization. 2. COMPARISON OF INDICATORS OF QUEUING SYSTEM FUNCTIONING FOR MULTIPURPOSE AND SPECIALIZED SERVICE PLACES In accordance with the defined objective, the following task has been defined: to explore whether number of customers waiting in a queue per time unit at all service places will be equal or significantly different, assuming that service places are multipurpose or specialized. By using the queuing theory, it is necessary to compare expected number of customers for a queuing system having a number of specialized service places with expected number of customers for a queuing system having an appropriate number of multipurpose service places. Assuming that two service places, each with respective given values for EMBED Equation  and EMBED Equation , i.e. having a number of customers arriving to the service place and a number of customers that can be served in an observed time unit at a given place, average number of customers in the queue waiting to be served at both places shall be equal to the sum of customers waiting at each service place   (1) If service places are of the same capacity EMBED Equation , and if customer arrival rates at both places are about equal EMBED Equation , average number of customers in the queue can be calculated as follows  (2) By analogy, formula for determining average number of customers in the queue can be derived for S service places operating independently (assuming the same customer arrival rate EMBED Equation  and service rate EMBED Equation )  (3) If service places differ by customer arrival and service rates, formula (3) can be nevertheless used, provided that EMBED Equation  and EMBED Equation  are assigned average values. If service places are not specialized, then each service place is not considered separately, but as an S-channel system of M/M/S/( type, where average number of customers waiting in the queue is calculated using the following formula  (4) where P0 is probability that all service places are unoccupied, i.e. probability that are no customers in a system. According to task defined in this paper it is necessary to compare formulas EMBED Equation  and EMBED Equation   (5) Since in single service place system EMBED Equation , where EMBED Equation  is traffic intensity, the demonstration continues by comparing formulas  (6) If formula (6) is to be greater than 0, the following inequation must be met  (7) i.e.,  (8) EMBED Equation.2 EMBED Equation.2 EMBED Equation.2 . (9) Value of EMBED Equation  in formula (9) is greater than 1, and EMBED Equation  is also greater than 1, since  (10) consequently, the left side of inequation (9) is greater than 1, meaning that EMBED Equation  is greater than EMBED Equation , which was to be proved. Since EMBED Equation , it is concluded that average number of customers and time a customer spends in the queue are considerably lower in one system with S service places, i.e., in multipurpose service place system than in S systems with one service place, i.e., specialized̋vތ&ZLDFh>$$$$$ & F $$ service places, whether customer arrival rates are equal for particular service place or not. SPECIALIZATION INDEX The previous part of this paper assumes that service rates for specialized service places equal to multipurpose places service rates. However, in real life, specialized service places should be more efficient than multipurpose service places since they are better equipped for providing intended services, and in that case the difference in number of customers in the queue will not be so high as mentioned above. In order to equalize specialized and multipurpose service places, considering number of customers in the queue, it is necessary to calculate service rate, using above proof, for specialized service places where number of customers in the queue will be equal (and not greater as shown) to multipurpose service places service rate. Border value of service rate for specialized service places is the solution of relation EMBED Equation , where A = S(LQ(1) and B = 1(LQ(S) . Considering above assumptions, service rate ((1) for specialized places is calculated using relation  (11) where: EMBED Equation  - multipurpose service place rate ks - specialization index. Specialization index for EMBED Equation  service places is defined as follows , (12) which leads to  (13) By analogy for  , (14) and for EMBED Unknown EMBED Unknown. (15) Above relations show that specialization index depends on EMBED Equation , EMBED Equation  and number of service places EMBED Equation , where ( represents customer arrival rate to specialized service place. Relations (12) (15) are valid when the number of service places in multipurpose system equals to the number of specialized service places. However, in reality, it is possible that implementation of specialization increases number of service places as compared to the multipurpose system. In such a case, specialization index ks is calculated using relation S1(LQ(1) = 1(LQ(S2), (16) where S1 stands for the number of specialized service places and S2 for the number of service places in one multipurpose system. Such, for S1= 4 and S2= 3, the following relation has been derived for specialization index EMBED Equation.2. (17) Defining value of ks becomes burdensome for higher values of S so, for convenience, it is suggested to enter mentioned values in a table. An example can be found in Table 1 for S = 2, ( = 0.1, & 0.9 and ( = 0.2,& 1.0. Table 1. Specialization index value (ks) for two service places according to values of EMBED Equation.2 and EMBED Equation.2  0.10.20.30.40.50.60.70.80.91.00.11.5001.8082.0662.2932.5002.6912.8693.0373.1970.21.3051.5001.6631.8081.9412.0662.1822.2930.31.2261.3751.5001.6111.7131.8081.8980.41.1811.3051.4081.5001.5841.6630.51.1521.2591.3481.4271.5000.61.1311.2261.3051.3750.71.1151.2011.2730.81.1031.1810.91.093 The empty part of the table does not contain calculated values of ks because ratio between EMBED Equation  and EMBED Equation  indicates the instability of a queuing system. Service rate for specialized service places can be calculated by multiplying ks (from Table 1.) (depending of EMBED Equation  and EMBED Equation ) by service rate of the multipurpose service place. If the service rate of a specialized place equals to EMBED Equation , then the number of customers in the queue will equal to that in system having multipurpose service places; if it is greater than EMBED Equation , queue will be smaller to that in system with multipurpose service places, and vice versa. Shown method can be used in planning service place types considering the service purpose. Of course, the final decision on specialization of service places should consider equipment required for specialized and multipurpose service places. 4. CASE STUDY - ORGANIZATION OF COUNTER SERVICE 4.1. Problem description The analysis of the service place specialization 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. The objective is to make business decision that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing clerks' idle time to minimum. 4.2. Problem definition 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 number of customers 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-first out). According to the task set, it is possible to make following business decisions: Organize multipurpose (universal) counters, meaning that any customer can be served at any counter Organize specialized counters, i. e., customers can be served at particular counter depending on service type required. Comparison of multipurpose and specialized counter indicators will yield an optimal solution of the observed problem. 4.3. 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 of M/M/3/( type and four systems of M/M/1/( type and shown in Table 2. Table 2. Comparison of Police Department counter service operation indicators by counter type No.IndicatorUnitMultipurpose service placesSpecialized service places1.2.3.1.Arrival rate (()cust./hour17.5745112.917652.658821.998042.Service rate (()cust./hour88883.Traffic intensity (()-2.196811.614710.332350.249754.Number of service places (S)counter32115.Utilization system coefficient ((/S)-0.732270.807350.332350.249756.Average number of customers in the queue (LQ)customer1.478453.022840.165440.083147.Average number of customers in the system (L)customer3.675274.637550.497800.332908.Average time a customer spends in the system (WQ)hour0.08413 (5.05 min)0.23400 (14.04 min)0.06223 (3.73 min)0.04161 (2.5 min)9.Average time a customer spends in the system (W)hour0.20913 (12.55 min)0.35901 (21.54 min)0.18723 (11.23 min)0.16612 (9.97 min)10.Probability that all service places are idles (P0)%8,1910.6666.7675.02 Results in Table 2. show that expected number of customers in the queue is 1.47845 and 3.27142 for multipurpose and specialized counters, respectively. This example supports the proof as per Chapter 2. of the paper, i.e., if the service rates are equal, the queue is greater in specialized than in universal service places. The same conclusion refers to the time customers spend in the queue as well as to busy/idle state of a counter. At last, the fact that cannot be neglected, is that specialization requires increased number of counters, specifically, four counter in this example. However, it is fair to assume that a specialized counter should have higher service rate, and according to relation (17) in Chapter 3. of the paper it comes out that for S = 3/4, (where the first number stands for number of counter in multipurpose system and the second for number of specialized counters), ( = 17.57451/4 = 4.39363, ( = 8, specialization index is ks = 1.21878 which requires specialized counter service rate of (1 = 9.75024 customers/hour. 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, or 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.Wsat0.17412 (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. 5. CONCLUSION In a queuing system, service places can be multipurpose or specialized. If both of service places have the same service rate, than the average number of customers and the time a customer spends in the queue are greater in specialized service places than in multipurpose ones. Using the queuing theory, above conclusion has been proven by comparing the number of customers in a queue for S systems with one specialized service place to one system with S multipurpose service places. However, considering better equipment of specialized service places, their service rate should be higher; specialization index is a number showing the smallest increase of service rate required in order to justify the implementation of specialized service places, i.e., to equal the system efficiency indicators that of multipurpose system. The presented method, recommended for planning of service place types considering its purpose, is illustrated through an actual counter service in a Police Department. REFERENCES D. Barkovi, Operacijska istra~ivanja, Ekonomski fakultet, Osijek, 2001. R. Bronson, Operations Research, McGraw-Hill, 1982. T. Pogny - Z. Zenzerovi, On the Parameter Shift Influence to the Total Waiting Time in M/M/S/( Queueing System, II, Proceeding of the 16th International Conference on Information Technology Interfaces, ITI 94, Pula, University Computing Center, Zagreb,1994,p.p. 397-402. Z.Zenzerovi, Optimizacijski modeli planiranja kapaciteta morskih luka, Ekonomski fakultet, dokt. disertacija, Sveu iliate u Rijeci, Rijeka, 1995. Z. Zenzerovi, Kapaciteti uslu~nih mjesta u funkciji efikasnosti sustava masovnog opslu~ivanja, Zbornik radova Ekonomskog fakulteta Rijeka, Rijeka, 1996. I. }upanovi, M. Gjumbir, Racionalizacija procesa izdavanja prijevoznih karata na autobusnom kolodvoru, Fakultet prometnih znanosti, Zagreb, 1978. 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A!"#$%IMPACT OF SERVICE PLACE SPECIALIZATION ON THE EFFICIENCY OF THE QUEUING SYSTEM FUNCTIONING Zdenka Zenzerovi Faculty of Maritime Studies at Rijeka Studentska 2, 51000 Rijeka, Croatia Telephone: 051/ 338-411;E-mail:  HYPERLINK mailto:zenzerov@pfri.hr zenzerov@pfri.hr Marija Marinovi Faculty of Philosophy Omladinska 14, 51000 Rijeka, Croatia Telephone: 051/345-034; E-mail: marinm@pefri.hr Abstract. This paper analyses the impact of service place specialization on efficiency of the queuing system operation. Using the queuing theory, it was proved that introducing service place specialization is only justified if the service rate of specialized service places is greater than those in multipurpose places. Specialization index has been introduced, showing the increase required of service rate, i.e., it is used to calculate the minimal capacity of specialized service places where the average number of customers in the queue and average time a customer spends in the queue will equal to the multipurpose service places. The method presented is useful in planning and organizing queuing system operation and it represents the basis when making decision as to whether to organize as specialized or multipurpose service place system. The results of analysis of the service place specialization impact are illustrated through example of actual organization of a counter service. Key words: queuing theory, service place specialization, specialization index, counter service INTRODUCTION The efficiency of a queuing system can be determined by the time customer spends in a system and unoccupancy of a service place. The values of these indicators are influenced by the number of service places and service rate of particular service place, considering given or expected number of customers. Depending on the service type offered to customers, service place could be multipurpose (universal) or specialized. Since in real life service places within a queuing system often offer various services, the main issue in organizing such a process is to decide whether service place should be multipurpose or specialized. The decision on service place type results in positive and negative impacts on queuing system functioning. In systems with specialized service places, customer is served at a service place specialized for a particular service; on the other hand, if service is organized with multipurpose service places, each service place handles any customer. If service rates of multipurpose and specialized service places are equal, it can be expected that the average number of customers and time of customers spend in queue would be greater/longer for specialized service places. So called specialization index has been introduced in this research, and it is used to determine limit value of service rate for a specialized service places. This value is used to equalize number of customers and time of customers in a queue for multipurpose system and it justifies implementation of specialized service places. The aim of this research was to present planning method and organization of queuing system considering the type of service place. The presented method was illustrated through a real life counter-service organization. 2. COMPARISON OF INDICATORS OF QUEUING SYSTEM FUNCTIONING FOR MULTIPURPOSE AND SPECIALIZED SERVICE PLACES In accordance with the defined objective, the following task has been defined: to explore whether number of customers waiting in a queue per time unit at all service places will be equal or significantly different, assuming that service places are multipurpose or specialized. By using the queuing theory, it is necessary to compare expected number of customers for a queuing system having a number of specialized service places with expected number of customers for a queuing system having an appropriate number of multipurpose service places. Assuming that two service places, each with respective given values for EMBED Equation  and EMBED Equation , i.e. having a number of customers arriving to the service place and a number of customers that can be served in an observed time unit at a given place, average number of customers in the queue waiting to be served at both places shall be equal to the sum of customers waiting at each service place   (1) If service places are of the same capacity EMBED Equation , and if customer arrival rates at both places are about equal EMBED Equation , average number of customers in the queue can be calculated as follows  (2) By analogy, formula for determining average number of customers in the queue can be derived for S service places operating independently (assuming the same customer arrival rate EMBED Equation  and service rate EMBED Equation )  (3) If service places differ by customer arrival and service rates, formula (3) can be nevertheless used, provided that EMBED Equation  and EMBED Equation  are assigned average values. If service places are not specialized, then each service place is not considered separately, but as an S-channel system of M/M/S/( type, where average number of customers waiting in the queue is calculated using the following formula  (4) where P0 is probability that all service places are unoccupied, i.e. probability that are no customers in a system. According to task defined in this paper it is necessary to compare formulas EMBED Equation  and EMBED Equation   (5) Since in single service place system EMBED Equation , where EMBED Equation  is traffic intensity, the demonstration continues by comparing formulas  (6) If formula (6) is to be greater than 0, the following inequation must be met  (7) i.e.,  (8) EMBED Equation.2 EMBED Equation.2 EMBED Equation.2 . (9) Value of EMBED Equation  in formula (9) is greater than 1, and EMBED Equation  is also greater than 1, since  (10) consequently, the left side of inequation (9) is greater than 1, meaning that EMBED Equation  is greater than EMBED Equation , which was to be proved. Since EMBED Equation , it is concluded that average number of customers and time a customer spends in the queue are considerably lower in one system with S service places, i.e., in multipurpose service place system than in S systems with one service place, i.e., specialized service places, whether customer arrival rates are equal for particular service place or not. SPECIALIZATION INDEX The previous part of this paper assumes that service rates for specialized service places equal to multipurpose places service rates. However, in real life, specialized service places should be more efficient than multipurpose service places since they are better equipped for providing intended services, and in that case the difference in number of customers in the queue will not be so high as mentioned above. In order to equalize specialized and multipurpose service places, considering number of customers in the queue, it is necessary to calculate service rate, using above proof, for specialized service places where number of customers in the queue will be equal (and not greater as shown) to multipurpose service places service rate. Border value of service rate for specialized service places is the solution of relation EMBED Equation , where A = S(LQ(1) and B = 1(LQ(S) . Considering above assumptions, service rate ((1) for specialized places is calculated using relation  (11) where: EMBED Equation  - multipurpose service place rate ks - specialization index. Specialization index for EMBED Equation  service places is defined as follows , (12) which leads to  (13) By analogy for  , (14) and for EMBED Unknown EMBED Unknown. (15) Above relations show that specialization index depends on EMBED Equation , EMBED Equation  and number of service places EMBED Equation , where ( represents customer arrival rate to specialized service place. Relations (12) (15) are valid when the number of service places in multipurpose system equals to the number of specialized service places. However, in reality, it is possible that implementation of specialization increases number of service places as compared to the multipurpose system. In such a case, specialization index ks is calculated using relation S1(LQ(1) = 1(LQ(S2), (16) where S1 stands for the number of specialized service places and S2 for the number of service places in one multipurpose system. Such, for S1= 4 and S2= 3, the following relation has been derived for specialization index EMBED Equation.2. (17) Defining value of ks becomes burdensome for higher values of S so, for convenience, it is suggested to enter mentioned values in a table. An example can be found in Table 1 for S = 2, ( = 0.1, & 0.9 and ( = 0.2,& 1.0. Table 1. Specialization index value (ks) for two service places according to values of EMBED Equation.2 and EMBED Equation.2  0.10.20.30.40.50.60.70.80.91.00.11.5001.8082.0662.2932.5002.6912.8693.0373.1970.21.3051.5001.6631.8081.9412.0662.1822.2930.31.2261.3751.5001.6111.7131.8081.8980.41.1811.3051.4081.5001.5841.6630.51.1521.2591.3481.4271.5000.61.1311.2261.3051.3750.71.1151.2011.2730.81.1031.1810.91.093 The empty part of̋.ȍD& @B*,<>@BV< ||56CJmH 6mH  6CJmH 0J6CJOJQJmH  5CJmH  5CJmH mH B*>*0JB*j?U jU5mH 0JCJH*OJQJmH  j0J6CJOJQJmH 0J6CJOJQJmH 0J5CJOJQJmH 0JCJOJQJmH /̋vތ&ZLDFh>$$$$$ & F $$ the table does not contain calculated values of ks because ratio between EMBED Equation  and EMBED Equation  indicates the instability of a queuing system. Service rate for specialized service places can be calculated by multiplying ks (from TablerdDocWord.Document.89qࡱࡱ> jNxNOP?P@PRRR&SSS TTTTTTVV@BV{56CJmH 6mH  6CJmH 0J6CJOJQJmH  5CJmH  5CJmH mH B*>*0JB*jU jU5mH 0JCJH*OJQJmH  j0J6CJOJQJmH 0J6CJOJQJmH 0JCJOJQJmH 0J5CJOJQJmH 0 &P . A!"#$%IMPACT OF SERVICE PLACE SPECIALIZATION ON THE EFFICIENCY OF THE QUEUING SYSTEM FUNCTIONING Zdenka Zenzerovi Faculty of Maritime Studies at Rijeka Studentska 2, 51000 Rijeka, Croatia Telephone: 051/ 338-411;E-mail:  HYPERLINK mailto:zenzerov@pfri.hr zenzerov@pfri.hr Marija Marinovi Faculty of Philosophy Omladinska 14, 51000 Rijeka, Croatia Telephone: 051/345-034; E-mail: marinm@pefri.hr Abstract. This paper analyses the impact of service place specialization on efficiency of the queuing system operation. Using the queuing theory, it was proved that introducing service place specialization is only justified if the service rate of specialized service places is greater than those in multipurpose places. Specialization index has been introduced, showing the increase required of service rate, i.e., it is used to calculate the minimal capacity of specialized service places where the average number of customers in the queue and average time a customer spends in the queue will equal to the multipurpose service places. The method presented is useful in planning and organizing queuing system operation and it represents the basis when making decision as to whether to organize as specialized or multipurpose service place system. The results of analysis of the service place specialization impact are illustrated through example of actual organization of a counter service. Key words: queuing theory, service place specialization, specialization index, counter service INTRODUCTION The efficiency of a queuing system can be determined by the time customer spends in a system and unoccupancy of a service place. The values of these indicators are influenced by the number of service places and service rate of particular service place, considering given or expected number of customers. Depending on the service type offered to customers, service place could be multipurpose (universal) or specialized. Since in real life service places within a queuing system often offer various services, the main issue in organizing such a process is to decide whether service place should be multipurpose or specialized. The decision on service place type results in positive and negative impacts on queuing system functioning. In systems with specialized service places, customer is served at a service place specialized for a particular service; on the other hand, if service is organized with multipurpose service places, each service place handles any customer. If service rates of multipurpose and specialized service places are equal, it can be expected that the average number of customers and time of customers spend in queue would be greater/longer for specialized service places. So called specialization index has been introduced in this research, and it is used to determine limit value of service rate for a specialized service places. This value is used to equalize number of customers and time of customers in a queue for multipurpose system and it justifies implementation of specialized service places. The aim of this research was to present planning method and organization of queuing system considering the type of service place. The presented method was illustrated through a real life counter-service organization. 2. COMPARISON OF INDICATORS OF QUEUING SYSTEM FUNCTIONING FOR MULTIPURPOSE AND SPECIALIZED SERVICE PLACES In accordance with the defined objective, the following task has been defined: to explore whether number of customers waiting in a queue per time unit at all service places will be equal or significantly different, assuming that service places are multipurpose or specialized. By using the queuing theory, it is necessary to compare expected number of customers for a queuing system having a number of specialized service places with expected number of customers for a queuing system having an appropriate number of multipurpose service places. Assuming that two service places, each with respective given values for EMBED Equation  and EMBED Equation , i.e. having a number of customers arriving to the service place and a number of customers that can be served in an observed time unit at a given place, average number of customers in the queue waiting to be served at both places shall be equal to the sum of customers waiting at each service place   (1) If service places are of the same capacity EMBED Equation , and if customer arrival rates at both places are about equal EMBED Equation , average number of customers in the queue can be calculated as follows  (2) By analogy, formula for determining average number of customers in the queue can be derived for S service places operating independently (assuming the same customer arrival rate EMBED Equation  and service rate EMBED Equation )  (3) If service places differ by customer arrival and service rates, formula (3) can be nevertheless used, provided that EMBED Equation  and EMBED Equation  are assigned average values. If service places are not specialized, then each service place is not considered separately, but as an S-channel system of M/M/S/( type, where average number of customers waiting in the queue is calculated using the following formula  (4) where P0 is probability that all service places are unoccupied, i.e. probability that are no customers in a system. According to task defined in this paper it is necessary to compare formulas EMBED Equation  and EMBED Equation   (5) Since in single service place system EMBED Equation , where EMBED Equation  is traffic intensity, the demonstration continues by comparing formulas  (6) If formula (6) is to be greater than 0, the following inequation must be met  (7) i.e.,  (8) EMBED Equation.2 EMBED Equation.2 EMBED Equation.2 . (9) Value of EMBED Equation  in formula (9) is greater than 1, and EMBED Equation  is also greater than 1, since  (10) consequently, the left side of inequation (9) is greater than 1, meaning that EMBED Equation  is greater than EMBED Equation , which was to be proved. Since EMBED Equation , it is concluded that average number of customers and time a customer spends in the queue are considerably lower in one system with S service places, i.e., in multipurpose service place system than in S systems with one service place, i.e., specializedRRRnSSUWRXxYzY|Y~YYYYYYLDFh>$$$$$ & F $$ service places, whether customer arrival rates are equal for particular service place or not. SPECIALIZATION INDEX The previous part of this paper assumes that service rates for specialized service places equal to multipurpose places service rates. However, in real life, specialized service places should be more efficient than multipurpose service places since they are better equipped for providing intended services, and in that case the difference in number of customers in the queue will not be so high as mentioned above. In order to equalize specialized and multipurpose service places, considering number of customers in the queue, it is necessary to calculate service rate, using above proof, for specialized service places where number of customers in the queue will be equal (and not greater as shown) to multipurpose service places service rate. Border value of service rate for specialized service places is the solution of relation EMBED Equation , where A = S(LQ(1) and B = 1(LQ(S) . Considering above assumptions, service rate ((1) for specialized places is calculated using relation  (11) where: EMBED Equation  - multipurpose service place rate ks - specialization index. Specialization index for EMBED Equation  service places is defined as follows , (12) which leads to  (13) By analogy for  , (14) and for EMBED Unknown EMBED Unknown. (15) Above relations show that specialization index depends on EMBED Equation , EMBED Equation  and number of service places EMBED Equation , where ( represents customer arrival rate to specialized service place. Relations (12) (15) are valid when the number of service places in multipurpose system equals to the number of specialized service places. However, in reality, it is possible that implementation of specialization increases number of service places as compared to the multipurpose system. In such a case, specialization index ks is calculated using relation S1(LQ(1) = 1(LQ(S2), (16) where S1 stands for the number of specialized service places and S2 for the number of service places in one multipurpose system. Such, for S1= 4 and S2= 3, the following relation has been derived for specialization index EMBED Equation.2. (17) Defining value of ks becomes burdensome for higher values of S so, for convenience, it is suggested to enter mentioned values in a table. An example can be found in Table 1 for S = 2, ( = 0.1, & 0.9 and ( = 0.2,& 1.0. Table 1. Specialization index value (ks) for two service places according to values of EMBED Equation.2 and EMBED Equation.2  0.10.20.30.40.50.60.70.80.91.00.11.5001.8082.0662.2932.5002.6912.8693.0373.1970.21.3051.5001.6631.8081.9412.0662.1822.2930.31.2261.3751.5001.6111.7131.8081.8980.41.1811.3051.4081.5001.5841.6630.51.1521.2591.3481.4271.5000.61.1311.2261.3051.3750.71.1151.2011.2730.81.1031.1810.91.093 The empty part of the table does not contain calculated values of ks because ratio between EMBED Equation  and EMBED Equation  indicates the instability of a queuing system. Service rate for specialized service places can be calculated by multiplying ks (from Table 1.) (depending of EMBED Equation  and EMBED Equation ) by service rate of the multipurpose service place. If the service rate of a specialized place equals to EMBED Equation , then the number of customers in the queue will equal to that in system having multipurpose service places; if it is greater than EMBED Equation , queue will be smaller to that in system with multipurpose service places, and vice versa. Shown method can be used in planning service place types considering the service purpose. Of course, the final decision on specialization of service places should consider equipment required for specialized and multipurpose service places. 4. CASE STUDY - ORGANIZATION OF COUNTER SERVICE 4.1. Problem description The analysis of the service place specialization 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. The objective is to make business decision that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing clerks' idle time to minimum. 4.2. Problem definition 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 number of customers 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-first out). According to the task set, it is possible to make following business decisions: Organize multipurpose (universal) counters, meaning that any customer can be served at any counter Organize specialized counters, i. e., customers can be served at particular counter depending on service type required. Comparison of multipurpose and specialized counter indicators will yield an optimal solution of the observed problem. 4.3. 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 of M/M/3/( type and four systems of M/M/1/( type and shown in Table 2. Table 2. Comparison of Police Department counter service operation indicators by counter type No.IndicatorUnitMultipurpose service placesSpecialized service places1.2.3.1.Arrival rate (()cust./hour17.5745112.917652.658821.998042.Service rate (()cust./hour88883.Traffic intensity (()-2.196811.614710.332350.249754.Number of service places (S)counter32115.Utilization system coefficient ((/S)-0.732270.807350.332350.249756.Average number of customers in the queue (LQ)customer1.478453.022840.165440.083147.Average number of customers in the system (L)customer3.675274.637550.497800.332908.Average time a customer spends in the system (WQ)hour0.08413 (5.05 min)0.23400 (14.04 min)0.06223 (3.73 min)0.04161 (2.5 min)9.Average time a customer spends in the system (W)hour0.20913 (12.55 min)0.35901 (21.54 min)0.18723 (11.23 min)0.16612 (9.97 min)10.Probability that all service places are idles (P0)%8,1910.6666.7675.02 Results in Table 2. show that expected number of customers in the queue is 1.47845 and 3.27142 for multipurpose and specialized counters, respectively. This example supports the proof as per Chapter 2. of the paper, i.e., if the service rates are equal, the queue is greater in specialized than in universal service places. The same conclusion refers to the time customers spend in the queue as well as to busy/idle state of a counter. At last, the fact that cannot be neglected, is that specialization requires increased number of counters, specifically, four counter in this example. However, it is fair to assume that a specialized counter should have higher service rate, and according to relation (17) in Chapter 3. of the paper it comes out that for S = 3/4, (where the first number stands for number of counter in multipurpose system and the second for number of specialized counters), ( = 17.57451/4 = 4.39363, ( = 8, specialization index is ks = 1.21878 which requires specialized counter service rate of (1 = 9.75024 customers/hour. 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, or 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.Wsat0.17412 (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. 5. CONCLUSION In a queuing system, service places can be multipurpose or specialized. If both of service places have the same service rate, than the average number of customers and the time a customer spends in the queue are greater in specialized service places than in multipurpose ones. Using the queuing theory, above conclusion has been proven by comparing the number of customers in a queue for S systems with one specialized service place to one system with S multipurpose service places. However, considering better equipment of specialized service places, their service rate should be higher; specialization index is a number showing the smallest increase of service rate required in order to justify the implementation of specialized service places, i.e., to equal the system efficiency indicators that of multipurpose system. The presented method, recommended for planning of service place types considering its purpose, is illustrated through an actual counter service in a Police Department. REFERENCES D. Barkovi, Operacijska istra~ivanja, Ekonomski fakultet, Osijek, 2001. R. Bronson, Operations Research, McGraw-Hill, 1982. T. Pogny - Z. Zenzerovi, On the Parameter Shift Influence to the Total Waiting Time in M/M/S/( Queueing System, II, Proceeding of the 16th International Conference on Information Technology Interfaces, ITI 94, Pula, University Computing Center, Zagreb,1994,p.p. 397-402. Z.Zenzerovi, Optimizacijski modeli planiranja kapaciteta morskih luka, Ekonomski fakultet, dokt. disertacija, Sveu iliate u Rijeci, Rijeka, 1995. Z. Zenzerovi, Kapaciteti uslu~nih mjesta u funkciji efikasnosti sustava masovnog opslu~ivanja, Zbornik radova Ekonomskog fakulteta Rijeka, Rijeka, 1996. I. }upanovi, M. 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(depending of EMBED Equation  and EMBED Equation ) by service rate of the multipurpose service place. If the service rate of a specialized place equals to EMBED Equation , then the number of customers in the queue will equal to that in system having multipurpose service places; if it is greater than EMBED Equation , queue will be smaller to that in system with multipurpose service places, and vice versa. Shown method can be used in planning service place types considering the service purpose. Of course, the final decision on specialization of service places should consider costs of equipment required for specialized and multipurpose service places. 4. CASE STUDY - ORGANIZATION OF COUNTER SERVICE 4.1. Problem description The analysisb[!&14ο`8?wZ0@penR~v60~a)F[.hsP;=}Ĥ\Y\`u;SDd|B  S A ? 2Ǽ-Y|#8|V_`!WǼ-Y|#8|V>`e 0%xcdd``gd``beV dX,XĐ )CRcgb v6znĒʂT ~35;a#L ! ~ Ay 9W;H&׀ of the service place specialization 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 - IE%"LPL0}27)?b.PH+22At 0py{v221)W2At1ufDd|B  S A? 2BEQU9ygFa`!YBEQU9ygFT@ `W0'xcdd``Vgd``beV dX,XĐ )ɁSRcgb x@؀깡jx|K2B* R .͔ `W01d++&104\ F\L 8@|Vj&2n Mp~ *_‡f#AClb(ΟŌwclE W&00aI+phae8ACwLLJ% ̀iSDd B  S A? 2,4CWNa`!,4CWNa`xJAg._g>DDEHRbaF,!੐*I  VVbb#H'2&<  ,2/:b2έLCI`r~V e^L 6KYK'4UO*hd%0p&h_Xv{d+V9K'20&>'(8q[{In JFV`9}Y9'yqDԐ{>/*JnU܃ɐA`0kgݝyUܒ|EW]kA?60n, Mne$VIp7eZ`}|7 ~hI9 L] RzΟv-4 `PcdbR ,.Iexf:uSDdB  S A? 2{<ŏW&n oWƠ`!O<ŏW&n o0 ؛1xOKA߼l4Yc4jҦsiCBBcj x= ^<'~C Xp;of3I;a6g{v-|V>1<3z2_;uv, Tk##p8W @ :jXXaq(8#⫭X}ﺫ`Vo3>L>' uA]\$1/c^g&(P{H>v.u.t}ZMS?0_,i/+(j|*V 'Y~2N]ykPrݰFF*ټiEIVn\g3&O&__*F4y~cMȦHDs2_ *SI/q'Ou>d/1ScFMiqVt⹠)/\|äi;\3Qգ+#ŷH k\cf1ZXoǩDd@<  C AYǀwRY8-yiؖ<=7 aURo8C+`x'F2^L j0BR-þ{ DdTB  S A? 2dט6>R: MP@ `!8ט6>R: MP@ 0= XJxUPKJA}U3M2(u5p@rEL.a mdv9.\u}Ne!VLtWUk>4vXPY"cdtb̀Rwy_sʣ֑DO)VlR<)xk-F2.¶F+ʚssuing ID and passports Counter 3 - Registration/deregistration of residence. Statistical observation in period from January 2, 2002 to May 2, 2002. produced data on total number of customers requiring one or more of above services or arrived at a counter, as follows: Counter 1 - 6588 customers Counter 2 - 1356 customers Counter 3 - 1019 customers. Considering working hours of six hour a day, the average number of customers per day and hour respectively is as follows: Counter 1 - 77.51 customers/day; 12.91765 customers/hour Counter 2 - 15.95 customers/day; 2.65882 customers/hour Counter 3 - 11.99 customers/day; 1.99804 customers/hour. It was also established that average processing time for any counter was 7.5 minutes. The objective is to make business decision that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing idle time of clarks to minimum. 4.2. Problem definition On the basis of above, it is possible to de2P|s7y nR,ף`!$|s7y nR, xcdd``cd``beV dX,$XĐ +7XRYaj`e~35[0d P@penR~^](;Vded( #6 *o.#duF9(0b2I $37X/\!(?71N飘xDF&2rc0X'ć`Q=cdbR ,.Ie`Ew3X?@r Equation@ rHP LaserJet 4/4MH Ma METAFILEPICT|> p .1  ` & MathType@ Symbol-!r"System-5Dd<  C A2gw|. JP"Mn#QM`!gw|. JP"Mn#Q`!x51 @EL\%$b K-P!`a0h<0^0Յa7)4s6hu$KT~L\wlGȱٷǭo*`Xۇ:܅rӗ>Oѻ0:b t׶bK~Y;a DdHB  S A? 2ƛyYjڨ`!ƛyYj@%9]xkAIblccֈE[miBŃ^*@ ƪ-J?E/"%zEmKn/6qgfwI un{3$B=OI6A6 p `argC/v`tgB Ỏ.lQng'3 wl7ƒ 8"A;f)w*˲a_vϗ}Y=.|-\,^$Zu޼\}bNfR Q?eAϊK~[`,)0_ͩ69P;t*;t1w跨2`|J:M@O/*dB{[VgKiՕ,0ؤ[I"}az nLweF=&9Hq,= ܄o~_٣4 jd nzݟQܻrG'~=#Yƫ^KVfbH8cCSy﹯y|yF7![)$/9JU9gD?oVP"t=ٽ?^qD痃ApZa ZBF6: %_&8DdB  S A? 22M}``!2M}ߌ` xJAgg/_`qTB*Ԃ"&`ψ X,|YYZYZ,}&eIrBc7B 4bj,KMWA+[bWJ(B42صNҋZ/@ G񋲞te&d:uivo=~:ηWh.7"^>1͖z0gDdB  S A? 2N- X1LWv-'TB`!N- X1LWv-'TXsxcdd```>0fine 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 number of customers and processing time in observed time unit follow a Poisson *q5j|wtPwEzws7}6]wP5Q5gd)]wޖb;lSt/;DdTB  S A? 2dAܖs @ `!8Aܖs   XJxUPAJP}3I b .)؝B/P PhZ.G…+oнx^wf{濙!4o цk+Kd sΘ+e*Mյ8<8~38-B$C7fia8Y=7/[kdm1QN3_ {$o/i^Wɬ75distribution, 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-first out). According to the task set, it is possible to make following business decisions: Organize multipurpose (universal) counters, meaning that any customer can be served at any counter Organize specialized counters, i. e., customers can be served a ĜL  312Ec21BUs30`|6GA?H0rCx|K2B* R\``w03@ d3DdB  S A? 2Dxv~b(`r{Ey`!qDxv~b(`r{E@'<]?xkA߼Mfun5TkwڦD"^Uڂ&^DO"=x_ ^=ɻV6ծc3g{y3ofo(>0Ɋ7/2zqzz< 3J5;r&!A 梵#iࡐ'>X\\Yw "OEXgZ{Xdk\~}[Sþ<Fp$. ݬtn&RY+>=_ET92ʑ" (f0a ?ǵ9g96}X#hz*, eho_A{GUa 􊡩EKբnjH^HH-KWΌt}leU5B<푳f4TsگL'(g|4$y͌惄3yknoy4{7ԿRQj:BiNFviLK2mvP5OI#=+ӶrNLO{β?fQ]'ɾ`TѮ<\C.Dq~]eDdB  S At particular counter depending on service type required. Comparison of multipurpose and specialized counter indicators will yield an optimal solution of the observed problem. 4.3. 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 of M/M/3/( type and four systems of M/M/1/( type and shown in Table 2. Table 2. Comparison of Police Department counter service operation indicators by counter type No.IndicatorUnitMultipurpose service placesSpecialized service places1.2.3.1.Arrival rate (()cust./hour17.5745112.917652.658821.998042.Serviceŷ;̇|Foo|)M]?,.: O}-Uws 2B`]!Q\%)6) g?j;[DdB  S A? 2e> `[/Y~r`!e> `[/Y~r `gxKK@g&+)|BAC g'+h  M[hAzZG(^ċC?@^ųqg6.!am;/`uQ` xe.[ڄ('HQI7J]W .D| <3e==swƍ5EWQ?x\ ^%XF}i;֊]n? 2,b;L \ܵ`!,b;L \Xqxcdd```>0 ĜL  312Ec21BUs30`|6GA?H0rCx|K2B* R\``w03^8Dd B  S A? 2SkTbl _c~A`!vSkTbl _cj18DxAkA{idK7M&AJcEЀ/BRcTjB)Z.DzY< D ׶83owI:a7{޼Y [J<=W<,HnWYj8,3:GG âwj`F+Q -./Zk \`C qyb#f~H돎ғQsfϽ<.wGYZM{lxGHOs%Oj.6Un"֍;@IxkNZZMZ}AfVBǤc:Kދyeޏ=2jZogz_:޳}Y |)|]$oȼ%gez$/vܰ_HolC' $/9. 8w\֮(!UU2! _ rate (()cust./hour88883.Traffic intensity (()-2.196811.614710.332350.249754.Number of service places (S)counter32115.Utilization system coefficient ((/S)-0.732270.807350.332350.249756.Average number of customers in the queue (LQ)customer1.478453.022840.165440.083147.Average number of customers in the system (L)customer3.675274.637550.497800.332908.Average time a customer spends in the system (WQ)hour0.08413 (5.05 min)0.23400 (14.04 min)0.06223 (3.73 min)0.04161 (2.5 min)9.Average time a customer spends in the system (W)hour0.20913 (12.55 min)0.35901 (21.54 min)0.18723 (11.23 min)0.16612 (9.97 min)10.Probability that all service places are idles (P0)%8,1910.6666.7675.02 Results in Tablm`2E2[ǻZ;̊_a+YPfE(fm ' y $|kR[WJ˸-$i*DFnfSx&50F.jFA0aIK)z<;:qXK QI7PY=b 92/|5 ʝ*+mk~PMtUZBTWGb\9lљِʻAT.p90,8/?%U)kt(RVY()ۮ()[W!_# o=y@o=z4E^=y5zUl&)?{zzYCk<'| Vǥݵ^2Y#RRf8>m`$wz2߼~:62H^>T*i?OUTT|Q;Kagʇ(8W: [ A]XM5LM3Nw"/a5 Equation@ XHP LaserJet 4/4MH Ma METAFILEPICT`N< u .1  @ & MathType DTimes New Roman-!X"System-Dd<  C A2 :xGv1.mѿ`!:xGv1.m@Rx51 @E%$ZE ;o`aZئX"*$"r o'"  +t/?<%PQ2P\%`Mv VYyU %Rs^R5w\s[k{K7<;yS> F}xh:E۪(mkQ}RDdXhB  S A? 2a>/h|YJ HXD=j`!5>/h|YJ HXDe 2. show that expected number of customers in the queue is 1.47845 and 3.27142 for multipurpose and specialized counters, respectively. This example supports the proof as per Chapter 2. of the paper, i.e., if the service rates are equal, the queue is greater in specialized than in universal service places. The same conclusion refers to the time customers spend in the queue as well as to busy/idle state of a counter. At last, the fact that cannot be neglected, is that specialization requires increased numbepAkRI*!nR~ gm~1?|Z]0FוYD; IilȜeE|%Ob"2?K&Tr"~Sr`m_7X83A/Pv(nm$S7^16=4cMr of counters, specifically, four counter in this example. However, it is fair to assume that a specialized counter should have higher service rate, and according to relation (17) in Chapter 3. of the paper it comes out that for S = 3/4, (where the first number stands for number of counter in multipurpose system and the second for number of specialized counters), ( = 17.57451/4 = 4.39363, ( = 8, specialization index is ks = 1.21878 which requires specialized counter service rate of (1 = 9.75024 customers/h@|xcdd`` @bD"L1JE `x0 Yjl R A@v n`f` UXRYvo`0L` ZZ]46Z` $~mՇX&1BM 6471@penR~.11pAÊ `@``b#RpeqIj.;t71HNDdB  S A? 2YWQ! ᵁo,F=5a`!-WQ! ᵁo,F=!`h(3x=OAg 㘏 E "A "i [QDV, c+ B(5KhHaQ"J"I+)ݻխd}g!"/x<'^$DA}_F^`NFu¼$iJ0 >'+%V'bn^)UׁT>ic,}>f(o3Ѽ17h9gLyN׌oѧ4π-ſ>GwZ_chooi7MJ&^zx3(Z *Ii[It#Qn[kQ,S PrG N => ͭrnODd|B  S A? 2 h:V'BLa^P`!V h:V'BLa>`e 0$xcdd``gd``beV dX,XĐ )CRcgb v6znĒʂT ~35;a#L ! ~ Ay 9W/;H& 7| p0泂4b[!&14ο`8?wZ0@penR~v60~a)F[.hsour. 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, or 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 :y"-x?$_DdB   S A ?  2ܩlg)C-W`!ܩlg)C- ``kx=K`.M?NЂढ)V(4Ђvtpե8*8:g{ox{.!,SQEEh*" ڦ?N3<!b'p(^݉0vbn{^]>(Se/,-*fstޢL/ Wa;l(yZCٸx5 E@RDxgj`fk?k#&=xywd3S"~/ml%##Ozp!'6Et32 }QeDdB   S A?  2\E$`!\E$@Hx5O ` ,NনS} - v$>C' $/9. 8w\֮(!UU2! _pAkRI*!nR~ gm~1?|Z]0FוYD; IilȜeE|%Ob"2?K&Tr"~Sr`m_7X83A/Pv(nm$S7^16=4cMӬ:y"-x?$ږDd  S2), while other services will be handled either at counters S3 or S4, or only at counter S3, thus reducing the number of counters to three. Variant 3 resulted from experience to date, where driving license were only handled at one counter. It is practicable if, on the basis of past number of customers as noted during observed period, average processing time ranges from 4 to 4.6 minutes, e.g. (=15 customers/hour. Table 4. shows results in comparison of indicators for proposed counter service variants. Table 4. Comparison of indicators for combined counter service variants IndicatorVariant 1.Variant 2.Variant 3. (( = 15)S432LQ2.122.886.15P037.6%27.47%27.8% On the basis of results shown in Table 4. and Table 2., one of the following business decisions should be made: Organize counter service as one system containing three multipurpose counters. Organize counter service with two counters, one specialized for driving licenses and the other multipurpose counter for other services. 5. CO t$/秆vSR Ή=*:)-q=#;㫧 A"ofc-ysiE=goiRDd< ! C A 2Cd$l@ (`B`!d$l@ (`B` !xcdd`` @bD"L1JE `xrYjl < %! `y&F=c2h@Hfnj_jBP~nbC7>]4;VdedpdL&x `6W&@| D@2_Ŝ0q[qaVt0&FNCLUSION In a queuing system, service places can be multipurpose or specialized. If both of service places have the same service rate, than the average number of customers and the time a customer spends in the queue are greater in specialized service plaB   S A ?  2ݹM*mgPMmU`!ݹM*mgPMmU̬9xO@߽BRm&dPbI qp0bl l:lb4NY޵4x͵s}?zeP`h!Y!>1u]2FJvc rk\5 .[W|ni&\je EZ )Q1üø YCܺI;?=3&4#۪$Wz.`e 0$xcdd``gd``beV dX,XĐ )CRcgb v6znĒʂT ~35;a#L ! ~ Ay 9W/;H& 7|ces than in multipurpose ones. Using the queuing theory, above conclusion has been proven by comparing the number of customers in a queue for S systems with one specialized service place to one system with S multipurpose service places. However, considering better equipment of specialized service places, their service rate should be higher; specialization index is a number showing the smallest increase of service rate required in order to justify the implementation of specialized service places, i.e., tM`-w 0py{Ĥ\Y\Nf:/ Dd$hB " S A ? !2tŦ/8FPi`!HŦ/8F@E |xcdd`` @bD"L1JE `x Yjl R A@2 N`c`x UXRYvo`0L`A $37X/\!(?71]<6j`dg)ց%Al 5p0T 1`_ +ss`.w+Hi.Ŀo 1@sAC `@ ;F&&\w~bf@dnJ  Equation@ m E METAFILEPICT=|=, p .1  `  & MathType@Symbol-!m"System-Dd< # C A"2- ^W1ڛRLx `!- ^W1ڛRLx `Ƚ!xcdd``.````b`V dX@`FE%"LPL0}27)?b.PH+22At 0py{v221)W2At1ufDd< $ C A!#257oB73"o equal the system efficiency indicators that of multipurpose system. The presented method, recommended for planning of service place types considering its purpose, is illustrated through an actual counter service in a Police Department. REFERENCES D. Barkovi, Operacijska istra~ivanja, Ekonomski fakultet, Osijek, 2001. R. Bronson, Operations Research, McGraw-Hill, 1982. T. Pogny - Z. Zenzerovi, On the Parameter Shift Influence to the Total Waiting Time in M/M/S/( Queueing System, II, Proceeding of the 16th International Conference on Information Technology Interfaces, ITI 94, Pula, University Computing Center, Zagreb,1994,p.p. 397-402. 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This example supports the proof as per Chapter 2. of the paper, i.e., if the service rates are equal, the queue is greater in specialized than in universal service places. The same conclusion refers to the time customers spend in the queue as well as to busy/idle  m( hat specialization requires increased number of counters, specifically, four counter in this example. However, it is fair to assume that a specialized counter should have higher service rate, and according to relation (17) in Chapter 3. of the paper it comes out that for S = 3/4, (where the first number stands for number of counter in multipurpose system and the second for number of specialized counters), ( = 17.57451/4 = 4.39363, ( = 8, s%LpI4I k s1 =2 3 + 4 6 "16 5 "8 4  2 +6 3  3 +12 2  4 +6 5 4 2  9service rate should be increased by 21.9%. 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