Application of ANFIS for the Estimation of Queuing in a Postal Network Unit: A Case Study

Transcription

1 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study Bojan Jovanovć 1, Tatjana Grbć 1, Nebojša Bojovć 2, Momčlo Kujačć 1, Dragana Šarac 1 1 Unversty of Nov Sad, Faculty of Techncal Scences Trg Dosteja Obradovća 6, Nov Sad, Serba bojanjov@uns.ac.rs, tatjana@uns.ac.rs, kujacc@uns.ac.rs, dsarac@uns.ac.rs 2 Unversty of Belgrade, Faculty of Transport and Traffc Engneerng Vojvode Stepe 305, Belgrade, Serba nb.bojovc@sf.bg.ac.rs Abstract: Regardless the level of technologcal development n a communty, the unavodable phenomenon s the appearance of queung. Ths stuaton wll contnue as long as there s the customer s need for the drect contact wth the servce supplers, as the case s n the Post of Serba. The am of the paper s to estmate the tme a customer wll spend n queung whle approachng the counter for fnancal servces n a postal network unt. The observed system comprses a sngle queue, three handlng channels and the servce accordng to the FIFO prncple. Ths paper presents a developed model that s realzed n the followng phases: recordng data, preparng data for tranng, tranng the neuro-fuzzy system, formng a data set for testng where the expected mean servce speed s obtaned usng the movng average method, and testng the neuro-fuzzy model. Observng the mass servce system has so far been drected towards, the evaluaton of ther behavour n the past whch presents a bass to evaluate whether the system provdes satsfactory performances. Ths paper moves a step n the drecton of the behavour evaluaton of a mass servce system, n the future, n order to observe whether t s possble to predct the servce qualty level to be provded to a customer. System customer n ths case s not lmted by the number of demanded servces. Keywords: Watng tme; Postal network unt; Fnancal servces; ANFIS 1 Introducton Inspred by the fact that the contemporary research of the mass servce systems has been drected towards the system analyss n the past, as well as the sgnfcance of the system behavour n the future and the possble applcaton n the estmated queung tme (for provdng the adequate level of servce and the 25

2 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study engagement of the necessary resources), we have decded to apply the combnaton of all the exstng methods to a postal network unt n the Post of Serba. Ths paper presents what can be consdered an nnovatve method presentng the combnaton of the two known methods, ANFIS approach and movng average method. Prevous approaches were only based on one method, e.g. on the mass servce theory or Monte Carlo method, see [1, 26]. Snce the method presented n the paper ntroduces the combnaton of the ANFIS approach and the movng average method, we expect that ths method or a modfcaton of t wll fnd the applcaton both n other postal unts and everywhere else where queung s a regular phenomenon. In order to realze the mass servce system management n the real tme, the dataset that traned the model n ANFIS has been modfed usng the movng average method wth the goal to estmate the system behavour, whch can be consdered a step forward n comparson to prevous research. Mass servce system managers are faced wth the ssue of provdng an optmal relatonshp between the engaged resources and the tme spent by the customer n order to realze ther own demands. Тhe satsfacton wth the watng tme s not only a determnant for the satsfacton wth the servce; t also moderates the relaton satsfacton loyalty [6]. The tradtonal standpont n the mass servce theory s amed at the optmal determnaton of the handlng channel, where the customer s seen as an external factor. However, more attenton s devoted recently to customer behavour and unfcaton of behavoural aspects, as well as studes dealng wth servce operatons. The mpact of watng on the evaluaton of servce presents the man focus n numerous papers [5, 9, 18, 28, 31]. The queston for queue management s not only the real tme spent by a customer n a queue, t s also the customer s percepton of that tme and the related level of ther satsfacton wth the servce [8]. Informaton on the watng tme has a postve mpact on the customer s satsfacton wth the watng tme [6]. After the realzed servce, the customer knows the watng tme spent and they use the experence to adjust ther vew on the average watng tme n a vsted faclty. Followng t, one can present the customer s degree of satsfacton as a quotent of the acceptable and the expected watng tme based on the prevous experence [30]. If a customer s provded wth the nformaton on the expected watng tme, ther acceptable watng tme wll be close to the expected tme, snce that nformaton would nfluence the decson whether to approach the system or not at a certan perod of tme. The problem that servce provders have to deal wth and that leads to alternate servce qualty s observed n the fluctuaton of servce demands. Solvng the problem can lead, n one hand, to the development of more flexble systems (durng the peak load addtonal resources are ncluded), or on the hand, t can nfluence the customer s motvaton by provdng more satsfactory servce 26

3 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 realzaton condtons durng low demand perods [6]. Lkewse, coordnatng demands and capactes wll allow for the utlzaton of dverse strateges for the queue organzaton, consderng ther confguraton or the formaton of a certan ambent, so that watng can be more nterestng and more bearable [34]. When watng tme estmaton s consdered, the research n call centres has been drected towards queung n the FIFO system [3, 10, 11, 14, 32]. Servces realzed by tradtonal postal operators are characterzed by heterogenety, whch s a consequence of dfferent contents of postal tems, as well as a wde range of fnancal servces (payments out and payments n from bank accounts, agents work for the beneft of banks, advce on recepts, etc.). The demand for servces provded by tradtonal postal operators has a stochastc character and t cannot be accurately determned snce t fluctuates n ntensty over tme and space. These oscllatons regardng servce demands hamper technologcal processes n terms of provdng adequate resources for ther mplementaton. Accordng to the Law on payment transactons [25], the publc postal operator n Serba s allowed, n addton to provdng postal servces, to provde fnancal servces as well. The bankng sector as a competton to the publc postal operator n the feld of fnancal servces s concentrated manly on the segment of customers who are credtworthy. It s very mportant to nclude the segments of populaton that are not of nterest to the banks. Consderng ths, the state has the venue for the fnancal ncluson through the publc postal operator. As a consequence, the fnancal avalablty s reflected n adjustng prces to the solvency of the populaton segments that are the target area of the fnancal ncluson. The predcton of the watng tme n postal network unts s of great mportance n order to ensure adequate capacty for provdng these servces. Lkewse, n order to offer a hgher qualty to the fnancal servce benefcares, t s essental for them as well to be provded wth the nformaton on the queung tme, whch s the research topc dscussed further n the paper. The paper s organzed n the followng manner: second secton contans the analyss of the exstng condtons n a postal network unt, the thrd secton contans a proposton for a model for watng tme estmaton, and the last secton provdes the concluson wth the drectons for future research. 2 Condton Analyss n a Postal Network Unt Data acquston was conducted by recordng queung n a postal network unt for provdng servces to customers. The observed postal unt ncluded counters for fnancal servces, as well as counters for the recepton and delvery of postal tems. The selected postal unt s located n a resdental neghbourhood, as well as 27

4 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study n the area wth a wealthy number of dverse contents (green market, shoppng mall and school). Accordngly, benefcares were both the people lvng n the area, as well as vstors to the green market, shoppng mall, etc. Based on the above, t can be consdered that the observed postal unt s a representatve unt for provdng customer servce snce t s not focused only on a specfc segment of customers and snce the users of fnancal servces form a sngle queue. The objectve of the research refers to the counters for the mplementaton of fnancal servces, so they were n the focus of the recordng. Consderng the fact that the volume of fnancal servces s the hghest n March and December [22, 23, 24], the recordng was completed n the perod of two weeks n December (from December, 6 to December, 18),.e. the perod of 12 workng days. Workng hours of the post offce s from 7:00 am to 7:00 pm on weekdays and from 7:00 am to 2:00 pm on Saturdays. The observed system ncludes three handlng channels, wth a sngle queue and the servce carred out accordng to the FIFO prncple. The recordng ncluded the number of 5727 system customers. Followng the Kendall s notaton queung theory, the system can be classfed as M/G/3. Testng the nput stream of customers was realzed accordng to days n Statstca 10 (StatSoft software package). An example for the frst day s provded n Table 1. It can be observed that the wdths of the classes are defned n the perods of 50 seconds. Table column Observed Frequency presents the recordng frequences, whle the column Expected Frequency presents the theoretcal frequences of the observed classes for exponental dstrbuton. Table 1 Testng the compatblty of the nput stream wth the exponental dstrbuton The header of the table presents the values of χ 2 test, the number of degrees-offreedom, and the resultng p value. Based on the provded p-value of , t can be observed that the dstrbuton of the nput stream for the frst day corresponds to the exponental dstrbuton at the sgnfcance level of Fgure 1, presents graphcal nterpretatons of recorded data for the presented day. 28

5 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 Fgure 1 Graphcal overvew of the nput stream of clents durng the frst day Investgatons were carred out for the remanng days as well, and the results obtaned are shown n Table 2. Based on the results shown n Table 2, t can be concluded that the md-ntervals of the nput streams of customers (tme flow between successve arrvals of customers) durng the observed days n nne cases correspond to the exponental dstrbuton wth the relablty level of 95%,.e. p- values are lower than 0.05 n three cases out of the observed 12 days. Table 2 p-values of the nput streams of customers n testng the compatblty wth exponental dstrbuton Days p-values <10-5 * * * By ntegratng the acqured data for the observed perod, t s clear that the tme ntervals of md-arrvals n the nput stream of clents correspond to the exponental dstrbuton wth the parameter λ = customer/mn, wth the relablty level of 95% (.e. p-value equals ). Fgure 2 Dstrbuton of servce speed for the observed perod of 12 days 29

6 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study After examnng the complance of the nput data stream, the next to be tested was the servce tself. It was establshed that the unfed servce for the observed perod tends towards the log-normal dstrbuton wth the parameters a = and b = (Fgure 2), wth the relablty level of 99% snce the low p-value of 0.02 was acqured. As for the data acqured for the observed days n 5 cases, after testng the compatblty wth the log-normal dstrbuton, t can be observed that the obtaned p-values are lower for 0.01 (Table 3), whereas n other cases t can be consdered that the servce level corresponds to the log-normal dstrbuton. Table 3 p-values of the nput streams of customers n testng the compatblty wth log-normal dstrbuton Days p-values * <10-5 * <10-4* 0.001* <10-5 * We can say that the precse analyss on the system wth the form M/G/n s extremely demandng snce the non-markovan processes are happenng nsde the system, hence the formulas for calculatng the system parameters that are wdely appled n the cases of, for example, M/M/n systems, cannot be appled n ths case. The observed system s characterzed by heterogenety, whch s a consequence of dfferent contents of postal tems, as well as a wde range of fnancal servces (payments out and payments n from bank accounts, agents work for the beneft of banks, advce on recepts, etc.). The demand for servces provded by tradtonal postal operators has a stochastc character and t cannot be accurately determned snce t fluctuates n ntensty over tme and space. In practce, the most commonly appled analyss s the approxmaton or smulaton (as n the case of call centres [33, 35]). Concernng the observed system, the stuaton s even more complcated wth the number of actve handlng channels altered durng the observed nterval. Dfferently from call centres, postal clerks are n a drect contact wth servce users and the overall ambent created n the counter hall. As a consequence, the workng performances of clerks are exposed to the nfluence of the phenomenon of congeston whch may lead to ther workng deteroraton (clerks fatgue) or mprovement (strvng to perform the job better to elmnate the occurrence of congeston). Lkewse, the customer s possblty to clam an unlmted number of servces further complcated the observance of the system. Consderng the complexty of the observed system,.e. the uncertanty and stochastcty as ts propertes, the research was focused on the development of a model based on the neuro-fuzzy approach for the evaluaton of the watng tme. 3 Proposton for a Model The man results are presented n ths secton of the paper. It descrbes the theoretcal bass for the model created wth ANFIS. After that, there s a proposton for a model for estmatng the watng tme. The preparaton of data for tranng ANFIS s realzed. Tranng ANFIS s carred out by the fuzzy logc 30

7 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 toolbox, Matlab R2007b. It s followed by the data preparaton for model testng. The test results are presented by RMSE (root mean square error), the coeffcent of determnaton and the dstrbuton of error values (dfferences between smulated and recorded values). At the end of ths chapter, the average watng tme per each day s observed, as well as the movement of the queue length n selected days. 3.1 Neuro-Fuzzy Systems Dfferent technques of artfcal ntellgence have been developed to solve problems of the real world usng ntellgent systems that possess sklls smlar to human sklls n certan domans. Among them, fuzzy logc and neural networks are the most popular and wdely appled n ndustral applcatons [4, 15, 16, 17, 20, 21]. Fuzzy logc systems are wdely appled n transportaton engneerng [29]. Adaptve Neuro-Fuzzy Inference System s a mult-layer adaptve network based on the fuzzy nference system [13]. ANFIS s a fuzzy nference system that can be traned on the bass of the acqured nput-output data. The tranng method allows the system to adjust ts parameters n order to perceve the nput-output relatonshp hdden n the data set. Snce t comprses two approaches (neural networks and fuzzy modellng), the approprate nference n the qualty and the quantty can be acheved [2]. In the case of the Sugeno fuzzy model of the frst type wth two nputs x and y, the rules are gven n the followng form: Rule 1: f x s A 1 and y s B 1 then f 1 =p 1 x+q 1 y+r 1 Rule 2: f x s A 2 and y s B 2 then f 2 =p 2 x+q 2 y+r 2 (1) where A 1, A 2, B 1, B 2 are membershp functons for the nputs x and y respectvely, whereas p 1, q 1, r 1, p 2, q 2, r 2, are the parameters of the output functons [27]. The correspondng ANFIS structure conssts of fve layers that are mplemented by dverse functons of the nodes for tranng and by adjustng the parameters of a fuzzy system (Fgure 3). Nodes located wthn the same layer have smlar functons. The output of the -th node wthn the layer l can be referred to as O 1,. The functonng of the presented ANFIS system can be graphcally represented as follows [13]: Fgure 3 ANFIS archtecture of the Sugeno fuzzy model wth two nputs [13] 31

8 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study - Layer 1: The nodes of ths layer generate membershp functons for nput varables. The output of the nodes O 1, s defned by the followng expressons: O O 1, 1, μ μ A B 2 (x) (y) for 1,2or for 3,4 where x and y are nputs nto the node, whle A and B are fuzzy sets related to the observed node and are defned by the shape of ther membershp functon. Membershp functons A and B can be presented by the generalzed bell functon: 1 μ (x) (3) 1 x c /a 2b where a, b and c are parameters that change the shape of the membershp functon μ (x) from the mnmum value 0 to the maxmum value 1. The parameters of ths layer correspond to the parameters of the premse (hypothess) of the fuzzy model. Outputs of the frst layer are values of the membershp functon of the premse. - Layer 2: The nodes labelled wth П make up the second layer, whch means that the nput sgnals n the node are multpled and the output of the node O 2, presents a strength of the -th rule w whch s calculated as follows: O w μ (x)μ (y) 1,2 (4) 2, A B - Layer 3: In ths layer the nodes labelled wth N calculate the rato of the strength of the -th rule and the sum of strengths of other rules, whle the normalzed power of the -th rule s obtaned as follows: O w w 1,2 (5) w w 3, Layer 4: The nodes of the fourth layer calculate the contrbuton of the -th rule to the output of the system wth the followng node functon: O w f w (p x q y r ) 1, 2 (6) 4, where s the output of the thrd layer, whle a set of parameters (p, q, r ) corresponds to the parameters of consequences. - Layer 5: Ths layer conssts of one node that s denoted by and calculates the total output of the ANFIS as follows: O 5, w f (7) w w f (2) 32

9 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 It may be noted that wthn the ANFIS archtecture there are two adaptve layers, those beng the frst and the fourth layer. The parameters that are set wthn the frst layer are connected to the nput membershp functon (n the explaned example those are parameters a, b and c ), the so-called premse parameters. Wthn the fourth layer, parameters that are set relate to the frst-order polynomal (p, q and r ) and are referred to as consequence parameters [12, 13]. Neuro-fuzzy nference system s optmzed by adaptng the premse parameters and the consequence parameters n a manner as to mnmze the defned objectve functon (most common, the dfference between model outputs and actual outputs). The methods for mprovng the ANFIS parameters may nclude the gradent descent and Least Square Error (LSE) [13]. Chen (1999) compared the algorthms for tranng parameters n ANFIS membershp functons [7]. The paper appled the hybrd learnng algorthm that s a combnaton of the least square estmaton and the back-propagaton algorthms [13]. 3.2 Model Development and Results Overvew Developng the model that s presented n ths paper has been mplemented through the followng stages (Fgure 4): recordng the mass servce systems, creatng a data set for tranng ANFIS, tranng ANFIS, establshng data for testng, and testng and evaluatng the model. Durng the recordng process, the followng varables were ncluded: the number of actve handlng channels, queue length (.e. number of customers) and the speed of the servce level (customer/mn). These values are used as nputs to ANFIS. Fgure 4 A model for the watng tme estmaton In preparng data for system tranng, t was necessary to modfy the nput sze of the handlng speed. Namely, the problem occurrng s reflected n the fact that, when a customer accesses the queue, t s necessary to predct the servce speed untl the arrval to the servce. Regardng ths, the decson was to observe the servce speed n segments of ten customers, for whom the average servce speed was calculated. It was decded not to tran the system wth data on speed servce that was recorded for each customer snce n ths manner the system would become too senstve to ths parameter. In other words, the predcton methods could not provde exact fgures of the level obtaned at the level of recordng. The next phase n the observed model s the system tranng. The tranng of the system s realzed n the ANFIS Edtor GUI, wthn the software Matlab R2007b. 33

10 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study Applyng the subtractve clusterng method, wth the default parameter values beng the range of nfluence 0.5, squash factor 1.25, accept rato 0.5 and reject rato 0.15, the ntal fuzzy tranng system was formed,.e. Sugeno fuzzy model wth three nputs (Fgure 8). Each of the three nputs s assocated wth two membershp functons (Fgure 5, 6, 7). Membershp functons n Fgure 5, are defned by the followng parameters: blue - and red -. Membershp functons n Fgure 6, are defned by the followng parameters: blue - and red -. And the parameters for the membershp functon n Fgure 7, are as follows: blue - and red -. Fgure 5 Fgure 6 Fgure 7 Number of actve handlng Servce speed Queue length channels (customer/mn) (number of customers) The system s defned by the two-rule base, as well as by two output membershp functons. The number of epochs for tranng the system s 300, whle the hybrd algorthm for tranng s mplemented. Features of the system are stablzed after 20 epochs already to the RMSE value of 80 seconds. Alterng the values of the ntal set parameters for applyng the subtractve clusterng method does not brng any sgnfcant mprovement related to the RMSE, even wth the ncrease n the number of the membershp functons (Table 4). In the case of decreasng the value of the range of nfluence, the number of generated membershp functons s ncreased. A lttle decrease n RMSE to 77 seconds s observed, though the surface of the transfer functon s dramatcally dsrupted. The smlar stuaton apples to the alteraton of other parameters n the drecton of ncreasng the number of the membershp functons. The graphcal representaton of the transfer functon of the obtaned system can be observed accordng to the pars of nput szes: number of handlng channels vs. servce speed (Fgure 9), number of handlng channels vs. queue length (Fgure 10), and servce speed vs. queue length (Fgure 11). A good feature of the obtaned surfaces s reflected n the fact that they do not have any peaks, that s, there s a bland transton on changng the nput values. 34

11 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 range of nfluence Table 4 RMSE as parameters n the subtractve clusterng method squash factor accept rato reject rato number mf RMSE Fgure 8 Fgure 9 ANFIS structure of the proposed model Transton functon for the nput of handlng channels vs. servce speed Fgure 10 Fgure 11 Transton functon for the nput of Transton functon for the nput of handlng channels vs. queue length servce speed vs. queue length After tranng the model, a new data set for the system testng was establshed. In addton to nput values, the number of handlng channels and the queue length, the value to be estmated was the speed of the handlng system when the n-th customer approaches. The estmaton of the speed rate was acheved usng the smple movng average method, n a manner that the speed from 3 prevous ntervals of ten recorded data, each was consdered. Data set for testng was reduced to 5376 due to the ntal segments for the movng average method for each day. 35

12 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study In order to examne the degree of smlarty of the developed model wth the recorded condtons n the postal network unt, absolute errors smulated vs recorded were calculated. The dstrbuton of errors s gven n Fgure 12, where t can be observed that the expected value of the error s seconds wth a standard devaton of 154 seconds. The level of smlarty of the smulated and the recorded data s expressed wth the coeffcent of determnaton (Fgure 13): (8) where S r are the recorded values, whle S s are smulated values. In comparson to the data recorded, the smulated data of the represented model have RMSE of 154 seconds and the coeffcent of determnaton wth the value of Lkewse, the model smulaton was mplemented wth the movng averages of 4 and 10 ntervals, where the results obtaned for RMSE were 158 sec and 167 sec, whle the coeffcents of determnaton were and 0.796, respectvely. A nderson-darlng Normalty Test A -Squared P-Value < Mean StDev V arance Skew ness Kurtoss N 5367 Mnmum st Q uartle Medan rd Q uartle Maxmum % C onfdence Interv al for Mean Mean 95% Confdence Intervals % C onfdence Interv al for Medan % C onfdence Interv al for StDev Medan Fgure 12 Dstrbuton of the error values n Mntab 16 Fgure 13 Comparson of recorded values and values obtaned n ANFIS 36

13 Acta Polytechnca Hungarca Vol. 12, No. 7, 2015 The model used n the paper s the one wth the movng averages from 3 prevous ntervals. It also possesses the advantage observed n the fact that too much tme would not elapse before the model begns the estmaton n comparson to the begnnng of the workng hours n the postal network unt for provdng servces to customers. Certanly, the sooner the estmaton begns, a lower number of unsatsfed customers s to be expected snce queung appears to last longer to a customer wth the lack of nformaton on ts duraton compared to the customer wth the known watng tme [19]. In contnuaton, the average watng tme per day was observed, as well as the values of average watng tmes obtaned by smulaton. Table 5, provdes the rato of average watng tmes that were recorded and average tmes obtaned by smulaton, where t can be observed that these values are very smlar (maxmum devaton s 17 seconds). The mnmum average watng tme of 176 seconds was observed on the fourth day of recordng,.e. n the nnth day of the month when dfferent nqures nfluencng the ncrease n volume of fnancal servces have not yet come to ther due dates. Durng ths day, the peaks related to the queue length reached the value of 7 customers at 9:50 and 11:30 n the mornng. In the afternoon, the queue length reached a maxmum value of 3 customers at 14:00, 15:00 and 18:00. Table 5 Average watng tmes recorded vs. smulated Day Average watng tme, recorded, sec. Average watng type of the model, sec The hghest average watng tme was observed on the 11 th day of recordng. The reason for ths s reflected n the fact that t was the 17 th day of the month when dverse nqures are due to be pad. Addtonal mpact was caused by the stuaton that t was the last workday for most customers, and also by the fact that busness polces of some companes provde ther customers a certan dscount for payments before the 20 th of the month. The maxmum queue length n the mornng reached a number of 24 customers at 10:30. In the afternoon, the longest queue was observed at 16:20, wth 28 customers. As an average day, there s the 8 th day of recordng. In the mornng, the queue length reached the maxmum value of 27 customers at 11:30, whle n the afternoon the maxmum queue length was wth 14 users at 13:45. Concluson Ths paper presents a model for estmatng the watng tme based on ANFIS. The developed model shows a satsfactory result consderng the value of the coeffcent of determnaton R 2 = 0.826, as well as the value of RMSE of

14 B. Jovanovć et al. Applcaton of ANFIS for the Estmaton of Queung n a Postal Network Unt: A Case Study seconds. The average watng tme recorded per day and the average watng tme obtaned by the proposed model provde smlar results (Table 5). Estmatng the servce tme as the nput value requred an adequate modfcaton (average value of 10 servng tmes, as well as the applcaton of the movng average method), so that the proposed model would be more relable to descrbe the observed queung system. The contrbuton of the model s n provdng nformaton to both the customers and the management of the postal network unt. Provdng nformaton on the watng tme presents an addtonal qualty of the servce snce customers can expect how much of ther tme to set asde, whch makes the watng process appear acceptable. The same nformaton provdes an opportunty for post offce managers to manage a number of actve counters n real tme. In other words, by changng the parameters n the proposed system an adequate smulaton can be mplemented, whch wll then ndcate whether a partcular procedure s justfed,.e. whether the avalable resources are utlzed n an optmal manner. Further development wll be focused on estmatng the number of consumers who wll ask for a servce at specfed tme ntervals and, thus, provde a tmely actvty n order to try to prevent the congeston of the system. Further consderatons may nclude the number of servces to be observed n order to obtan an average speed of servce n the observed tme nterval, as well as the number of sample ntervals n estmatng the servce speed by usng the movng average method. Acknowledgements Ths research s supported by the Mnstry of Scence of Serba, Grant Numbers 36040, , References [1] I. Adan, J. Resng, Queueng theory, Dept. of Mathematcs and Computng Scence, Endhoven Unversty of Technology, Netherlands, 2002 [2] M. Alzadeh, F. Jola, M. Amnnayer, R. Rada, Comparson of Dfferent Input Selecton Algorthms n Neuro-Fuzzy Modelng, Expert Systems wth Applcatons, 39(1), , 2012 [3] M. Armony, C. Maglaras, Contact Centers wth a Call-Back Opton and Real-Tme Delay Informaton, Operatons Research, 52(4), , 2004 [4] G. Athanasopoulos, C. R. Rba, C. Athanasopoulou, A Decson Support System for Coatng Selecton Based on Fuzzy Logc and Mult-Crtera Decson Makng, Expert Systems wth Applcatons, 36(8), , 2009 [5] J. Baker, M. Cameron, The Effects of the Servce Envronment on Affect and Consumer Percepton of Watng Tme: an Integratve Revew and Research Propostons, Journal of the Academy of Marketng Scence, 24(4), ,

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