CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY

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1 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Sidonia Otilia Cernea Mihaela Jaradat 2 Mohammad Jaradat 3 ABSTRACT: Thi paper i dedicated to the preentation of the ingle-channel waiting line ytem with Poion arrival and exponential ervice time.they repreent tage of cutomer flow management procee. Waiting ytem are tochatic mathematical model and they repreent the decribing bae of the waiting phenomena, ervice procee, prioritization, etc. Mathematical model of queuing theory preent interet in modeling, deigning and analying of nowaday information network. Increaing trend of their development and emergence of new network technologie, impoe new requirement regarding the development of new mathematical waiting model. In thi paper we merely preent the ingle-channel waiting line model, with example on a fat-food retaurant. Key word: cutomer flow, queuing theory, waiting ytem, waiting line, efficiency JEL code: A2, C4, C46, L84 Introduction There are many cae when we face the waiting ituation. We find ourelve in uch ituation everyday at checkout, in upermarket, bank, retaurant, etc. In order to reduce the time pent in waiting ytem, one olution would be to upplement the checkout clerk, but thi i not alway the mot economical trategy to improve ervice. One of the factor influencing conumer' perception on ervice quality i the efficiency of waiting ytem. The waiting time i inevitable in the cae of random requet. Thu, providing the capacity for a ufficient ervice i needed, but it i involving high cot. Thi i the premie from which the queuing theory tart in deigning ervice ytem (Alecu, F., 2004). Cutomer flow management refer to cutomer flow handling a well a to their experience from the firt contact with the company until the delivery of good/ervice. Cutomer flow management play a key role in increaing the productivity, ale and alo in reducing cot, ince each cutomer will be directed to the right place, at the right time and will be erved by the adequately operator. Thu, the tep of the cutomer flow are a follow 4 : pre-reception involve uing programming in advance, thu reulting a horter waiting time. It can be made by phone or uing the Internet; reception cutomer flow management i opting to place cutomer in different waiting line, depending on their need; Bogdan Vodă Univerity, Gr. Alexandrecu treet, no. 26A, Cluj-Napoca, România, 2 Bogdan Vodă Univerity, Gr. Alexandrecu treet, no. 26A, Cluj-Napoca, România, 3 Bogdan Vodă Univerity, Gr. Alexandrecu treet, no. 26A, Cluj-Napoca, România, 4 *** 66

2 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 waiting waiting time optimization can be achieved by improving the taff planning and increaing the procee flexibility; ervice once the cutomer are waiting in a line, the taff can tart the required ervice; pot-ervice after erving, waiting and proceeding time are recorded by the taff; adminitration the record can be ued to the current procee aement, by generating report, in order to determine the operational efficiency. Literature review In the literature, there were developed ome model in order to upport and ait manager in making the bet deciion on waiting line (Pang, P., 2004), (Sweeney, D. et. al. 200). In the management terminology, a waiting line i alo called the tail and their characteritic concept form the queuing theory (Shim, J., K., Siegel, J., G., 999), (William, A., S., 2003). Thi theory i underlying the analyi of ome communication, logitic, manufacturing and ervice ytem (Bejan, A., 2007). The main advantage of queuing theory reide in determining very important information about waiting time, arrival and ervice tation characteritic and about the ytem dicipline (Alecu, F., 2004). Waiting line model conit of mathematical formula and relation ued to determine the operating characteritic of thee line. Among thee feature we mention (William, A., S., 2003): the probability that there i no item in the ytem; the average of the item in the waiting line; the average of the exitent item in the ytem (the item in the waiting line and the item being erved); the average time an item pend in the waiting line; the average time an item pend in the ytem (conit of the waiting time beide the ervice time); the probability that an item ha to wait for the ervice. Reearch methodology In order to highlight the characteritic of a waiting ytem, we conider the example of a fat-food retaurant. Although every fat-food retaurant want to provide a ervice a prompt a poible, there are many cae when the peronal can t handle the cutomer. Thu the waiting ituation arie. Single-channel waiting line The way the cutomer are erved in a fat-food retaurant i an example of a ingle-channel waiting line. That i, the cutomer order i taken and a the tranaction i completed, the order of the next cutomer in the waiting line i taken. Thu, each cutomer of the retaurant goe through a ingle-channel where he place the order, pay and pick-up the product. If there are more cutomer than can be erved, a waiting line arie. The diagram below how a ingle-channel waiting line for the fat-food retaurant: 67

3 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 Sytem Server Cutomer arrival Waiting line Order taking and order filling Cutomer leave after order i filled Fig. no. The fat-food ingle-channel waiting line Source: William, 2003 Ditribution of arrival A feature of the arrival proce i the probability ditribution of arrival in a given time period. In many ituation, arrival occur randomly and independently of other arrival, uch that the etimation of an arrival occurrence i difficult to determine. Thu, the Poion ditribution i the bet olution to decribe the arrival pattern. Starting from the definition of a Poion ditribution of random variable [4], the probability ditribution function of x arrival in a pecific time period, i: P( x) = x e x! for x = 0,, 2,... () where: x the number of arrival in a time period; the mean number of arrival per time period; e = Suppoing that in our fat-food, the mean number of arrival i 45 cutomer per hour, the mean number of cutomer arrived in one minute i = 45/60 =. The probability that x cutomer arrive in one minute period will be: x x e e P( x) = = x! x! (2) The probabilitie of 0,, 2 arrival in one minute period will be: 0 () e P(0) = = e = ! (3) () e P() = = e = ! (4) 2 () e P(2) = = e = ! (5) 68

4 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 Service time ditribution Service time tart when a cutomer place an order and finihe when the cutomer receive the order. Service time i not contant, but it depend on how large the order i. The quantitative analyi highlighted the fact that the exponential ditribution of the ervice time provide the bet information regarding the operation of waiting line. If the exponential probabilitic ditribution i ued then the probability that the ervice time i le than or equal to a ime t will be (Jaradat M., Mureşan, A., 2004), (William, A., S., 2003): where: t the time for ervice; t the length of the pecified time period; the mean number of item that can be erved in a period; e = t P( t t) = e (6) For example, uppoe that in our fat-food, an operator erve an average of 60 cutomer per hour. Then, in one minute, the operator will erve = 60/60 = cutomer. If =, we can determine what i the probability of an order to be proceed in ½ minute or le, in minute or le, or in 2 minute or le. Thu, from (6) reult: P t 0.5 ( 0.5) e = = (7) P t ( ) e = = (8) P t 2 ( 2) e = = (9) Waiting line dicipline When talking about waiting ytem and line, it i neceary to pecify the way the element are arranged for erving, in other word it ha to pecify the waiting line dicipline. In cae of fat-food retaurant, a in other many cae, the queue dicipline i the FIFO type (Firt In Firt Out). In other word, the cutomer are erved in order of their arrival in the waiting line. There are ituation when the waiting line dicipline i type of LIFO (Lat In Firt Out). Such an example i given by peron who are going to ue an elevator. Thu, the lat peron entered the elevator will be the firt peron to leave it. There are real ytem in which the requet need to be erved before other baed on ome prioritie (Benderchi O., 2009). Single-channel waiting line model, with Poion arrival and exponential ervice time In order to be able to highlight the way the exiting formula can provide information related to the waiting line characteritic, we turn to our fat-food retaurant example. Next, we preent the characteritic of the waiting line operation, taking into account the following: the average number of arrival in a given period of time and - the average number of ervice in a given period of time (William, A., S., 2003): the probability that there i no item in the ytem: P(0) = (0) 69

5 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 the average number of item exiting in the waiting line: L q 2 = ( ) () the average number of item exiting in the ytem: the average time pent by an item in the waiting line: L = L q + (2) W L q q = (3) the average time pent by an item in the ytem: W = W q + (4) the probability that an item i waiting to be erved: P w = (5) the probability that there are n item in the ytem: Pn = n P 0 (6) It i worthy to note that the formula from (0) to (6) can be applied only if i greater than. In other word, they can be applied only if / <. Failing to meet thi condition lead to a growing of the waiting line, becaue the ervice capacity i inufficient. Next, we are going to determine the operation characteritic in cae of fat-food retaurant. Baed on previouly determined value for = and = and uing formula (0) (6), we get: 620

6 Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 P(0) = = () Lq = = 2.25 cutomer ( ) L = = 3 cutomer 2.25 Wq = = 3 min ute W = 3 + = 4 min ute Pw = = (7) Waiting line model utility In cae of the conidered fat-food, the reult highlight ome important iue about the waiting line operation mode. Thu, the cutomer have to wait an average of 3 minute to place an order; the average number of cutomer who have to wait i 2.5 and 75% of arriving cutomer have to wait for the order. Thee value how that it i neceary to improve the operation occurring within the waiting line. If the manager continue to ue the ingle-channel waiting line, the number of waiting cutomer will be increaingly higher. Concluion The waiting line model play a key role in highlighting the operation effectivene and hence the need of improving their characteritic. The analyt are thoe who decide if there will be any change regarding the waiting line configuration. In general, in order to improve the operation within the waiting line, they appeal to improve the ervice rate. Thi i poible by adopting one or both olution lited below: the increae of the average ervice rate thi i poible by either redeigning the waiting line or uing the new technologie; the addition of new ervice channel o that more cutomer can be erved imultaneouly. Reference. Alecu, F., Siteme de aşteptare pentru calculul paralel şi ditribuit, Romanian Journal of Information Technology and Automatic Control, Vol. 4, No Bejan, A., Modelarea timpului de orientare în iteme de aşteptare cu priorităţi, Paper of the PhD thei in phyical-mathematical cience, Chişinău, (available online at 3. Benderchi O., Analiza itemelor de aşteptare cu priorităţi şi trafic critic, Paper of the PhD thei in phyical-mathematical cience, Chişinău, (available online at 4. Jaradat M., Mureşan, A., Matematici pentru economişti. Teorie şi aplicaţii, Rioprint Publihing, Cluj-Napoca. 5. Pang, P., Eential of Manufacturing Engineering Management, iunivere Inc., USA. 6. Shim, J., K., Siegel, J., G., 999. Operation Management, Barron Educational Serie, USA. 7. Sweeney, D., J., Anderon, D., R., William, T. A., Camm, J., D., Kipp, Martin, R Quantitative Method for Buine, Cengage Learning, USA. 62

7 Annale Univeritati Apuleni Serie Oeconomica, 2(2), William, A., S., An Introduction to Management Science Quantitative Approache to Deciion Making. Tenth Edition, South-Wetern, USA. 9. *** 622

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