CONSIDERATION OF SOME PERFORMANCE OF CONTAINERS FLOW AT YARD

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1 CONSIDERATION OF SOME PERFORMANCE OF CONTAINERS FLOW AT YARD Maja Škurić (a), Braislav Dragović (b), Nead Dj. Zrić (), Romeo Meštrović (d) (a), (b), (d) Maritime Faulty Kotor, Uiversity of Moteegro () Faulty of Mehaial Egieerig, Uiversity of Belgrade (a) (b) () (d) ABSTRACT Performae of otaiers flow at yard represets very importat poit of otaier termials. The orgaizatio of otaier trasport stakig poliies leads to less ogestio lower osts. Otherwise, otaiers wait i queue before they are servied. This study presets a aalytial approah for obtaiig the average umber of otaiers i queue. We proposed two models (ostat geometri) of bulk arrival multi-server queuig system. Traffi itesity utilizatio fator are very importat parameters that osist of data for arrival servie rate umber of servers (yard raes). I this paper, we assume that there are three yard raes that operate at otaier yard. A give umerial example for two models will improve the best values for performae of otaiers flow at otaier yard. Keywords: otaier yard, average umber of otaiers i queue, bulk arrival, umerial example 1. INTRODUCTION For determiig the optimal apaity of a otaier yard, a maximum attetio should be paid to the stakig poliies defied by yard raes. This is due to the aommodative apaity of the yard, expressed by the umber of yard raes, determiig the required apaity of otaier yard as a whole. Applyig queuig models, otaier yard a be treated as: a system with the ifiite waitig apaity determied umber of yard raes, a sigle or multi-server system (depedig o the umber of yard raes), a system i whih serviig is most ofte arried out aordig to the FCFS rule (first ome, first served), but it is possible that there are ertai otaiers whih have priority i serviig a system where otaiers must be servied at oe (Škurić, Dragović, Meštrović 011). I this paper we alulate the average umber of otaiers i queue at otaier yard. We assume that the otaiers are arrivig i group at yard follows ostat or/ geometri distributios. Their servie time follows the expoetial distributio. I aordae to the exteded Kedall s queuig otatio, these two X ost models may be deoted as ( M / M / ( ) ) for k 1 X (1 a) a ostat ( M / M / ( ) ) geometri distributios of otaier group arrivals. The umber of assumed yard raes is three. The level of traffi itesity values of some other parameters are also stated. The objetive is to desribe these models for defiig the strategies at yard alulate the average umber of otaiers i queue. This paper is orgaized as follows. Literature review is give i Setio, while i Setio 3 the aalytial formulatios are provided. Related umerial results aalysis for obtaiig the average umber of otaiers i queue with orrespodig graphial results is show i Setio 4. Fial olusios are give i Setio 5.. LITERATURE REVIEW Geerally speakig, authors used queuig models (sigle or multiple) to desribe the arrival servie proesses of ustomers (ships) i ports. They are used to aalyze omplex dyami stohasti situatios (see e.g. Dragović, Park, Zrić, Meštrović 01). The models otai aalyti formulatios umerial solutios for the performae evaluatio of port systems. Various models from simple queues to omplex queuig etwork models have bee suggested to aalyze: movemet of ships i port, ship traffi modellig, mehaism of ogestio ourree, ompositio ogestio osts, evaluatio method for optimal umber of berths, optimum alloatio size of ports, optimal berth rae ombiatio i ports, average ost per ships served, the ship turaroud time at the port so o. Regardig multiple queuig system, the authors preseted i Table 1 have ivestigated bulk arrivals of the ustomers. Their osidered problems are based o the followig statemets: a ompariso of aalytial simulatio plaig models, the aalysis of a queue with bulk arrivals bulk-dediated servers, a aalytial methodology of bulk queuig system that determies the apaity of berths withi seaports river ports, port storage loatios as queuig systems with bulk arrivals a sigle servie, the optimal umber of servers with bulk arrivals by miimizig the total osts of system, the ahorage-ship-berth lik at the port utilizig queuig theory with bulk arrivals, a multi-server queue with bulk arrivals fiite-buffer spae queuig approahes at otaier yard with Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, Proeedigs ISBN of ; the It. Cof. o Harbor Bruzzoe, Maritime Groalt, Multimodal Merkuryev, Logistis Piera Eds. M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 119

2 detailed aalytial expressios real ase study (see e.g. Škurić, Dragović, Meštrović 011). The importae of bulk queuig models for otaier termial problem is explaied i Koza (1997). The aalysis of a queue with bulk arrivals bulk-dediated servers is speified i Gullu (004). I this paper, it is osidered the M / G / queuig system with bulk arrivals whose jobs belog to a bath have to be proessed by the same server. Similarly to this study, bulk arrivals are preseted by pushed pulled ovoys of barges i Radmilović (199). The queuig system desribes that barges i ovoy have a ostat or geometri probability distributio. O the other h, i Radmilović, Čolić, Hrle (1996), the authors deal with the port storage loatios with bulk arrivals a sigle servie. Fially, i Radmilović, Dragović, Meštrović (005) the aim was to miimize the total ost of system by determiig the optimal umber of servers. The proesses of ahorage-ship-berth lik at the port desribed by the o-statioary multi-server queuig system are preseted i Dragović, Zrić, Radmilović (006) i Zrić, Dragović, Radmilović (1999). Likewise, partial total bulk rejetios the distributios of the umbers of ustomers i the system for multi-server queue are explaied i Laxmi Gupta (000). Agai, the related literature overview is give i Table 1. Table 1: Related Literature Overview Referees Cosidered problem Results (Radmilović 199) (Radmilović, Čolić, Hrle 1996) (Koza 1997) (Zrić, Dragović, Radmilović 1999; Dragović, Zrić, Radmilović 006) (Laxmi Gupta 000) (Gullu 004) (Radmilović, Dragović, Meštrović 005) (Škurić, Dragović, Meštrović 011) (Dragović, Park, Zrić, Meštrović 01) Aalytial methodology of bulk queuig system. Bulk arrivals a sigle servie. A ompariso of aalytial simulatio plaig models of otaier termials. Aalyzed the ahorage-ship-berth lik utilizig queuig theory with bulk arrivals. A multi-server queue with bulk arrivals fiite-buffer spae. The aalysis of a queue with bulk arrivals bulk-dediated servers. Optimal umber of servers i with bulk arrivals. A multi-server queue with bulk arrivals. Disuss dyami system performae evaluatio i the river port utilizig queuig models with bath arrivals. Determied the optimum umber apaity of berths withi seaports river ports. Port storage loatios as queuig systems are solved. The advatages of simulatio are show beause it is able to apture all details the omplexity of a real system. Determied ost ratio total system ost. Partial total bulk rejetios the distributios of the umbers of ustomers i the system are obtaied. Cosidered the M/G/ queuig system with bulk arrivals whose jobs belog to a bath have to be proessed by the same server. Miimized the total osts of system. Obtaied average umber of otaiers i queue related ost ratio. The results have revealed that aalytial modellig is a very effetive method to examie the impat of itroduig priority, for ertai lass of ships, o the ahorage-ship-berth lik performae. 3. ANALYTICAL FORMULATIONS I this Setio, we preset the methodology that otais aalytial formulatios of parameters for alulatig the average umber of otaiers i queue. First, we start with obtaiig the expliit formulae for steady-state probability that otaiers are at the yard. After providig the probabilities of ostat geometri distributio of X i ase whe there is speified umber of yard raes, we give formulae for related umbers of otaiers i queue that orrespods to the metioed probabilities. Fially, umerial example is used for sesitivity aalysis of average umber of otaiers i queue i relatio to three parameters (umber of otaiers i group, umber of yard raes traffi itesity). Traffi itesity of otaiers at yard is i depedee of their arrival servie rate, deoted as / where is the average arrival rate of otaiers i group represets the average servie rate of otaiers. These are servied by yard raes ( ) whih represet the umber of servers. The average umber of otaiers i group is give as a while the utilizatio fator for bulk queuig system is defied as ( a) /( ). Notie that ( a ) /. We osider a bulk arrival multi-server queue M X / M / where the bulk size X is a ostat or geometrially distributed rom variable. The yard raes have idepedet, expoetially distributed servie times. The otaiers that arrive for servie i groups X the mea of X is equal to E ( X ) a 1/ a the variae of X is equal to var X a 1/ a. The ase whe X is a ostat that Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, Proeedigs ISBN of ; the It. Cof. o Harbor Bruzzoe, Maritime Groalt, Multimodal Merkuryev, Logistis Piera Eds. M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 10

3 is X b 1 P for some fixed b {1,,3,...}, the E( X ) a b ( X ) 0. The iter-arrival times, the bulk sizes servie times are mutually idepedet. It is kow that a probability that otaiers are preset i queuig system is (Chaudhry Templeto 1983) 1 0 P Q (1) 0 where Q0 is a probability that average umber of otaiers that are preset i a queuig system, L, are busy. The above formula immediately yields P Q () 0 P 0 whih i view of the fat that 1 0 P 1 Q 0 P beomes (3) where a /( ) is the utilizatio fator. Followig Chaudhry Tampleto (1983), L L q 0 P where P is steady-state probability that otaiers are at the yard, i.e., that otaiers are just beig servied or are waitig i a queue to be servied. O the other h, it is suitable to determie the probabilities P0 P usig Kabak s reurree formulae (Dragović, Zrić, Radmilović 006; Kabak 1970; Škurić, Dragović, Meštrović 011). These probabilities follow reurree relatios: 1 k k0 k (4) P y( ) P A, 1,,... (5) with, ( ) mi, y( ) / ( ) k k 1 ai i0 (6) A 1 A 1 1 (7) where ai is a probability that a group of i otaiers arrives i the bulk queuig system, PX i ai, i 1. Substitutig (6) (7) ito (5), the probability P0 is obtaied as P 1 P0 1. (8) Furthermore, the average umber of otaiers preset i queuig system with yard raes is L P (9) 0 it also holds for the bulk arrivals queuig system. Followig Chaudhry Templeto (1983), a a a L a 1 0 P. (10) a I the ase whe there are three yard raes i.e. 3, it is eessary to substitute a 1/ a, a 1/ a ito (10) (6), respetively. Also, for takig the values for geometri distributio of X with k 1 X k a 1 a a P k, k 1,, 3..., where 0 a 1, puttig k 1 k 1 i1 i1 k 1 1 a 1 a A 1 a 1 a, k 1,,... i (7), k i i1 we obtai the formulae for Pk related to geometri distributio of X. The orrespodig formulae for ost g average umber of otaiers i queue, Pk P k related to the ostat geometri distributio of bath size X, respetively, are as follows (Škurić, Dragović, Meštrović 011): The probabilities of ostat distributio of X i ase whe 3 yield (1 ) if b 1 4 3, P0 ost b (1 ) b 5b 3 if b 1 6(1 ) if b 1 b( 4 3 ) P1 ost, 6b(1 ) if b 1 b 5b 3 9 (1 ) if b 1 b ( 4 3 ) P ost 9 (1 ) if b 1 b 5b (1 ) if b 1 3 b ( 4 3 ) P3 ost. (11) 3 9 (1 ) if b 1 3 b 5b 3b P g 0 The probabilities of geometri distributio of X i ase whe 3 are (1 ), a(5 a) 3a Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 11

4 P g 1 P g P g 3 6a(1 ), a(5 a) 3a 3a(1 )(1 a 3a) a(5 a) 3a 9a(1 a) 9 a 1 a P0 a. (1) Related umbers of otaiers i queue that orrespods to the probabilities give by (11) (1) are L ost 3 if b 1, ( 4 3 )(1 ) b ( b1) (13) 18 L ost 3 if b 1 b 5b 3 (1 ) 6a(3 3a a) ( a) L g 3. (14) a(5 a) 3a a(1 ) 4. NUMERICAL RESULTS ANALYSIS The umerial example is i relatio to otaier termial i port of Bar, Moteegro. The otaier termial throughput from 008 to 01 is give i Figure 1 (PBR 01). The biggest throughput of TEU is reahed i 008. We observe that the otaier yard is osisted of three yard raes that serve for otaier stakig. Trator trailer system fork lifters are used for termial trasport of otaiers from berth to yard vie versa. Cotaier termial throughput Figure 1: Cotaier termial throughput i port of Bar At time, otaier termial i port of Bar is osisted of oe berth oe quay rae for serviig the otaier ships. The maximum arryig apaity of otaier ships is 4000 TEU. There are also the rail road vehiles for il oetio (Škurić, Dragović, Meštrović 011). The iput data for the aalytial models are based o the atual otaiers arrivals at the termial of port of Bar where we assumed that the otaiers arrivals fit ostat or geometri distributio. Usig formulae (13) (14), i Figures 3 we ompare the values for the average umber of otaiers i queue whih are i futio of traffi itesity for a b a b 4 with ostat geometri distributio. The graphs are obtaied i Mathematia 8. Cosiderig Figure, the iput data suh as average umber of otaiers i group a b is speified as well as the differet values of traffi itesity are from 0.1 to 1. 3 while the umber of yard raes is ost g L 3 ; L 3 Geometri Distributio Costat Distributio Traffi itesity Figure : Average Number of Cotaiers i Queue for Costat Geometri Distributios of X with a b, 3 (0,1.3) From Figure, we a otie that the results for average umber of otaiers i queue are i futio of three differet parameters (average umber of otaiers i group, traffi itesity umber of yard raes) may serve to see related parameters for omparig results for ostat geometri distributios. The irease of average umber of otaiers i queue auses higher traffi itesity of otaiers at yard. Therefore, for the same traffi itesity, the average umber of otaiers i queue for geometri arrivals of otaiers implies higher values the those for ostat distributio. The results for average umber of otaiers i queue i the ase of a b 4 are give i Figure 3 with the same umber of yard raes as i the first ase traffi itesity values are from 0. 1 to Obviously i Figure 3, the values for geometri distributios are more dyami are ireasig faster the those for ostat distributios. It meas that i ase that the group arrivals of otaiers have its behaviour by ostat distributio; it implies that the average umber of otaiers i queue is lower i ompariso to the geometri distributed otaiers arrivals with the same value of traffi itesity ost g L 3 ; L 3 Geometri Distributio Costat Distributio Traffi itesity Figure 3: Average Number of Cotaiers i Queue for Costat Geometri Distributios of X with a b 4, 3 (0,1.3) Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, Proeedigs ISBN of ; the It. Cof. o Harbor Bruzzoe, Maritime Groalt, Multimodal Merkuryev, Logistis Piera Eds. M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 1

5 5. CONCLUSION The umerial results aalysis for differet values of parameters for presetig the performaes of otaiers flow at otaier yard i port of Bar leads to the followig olusios: The dyamial arrivals of otaiers i aordae to ostat distributio showed better results i the view of average umber of otaiers i queue whih may lead to less ogestio i ompariso to the geometri distributed otaiers arrivals. As a matter of fat that Figure 1 implies that there would ot be huge flutuatios i otaier termial throughput i omig years, this suggests that the level of traffi itesity will ot be drastially haged that assumed values i umerial example represet the real situatio i port. The values of average umber of otaiers i queue diretly impat o speifi ost ratio of total aual ost for queuig system to the aual otaier ost total system osts. The obtaied results suggest that the operatioal strategy at otaier yard a be improved by reduig the average umber of otaiers i queue. This a be evaluated through the employmet of aother yard rae or to observe other group arrivals of otaiers. O the other h, this aalysis also has some limitatios. There are a lot of parameters that did ot take ito aout, but o matter to that, we suppose that it represets a oveiet approah for implemetig some other modellig tehiques i some further ivestigatios simulatio model employmet would be able to apture the omplexity of a real system suh as otaier yard. REFERENCES Chaudhry, M.L. Templeto, J.G.C., A first ourse i bulk queues. 1st ed. New York: Joh Wiley Sos, I. Dragović, B., Zrić, Dj. Radmilović, Z., 006. Ports & otaier termials modelig. 1st ed. Uiversity of Belgrade: Faulty of Trasport Traffi Egieerig. Dragović, B., Park, N.K., Zrić, Dj. N. Meštrović, R., 01, Mathematial models of multi-server queuig system for dyami performae evaluatio i port. Mathematial Problems i Egieerig, Volume 01 (01), Artile ID , 19 pages, doi: /01/ Gullu, R., 004. Aalysis of a M / G / queue with bath arrivals bath-dediated servers. Operatios Researh Letters, 3, Kabak, I.W., Blokig delays i M X / M / bulk arrival queuig systems. Maagemet Siees, 17 (1), ( ) Koza, E., Compariso of aalytial simulatio plaig models of seaport otaier termials. Trasportatio Plaig Tehology, 0 (3), Laxmi, P. V. Gupta, U.C., 000. Aalysis of fiitebuffer multi-server queues with group arrivals: GI X / M / / N. Queueig Systems, 36, Port of Bar Reports (PBR), 01. Radmilović, Z., 199. Ship-berth lik as bulk queuig system i ports. Joural of Waterway, Port, Coastal Oea Egieerig, ASCE, 118 (5), Radmilović, Z., Čolić, V. Hrle, Z., Some aspets of storage bulk queueig systems i trasport operatios. Trasportatio Plaig Tehology, 0, Radmilović, Z., Dragović, B. Meštrović, R., 005. Optimal umber apaity of servers i M X a M / / queuig systems. Iteratioal Joural of Iformatio Maagemet Siees, 16 (3), 1-. Zrić, Dj., Dragović, B. Radmilović, Z., Ahorage-ship-berth lik as multiple server queuig system. Joural of Waterway, Port, Coastal Oea Egieerig, ASCE, 15 (5), 1-7. Škurić, M., Dragović, B. Meštrović, R., 011. Some results of queuig approahes at otaier yard. It. J. Deisio Siees, Risk Maagemet, 3 (3/4), AUTHORS BIOGRAPHY Maja Škuri is a PhD studet at Maritime Faulty, Uiversity of Moteegro. After graduatios, she started to work as a Teahig Assistat at the same faulty. She reeived her Master of Siee degree at Faulty of Trasport Traffi Egieerig, Uiversity of Belgrade. Up to ow she published more the 30 oferee papers few joural papers. Her mai researh iterests are related to modellig optimizig proesses at otaier termials lastly she iluded the ruise ship traffi modellig i her iterests. These topis are maily oered with the ivestigatios of queuig systems, plaig trasportatio systems operatios researh i trasport. Braislav Dragović is a Professor at the Maritime Faulty, Uiversity of Moteegro. He reeived his BS MS at Maritime Trasport Traffi. Further, he reeived his PhD from Faulty of Mehaial Egieerig, Uiversity of Belgrade with speializatio o plaig desig of trasport systems. He was a Visitig Professor of the Korea Maritime Uiversity from 006 to 007. He was ivited as a Leturer i the outry abroad may times. He has published over 130 sietifi papers iludig oes i jourals suh as Joural of Waterway, Port, Coastal Oea Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 13

6 Egieerig; Maritime Eoomis Logistis; Maritime Poliy Maagemet; Mathematial Problems i Egieerig amog others. He has writte four books, three atioal two iteratioal moographs. He was quoted may times. He is a member of IAME WCTRS. Nead Zrić is Assoiate Professor at the Faulty of Mehaial Egieerig of the Uiversity of Belgrade. He reeived egieerig degree Dipl.-Ig. (equivalet to MS) from the UB, FME i 199, MagTehS (equivalet to CS) i 1996 at the same istitutio. He reeived his DrS from the Faulty of Mehaial Egieerig, Uiversity of Belgrade i 005 with i material hlig equipmet i ports termials. He is a reviewer i the jourals Mehais Researh Commuiatios (SCI), Mehatrois (SCI), Natural Hazards Earth System Siees (SCI), amog others. He has published over 10 sietifi papers i the outry abroad, amog them 15 i world s leadig joural (SCI). He preseted 7 ivited pleary letures at iteratioal oferees. He wrote 1 atioal moograph, 1 iteratioal moograph 1 book. He was quoted several times. Romeo Meštrović is a Professor of Mathematis at Uiversity of Moteegro. He reeived his BS from the Uiversity of Zagreb his MS PhD from the Uiversity of Moteegro, i the area of mathematis. His topis are omplex aalysis, ombiatorial umber theory, operatio researhes queuig theory i trasport systems. He has published more tha 0 papers i aademi jourals suh as Joural of Mathematial Aalysis Appliatios, Joural of Iequalities Appliatios, Ars Combiatoria, Ameria Mathematial Mothly et. Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. Proeedigs of the It. Cof. o Harbor Maritime Multimodal Logistis M&S, 013 ISBN ; Bruzzoe, Groalt, Merkuryev, Piera Eds. 14

Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness

Construction of Control Chart for Random Queue Length for (M / M / c): ( / FCFS) Queueing Model Using Skewness Iteratioal Joural of Sietifi ad Researh Publiatios, Volume, Issue, Deember ISSN 5-5 Costrutio of Cotrol Chart for Radom Queue Legth for (M / M / ): ( / FCFS) Queueig Model Usig Skewess Dr.(Mrs.) A.R. Sudamai