Application of queuing theory to the container terminal at Alexandria seaport

Transcription

1 Joural of Soil Sciece ad Evirometal Maagemet Vol 1 (4), pp 77-85, Jue 2010 Available olie at ISSN Academic Jourals Full Legth Research Paper Applicatio of queuig theory to the cotaier termial at Alexadria seaport M E El-Naggar Civil Egieerig Departmet, Faculty of Egieerig, Alexadria Uiversity, Alexadria, Egypt aggarmo@gmailcom Accepted 29 April, 2010 This paper describes a methodology desiged to support the decisio-makig process by developig seaport ifrastructure to meet future demad I order to determie a optimum umber of berths at a sea port, the queuig theory is applied i the light of port facilities ad activities The aim is to avoid iadvertet over ad uder-buildig Withi this methodology, the movemets i port should firstly be aalyzed The waitig time of vessels outside the port ad i queue is calculated i accordace with the cosidered queuig model The theoretical fuctios represetig the actual vessel arrival ad service time distributios are determied For the ecoomic cosideratios, cost estimate studies icludig cost of port ad waitig vessels are carried out Fially, the optimum umber of berths that miimizes the total port costs ca be decided Both proposed mathematical ad ecoomical models are applied to Alexadria port i Egypt Key words: Port capacity, port ecoomy, port modelig, queuig theory INTRODUCTION The port trasportatio system icludes differet physical elemets, eg berths, hadlig equipmets, storage ad traffic facilities Although the capacity of ay sigle elemet may be expressed as a absolute figure, such as the umber of cotaiers loaded per hour by a certai crae, the aggregate capacity of the whole port caot be so simply described Each elemet ca limit the overall port productivity Port productivity ca be viewed from two stadpoits To ship operators, productivity implies the time eeded at the port to serve ships, while at atioal level, port productivity ca be defied as the amout of cargo trasported through the port durig a certai time period Port developmet is ofte affected by operatig policies as well as by the traffic demad imposed i the port i terms of the volume of cargo expected to be accommodated, the service time at the available berths withi which this volume should be hadled, ad the frequecy of ships arrivals It would be possible to develop the port facilities so that its capacity is fully utilized at all times I this maer, chages i demad have to be accommodated by forcig ships to wait (at achorage) util ships that arrived previously had bee serviced This policy would be iefficiet ad uecoomic due to the delay costs of waitig ships Coversely, developig the port so that ships are ever forced to wait also represets a uecoomic use of port resources The ideal situatio is oe i which all berths are occupied at all times ad o ship is ever kept waitig This situatio is impossible to achieve i practice because of the radom arrivals of cargo ships ad the variatios i service time of ships of differet sizes Therefore, decisios cocerig port developmet ca be made by tradig-off the cost of icreasig the port capacity ad the costs of both waitig ad service times The purpose of this paper is to itroduce a methodology which ca be used to facilitate the decisio-makig process of port developmet The proposed methodology covers two pricipal areas: a) Ivestigatio of the patter of ship traffic at a seaport from the stadpoit of queuig theory, ad to use the fidigs to draw some hypotheses regardig its applicatio to the overall operatio i sea ports b) Determiatio of the optimum umber of berths eeded i a sea port that will miimize the total port usage costs

2 78 J Soil Sci Evir Maage Ships waitig for a berth w S umber of berths umber of waitig ships w Figure 1 Ship queue at a seaport S Ships served at a berth tools for aalyzig the behavior of waitig uits (ships i this case), for ivestigatig the compoets of a multiple operatio system (Braislav ad Nam, 2006) Thus, queuig theory may be adequate for studyig ship movemet i sea ports Two basic elemets are ecessary for the applicatio of queuig theory to a waitig lie problem: a arrival fuctio ad a service fuctio These fuctios should first be modeled Oce the validity of these models is tested, the differet characteristics of the theoretical models, which describe the actual system with the accuracy that may be realized i estimatig future traffic, ca the be determied To aalyze the movemet of ships i a sea port usig the queuig theory, the followig coditios are assumed: i) Ships arrivals ad service times coform to the patter of radom occurreces ii) Ships are processed o the first-come first-served queue disciplie iii) The queue legth is ulimited, that is, if a ship arrives ad fids a log queue, it jois the waitig ships ad does ot leave the port Modelig ship arrival Figure 2 Ships arrival distributio as poisso fuctio, hypothetical port ANALYSIS OF SHIPS' MOVEMENT IN A SEAPORT A importat parameter measurig the performace of a seaport is the delays that ships experiece while waitig to be processed Two factors affect these delays: (a) the patter of ships arrival, ad (b) the berth time requiremet for cargo hadlig The arrival of a cargo ship i a port is ofte irregular, ad whe it arrives, it may be able to move directly oto a berth or has to wait util a berth becomes empty, if all berths are occupied The berth time eeded to serve a ship is also variable, as it depeds o the amout of cargo which the ship carries ad the capacity of the preset facilities for hadlig ad storig cargo (Gokkup, 1995) Figure 1 shows ship behavior at a seaport The ivestigatio of such radom occurreces requires a complex ad detailed aalysis The cocept of Queuig theory-waitig lie problem ca successfully be applied Queuig Theory is oe of the most useful Probably the two most commoly ecoutered arrival patters of ships i a sea port are the radom ad scheduled arrivals with cosiderable delays Thus, to predict the umber of ships preset i a port i a certai time period (usually a day), the arrival patter of ships may be approximated by a Poisso fuctio (Tadashi, 2003) I this way, the probability P of the arrival of ships i the port i a give time ca be expressed as i Figure 2: P ( λ) λ e! Where λ average arrival rate of ships durig the give time (oe day, for example), e base of the atural logarithm (e ), λ the average arrival rate of ships, ad;! the factorial of the ship umber The distributio of ships arrivals with Poisso fuctio ca be calculated, oly if the average arrival rate durig a etire period is kow The expected frequecy F of ships i port i a give time T is: F T P T is the cosidered time period of the port operatio (ofte expressed o a aual basis as 365 days)

3 El-Naggar 79 Figure 3 Service time distributio as Erlag Fuctio, hypothetical port Modelig of service time The duratio of ships at a berth for hadlig cargo may be described as a Erlag-fuctio (So ad Kim, 2004) which is usually used to preset service times that are more regularly spaced i time tha those represeted by the Poisso distributio There are purely theoretical curves (Erlag-fuctios), each of which is based o the assumptio that the service time is split ito two or more operatig phases followig oe aother, ad that the ship does ot leave the berth util all phases are completed k is the umber of Erlag Phases of ships service time distributio at a berth Each fuctio has a egative expoetial distributio As k icreases, the total service times become more uiform, util fially with k all service times are idetical I the geeral case the total service time probability P 0 is give i Figure 3 o kb P e k 1 ( kb) 0! Where b Average berth service time (i days), k Erlag umber (k 1, 2, 3,, ), ad; Couter b P for k 1, o e 2b(1+ 2b) P for k 2, o e for k 3, P o e 2 3b(1+ 3b+ 9b / 2) Through the choice of k, a service time fuctio may be described as aythig from the purely radom expoetial type (k 1) to the completely regular costat service time type (k ), the value of k should be selected ad tested to provide the best fit to the observed data QUEUING PHENOMENON As the ature of the problem is defied, i this paper, as multi-chaels (berths), with expoetial arrivals (Poisso), ad multiple expoetial services (Erlag), o feasible mathematical solutio is possible (Zora ad Braislav, 2005) The theoretical models available i the literature for multi-chael systems are iflexible for other tha expo-etial distributio of arrivals ad multiple expoetial service time distributio For ivestigatig queuig situa-tios of multi-chael systems, models are accessible oly for the followig two cases: Case : Expoetial distributios for both arrivals ad service times Case II: Expoetial arrivals ad a costat service time A approximate method has recetly bee proposed regardig the queuig model of case II (We-Chih et al, 2007) The essetial parameters are derived as follows: λ Average arrival rate i ships/day (Poissodistributio), µ Average service rate i ships/day (Erlag-distributio) 1/ average berth service time 1/ b, ad; S Number of berths The ratio of the arrival rate to the service rate is usually kow as the traffic itesity, thus: λ σ µ I this case, it ca be oted that the average waitig time before service w k is give by (Erlag fuctio) (We-Chih

4 80 J Soil Sci Evir Maage et al, 2007): Where, w 1 is the correctio of the average costat service time obtaied by selectig a Erlag-fuctio with costat k umber w 1 ca be calculated from the followig fuctio, (We-Chih et al, 2007): w 1 ( ( S 1)! µ S 1) S λ 2 S ( S µ λ ) λ! µ S + λ ( S µ λ) Where, actual umber of ships preset i a port i a certai time period Thus the average time that a ship speds i seaport t s ca be determied as follows: t b + s w k From the above aalysis of delays i the queue, computatio ca readily be made of the average legth of queue, that is,for average umber of ships waitig for a berth w, the appropriate expressio is: η λ The average umber of ships s preset i port with S berths i a certai time period ca be determied usig the followig formula: η η + η Where, b average umber of ships served at S berths S berth utilizatio factor S (λ/(µs) σ Thus, it is see that the traffic itesity, σ defied i the queuig theory equals the average umber of ships served at berths b ANALYSIS OF PORT CAPACITY Miimum capacity The miimum umber of berths S mi eeded i a seaport to hadle a certai amout of cargo ca be calculated usig the followig procedure: Let Q the total amout of cargo (i tos) hadled i a port sectio i a time period T (for example, T oe year 8760 h), ad R average rate of cargo trasfer betwee ship ad berth (i tos per hour) The, Thus, the gross berth time available is S mi T The, let β (berth utilizatio) equal the % of berth usage throughout the period T β (berth time required)/ (berth time available), or Q S mi ( R T ) I this maer, the calculated umber of berths is based o average values; regardless of the radom arrivals of ships ad the variatio i berth service times Optimum capacity If the umber of berths i a port is S, the total cost spet i the port durig a certai period, C equals the sum of two differet types of costs: cost related to berths ad cost related to ships preset (Ja ad Robert, 2002) Thus, it ca be expressed as (Figure 4): C C T S + b C s η s I which, C total cost of a port with S berths durig the period T, usually oe year 365 days, (i LE), C b average cost of a berth; that is, costructio ad maiteace costs (LE/day/berth), c s average delay cost of a waitig ship (LE/day/ship), ad; s average umber of ships preset i port Accordigly, if the amout of cargo that must be dealt with at a port durig the period T is give as time plaig target, the such umber of berths S becomes the optimum that miimizes the total cost C Therefore, C is a proper measure to examie the optimality of a port system Now, both sides of the above equatio are divided by C s t i order to decrease the umber of the parameters ivolved Thus, R s C / (C S T) ) (c b /c S )S + η s (r bs S) + η s I which, r s ratio of the total aual cost for port to aual ship cost, ad r bs berth-ship cost ratio Assumig that S is optimum, the the followig optimizatio coditio must be held: r s < r s + 1, ad r s < r s-1

5 El-Naggar 81 hadled i the port i that year was 4326 millio tos (Egyptia Maritime Data Bak, 2008) Alexadria port is costituted of a old ad complicated layout with short quays ad too arrow or too log piers A large umber of quays has limited drought less tha 80 meters, ad oly a lower umber of berths is capable to receive ships with more tha meter legth The umber of berths available for geeral cargo i the port is 32 berths Data base Figure 4 Total usage cost, hypothetical port Thus, r s will be adopted hereafter as a measure to determie the optimum umber of berths From the precedig iformatio the procedure ca be stadardized as follows whe give the data Q, R, c b, c s, λ, µ, k: Step 1 Calculate the miimum umber of berths from the Q S mi R T equatio, Step 2 Determie the value of traffic itesity σ as; λ σ ) µ ( Step 3 Compute the value of berth-ship cost ratio r bs from the give data c b ad c s Step 4 For each umber of berths, with S greater tha the miimum value, estimate the umber of ships preset i port s, ad predict the ratio r s Step 5 The umber of berths which satisfies the optimizatio coditio (r s < r s+1, ad r s < r s-1 ) is optimum Step 6 Compute the average berth utilizatio, β: σ β Step 7 Summarize the queuig results (average umber of ships preset i the port, average umber of ships at berths, average umber of waitig ships, average waitig time) APPLICATION OF THE PROPOSED METHODOLOGY TO ALEXANDRIA SEAPORT The foregoig methodology is applied to ivestigate the movemets of ships i Alexadria Port ad to predict the future capacity The applicatio is restricted to geeral cargo ships, excludig fullcotaier, bulk, ad RO/RO ships which have particular berths at the port Alexadria Port is the major port i Egypt About 4080 millio tos passed through the port i the year 2007/2008, that is, 36% of the total volume of the foreig trade The amout of geeral cargo The daily log books of the traffic departmet of the Alexadria Port Authority iclude (amog others) the arrival time of each ship at the pilot vessel I additio, detailed iformatio cocerig the movemet of each ship i the port is also available i the so-called ship log sheets Every sheet is a ship report, ad it cotais the followig data: a) Ship ame, atioality, type of cargo, ad total toage b) Berth occupacy, icludig berth chages durig the period i port c) Date ad time of arrival, berthig, ad quittig the port Ships arrivals If the distributio of ships arrivals ca be predicted reliably, port plaer ca proceed with great cofidece i makig developmet plas that may avoid over-buildig or uder-buildig the port facilities The actual patter of ship arrivals at the port of Alexadria is compared with the theoretical fuctio progosticated mathematically by Poisso distributio of radom occurreces The applicatio icludes a specific aalysis of the umber of ships preset, day by day, over a period of oe year (from July 1, 2007 to Jue 30, 2008) The umber of ships preset i the port, each day, was trascribed from the port log books ad the summarized to obtai the umber of days, that various umber of ships were preset durig the period studied The theoretical distributio, Poisso is computed Table 1 compares the predicted distributio with the actual oe The average arrival rate was 568 ships per day Table 1 shows a good agreemet betwee actual ad predicted distributios The umber of days that various umbers of ships are predicted to be preset i the port is i agreemet with the actual distributio o 336 days of 360 days, that is, o 92% of days To judge whether the observed frequecies of ship arrival distributio is compatible with the predicted theoretical frequecies, Chi-square is computed, ad the result, x with 10% probability, idicates a good fit From the statistical stadpoit, probability values betwee 5 ad 95% desigate good fit from which it is cocluded that this theoretical distributio is plausible (Tadashi, 2003) Figure 5 demostrates the goodess of fit betwee actual ad predicted distributios Berth service times Iformatio givig the date ad time of arrival at a berth ad the date ad time of departure from the berth were obtaied from the ship log sheets A total of 315 observatios, icludig those geeral cargo ships which were tied up at the berths betwee July 1, 2007 ad Jue 30, 2008 were radomly selected to be aalyzed A class iterval of 15 hours was selected for such aalysis Search for a suitable model for the distributio of the duratios at berths led to a Erlag distributio givig K 3 The mea time

6 82 J Soil Sci Evir Maage Table 1 Compariso of actual versus predicted ship arrival distributio Arrival rate (Ships/day) Actual umber of days (A) Predicted umber of days (B) Miimum (A) or (B) Total spet at a berth was foud 558 days for the 315 observatios The stadard deviatio of the distributio was computed ad foud to be ± 143 days Figure 6 presets the frequecy ad the cumulative distributios of the observed data ad compares the values of the cumulative distributio with those of the Erlag fuctio havig K 3 A Chi-square test was also performed to test the goodess of fit betwee the observed frequecy distributio ad the postulated Erlag fuctio, ad a value X for 42% probability was foud Compariso with other Erlag fuctios (K 1, K 2, ad K 4) idicates that K 3 is the best choice for this distributio fuctio Figure 6 also shows the observed data poits ad a plot of the selected fuctio Figure 5 Frequecy Distributio of Ships Arrivals, Alexadria Port 2007/2008 Optimum umber of berths To establish the optimum umber of berths eeded for geeral cargo hadlig at Alexadria port i the year 2017, applyig the proposed procedure, the followig iput data are used: i) Due to the further developmet of the Egyptia ports, particularly the Dekheila port, the aual geeral cargo toage to be hadled at the berths of Alexadria port will be oly about 400 millio tos at the target year (toage i year 2007/ millio tos) (Egyptia Maritime Data Bak, 2008) ii) The average arrival rate of geeral cargo ships will be 568 ships /day, assumig that the average ships load equals 2084 tos (the preset value) iii) The average rate of cargo hadlig at a geeral cargo berth R 3735 tos per day (the existig rate) iv) The average cost of a berth c b 2000 per day (approximately $600 per day), based o the developmet program of the Alexadria port (Egyptia Maritime Data Bak, 2008) v) The average delay cost of a geeral cargo ship c s $ 6000 per day The calculatios are carried out as follows: Figure 6 Frequecy distributio of berth service time, Alexadria port 2007/2008 S mi /( ) berths λ 568 ships/day

7 El-Naggar 83 Figure 7 Determiatio of optimum umber of berths,, Alexadria port, case study Figure 8 Determiatio of Optimum Number of Berths, due to differet cost ratio, r bs, Alexadria Port, Case Study µ 1/ ships/day 568/ r bs 600/ Figure 7 shows the relatioships betwee traffic itesity ad the cost ratio r s for a proper umber of berths (from S 29 to S 34) The optimum port capacity is 33 berths I this istace, r s 3434, ad the total port costs C $75200 millio (the developmet of 33 berths plus the aual maiteace costs) The average berth utilizatio 2935/ Queuig results: Average umber of ships preset i port s 3104 Average umber of ships served at berths b 2935 Average umber of waitig ships w 169 Average waitig time per ship w k 032 days The relatioships i Figure 7 are prepared as desig curves derived to determie the optimum umber of berths for Alexadria port by chagig the traffic itesity ad/or the cost ratio r bs The optimum umber of berths correspodig to a suitable cost ratio r bs values (from 010 to 030) is oted i Figure 8 It ca also be see that a 33-berths set is the optimum port capacity i case of traffic itesity values varyig betwee 2758 ad 2960 Table 2 shows the calculatio of the costs of idle berths ad idle ships for 33 berths i view of the expected frequecy (umber of days per year) It also presets the combied costs (vacat berths ad ships) i case of port size 31, 32, 33 ad 34 berths The cost compariso idicates that the total port cost is least whe there are 33 berths This coclusio cofirms the result previously obtaied by applyig the proposed methodology RESULTS CONCLUSIONS This paper presets a methodology proposed to predict the optimum umber of berths required i a sea port to meet the future traffic volumes The methodology is based o the hypothesis that the umber of berths ca

8 84 J Soil Sci Evir Maage Table 2 Cost calculatio i case of 33 berths, ad the compariso of the resultig value with those for 31, 32 ad 34 berths Arrival rate (ships/day) Predicted frequecy (i days) Berth utilizatio Required umber of berths Over-buildig Number of berths Berthsdays Uder-buildig Number of ships Ships-days µ F λ µ 33 F (33 ) λ - X * F (λ - X * ) Total Cost i Millio $ (usig c b 600, c s 6000) Total costs for 33 berths i Millio dollars 289 Total costs for 31 berths i Millio dollars 315 Total costs for 32 berths i Millio dollars 308 Total costs for 34 berths i Millio dollars 293 X * Number of available berths maximum berth utilizatio / average service time 33 1/ be icreased as log as the margial cost of berths (costructio ad maiteace) is less tha the delay costs of waitig ships The Queuig theory has bee employed to derive the umber of waitig ships ad the average ship delays The usage of queuig theory is subjected to the followig two assumptios: i) Ships arrivals at a sea port ca be described as a egative expoetial distributio, ad, ii) Berth service time yields to a multi-expoetial fuctio The employmet of the queuig theory to study the movemets of geeral cargo ships at Alexadria port was profitable The observed patter of ships arrivals appears to agree with Poisso s law of radom distributio I additio, the berth service time for 315 ships was foud to coform most closely to a Erlag distributio with K 3 The usage of a approximate model of queuig theory led to acceptable results The criterio for acceptace of this model was the reasoable agreemet achieved betwee the computed ad observed values of average waitig time ad average umber of waitig ships i queues at berths Thus, there is o doubt that ships arrive at Alexadria port i accordace with a radom patter ad that the degree of accuracy compares favorably with the accuracy that may be realized i estimatig future traffic The applicatio to Alexadria port verifies the aticipated beefit of usig the suggested methodology to evaluate the port size i the best iterests of both ship operators ad the port authority The evaluatio is settled o the premise that maximum port efficiecy results whe the total port cost is miimum, that is, the cost of vacat berths over a substatial period plus the time cost of ships waitig for a berth durig the same period REFERENCES Gokkup U (1995) Applicatio of queuig theory o the desig of fishig Harbors, Trasactios o the Built Eviromet 11: Braislav D, Nam KP (2006) Modelig of ship-berth-yard Lik Performace ad Throuput Optimizati, IAME 2006 Coferece, Melboure, Australia, Tadashi Y (2003) Optimizig the hadlig Capacity i a Cotaier Termial for Ivestigatig efficiet Hadlig Systems, J Easter Asia Society for Trasportatio Studies, 5: So JH, Kim MH (2004) A aalysis of the optimal umber of servers i distributed cliet/servers, Eviro Decisio Support Syst, 36: Zora R, Braislav D (2005) Optimal Number ad Capacity of Servers i Queuig Systems, If Maage Sci, 16(3): 1-16

9 We-Chih H, Tu-Cheg K, Sheg-Chieh W (2007) A Compariso of Aalytical Methods ad Simulatio for Cotaier Termial Plaig, 200 J Chiese Istitute of Idus Egieers, 24(3): Ja OJ, Robert E (2002) Uificatio of Accouts ad Margial Costs for Trasport Efficiecy Swedish Seaport Case Study: Price-relevat margial cost of Swedish seaport services, ITS, Uiversity of Leeds, El-Naggar 85 Fuded by the Europea Commissio, 5th Framework Trasport RTD Egyptia Maritime Data Bak (2008) Aual Statistical Report, Natioal Plaig Istitute, Cairo