Analysis of an M/G/1 system for the optimization of the RTG performances in the delivery of containers in Abidjan Terminal
Bakary Kone, Salimata Gueye Diagne, Dethie Dione, Coumba Diallo
AAnalysis of an M/G/1 system for the optimizationof the RTG performances in the delivery ofcontainers in Abidjan Terminal
Bakary Kon´e , Salimata Gueye Diagne , D´ethi´e Dione , Coumba Diallo [email protected]@gmail.com Abstract
In front of major challenge to increase its productivity while sat-isfying its customer, it is today important to establish in advance theoperational performances of the RTG Abidjan Terminal. In this arti-cle, by using an M/G/1 retrial queue system, we obtained the averagenumber of parked delivery trucks and as well as their waiting time.Finally, we used Matlab to represent them graphically then analyzethe RTG performances according to the traffic rate.
Key words
RTG, Embedded Markov chain, Retrial Queue, Stationary Distribution, Performance Measures
In today’s maritime world, marked by increasing modernity, containers ter-minals do not only have the obligation to straighten their infrastructures,but also they new ones have to improve to keep their customer and acquire.This task is not easy given the deep international competition. A task Ter-minals with containers show awareness by trying to satisfy ever increasingtheir customer. This report was made at Container Terminal of Abidjan(Abidjan Terminal), where one of the priorities is to satisfy the customerin particular trucks in mission of delivery to the city [12]. A vast distri-bution platform of containers of any singles and the modern installations,allow Abidjan Terminal to receive and to handle a high volume of importedcontainers and a more and more demanding customer. Given its importantrole in the Ivory Coast maritime exchanges, it is important to analyze theRTG performances which establish its most used porticoes of courses in theoperations of containers deliveries [11].By using the queue theories, several researchers have already carried out1 a r X i v : . [ c s . PF ] A ug tudies which analyses the performances in various domains. It is the caseof Mohamed Boualem and al. (2013) who deal retrial queue system andBernoulli feedback [6]. They considered the type M/G/
M/G/
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As well as its role of storage space, the zone under RTG constantly receivescontainers in import, export and transshipment. But in this section we willpresent the process of containers delivery.Figure 1: Map of Abidjan Terminal (cid:7)
Delevery process
The driver arrives at the sentry box of the Terminal provided with a de-livery slip. After routine check of the number of the container, the sentryagent affects on Oscar (Operating Software Container and adjournment isa management system of exploitation of the operations of a Container ter-minal) a token with a number to this container then confirms (putting indelivery). Then he returns the token to the driver. The latter leads thetruck on the park and parks at the address of the container indicated onthe token. The RTG operator having received the mission loads the truckafter checking its numbers (container and the token) (mission in processes).The loaded truck returns to the exit sentry box for the end of the deliveryprocedure. (cid:7)
The RTG operational function
The zone under RTG which constitutes our workspace is formed by 18 blocksparallel to the linear band of the Terminal Quays and the containing eachseveral bays of various types of containers. Containers are piled in stan-dard ISO there, according to their statuses at using addresses to allow theirlocalizations. During the delivery phase of containers, trucks make theirentrance by the procedure and park at this address of the container. TheRTG operator who covers the zone receives immediately the number of thetoken from a small screen placed in his cabin. This list reached him in ar-rival order of trucks. So, after checking the information, (the numbers ofthe container and that of the token) the RTG operator loads the container.Contrary to arrival spontaneous of the trucks, the RTG being in lower num-ber cannot handle all the containers at reasonable time. The loading is doneat a disproportional rhythm in the flows of parked trucks. In order to in-crease the operational and competitive qualities of the Abidjan Terminal,it is important to analyze the performance of its machines to make it moreattractive. In this article, by using the generative functions, we analyze theRTG performances which allows to identify the critical elements or, again,to grasp the effects of a modification of the operating conditions.
It is composed of several stages:
Delivery trucks make their orderly entry and park at the precise addressesof containers. The zone can contain several trucks which makes its structurelook like a classic queue where the server is the RTG. The parked trucks arethe considered first category or the second category customers according totheir waiting times. While a truck parks and is not served for the momentthe service of piloting makes retrials to the RTG operator so that this truck4s loaded. Once served, it immediately leaves the Terminal with the con-tainer. • The primary trucks are those which are immediately loaded upon theirarrival. • The secondary trucks are the ones which wait for a whole. These trucksare loaded further to one or several retrials made by the piloting. Indeed,to avoid of lengthy waiting, the piloting follows and coordinates the move-ments of machines on the Park. He constantly reminds the RTG operatorsto handle first and foremost the parked trucks having spent a long timewaiting. This principle of reminder is defined according to the number ofparked trucks and the duration of their waiting time. That increases inintensity when the zone of the RTG is saturated. The intervals the retrialstime are they follow an exponential distribution rate kθ where ( k ∈ N isthe number of waiting trucks and θ > al. b ( t ) istransformed by Laplace-Stieltjes B ( t ) for t at a given moment. That meansof τ si is the service which is between the i th and ( i + 1) th service, then P( τ si ≤ t ) = B ( t ). If we are in the i th stage, the RTG executes the i th servicethere which corresponds to the loading of the i th delivery truck. The time of( i +1) th begins just at the end of the loading of the i th truck. It’s establishedby the travel time made by the RTG between these two trucks and by theunproductive movements time as well as the loading of the container. Thefirst two moments finished of this distribution are β and β , respectively.We suppose that the RTG is free every time the system becomes empty(withno trucks in the zone)Either t represents any moment. The state of the system at this arbitrarymoment is given by: M ( t ) = { C ( t ) , N ( t ) , ζ ( t ) , } With, C ( t ) = ( N ( t ) represents the number of trucks parked at a given moment tτ ( t ) represents the duration of the RTG service to handle the truck if C ( t ) =1. The Poisson distribution of an
M/G/ i The stage in which a service was made by the RTG. Aservice corresponds to a stage which amounts to a truckloaded. k, j
The number of trucks in waiting. ζ i The exact time from which the i th service is ended. τ si The elapsed time by the RTG for the i th service N ( ζ i ) The total number of waiting trucks parked after the i th service.Let us consider { ζ i , i ∈ N } an increasing continuation of moment ofthe completion of a service time. Or N ( ζ i ) the number of trucks parkedwaiting in the system just after the loading of the i th truck. We then definea stochastic process by the sequence of the random variables: Q i = { N ( ζ i ); i = 0 , , , , · · · } where ζ i is the exact moment of the end of the treatment of the i th truck.Let us pose: N ( ζ i ) N i And A i +1 the number of parked trucks while ( i + 1) th truck is being served.Then, the random variables A i are independent between them; their commondistribution is given by: P ( A i +1 = k ) = q k = Z + ∞ ( λt ) k k ! e − λt b ( t ) dt ; q k ≥ , ( k = 1 , , ... ) (1)Its generative function is: A ( z ) = + ∞ X k =0 q k z k = B ( λ (1 − z ))The fundamental equation of embedded Markov chainWhen the RTG is handling ( i + 1) th truck, if A i +1 arrive, then at the endof his load we shall have: N i +1 = ( N i − A i +1 si N i ≥ A i +1 si N i = 06ence the fundamental equation of embedded Markov chain is given by: N i +1 = N i − δ N i + A i +1 (2)The variable of Bernoulli δ N i is defined by: δ N i = ( i + 1) th Served truck is from the second category0 Otherwise de-pend on N i . It’s translated by in most 1 truck left and A i +1 trucks arrivedat the end of ( i + 1) th service.Conditional distributionFor k trucks waiting in the stage i , the following state is determined ac-cording to the value taken by δ N i . Then the conditional distribution isgiven by (Boulem 2011) P ( δ N i = 1 /N i = k ) = kθλ + kθP ( δ N i = 0 /N i = k ) = λλ + kθ (3)With θ the flow of truck loaded from the second category.Transition rate.The probability that there is j trucks parked in ( i + 1) th stage knowingthat it had k the i th is noted: ∀ j ≥ ≤ k ≤ j , we have: P kj = P ( N i +1 = j/N i = k ) (4)Which gives: P kj = P ( A i +1 = j − k ) P ( δ N i = 1 /N i = k )+ P ( A i +1 = j − k +1) P ( δ N i = 0 /N i = k )According to (1) and (4) we have: P kj = ( P ( N i +1 = j/N i = k ) = q j − k kθλ + kθ + q j − k +1 λλ + kθ , ≤ k ≤ j ζ i is the precise moment in which the treatment of the i th truck parked in the line ended.We verify that this continuation of random variable is an actually Markovchain in discrete time.Yet A i the number of parked trucks while the i th truck is being served.7xistence condition of the stationary distribution of the number of trucksThe chain is Markovian because the state of the system in the stage ( i + 1),depends on the state i . So that the line does not lengthen infinitely, theaverage number of arrivals during the duration of a service must be strictlylower than 1. Then, we have: E ( A i ) = i X k =0 Z + ∞ P ( A i = k ) b ( t ) dt = i X k =0 k Z + ∞ ( λt ) k k ! e − λt b ( t ) dt = Z + ∞ e − λt b ( t ) λt { + ∞ X k =0 ( λt ) i k ! } dt = λ Z + ∞ tb ( t ) dt = λβ With β the moment of order 1 of the distribution. Hence the existence ofthe stationary distribution is conditioned for: ρ = λβ ≤ N i and thus is the trans-formed z of its function of density which is defined by: f i ( z ) = E ( z k ) = + ∞ X k =0 z k P [ N i = k ]Yet N i +1 = N i + δ N i + A i +1 f i +1 = E [ z N i − δ Ni + A i +1 ]As A i and N i are independent then, f i +1 = E [ z N i − δ Ni + A i +1 ]= E [ z N i − δ Ni ] E [ z A i +1 ]8iven that the condition of stationarity is then verified A ( z ) = E ( z A i )independent of i . Thus we have: f i +1 = E [ z N i − δ Ni ] E [ z A i +1 ]= E [ z N i − δ Ni ] A ( z )= [ + ∞ X k =0 z k − δ k P ( N i = k )] A ( z )= [ + ∞ X k =0 z k P ( N i = k ) × P ( δ N i = 0 /N i = k )+ + ∞ X k =1 z k − P ( N i = k ) × P ( δ N i = 1 /N i = k )] A ( z )= + ∞ X k =0 z k P ( N i = k ) λλ + kθ + + ∞ X k =1 z k − P ( N i = k ) kθλ + kθ In the stationary state,lim i + ∞ f i +1 ( z ) = lim i + ∞ f i ( z ) = f ( z ) and the stationary distribution asdefined by: π k = lim i + ∞ P ( N i = k ) By using them in the equation, weobtain: f ( z ) = + ∞ X k =0 z k π k λλ + kθ + + ∞ X k =1 z k − π k kθλ + kθ (7)We pose: ψ ( z ) = + ∞ X k =0 π k z k λ + kθ Then ψ ( z ) = + ∞ X k =1 kπ j z k − λ + kθ So, the equation (10) allows us to obtain the following relation: f ( z ) = A ( z )[ λψ ( z ) + θψ ( z )] (8)In a different way, we notice that: f ( z ) = + ∞ X k =0 π k z k = + ∞ X k =0 π k z k λ + kθλ + kθ = λψ ( z ) + θψ ( z )9ence the relation: f ( z ) = λψ ( z ) + θψ ( z ) (9)Consequently, the use of the equations (11) and (12) entails: λψ ( z ) + θψ ( z ) = A ( z )[ λψ ( z ) + θψ ( z )] θψ ( z )[ A ( z ) − z ] = λψ ( z )[1 − A ( z )] (10)The function h ( z ) = A ( z ) − z positive, increasing for z ∈ [0 ,
1] and ρ < z < A ( z ) < h (1) = 1 h ( z ) = A ( z ) − z , and h (1) = ρ − <
0. Besides by using the results (4)and (5), we have: ψ ( z ) = λθ { − A ( u ) A ( u ) − u } ψ ( z )For ρ <
1, The resolution of this differential equation gives us: ψ ( z ) = ψ (1) exp { λθ Z z (1 − A ( u )) A ( u ) − u d u } (11)Yet f ( z ) = λψ ( z ) + θψ ( z )Therefore f ( z ) = λψ ( z ) + { θ λθ z (1 − A ( z )) A ( z ) − z } ψ ( z )= λψ ( z ) A ( z ) 1 − zA ( z ) − z The traffic intensity is ρ = λβ and f (1) = B (0) = 1. So a simple calcula-tion gives us − zA ( z ) − z = h (1) = ρ − ψ (1) = − ρλ .By replacing ψ (1) by its expression we obtain the generative function of thestationary distribution of the number of trucks waiting is: f ( z ) = (1 − ρ ) B ( λ − λz )(1 − z ) B ( λ − λz ) − z × exp { λθ Z z [(1 − A ( u ))] A ( u ) − u d u } (12) In this section, we establish the explicit expressions of the RTG performancemeasures during the operations of delivery of containers in this form.10 roposition 5.1
In the stationary state, the average number of trucks inwaiting in the RTG zone and the average waiting time of the delivery trucksof containers are respectively expressed by: N = ρ + λρθ (1 − ρ ) + λ β − ρ ) (13) W = β + λβ − ρ ) + ρθ (1 − ρ ) (14) Demonstration • Let us determine the average number of trucks waiting in the RTG zone.In the stationary state, the average number of trucks being in the RTGzone corresponds in numbers of trucks which were already parked and thosewhich have just arrived. It thus expresses itself by: N = lim i + ∞ E ( N i +1 ) = lim i + ∞ E ( N i )Because f ( z ) of its generative function, This allows to say that N = f (1)With f ( z ) the derivative of the generative function.So we calculate f ( z ), then we replace z by (1) to obtain the different per-formances.Calculation of the derivative of the generative function f ( z ) = (1 − ρ ) B ( λ − λz )(1 − z ) B ( λ − λz ) − z × exp { λθ Z z [(1 − A ( u ))] A ( u ) − u d u } We pose: G ( u ) = [(1 − A ( u ))] A ( u ) − u , where 1 ≤ u ≤ zT ( z ) = (1 − ρ ) B ( λ − λz )(1 − z ); P ( z ) = exp { λθ R z g ( u ) du } and Q ( z ) = B ( λ − λz ) − z So T (1) = 0 , P (1) = 1 , Q (1) = 0 (15) T ( z ) = (1 − ρ ) B ( λ − λz )(1 − z ) + ( ρ − B ( λ − λz ) P ( z ) = λθ × G ( z ) exp { λθ Z z g ( u ) du } Then T (1) = ( ρ − , P (1) = − λθ × ρ − ρ , Q (1) = ( ρ −
1) (16)11et us now notice that: f ( z ) = T ( z ) P ( z ) Q ( z )Because we have: 1 − zB ( λ − λz ) − z = h (1) = 1Then f (1) = 1 (17)Therefore f ( z ) Q ( z ) = T ( z ) P ( z ) f ( z ) Q ( z ) + f ( z ) Q ( z ) = T ( z ) P ( z ) + T ( z ) P ( z )By replacing each function by its expression Z = 1, we have an indefiniteshape. We raise it by the application of the hospital theorem.The second derivative gives: f ( z ) Q ( z )+2 f ( z ) Q ( z )+ f ( z ) Q ( z ) = T ( z ) P ( z )+2 T ( z ) P ( z )+ T ( z ) P ( z )The equations (13) allow to obtain: f ( z ) = T ( z ) P ( z ) + 2 T ( z ) P ( z ) − f ( z ) Q ( z )2 Q ( z ) (18)Yet T ( z ) = 2 ρ ( ρ − β k = ( − k β k (0) β = − β (0) , β = β (0)The equations (14) and (15) entail: f (1) = 2 ρ ( ρ −
1) + 2 ( ρ − × λ × ρθ × (1 − ρ ) − λ β ρ −
1) (19)In conclusion, • The average number of trucks waiting in the zone of the RTG is given by: N = ρ + λρθ (1 − ρ ) + λ β − ρ )This expression is the total sum of the trucks which were parked just atthe beginning and those which arrived during ( i + 1) th service, the averagenumber of trucks which parks during the duration of its service is ρ . • Let us determine the average waiting time of the delivery trucks of con-tainers 12et us apply the formula of Little. So this number expresses itself by: W = Nλ Which gives: W = β + λβ − ρ ) + ρθ (1 − ρ ) To simulate our results, we limit our work to an RTG zone. We analyze theinfluence of the traffic rate on the stationary characteristics of the systemto obtain a tendency of the variations of the number of trucks and theirwaiting time. For that purpose, we consider the variation of the traffic inthe
M/G/ β
1. So, the rate of retrials is 1 , . In this article, we made our works on the operation zone of an RTG in Abid-jan Terminal. By using the method of the generative functions, the theoryof quasi Markovian retrial queue allowed us to obtain the performances suchas the number of trucks and their waiting time. We used the software ofprogramming 13 .
10 version of the Matlab for graphical representations toanalyze and release the sensitivity of the operations processing regarding todelivery trucks in the Terminal. The obtained results allowed us to say thatthe waiting times of delivery trucks depend widely on the traffic rate. Theoperational performances of the RTG depend widely on the traffic rate.In perspective, we plan to estimate the unproductive movements then theprobability of breakdowns of the RTG by using the same factor of sensitiv-ity.
Acknowledgements
I thank the Director of the Abidjan Terminal for welcoming me as a trainee.I am very grateful to Mrs Asta Rosa CISSE for her moral support.I am extremely gratitude to Mr Laurent Bodji KASSI Chief Operating Offi-cer and all his team for their simplicity, their loyalty and their professionalquality . 14 eferences
1. Choo. H. Q and Conolly. B (1979). New results in the theory of re-peated orders queueing syst`ems.
Journal of applied probability
16 :631 − Naval Research logistics quarterly , 9 : 31 − Managementscience , 1 : 18 − Journal europ´een des syst`emes automatis´e , 45 : 253 − Journal eu-rop´een des syst`emes automatis´e , 47 : 181 − https://tel.archives-ouvertes.fr/tel-00829089 ,PP: 35 −−