Port's Bulk Loading Terminal

ISEP 2006

STOCHASTIC PROCESSES OF

PORT'S BULK LOADING TERMINAL

Mirano Hess, Svjetlana Hess

Faculty of Maritime Studies

Studentska 2, 51 000 Rijeka, Croatia

e-mail: ,

ISEP 2006

Abstract

Necessity for studying the processes in bulk cargo ports, which are the objects of research in this paper, follows from the fact that many real processes, which are characterized with unpredictability and changeability, are called stochastic because the parameters that determine these processes are the stochastic ones. In this paper the theoretical features of the Poisson process and a non-Poisson process, related to the queue will be given. This type of the queue, marked with M/D/1 by the Kendall notation, characterizes Poisson distribution of entity arrivals and deterministic distribution of service time. The main goal of this paper is to define port's bulk cargo terminal as the queuing system and then set up the appropriate model in which the loading terminal is determined as queuing system of type M/D/1. Subsequently, the model set up will be tested on the real example of the port's bulk cargo loading terminal in Bakar. On the basis of this model, the operating of the port can be determined with the purpose of making decisions for further running of the business.

Keywords

stochastic processes, queuing system M/D/1, loading terminal of the bulk cargo port

INTRODUCTION

Many real processes, which are characterized with unpredictability and changeability, are called stochastic because the parameters determining these processes are the stochastic ones. A part of the probability theory dealing with the stochastic processes is called the theory of stochastic processes.

The queuing theory is one of the operational research methods dealing with the servicing processes of the entities that randomly arrive at the system and demand the service. By the mathematical models of queuing theory the interdependence between entity arrivals, waiting on service and their leaving the system are determined, with the purpose of system functionality optimization. The basic features of the queuing phenomenon are the mass character and random property, because the demand for service and the service time are stochastic variables.

Through statistical data analysis on the number of ship arrivals per day and month of a chosen bulk cargo terminal, it has been established that no significant dependence exists in the sequence of daily arrivals of bulk ships, i.e. that arrivals are statistically random.

From the previous statement, it follows that the number of ship arrivals can be taken as random variable and, in addition, the empirical distribution of this variable approximated with the appropriate theoretical distribution. The queuing theory can be applied in such cases for computing indices of port's bulk cargo terminal operations.

In this paper the theoretical features of the Poisson process and a non-Poisson process, related to the queue will be given. This type of queue, marked with M/D/1 by the Kendall notation, characterizes Poisson distribution of entity arrivals and deterministic distribution of service time. The main goal of this paper is to identify port's bulk cargo terminal as the queuing system and then set up the appropriate model in which the loading terminal is determined as queuing system of type M/D/1. Subsequently, the model set up will be tested on the real example of the port's bulk cargo loading terminal in Bakar. Through application of the proposed model it should be possible to make decision on how to make optimization of the transhipment processes on a bulk loading terminal to increase its efficiency.

THEORETICAL FEATURES

The M/M/1 system is made of a Poisson arrival, one exponential (Poisson) server, FIFO (or not specified) queue of unlimited capacity and unlimited customer population. Note that these assumptions are very strong, not satisfied for practical systems (the worst assumption is the exponential distribution of service duration - hardly satisfied by real servers). Nevertheless the M/M/1 model shows clearly the basic ideas and methods of queuing theory. Next part summarizes the basic properties of the Poisson process and gives derivation of the M/D/1 theoretical model [6].

Most of the queue models that are not made of a Poisson arrival and exponential (Poisson) server, because of its complexity demand special approach. One of the models with Poisson arrivals and general independent service times has notation M/G/1. A modification of this model, with the assumption of constant service time is the model M/D/1, which will be used in this paper and applied at the real example.

The Poisson process

The Poisson process satisfies the following assumptions, where P[x] means "the probability of x":

1) P[one arrival in the time interval (t, t+h), for h 0] = h+o(h), where  is a constant, t 0.

2) P[more than one arrival in the interval (t, t+h)] .

3) The above probabilities do not depend on t ("no memory" property - time independence - stationarity).

Let pn(t) = P[n arrivals in the time interval (0, t)]. Using the above assumptions 1) and 2), it is possible to express the probability pn(t+h), h  0:

pn(t+h) = pn(t) [1h] + pn-1(t)h(1')

(n arrivals by t, no more arrival or n-1 arrivals by t, one more arrival)

p0(t+h) = p0(t)[1h] (1'')

(no arrival by t, no more arrival)

The equations (1') and (1'') may be written in this way:

, .(2)

Because of small h the terms at the left sides of (2) may be considered as derivatives:

, ,

that is:

, .(3)

Equations (3) represent a set of differential equations, with the solution

, n=0, 1, 2, ...(4)

Because of the assumption 3) the formula (4) holds for any interval (s, s+t) from R+. In other words the probability of n arrivals during some time interval depends only on the length of this time interval (not on the starting time of the interval).

Number of arrivals Nt during some time interval tis a discrete random variable associated with the Poisson process. Having the probabilities of the random values (4), it is possible to find the usual parameters of the random variable Nt. Let E[X] be the mean (average) value, Var[X] the variance, and Std[X] the standard deviation of the random variable X:

, t0 ,(5)

where (5) gives the interpretation of the constant , that is the average number of arrivals per time unit. That’s why the  is called arrival rate.

.(6)

.(6')

Another random variable associated with the Poisson process is the random interval between two adjacent arrivals. Let X be the random interval. To find its distribution, let’s express the distribution functionF(x):

F(x)=P[Xx] = P[at least one arrival during the interval x] = 1 -p0(x) = 1 e-x.

Because the interval is a continuous random variable, it is possible to compute the probability density as a derivative of the distribution function:

f(x) = dF(x)/dx .

Subsequently, distribution function of the random variable X is:

.(7)

The distribution (7) is called exponential distribution, with parameter >0. For the exponential distribution, probability density is:

.(8)

If the random variable X is distributed by the exponential distribution with parameter >0, then:

,,

.(9)

The expression (9) gives another interpretation of the constant . Its inverted value is the average interval between arrivals. Like the number of arrivals, the distribution of intervals between arrivals does not depend on time.

When applied to a service, the rate is called service rate(). The parameter  is the average number of completed services per time unit (provided there are always customers waiting in the queue). Its inverted value 1/ is the average duration of the exponential service. The variance and the standard deviation can be computed by replacing  by  in the formulae (9). Unlike arrival, exponential service is an abstraction that is hardly satisfied by real systems, because mostly it is very unlikely to have very short and/or very long services. Real service duration will be typically "less random" than the theoretical exponential distribution.

Another very important parameter of queuing systems is the ratio  of the arrival and the service rates called traffic rate(sometimes called traffic intensity or utilisation factor), representing the ratiobetween the arrival and service rate:

(10)

The value of  shows how "busy" is the server. It is obvious, that for  1 the queue will grow permanently. Therefore, the basic condition of the system stability is  < 1, for the cases with the unlimited customer population.

NonPoisson models - type M/D/1

Most of the queue models that are not made of a Poisson arrival and exponential (Poisson) server, because of its complexity demand special approach. In these cases, simulation methods are being used with which it is relatively simple to simulate the process in the queuing model, or the analytical methods are being used.

One of the models with Poisson arrivals and general independent service times, with the mean (average) value E[t] and the variance Var[t], t0, has Kendall notation M/G/1. The inadequacy of this model is limitation regarding obtained results. It's impossible to compute the probabilities pn , therefore only basic parameters are being determined – the number of entities in queue LQ and in system L and the waiting time in queue WQ and system W .

Let  be the expected number of Poisson arrivals, and service time distribution with E[t] and Var[t], where t is nonnegative random variable, t0, then the model M/G/1 has following formulae [1, pp. 428]:

 the expected number of entities in system, known as Pollaczek-Khintchine (P-K) formula:

,where E[t] <1, t  0 ,

and by insertion the service rate μ=1/E[t] next is obtained:

,

 the expected number of entities in queue:

,

 the expected waiting time in system:

,

 the expected waiting time in queue:

.

The modification of this model, with the assumption of constant service time is the model M/D/1. According to mentioned assumption Var[t] =0. In that case a Pollaczek-Khintchine (P-K) formula is simplified [1, pp. 430]:

,

where ρ=λ/μ, and μ is constant rate of service.

If the service time t is distributed by Erlang with parameters k and μ (so E[t] =1/μ, and Var[t] =1/kμ2), which has notation by Kendall M/Ek/1, Pollaczek-Khintchine (P-K) formula transforms its form into following:

.

The rest indices of the model are:

 average number of entities in queue:

,

 expected waiting time in queue:

,

 expected waiting time in system:

.

For the system M/D/S, where S>1, the other formulae would be used [7 and 8].

MODEL FOR A PORT'S BULK CARGO LOADING TERMINAL

In order to define port's bulk cargo loading terminal as the queuing system and then set up the model M/D/1 it is necessary to obtain basic features of the bulk cargo port, technical characteristics of the facilities in the port and describe working technology and organization.

Basic features of the bulk cargo port Bakar

The systems which functioning is subjected to random changes are analyzed by queuing theory. For analyzing port as a mass servicing system, the following is significant [3, p. 493]:

- arrival time of ships can not be predicted with certainty,

- servicing time, i.e. duration of transshipment, is random variable dependent on facility's capacity, transhipment line mechanization, ship deadweight, weather, and so on,

- facility doesn't have constant working load since there are maintenance, inspection and repair breaks, breakdowns, breaks due to bad weather causing queues and intervals of low degree of utilization.

Through analysis of present state of the facility the question that emerges is whether it is justifiable to invest in modernisation and reconstruction of the port in order to produce better business effects or it would be more appropriate to build a strategy for optimization of usage of existing resources.

Through analysis of the bulk cargo port Bakar the next is observed [4, p. 54, 55]:

- port is an open system since the ship entries are not part of it,

- the bulk cargo port Bakar has two specialized quays for which ship queuing lines are eventually formed at anchorage,

- unlimited number of ships waiting on service,

- ships are patient clients, they don't abandon queue,

- arrival rate is Poisson distributed which can be determined with statistical tests and is the most frequent case in practice,

- servicing time, that is time that ship spends at the quay for loading has deterministic distribution because loading is continuous without breaks,

- mutual assistance between loading and unloading terminals does not exist,

- FIFO service rule is applied, without priority.

Course of ship arrivals is stationary Poisson course with the following properties [3, p. 495]:

- time independence, in arbitrary short time probability to arrive more than one ship is very small, i.e. ships enter the port one by one,

- "no memory" property, arrivals of a ships are independent,

- stationarity, intensity of a ship course is time independent since it is constant value dependent only on length of the observed period.

For the bulk cargo terminal system, parameter  represents the average number of bulk ships or quantity of bulk cargoes that arrive at the terminal during an observed time unit (e.g. during a year, month or day). However, in this paper arrival entities are the average quantity of bulk cargoes arrived by ships into port on yearly basis.

The average number of bulk ships (in this case the average quantity of bulk cargo) that can be serviced in a time unit at certain berths is service rate ..

ISEP 2006

Table 1: Facility capacities of the bulk cargo terminal Bakar

ISEP 2006

Quay Podbok
length: 394 m / maximum depth: 18.5 m / ship: max 160,000 DWT
Storage
length x width (m) / capacity in tons / Equipment
iron ore / coal
Podbok / 330 x 27 / 300,000 / 100,000 / discharging equipment with conveyor
Dobra / 340 x 19 / 80,000 / 25,000 / conveyor
Plato / 160 x 36 / 80,000 / 26,000 / unequipped
Cranes
capacity in t/h / year of manufacture / quantity
discharging equipment no.1 / 800 / 1967. / 1
discharging equipment no.2 / 1,600 / 1978. / 1
loading equipment / 600 / 2001. / 1
storage conveyor / 500 / 1967. / 1
Conveyors
capacity for iron ore / 1,600 t/h
capacity for coal / 1,300 t/h
Capacities according to cargo type in tons
Ore / Coal
single storage capacity in tons / 400,000 / 150,000
tech. capacity ship-storage for crane 45+16 t/shift / 8,167 / 3,268
tech. capacity ship-storage for crane 45 t/shift / - / 2,400
tech. capacity storage-wagon for storage gantry crane in t/shift / 2,500 / -
theoretical max. capacity in tons / 4,960,000 / 3,000,000
real capacity in tons / 3,500,000 / 2,000,000
technological-market capacity in tons – loading terminal / 1,100,000

ISEP 2006

Source: Port of Rijeka [5]

The ratio between arrival rate and service rate of cargo quantity is traffic rateor utilisation factor, that is traffic intensity of the berth (=/).

If , one berth is insufficient as the employment rate is greater than 100%. In this event, the number of berths should be increased until service system is brought to stability condition, that is the system employment coefficient S< 1 has been satisfied.

In practice, parameters values  and  are determined on the basis of empirical data or assessment depending on the goal and subject of research.

Based on a definition of bulk cargo terminal as a service system and basic parameters of a terminal, operation indices of a loading bulk cargo terminal are computed.

In this paper loading bulk cargo terminal Bakar, as part of port of Rijeka, has been analyzed. This terminal is capable to handle various types of bulk cargoes, iron ore, coal, bauxite, phosphate.

Loading terminal has maximum degree of utilization for cargoes with bigger specific gravity, for example iron ore. Limiting possibility for expanding port capacities, so far as forwarding is concerned, rest on the number of stationed wagons per day. The amount of cargo transported by wagons amounts to 7,000 - 8,000 tons a day, and the maximum capacity of the wagon distribution center is 14,000 tons. The facility capacities are presented in table 1.

Since the transhipment process consists of several technological operations (weighing of cargo, transport of cargo with conveyor from storage to ship, ship loading), the theoretical maximum capacity of loading terminal includes the maximum capacity of every single equipment that take part in transhipment (cranes, conveyors, distribution station, weighbridge, storage, wagons). Theoretical maximum capacity implies the maximum capacity of the equipment with the minimum capacity in transhipment chain.

Terminal capacity is the maximum terminal capacity reduced for the cargo that has not been transhipped during breaks which includes breaks caused by mechanical failures of equipment, breaks caused by maintenance and cleaning of facilities, working breaks and time required for ship mooring and unmooring.

Finally, quantity of cargo transshipped in port doesn't depend only on equipment, transport and storage capacities, but also on external factors. These are:

- transport of cargo in port and out of port that depends on railway flow rate, flow rate of the railway hub and inland storages,

- cargo demand,

- breaks caused by weather or strikes.

Technological-market capacity of the terminal includes the above factors, and is calculated taking into account the terminal capacity and several-years record of cargo flows.

Modeling the port's bulk cargo loading terminal Bakar - queueing model M/D/1

Bulk cargo port in Bakar contains unloading and loading bulk cargo terminal. The procedure of loading cargo on ships consists of several technological processes:

- cargo load with storage gantry cranes or bulldozer from storage on conveyor belts,

- cargo transport by conveyors from storage to port's loading equipment, and

- cargo load on ship.

The assumption is that loading is continuous without any breaks and bottlenecks and for this reason loading time is constant. Subsequently, duration of ship service, i.e. time of ship's stay at the loading terminal, has deterministic distribution.

From statistical data of port of Rijeka follows that the loading terminal's arrival rate of cargo (coal)  is 918,518 tons for year 2005.

It can be seen from table 1, that the yearly capacity of the loading terminal, representing the loading terminal service rate , amounts to 1,100,000 tons (technological-market capacity). This data takes into account capacity of storage equipment and capacity of conveyors, which yearly capacity separately regarded is well over,. In contrast the storage capacity is here a key limitation factor.

According to the queuing systems classification, the loading terminal is the queuing system with one service place and unlimited number of entities in queue, where service time is deterministically distributed with notation M/D/1/.

Consequently, the parameters for the observed model M/D/1/ are: