We study a situation in which, owing to the exhaustion of non-renewable energy sources, conventional motor vehicles will turn out of use. We consider two scenarios: recycling or dismantling these motor vehicles. M|G|∞ queue system is used to study the process. Through it, we conclude that if the rate of dismantling and recycling of motor vehicles is greater than the rate at which they become idle, the system will tend to get balanced. The model allows also performing a brief study about the recycling or dismantling economic interest.

José A. Filipe

Manuel A. Ferreira

Manuel Coelho

English

Research Papers in Economics

 
 
 
  
 
 
 
Carlos Pestana Barros & Nicolas Peypoch  
 
 
A Comparative Analysis of Productivity Change in Italian and 
Portuguese Airports  
 
 
 
 
 
WP 006/2007/DE 
_________________________________________________________ 
 
 
 
Manuel A. Ferreira, José A. Filipe and Manuel Coelho 
 
 
A queue model for recycling and dismantling motor 
vehicles 
 
 
WP 18/2009/DE/SOCIUS 
_________________________________________________________ 
Department of Economics 
WORKING PAPERS 
 
ISSN Nº 0874-4548 
School of Economics and Management 
TECHNICAL UNIVERSITY OF LISBON  
 
1
 
A queue model for recycling and dismantling motor vehicles 
 
 
 
Manuel Alberto M. Ferreira 
Departamento de Métodos Quantitativos  
UNIDE/ISCTE  
Full Professor 
Edificio ISCTE  
Av. das Forças Armadas, 1649-026 
LISBOA, Portugal 
Phone: +(351) 914970560, +(351) 217903240.  
Fax: +(351)  217903941. 
Email: manuel.ferreira@iscte.pt 
 
José António Filipe 
Departamento de Métodos Quantitativos 
UNIDE/ISCTE 
Assistant Professor 
Edificio ISCTE  
Av. das Forças Armadas, 1649-026 
LISBOA, Portugal 
Phone: +(351) 966058963, +(351) 217903401.  
Fax: +(351)  217903941.  
Email: jose.filipe@iscte.pt 
 
 
Manuel Coelho 
Departamento de Economia 
ISEG 
Assistant Professor 
Rua do Quelhas, 6, 1200-781 
LISBOA, Portugal 
Phone: +(351) 213925800.  
Fax: +(351)  213922808. 
Email: coelho@iseg.utl.pt 
 
  
 
2
A queue model for recycling and dismantling motor vehicles 
 
 
Abstract 
 
We study a situation in which, owing to the exhaustion of non-renewable energy sources, 
conventional motor vehicles will turn out of use. We consider two scenarios: recycling or 
dismantling these motor vehicles. M|G|∞ queue system is used to study the process. Through it, 
we conclude that if the rate of dismantling and recycling of motor vehicles is greater than the rate 
at which they become idle, the system will tend to get balanced. The model allows also 
performing a brief study about the recycling or dismantling economic interest. 
 
 
 
Introduction 
 
Many resources on Earth got overexploited. Oil, gas and coal exploitation posed many questions 
about the way humans use resources. The exploitation of non-renewable sources of energy and 
their use brought serious problems. We are now observing strong climatic changes, a fast 
extinction of species and many other critical problems that may have an end. Governments and 
institutions may change things. If not, very strong changes in our way of living and in the balances 
of the planet will happen soon. 
 
The generalized access to all goods by the majority of populations around the world has created 
new needs to people and has created a global consumption economy. In general, a significant 
part of the countries’ populations got access to the generality of industrial goods and it seems that 
people would not be prepared in the future to abdicate from it. 
 
However, the non-renewable energy, as we know it today, may not be enough for many decades 
more, from now on. And even if it would be possible to keep the production of this kind of energy, 
the effective changes on Earth balances would establish that life would not be compatible with the 
existing way of life for people. Consequently, people need to adjust behaviors and governments 
need to prepare their economies for the new era of globalization. 
 
New situations may occur very suddenly. The oil production has already got its peak and new oil 
productions will occur in the future with decreasing rents until its complete depletion. The other 
non-renewable sources of energy will have the same end. 
 
Therefore we may have to reorder the priorities and to reorganize structures in societies. We 
have now to produce the new kind of energies (clean energies) at a major scale. The big problem 
that remains is to know if the transition period from non-renewable sources of energy to the 
renewable sources is enough to overcome the big problems related with the destruction of Earth. 
All the wastes that people have made for so many decades must be overcome, as well. Anyway 
many kinds of new problems will occur. However what is important now is to know how quickly 
changes may happen while we develop the new sources of energy in order to create a new 
economy and a reorganized society. 
 
Too many motor vehicles will become idle if conventional energy misses or even when 
conventional energy becomes replaced by a renewable one. Motor vehicles dismantling or 
recycling will become very usual because there will not be a way to get them functional with 
conventional oil, since the moment it gets depleted.  
 
Just before oil becomes really scarce, it will get very high prices. At that moment, many motor 
vehicles’ owners must find out an alternative solution for energy. In our study we’ll see how to 
model a situation in which motor vehicles become idle and then recycled or dismantled,  
 
3
depending on their specific situation and depending on the will of motor vehicles’ owners. Our aim 
is to show that any kind of equipments that work on the basis of oil may have an alternative use 
when this conventional energy collapses; or simply they may become dismantled. Then other 
uses for their materials and components will be made.  
 
We apply our model to motor vehicles.  
 
A motor vehicles’ dismantling and recycling queue model 
 
We consider the M|G|∞ queue system (Kelly, 1979) where customers arrive according to a 
Poisson process at rate λ (Ferreira, 1996). They receive a service whose length is a positive 
random variable with distribution function G(.) and mean α. Each customer as soon as it arrives 
at the system, immediately finds an available server. Each customer service is independent from 
the others customers’ services and from the arrival process. The traffic intensity is given 
by λα ρ = . 
 
With this model we intend to analyze a situation in which motor vehicles arrive at the system 
getting idle and leave the system as soon as they are recycled or dismantled. Both situations are 
modeled with the same purpose in the model. Our interest in studying this situation is precisely to 
see how the system may recover to a balanced situation in which motor vehicles get operational 
or get dismantled (in this situation materials would become employed as components in other 
applications). 
 
Let  () t N  be the number of busy servers (or, what is the same, the number of customers being 
served) in the instant t, in a M|G|∞ system. If we consider ( ) ( )( ) [ ], 0 0 0 = = = N n t N P t p n   
,..., 2 , 1 , 0 = n we may have (Carrillo, 1991): 
()
() [] () []
,
!
1
0
1 0
0
∫


 

 −
=
− − ∫
t
dv v G
n t
n e
n
dv v G
t p
λ λ
     ,... 2 , 1 , 0 = n      (1). 
So if the initial instant is a moment at which the system is empty, the transient distribution is 
Poisson with mean () [] dv v G
t
∫ −
01 λ . 
 
The stationary distribution is the limit one: 
() ,
!
lim 0
ρ ρ −
∞ → = e
n
t p
n
n t
    ,... 2 , 1 = n         ( 2 ) .  
 
This queue system, as any other, has a sequence of busy periods and empty periods. A busy 
period begins when a customer arrives at the system, finding it empty. 
 
Let’s see the distribution of the number of customers being served in the instant t in the M|G|∞ 
system, when the initial instant is the moment at which a busy period begins, that gets relevant 
for our purposes. 
 
Be  () () [] , ' 1 0 ' 1 = = = N n t N P p n ,..., 2 , 1 , 0 = n and  ( ) ' 1 0 = N  the initial instant at which a customer 
arrives at the system and the number of customers being served turns from 0 to 1. This means 
that a busy period has just begun (Ferreira, 1988). 
  
 
4
So at the instant  0 ≥ t , we may have a situation that represents (Ferreira, 1998): 
1.  the customer that arrived at the system at the initial instant has left the system with a 
probability  () t G , or he remains in the system, with probability  ( ) t G − 1 ; 
2.  the other servers, which were empty at the beginning (initial instant), may be now empty or 
busy with 1, 2, ... customers, with probabilities given by  ( ), 0 t n p   ,.... 2 , 1 , 0 = n  
 
The two subsystems, the one of the initial customer and the one of servers initially empty, are 
independent. Consequently: 
() () () t G t p t p 00 0 ' 1 =  
() () () () () () , 1 1 0 0 ' 1 t G t p t G t p t p n n n − + = − ,... 2 , 1 = n      ( 3 ) .  
 
We also have for this situation: 
() ,
! ' 1 lim ρ ρ − =
∞ →
e
n
n
t n p
t
  ,... 2 , 1 , 0 = n         ( 4 ) .  
For the M|M|∞ system (exponential service times), the equations 3 are applicable even when 
() 1 0 = N  (when the initial instant is a moment at which there is a customer in the system; that 
does not enforce that this is the moment at which we are turning from 0 to 1 customer to be 
served). This results from the lack of memory of the exponential distribution. 
 
If  ) (t g  is the probability density function related to ) (t G , and if we call  ) (t h  the hazard rate 
function, we’ll have (Ross, 1983): 
() ()
() t G
t g
t h
−
=
1
           ( 5 ) .  
The function  ) (t h  is the rate at which services end. 
So, 
Proposition 1: 
If  () 1 < t G ,   0 > t , continuous and differentiable and if 
() ) (t G t h λ ≥ ,  0 > t           ( 6 )  
() t p 0 ' 1  is a non-decreasing function. 
Dem: 
It’s enough to observe that  () () ()  


 


−
−
− = ) (
) ( 1
) (
1 ) ( 00 0 ' 1 t G
t G
t g
t G t p t p
dt
d
λ . 
 
Besides, we may note that 
() , λ ≥ t h    0 > t            ( 7 )  
is a sufficient condition for the result in 6. 
So, if the rate at which the services end is greater or equal than the rate of arrivals, we conclude 
that  ) ( 0 ' 1 t p  does not decrease. 
 
For the system M|M|∞, the equation 7 is equivalent to  
 
5
1 ≤ ρ            ( 8 ) .  
 
Considering  () t , ' 1 µ  and  () t , 0 µ  the mean values of the distributions given by 3 and 1, 
respectively, we’ll have 
( ) () () () () () () = − ∑
∞
=
− + ∑
∞
=
= ∑
∞
=
= t G t
n
n np t p t
n
nG t
n
n np t 1
1
1 0 00
1 1
' 1 , ' 1 µ  
() ( ) () ( ) ( ) () ( ) () () . 1 , 0 1 1 , 0 0
0
t G t t p j t G t t G j
j
− + = + − + = ∑
∞
=
µ µ  
So, 
() ( ) ( ) [] dv t v G t G t ∫ − + − = 0 1 1 , ' 1 λ µ          ( 9 ) .  
 
Proposition 2: 
If   () 1 < t G ,   0 > t , continuous and differentiable and if 
() , λ ≤ t h    0 > t                               ( 1 0 ) .  
() t , ' 1 µ  is a non-decreasing function. 
 
Dem: 
It’s enough to observe that, considering equation 9,   () ( ) () ( ) () t h t G t
dt
d
− − = λ µ 1 , ' 1 . 
 
Besides, if the rate at which services end is lesser or equal than the rate at which customers 
arrive () t , ' 1 µ  is a non-decreasing function. We can note additionally that, for the M|M|∞ system, 
the equation 10 is equivalent to 
1 ≥ ρ                              ( 1 1 ) .  
 
Results and comments 
 
According to our study interests, the customers are the motor vehicles that become idle.  The 
arrival rate is the rate at which the motor vehicles become idle. The service time for each one is 
the time that goes from the instant they get idle until the instant they become recycled or 
dismantled. The service time hazard rate function is the rate at which the motor vehicles become 
recycled or dismantled.  
 
An idle period for our M|G|∞ system should be a one at which there were no motor vehicles idle.  
In a busy period there are always continuously idle motor vehicles. 
 
The equation 6 shows that if the dismantling and recycling rate is greater or equal than the rate at 
which motor vehicles get idle, the probability that the system gets empty (that is, there are no idle 
motor vehicles) does not decrease with time. This means that the system has a tendency to 
become balanced as far as time goes on. 
 
The equation 10 shows that if the dismantling and recycling rate is lesser or equal than the rate at 
which motor vehicles get idle, the mean number of motor vehicles in the system does not 
decrease with time. This means that the system has a tendency to become unbalanced as far as 
time goes on. 
 
Consequently, we conclude that when the rate of dismantling and recycling of motor vehicles is 
greater than the rate at which they become idle, the system has a tendency to get balanced. In 
this situation, the motor vehicles that become unused with the conventional energy turn useful 
with another kind of energy or get included in other useful devices.  
 
6
 
We must note that it is important the recycling or the dismantling of motor vehicles but, more than 
that, it is essentially relevant the cadence at which these actions are performed. Moreover, we 
give a reference for this cadence: λ, the rate at which motor vehicles get idle. 
 
An economic analysis as a complement to the model 
 
We have seen that rates λ and h(t) are determinant to monitor the way that the system of motor 
vehicles recycling and dismantling may be managed. 
 
We consider now additionally p as the probability for the motor vehicles arrivals destined to 
recycling and (1-p) as the probability for the motor vehicles arrivals destined to dismantling. Let 
hi(t), ci(t) and bi(t), i=1,2  be the hazard rate function, the mean cost and the mean benefit, 
respectively for recycling when i=1 and dismantling when i=2. 
 
With these new variables we can perform an economic analysis (beyond other considerations 
that may be posed) to evaluate about the interest of recycling and dismantling. 
 
We will analyze this situation in a global approach and not in a selfish way, considering the 
individuals point of view. 
 
So, we may consider the total cost per unit of time for motor vehicles recycling and dismantling 
as: 
λ λ ) ( ) 1 ( ) ( ) ( 2 1 t c p t pc t C − + =          ( 1 2 ) .  
 
Furthermore, the benefit per unit of time resulting from recycling and dismantling is given by: 
) ( ) ( ) ( ) ( ) ( 2 2 1 1 t h t b t h t b t B + =          ( 1 3 ) .  
From an economic point of view, it must be B(t)>C(t). 
 
To conclude about the advantage of recycling, we have the following: 
[ ] 0 ,
) (
) ( ) ( ) ( ) 1 ( ) (
max ) (
1
2 2 2 1
1 t h
t h t b t c p t c p
t b
− − +
>
λ λ
                    (14). 
G1(t) and G2(t) are both exponential, equation 14 becomes: 
[ ] 0 ), ( )) ( ) 1 ( ) ( ( max ) ( 2
2
1
1 2 1 1 t b t c p t pc t b
α
α
ρ − − + >                    (15). 
 
To conclude about the advantage of dismantling we have, in the same conditions: 
[ ] 0 ,
) (
) ( ) ( ) ( ) 1 ( ) (
max ) (
2
1 1 2 1
2 t h
t h t b t c p t c p
t b
− − +
>
λ λ
                   (16) 
and 
[ ] 0 ), ( )) ( ) 1 ( ) ( ( max ) ( 1
1
2
2 2 1 2 t b t c p t pc t b
α
α
ρ − − + >                    (17). 
  
 
7
So, there are minimum benefits above which, from an economic point of view both, recycling and 
dismantling, are interesting. The most interesting is the one for which this minimum benefit is the 
least. By other words: in a global perspective, it is more efficient the activity that corresponds to a 
lower level for the minimum interesting benefit. 
 
Recycling seems to be as much interesting as far as it is more economically profitable and our 
inequalities 14 to 17 are tools that may be applied to evaluate this interest. 
 
Strengths and limitations  
 
Our model contributes for a better understanding of this kind of problems and it (or some modified 
versions of it) may be applied to study some other social and economic phenomena, such as 
unemployment, health or projects of investment, for example, with interesting results. 
 
An extension of our model has also permitted to get conclusions about the economic advantages 
of recycling and dismantling. 
 
The model application to the phenomenon studied in our paper shows that it is very useful and 
that its conclusions and results are quite simple to understand. Just through its theoretical 
analysis, we evidence some remarkable topics in analyzing the evolution of the studied system. 
Or another one, whichever it is, since it is according the assumptions of the model. 
 
In practice, it is essential to estimate λ and h(t) to get conclusive particular results for the 
available data about the system. This will give us the tools to monitor the situation and to suggest 
solutions. A correct estimation of λ will depend on the arrivals process to be Poisson, in real.  
 
Additionally, in general, it is correct to admit that with very large populations, such as the one we 
are dealing with, the estimation of h(t) is usually technically complicated. So, frequently, the best 
to do is to estimate directly h(t) instead of estimating first the service time distribution and then 
computing h(t). 
 
A particular situation at which the computation is easier is the exponential service time one, for 
which h(t) = 1/ α. 
 
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Dolsak, P. C. Stern, S. Stonich, & E. U. Weber (Eds.), The drama of the commons (pp. 42-53). 
Washington DC: National Academy Press  
 
Carrillo, M. J. (1991), Extensions of Palm’s Theorem: A Review. Management Science. Vol. 37 
N.º 6, 739-744. 
 
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problemas de doença e de emprego. Revista de Estatística. Vol. 3., 3.º Quadrimestre. INE, 23-
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aos Comuns da Pesca. Estudo da Cooperação aplicada à pesca da sardinha nas Divisões VIIIc e 
IXa do ICES. PhD Thesis presented at ISCTE. Lisboa.  
 
8
 
Hardin (1968), The Tragedy of the Commons, Science,162, pp.124-148. 
 
Kelly, F. P. (1979), Reversibility and Stochastic Networks. New York. John Wiley and Sons. 
 
Kunstler, J. H. (2006), O fim do petróleo – O grande desafio do século XXI. Editorial Bizâncio. 
Lisboa. 
 
Ross, S. (1983), Stochastic Processes. Wiley. New York. 
 

Aplicação da Equação de Ricatti ao Estudo do Período de Ocupação do Sistema M|G|∞.

Common resources and institutional sustainability. In

Comportamento Transeunte do Sistema M|G|∞ - Aplicação em problemas de doença e de emprego.

Extensions of Palm’s Theorem:

O Drama dos Recursos Comuns. Um caso de Aplicação da Teoria dos Jogos aos Comuns da Pesca. Estudo da Cooperação aplicada à pesca da sardinha nas Divisões VIIIc e IXa do ICES. PhD Thesis presented at ISCTE.

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