Inductive Loop
Detection (ILD) systems are one of the important systems in Intelligent
Transportation Systems (ITS). Its
daily operation requires the sustained availability of the system. Frequent malfunctions of the system
will degrade the quality of the service provided by the system. Therefore, it is necessary to maintain
the system in an adequate availability level.
In the current
practice of maintaining ITS, different policies have been adopted with the
discretion of different ITS. One
policy provides repair maintenance only for the component(s) that fails on a
system. The other components of
the system at the same location are not provided with maintenance for
preventive purpose. Under another
policy, the live components in a system are checked for preventive maintenance
during repair maintenance. A third
policy is to provide preventive maintenance to a system even though there is no
repair maintenance needed. There
has been no guidance on policy adoption for a particular ITS. Because of the lack of guidance,
decisions on adopting maintenance policy are ad hoc in nature.
The objective of
this study is to evaluate the typical maintenance policies through which
guidance on adopting maintenance policies can be developed. As the first step in this study, it is
important to investigate the deterioration of the loop detection system. Due to the availability of maintenance
records for the inductive loop detectors in a traffic monitoring system, it is
easy to derive the deterioration data for the system components in an ILD,
which makes this study to focus on loop detector systems’ maintenance
policies. It is expected that the
results from this study can be applied to other ITS systems such as CCTV,
variable message sign, etc.
Historically,
inductive loop detectors have been used to measure traffic for decades. The improvements on their performance
have been continued such that they have been continuously installed in
ITS. In addition, more
applications have been developed based on the inductive loop detector
system. One recent application is
to detect and classify vehicles based on the signature that can be recognized
on the signals from the inductive sensors. Extensive studies have been done on tapping on such detected
vehicle information to derive origin-destination information for advanced traffic
control and management. On the
other hand, loop detection systems have been suffering from the lack of
maintenance. The usage of
detection systems for automatic incident detection has been stopped in some
traffic management centers in the U.S., simply due to the frequent malfunctions
of the traffic detection systems.
The traffic detection systems cannot meet the requirement of incident
detection in terms of data quality due to their malfunctions. Even thought the data from traffic
detection and monitoring systems may not be used for incident detection
nowadays, they may be used for other system functions such as traffic flow
condition display, travel time estimation, and forecasting. In addition, they can be archived to
derive system performance measures such as travel time over a corridor and
vehicle miles traveled, which are important for decision-making in
transportation planning, design, and operations. Thus, it is necessary to maintain the detection systems to
produce data with a certain quality level satisfying the requirements for these
functionalities.
From the
perspective of combining data from different loop detection systems to derive
vehicle miles traveled, Wang (2004) looked into the maintenance issues of loop
detection systems. It was believed
that different loop detection systems should be maintained to the comparable
levels such that the data from these systems can be combined. The loop detector system studied in
Wang (2004) is the Traffic Management System (TMS) in the Virginia Department
of Transportation (VDOT). A
continuous count station (CCS) of the system is represented as Figure 1.1. It can be seen that the system consists
of inductive loop detectors (loop), piezoelectric sensors (piezo), automatic
data recorder (ADR), and modem & telephone line (communications).
Figure 1.1
Typical Layout of the Detection System
Inductive loops
work by detecting a change of inductance caused by the vehicles running over
the loops. In a CCS, two 6 feet by
6 feet loops are installed in a squared saw-cut in each lane and used to
collect traffic volume. After the
loop wires are placed, the saw cut will be filled with loop sealant. Between the two loops in the same lane,
there is a piezoelectric sensor used to classify vehicles. The physical touch between the vehicle
axles and piezoelectric sensor is converted into electronic signal, which is
recorded by the ADR in the control cabinet.
An ADR is held in the control
cabinet. The ADR works together
with the loop detectors and piezoelectric sensors to count and classify
traffic. Traffic data collected
can be stored internally in the equipment and transferred to the database in
the VDOT central office. The
traffic data stored in the ADR are transferred to the VDOT TMS central office
through modems and phone lines.
The modem is placed in the cabinet and relies on the phone line voltage
for power. Their malfunctions will
terminate the communications and transmissions of traffic data with the database
in the central office. However,
the collected data can be temporarily stored in the equipment on site and
retrieved by the central office after the problems are fixed.
The conditions of
the CCSs are monitored remotely by VDOT on a daily basis. When a problem is identified with the
traffic counting equipment, a contractor will be notified for repair
service. A crew will be dispatched
to check the equipment and fix the problem if necessary. VDOT TMS does inspection for each
location on the anniversary dates of the installation of the stations. Equipment problems identified during
the inspections are also noted for maintenance by a contractor.
In the study by
Wang (2004), loop detection system deterioration data such as life and failure
times were collected from VDOT TMC.
Linear regression models were developed to identify the factors that
influence the deterioration of loop detection systems. Results show a negative
relationship between the life time and heavy vehicle traffic volume, which
implies that heavy vehicles may contribute mostly substantially in reducing the
life durations of loops. Some
difficulties were encountered during the analysis using linear regression which
suggests applying better modeling approaches.
In this study, the
survival theory-based duration modeling approach is proposed by which more
realistic probability distributions can be fitted to the deterioration data of
loop detection systems. From the
modeling, factors that influence the deterioration of loop detectors can be identified. A microscopic simulation model is
proposed to evaluate three typical maintenance policies. The simulation has the capability of
emulating the deterioration of loop detection systems and the maintenance
practices according to the maintenance policies evaluated.
In the following
sections, Chapter 2 is devoted to a literature review on infrastructure
deterioration models and optimization models for maintenance. Chapter 3 describes the survival
theory-based modeling approach to identify the influencing factors to the life
time of a loop detection system and the framework of the simulation model built
in this study. Chapter 4 presents
the calibration of the survival theory-based duration models. Chapter 5 describes the analysis of
maintenance policies based on the results from the simulation model. Chapter 6 summarizes findings, draws conclusions
and suggests further research.
The reliability
has been a major issue with many manufactured items. Estimation of mean time of successful operations and the
entire life time distribution is an important element of many quality
improvement efforts.
In the late 1930s
and 1940s, Weibull analyzed fatigue life in materials, leading to the
probability distribution bearing his name. The advent of the electronic age, accelerated by the Second
World War, led to the need for more complex mass-produced component. The experience of poor field
reliability of military equipment throughout the 1940s and 1950s focused attention
on the need for more formal methods of reliability engineering. During the 1950s, the researches were
focused on the use of exponential distribution to represent failure times. The interestes in and the importance of
the Weibull distribution increased by the end of the 1950s.
Since 1960s, the
failure rate data for the components of electronic devices in many categories
were collected and published (Smith, 1997). The information has been used by many companies to improve
the quality of their products. It
has also been used by decision makers to choose the right maintenance
strategies.
In the
transportation engineering area, deterioration models have been developed for
the infrastructures, such as bridge deck and pavement. In modeling the deterioration for
bridges, the focus is placed on the service conditions of the bridge
decks. The Federal Highway
Administration (FHWA) uses an ordinal bridge rating system (FHWA, 1979) to
measure the deck conditions. In
modeling the deterioration of pavement, measurements such as rutting and
roughness are usually used to describe the pavement conditions.
Basically, the
deterioration of a bridge deck or a pavement segment is a continuous, gradual,
and relatively slow process that is affected by traffic loading, environmental
factors, bridge or pavement designs, and material properties, many of which are
generally not captured by available data.
Therefore, probabilistic models are often used to characterize
deterioration (Mauch and Madanat, 2001).
Two types of probabilistic models have been used for infrastructure
facility deterioration modeling: discrete-time,
state-based models and time-based
models. According to Mauch and
Madanat (2001) and Mishalani and
Madanat (2002), discrete-time, state-based models, such
as Markov chains, characterize the probability that a facility undergoes a
change in condition state at a given discrete time, given a set of explanatory
variables such as design attributes, traffic loading, environmental factors,
age, and maintenance history. Time-based models, on the other hand,
characterize the probability density function of the time it takes an
infrastructure facility to leave a particular condition state once entering
(this time is referred to as state duration), given the same set of explanatory
variables. This approach requires
more accurate data in general. As
Mauch and Madanat (2001) observed, it is possible to use one model to determine
the dependent variable of the other.
More specifically, condition state transition probabilities can be
determined from the probability density function of state duration and vice
versa.
Mauch and Madanat
(2001) discussed the appropriateness of using each of the two types of models
based on empirical considerations relating to the specific nature of the
condition data available for model development and estimation. For example, the estimation of
time-based models would be practically feasible if frequent observations over
long periods of time are available.
The model has been used to predict the time to cracking initiation
(Patterson & Chesher, 1986).
On the other hand, if inspections are made infrequently, the measurement
of the time between condition-state transitions will suffer from potential
large measurement errors. In such
a situation, a state-based model will be a better approach. Note that the maintenance data
collected for the loop detection systems in this study include the maintenance
records over about six years, which makes it possible to adopt the time-based
modeling approach.
Madanat and
WanIbrahim (1995) is a typical study where state-based models were used to
forecast facility conditions. The
Poisson and negative binomial regression models were used to forecast the
number of state changes between two discrete times. Madanat, Mishalani and WanIbrahim (1995) took a further step
by applying ordered probit model to link the deterioration to the relevant
explanatory variables. Recognizing
the fact that most deterioration models were developed using panel or
longitudinal data sets, Madanat,
Karlaftis and McCarthy (1997) adopted a random-effects specification in the
probit model to address the heterogeneity in different panel units.
The proposed model yields improved results in comparison with a simple
probit model.
In Mauch and Madanat (2001), the
semiparametric hazard rate duration model (Cox proportional hazards model) is
developed to predict the distribution of times between changes in
condition-state for reinforced concrete bridge decks. The coefficients of
explanatory variables were estimated using Cox regression. Mishalani and Madanat (2002) applied
the parametric duration model to model the time spent in a condition state. Two commonly used distributions are
introduced: exponential distribution and Weibull distribution in the
study. Coefficients of explanatory
variables are estimated using accelerated
failure time models.
Two conflicting
objectives exist in making maintenance and rehabilitation decisions: maximizing
the service quality of facilities (or facilities’ performance, etc.) and
minimizing the total maintenance cost.
In optimization modeling, minimizing the maintenance cost has been
widely used as the objective while a fixed level of the facilities’
serviceability is used as a constraint.
Both linear and
non-linear optimization models have been applied in modeling the maintenance
and rehabilitation decisions.
Usually, linear optimization models can be solved by using software
packages available in the market, while non-linear optimization models most
likely have to be solved by developing case-by-case based algorithms.
Arizona Pavement
Maintenance System (PMS) is the first modern network-level system, serving as
decision-aid tools to the policymaker (Ferreira et al., 2002). It has been adopted by several other
State Departments of Transportation (DOTs) in the United States. The objective of the optimization model
in the system was to minimize the costs of M&R actions over a given
planning time span. The linear optimization
model was applied separately to each road category, considering traffic
loadings and climate conditions as independent variables in the integrated
deterioration models. To solve the
linear optimization problem, the Arizona DOT used NOSLIP, a computer program
was developed for this system.
Singapore PMS is
another type of system. They
realized that it was more realistic to apply the system in a network level
rather than the road category level.
“The objective of the system may be to minimize agency and/or user
costs, or to maximize service quality.
It may also be any combination of both.” A nonlinear integer optimization model was defined in the
Singapore approach. A
genetic-algorithm heuristic called PAVENET-R was used to solve the problem.
(Ferreira et al. 2002)
Markov decision
process is another widely used methodology in the infrastructure management
systems. Li and Madanat (2002)
claims a simple but exact solution method given the pavement deterioration
model is deterministic. The deterministic
model used here is a Markov decision process (MDP). The result from the research suggests that the decision
problem “be focused on obtaining robust policies that can sustain a certain
amount of modeling errors”.
Ouyang and Madanat
(2004) presented another programming model, a mixed-integer nonlinear
programming (MINLP). It is a
nonlinear pavement performance model with integer decision variables. Two approaches, a branch-and-bound
(B&B) algorithm and a greedy heuristic algorithm, were proposed in the
paper. A deterministic model was used to describe the deterioration
process. The paper also raised one
drawback of the MINLP, that the nonlinear programming incurs unaffordable
computational cost when the problem scale increases.
Based on the
literature review, the following observations can be derived.
First, time based
deterioration models are more appropriate to model the behaviors of loop
detection systems because the health conditions of ILDs are not evaluated based
on discrete states as does for pavement and bridge systems. In addition, there
is at least one lifetime data point available for a station which makes the
modeling using time based duration models feasible.
Second, the
maintenance policies adopted for ILDs are different from those for pavement and
bridge systems. Specifically, instead of emergency and preventive maintenance
that are adopted in pavement and bridge systems separately at different times,
a combination of emergency and preventive maintenance practices is possible to
be used simultaneously at the same time at an ILD station because of the
existence of more than one system component at one detector station. The
adoption of such a mix of policies makes it difficulty to relate the cost and a
maintenance policy in an one-on-one basis analytically. Thus, simulation
modeling approach tends to be a better approach to evaluating the maintenance
policies in ILDs. It is because simulation model is a powerful tool to tackle
the complexities caused by the interactions between the deteriorations of
systems components and maintenance policies.
The deterioration
data of loop detector systems posts the following challenges for modeling: (1)
unidentifiable life times for the two loops in the same lane, (2) existences of
censoring data, (3) existences of unique characteristics for the data from a
same loop detector station. First,
the maintenance records obtained from
VDOT for this study only indicate the lane at the station where a loop needs
maintenance, but not the specific loop (upstream or downstream) in the
lane. Thus, what can be
observed are the time intervals between two consecutive failures, the
identities of which are not known.
This study proves that it is still possible to identify explicitly the
probability distributions for the lifetimes of these two loops in the same lane
at one location (see Figure 1.1), if an appropriate assumption is made. Second, the maintenance records stop at
the time when the data were collected for this study. Thus, the derived life times associated with this “cutting
line” are not complete for the stations which were still in the live conditions
when the records were collected.
These incomplete duration data are called censoring data, and will be
considered in this study. Third, a
loop detector station may be reinstalled several times since its first
installation. Note that the
average lifetime of a system component in a loop detection system is 1.5 year,
while most of the stations were initially installed at least five years
ago. Thus, several lifetime data
may be obtained for one station, and they may share the same influencing
factors such as traffic volume.
Such data are called repeated observations in econometrics and need to
be taken special care in modeling.
In the following sections, the basic duration model and the ways by
which these three special features of loop detector data are treated are
described.
Let T be the time until some specified event. This event may be death, component
breakdown, and so forth. Clearly,
it is a nonnegative random variable ‘time until transition’, which can be
described by a continuous distribution f(t),
where t is a realization of random
variable T. The cumulative density function (c.d.f.) of this random
variable can be expressed as (Greene, 2003):
Based on this
c.d.f., the probability of an individual surviving beyond time t, which is
called survival function, can be written as (Greene, 2003):
Another basic
quantity, which is the hazard function or hazard rate, can be mathematically
defined as (Greene, 2003):
Because 
can be viewed as
the “approximate” probability of an individual of age t experiencing the transition in the next instant, the hazard
function is usually more informative about the underlying mechanism of failure
than the survival function. The
hazard rate is usually increasing because of natural aging and fatigue. However, the mathematical models allow
the existence of decreasing hazard rate, which may be reasonable under some
circumstances.
Two commonly used
distributions in survival analysis are exponential distribution and Weibull
distribution. Exponential
distribution has a constant hazard rate; while Weibull distribution’s hazard
rate can be either monotone increasing, decreasing or constant. Weibull distribution turns into
exponential distribution when p=1
(See Table 3.1).
Table 3.1 Hazard Rates and Survival Functions for
Two Common Distributions
|
Distribution
|
Hazard Rate,
|
Survival Function,
|
|
Exponential
|
|
|
|
Weibull
|
|
|
To identify the
factors that influence the lifetime of a system, it is assumed that
where 
is expressed as
a function of a set of explanatory variables (regressors). (Greene, 2003)
Note that the
effect of the explanatory variables for survival function is to change the time
scale by a factor 
(see Table
3.1). So, the model in Equation
(3.4) is called the accelerated failure-time model. In the estimation of the coefficients and parameters in Equation
(3.4), the following log-likelihood function is commonly used (Greene, 2003):
where 
.
In this study, the two loops in a same lane
at one station are viewed as identical from the perspective of health performance. It is because these two loops are
contiguous with just a few feet apart in a same lane and thus the factors such
as traffic volume, pavement type, and weather that influence their lives can be
viewed as the same. It is
necessary to make such an assumption because it is difficult to reveal the
identities of these two loops in one lane at one station from the maintenance
records. Specifically, the
maintenance records only indicate the lane at a station where a loop needs
maintenance, but not the specific loop (upstream or downstream) in the
lane. As a result, the two
consecutive failures that defines a lifetime data may happen on two different
loops (e.g., the first failure is the upstream loop while the second failure is
the downstream loop, or vice versa).
Thus, the resultant lifetime data cannot be clearly related to any
particular loop. The following
mathematical derivation indicates that it is still possible to derive the
probability distribution of the life duration for a specific loop if the two
loops in the same lane at one station are viewed as the same in deterioration
process.
Let 
be the random
variables of the lifetimes for the n
(n≥2) loop detectors in a same lane at a station. n=2 for the case
in this study. These random
variables can be viewed as independent
and identically
distributed, with the
cumulative probability distribution and the survival functions 
and 
. Let 
represent the
value of the observed time interval between two consecutive failures occurred
to the n loops in the same lane at
one station. Then, the
relationship between the underlying lifetime of the n loops 
and that the
observed failure intervals can be expressed as:
,
The
probability distribution of the observed intervals can be written as:
Let 
and 
be the
cumulative probability distribution functions of the observed intervals and the
corresponding survival function, respectively. Then, we can have the following relationship:
In this study, the
two loops in the same lane at one station can be viewed as going through two
identical deterioration processes with Weibull distributions, for which the
survival function of the combined process can be written as:
From this
expression, it can be derived that
.
Then, we can have:
,
where 
represent the
factors that influence the lifetime of loops. From this equation, it can be seen that the parameters of
the probability distributions representing the life duration of the system
component can be estimated with the data observable from the maintenance
records. In addition, the only
difference existing between the coefficients for the observable and
unobservable life duration distribution is a constant 
.
From the
maintenance records, incomplete (censoring) duration data can also be derived
for each station. These censored
duration data is the results of the cutting time December 31, 2003 for
maintenance records, at which time the maintenance records were obtained from
the Mobility Division in the Virginia Department of Transportation. These data is right censored in nature
because no failures have occurred yet up to the cutting time when the
maintenance records were obtained.
What we get of these duration data are the times these stations have
survived so far after their last failures. It is worthwhile to incorporate censored data in estimating the
models described above. It is
because they contain some valuable information: the lifetime of the system
component is longer than the time up to the day when the maintenance records
were obtained.
By nature, the
classical linear regression model cannot differentiate the censoring data from
complete life time data, so it cannot deal with the censoring data. It is at this point that duration model
shows its advantage. In estimating
the coefficients and needed parameters in Equation (3.4) considering the
censoring data, the following log-likelihood function can be used (Greene,
2003):
where 
. Assuming that
the time durations follows Weibull distribution, the
log-likelihood function can be transformed as (Greene, 2003):
where 
, and
.
The maintenance
records indicate that most components of the stations have experienced more
than one failure before the records were collected for this study in 2003. It is reasonable to perceive that some
unique attributes exist for these lifetime data from one component in the
station. In econometrics, this
kind of lifetime data can be viewed as panel data where the data from one
station can be viewed as one panel.
The lifetime data from one component is usually called repeated
measurements. The unique attribute
is referred to as heterogeneity.
There have been several approaches to address
the heterogeneity, and one of them is called the random effect
model. In essence, a random constant
term is added to the regression model (Greene, 2003):
,
where 
;
is the i-th observation for the k-th unit (station) which doesn’t
include a constant term; 
is the random
constant term;
is a random term
with zero mean, distributed as a Normal, Tent, or Uniform distribution;
is normal
distributed.
In estimation, the
following likelihood function can be used (Greene, 2003):
where
,
and 
is the CDF for 
with the
parameter 
. Usually we set 
to be normally
distributed. Then, we have (Greene, 2003):
,
where 
is the CDF of
the standard normal distribution.
Given the development of the
deterioration models, it is possible to evaluate the impacts of maintenance
policies on the system health performance. It is very difficult to relate the maintenance policies to
the system health performance directly using analytical equations because the
lifetime of each system component is a random variable. A microscopic simulation model is
developed in this study. Three
elements are included in the simulation: system deterioration and failure, service
crew, and maintenance policies. In
the following sections, each of these three elements is described in the
context of simulation.
System
Deterioration and Failure
In the simulation model, the
deterioration of each component is simulated based on the deterioration models
developed in this study. Because
the lifetime of each system component follows Weibull distribution, a random
number for the lifetime can be generated following this probability distribution. This generated lifetime is checked against
the day “clock” in this simulation.
As the “clock” advances in a daily basis, it will reach the simulated
lifetime. It is at this time that
a failure is generated from which an emergency maintenance call is triggered
simultaneously. If there is a
service crew available, the maintenance call will be responded immediately
without any delay. Otherwise, it will need to wait until a service crew becomes
available. After the arrival of a
crew, the simulation will monitor the progress of the maintenance work on a
daily basis against a fixed duration that is predetermined for the system
component. If the time a crew
spends on the failed component exceeds the fixed duration, the maintenance work
is assumed done and the state of the component is back to health. From this point of time, a new lifetime
is generated for that component and one maintenance job is accomplished.
Service Crew
A dispatcher is simulated in
the model who is responsible to determine which service crew is assigned for an
emergency maintenance request.
When there is more than one crew available, the one having been in the
state of idle for the longest time will be assigned to the service call.
Service crews are
simulated. Each crew has two
states: idle or busy. There are
probabilities that the number of crews is run out, which causes the components
to be fixed to wait. The time each
crew stays in different states is recorded and accumulated for deriving the
work load that is used in evaluating the requirement for the number of crews
versus the system health performance.
Maintenance
Policies
Three maintenance policies
(Policies 1, 2 and 3) are simulated in this study where the extent of
preventive maintenance varies for three different levels. Policies 1 and 3 represent two extremes
where preventive maintenances are either not adopted at all or adopted in
full. In Policy 2, preventive
maintenance is partially used.
Under Policy 1, only the
failed component(s) receives emergency repair service during a maintenance
visit. The other components at the
same station are left untouched.
No preventive maintenance is provided for them.
Under Policy 2, preventive
maintenance is provided to other components during an emergency repair visit to
a failed component. Whether a
component receives a preventive maintenance depends on the lifetime up to the
time of the emergency repair. It will receive preventive maintenance only when
its life has exceeded a certain value like the 90th percentile of their life
distribution adopted in this study.
Figure 3.1 illustrates the 90th percentile based on a
presumed probability distribution.
In simulation, the lifetime of each component will be updated with the
advance of time clock. When there
is an emergency maintenance service under Policy 2, the life of other
components will be compared with the predetermined 90th percentile
life duration to determine whether a preventive maintenance is needed.
Under Policy 3, a visit to a
station can be initiated by either an emergence or a preventive
maintenance. When an emergency
service call initiates a visit, preventive maintenances may be performed to
other healthy components based on their preventive maintenance schedule. When a preventive maintenance service
initiates a visit, there will be no repair work involved. The scheduled preventive maintenance
can be twice a year (Policy3_0.5yr), once a year (Policy3_1yr), once per 2
years (Policy3_2yr), etc., where the preventive maintenance intervals are once
in half a year, one year, two years, etc., respectively.
Figure 3.1 90th Percentile of the Weibull
Distribution
Flow Chart
A flow chart is developed for
the simulation and presented in Figure 3.2. It can be seen from the flow chart, there is a fixed time
frame simulating the deterioration and process. A simulation run is complete when the simulation clock
reaches this fixed time frame. In
each simulation day, all the stations are checked for their states, which can
be either healthy, failed and in repair, or failed and waiting for repair. Each crew is also checked for states:
busy or idle.
Figure 3.2 Simulation Flow Chart
Six system performance indexes
are proposed to evaluate different maintenance policies: system availability,
annual maintenance cost per station, number of crews employed, working load,
annual maintenance delay time, and cost to availability ratio. The definitions
of these indexes are described below.
System availability (Index 1): System availability can be defined
on either traffic count station level or the whole system level. The availability of a station is
defined as the percentage of time that the station works properly. System availability is the average of
availabilities of all the stations in the system. Note that each station consists of a number of components
(e.g., two loops and a piezo in each lane, ADR and Communications). The failures of communications do not
affect the availability of a station because traffic count data can be stored
in ADR temporally and rescued later after the communications are back to
work. However, their failure needs
immediate maintenance, thus would impact on the other indexes of system
performance such as costs and crew working load. If any of the other components fail, the station stops
working. A repair is needed
immediately.
Annual maintenance cost per station (Index 2) is defined as the
annual total system maintenance cost divided by the number of stations in a
traffic monitoring system. It includes the costs for facility replacement,
maintenance crew relocation and wages.
Number of crews employed (Index 3) is referred to the number of
groups of employees that are involved in the response to maintenance
service. It is assumed that the
crews work independently. One crew
alone is responsible for one maintenance job alone. No cooperation between the crews is simulated in this study.
Working load (Index 4) is defined as the average percentage of days
that a crew spends in its maintenance job on site.
Annual maintenance delay time (Index 5) is calculated as the
average days in one year that one station waits for a crew becoming available
after its failure. It can be
viewed as the number of days that a station doesn’t work because of the
unavailability of crews.
Cost to availability ratio (Index 6) can be derived by dividing the
annual maintenance cost by system availability. It is used to measure the average cost to gain 1%
availability.
These indexes are
interdependent in nature. The
evaluation of the maintenance policies will be based on the investigation of
the interdependences between these indexes.
The number of runs
or iterations (i.e., sample size) that is needed to get reliable outputs from a
simulation model has to be known before the simulation model is used for any
analysis. In this study, this
number of runs is determined based on an assumed mapping from one set of random
variables (A), which are the simulated life times of components, to another
sets of random variables (B), which are the performance indexes:
If we take enough
samples of A, we can get enough samples of B as we want. If all the samples are independent, the
weak laws of large numbers (WLLNs) (Feller, 1968) guarantees that the difference between sample mean and true mean
goes to zero as the sample size goes to infinite. The difference between sample variance and population
variance also goes to zero as the sample size increases. Thus, the mean and variance of these
performance indexes (random variables B) can be estimated based on the repeated
trials of life times (random variables A).
According to
Levy-Lindeberg central limit theorem
(CLT), the samples from B can be represented as a sequence of iid random
variables 
. Let 
and 
, where 
Also let 
. Then, for every value x
we have:
In another word,
the value 
is standard
normal distributed if sample size n
is large enough. In our case, the
required accuracy can be defined as 
, which indicates the difference between the sample mean and
true mean of a performance index.
Usually the three-sigma standard
is used in practice, which promises that the chance for any sample value from a
normal distribution falling within 3 standard deviation is 0.997. To apply this standard, we can write:
Substituting 
in Equation , we can have:
Once setting 
as the maximum
acceptable level of error, the number of iteration can be computed as:
.
In practice, 
is unknown. An estimation of 
can be derived
by running the simulation with a reasonable small number of runs. With the estimated 
and the
acceptable level of error, the required number of iterations or runs can be
calculated.
Among the four
components of an inductive loop detector system, only loops and piezos were
considered for developing duration models in this study because the factors
such as traffic volume that influence their lifetimes are available. The data for the factors such as
weather that presumed to influence the lives of ADR and Communications are not
available to this study. Thus,
duration models are not developed for them. Instead, only probability density functions are fitted for
these two system components, which are important inputs to the simulation model
developed in this study.
The following
factors and the associated variables are considered in the modeling for the
loops and piezos: traffic (traffic volume and vehicle type), geometric
conditions (number of lanes), and road surface type. Other factors such as maintenance practice and quality, as
well as weather are also important contributors to the deterioration of loops
and piezos. However, their impacts
can only be identifiable when data from different areas of the Nation is
collected. Thus, these factors are
not considered in this study.
Intuitively,
different types of vehicles will influence the deteriorations of loops and
piezos differently. Based on the
thirteen vehicle classifications by the Federal Highway Administration (FHWA),
four vehicle groups are formed which are (1) automobiles and other four-tire
vehicles, motorcycles, (2) other single-unit vehicle, (3) single-trailer
combination, and (4) double-trailer combination. The reclassification of the
vehicles is necessary because it would be tedious to create thirteen variables,
each representing one vehicle class, in duration modeling. The match between the FHWA
classification and these four groups can be found in Table 4.1. Based on the presumed weights of
vehicles in each group, these groups can be listed in an increasing order as:
Groups 1 (X1), Group 2 (X2), Group 3 (X3), and Group 4 (X4).
Table 4.1 Vehicle Reclassification
|
Class
|
Classification
|
Major Configuration Groups
|
|
1
|
Motorcycles
|
automobiles and
other four-tire vehicles (X1)
|
|
2
|
Passenger Cars
|
|
3
|
Other Two-Axle,
Four-Tire Single Unit Vehicles
|
|
4
|
Buses
|
other single-unit
vehicle (X2)
|
|
5
|
Two-Axle, Six Tire,
Single Unit Trucks
|
|
6
|
Three-Axle, Single Unit Trucks
|
|
7
|
Four or More Axle
Single Unit Trucks
|
|
8
|
Four or Less Axle
Single Trailer Trucks
|
single-trailer
combination (X3)
|
|
9
|
Five Axle Single
Trailer Trucks
|
|
10
|
Six or More Axle
Single Trailer Trucks
|
|
11
|
Five or Less Axle
Multi-Trailer Trucks
|
double-trailer
combination (X4)
|
|
12
|
Six or Less Axle
Multi-Trailer Trucks
|
|
13
|
Seven or More Axle
Multi-Trailer Trucks
|
In addition to the
reclassification of vehicles, lane distribution of traffic volumes is also
investigated in this study, the result of which is used as a justification for
developing different duration models separately for different lanes. Figures 4.1, Table 4.2, Figure 4.2,
Table 4.3, Figure 4.3, Table 4.4, Figure 4.4, and Table 4.5 present the
histograms for each vehicle group in Lanes 1 and 2 and the corresponding test
(e.g., Wilcoxon Two-Sample
Test) for the traffic volumes in these two lanes. The data used to develop these figures and tables are
those for the roadway segments that have two lanes only and they spanned from
1997 to 2002. The results indicate that
the traffic volumes in Lanes 1 and 2 are different significantly. Note that there are a few roadway
segments that have more than two lanes.
The analyses on their traffic volume lane distributions were not
performed in this study because the focus of this study is on four lane
highways (two lanes in each direction).
Figure 4.1
Histograms of Group 1 Traffic Volumes (X1) in Lanes 1 and 2
Table 4.2
Test for the Difference between Group 1 Traffic Volumes in Lanes 1 and 2
|
Wilcoxon Two-Sample
Test
|
|
Normal
Approximation
|
|
Z
Statistics
|
2.7895
|
|
One-Sided
Pr > Z
|
0.0026
|
|
Two-Sided
Pr > |Z|
|
0.0053
|
Figure 4.2
Histograms of Group 2 Traffic Volumes (X2) in Lanes 1 and 2
Table 4.3
Test for the Difference between Group 2 Traffic Volumes in Lanes 1 and 2
|
Wilcoxon Two-Sample
Test
|
|
Normal
Approximation
|
|
Z
Statistics
|
9.7128
|
|
One-Sided
Pr > Z
|
<.0001
|
|
Two-Sided
Pr > |Z|
|
<.0001
|
Figure 4.3
Histograms of Group 3 Traffic Volumes (X3) in Lanes 1 and 2
Table 4.4
Test for the Difference between Group 3 Traffic Volumes in Lanes 1 and 2
|
Wilcoxon Two-Sample
Test
|
|
Normal
Approximation
|
|
Z
Statistics
|
8.0622
|
|
One-Sided
Pr > Z
|
<.0001
|
|
Two-Sided
Pr > |Z|
|
<.0001
|
Figure 4.4
Histograms of Group 4 Traffic Volumes (X4) in Lanes 1 and 2
Table 4.5
Test for the Difference between Group 4 Traffic Volumes in Lanes 1 and 2
|
Wilcoxon Two-Sample
Test
|
|
Normal
Approximation
|
|
Z
Statistics
|
9.0021
|
|
One-Sided
Pr > Z
|
<.0001
|
|
Two-Sided
Pr > |Z|
|
<.0001
|
The correlation matrixes for these 4 groups in Lanes
1 and 2 have been derived separately and presented in Tables 4.6 and 4.7,
respectively. It can be seen that
the correlations between Groups 1 and 2, and Groups 3 and 4 are strong in Lane
1. The correlations between Groups
1 and 2, Groups 2 and 3, and Groups 3 and 4 in Lane 2 are very strong. These characteristics will be taken
into account in the calibration of duration models.
There are two
pavement types: asphalt and concrete for the roadways in the maintenance
records. The number of lanes in a
CCS ranges from 2 to 4. The
distribution of the pavement type and the number of lanes is presented in Table
4.8.
Table 4.6 Correlation Coefficient between Four Groups of Traffic
Volumes in Lane 1
|
|
X1
|
X2
|
X3
|
X4
|
|
X1
|
1.0000
|
|
|
|
|
X2
|
0.5924
|
1.0000
|
|
|
|
X3
|
-0.1234
|
0.3932
|
1.0000
|
|
|
X4
|
-0.0946
|
0.2839
|
0.9022
|
1.0000
|
Table 4.7
Correlation Coefficient between Four Groups of Traffic
Volumes in Lane 2
|
|
X1
|
X2
|
X3
|
X4
|
|
X1
|
1.0000
|
|
|
|
|
X2
|
0.7832
|
1.0000
|
|
|
|
X3
|
0.5159
|
0.7359
|
1.0000
|
|
|
X4
|
0.4950
|
0.5899
|
0.7901
|
1.0000
|
Table 4.8
Summary for CCS Stations in Terms of Pavement Type and
Number of Lanes
|
|
|
I-77
|
I-81
|
I-381
|
I-581
|
I-64
|
I-264
|
I-464
|
I-664
|
I-66
|
I-85
|
I-95
|
I-195
|
I-295
|
I-495
|
Grand Total
|
|
Pavement
Type
|
Asphalt
|
4
|
14
|
2
|
2
|
5
|
2
|
1
|
|
6
|
2
|
18
|
1
|
|
4
|
61
|
|
Concrete
|
|
|
|
|
6
|
|
|
2
|
|
2
|
|
1
|
6
|
|
17
|
|
Number
of Lanes
|
2 Lanes
|
4
|
13
|
2
|
|
5
|
|
|
|
6
|
4
|
4
|
|
2
|
|
40
|
|
3 Lanes
|
|
1
|
|
2
|
4
|
2
|
1
|
2
|
|
|
