# A Deterministic Simulation Model for Sojourn Time in  Urban Cells with Square Street Geometry

Authors: Georgios S. Paschos; Ioannis G. Tsoulos; Efstathios D. Vagenas; Stavros A. Kotsopoulos; George K. Karagiannidis
Journal: International Journal of Vehicular Technology
 (2008)

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## Abstract

A deterministic simulation model for the calculation
of sojourn time statistics is proposed. The model is designed for
urban cells taking into account city movement in street pattern,
traffic lights, and crossings. Sojourn time and remaining sojourn
time are assumed to be random variables that follow a gamma
distribution. The parameters of gamma are found for a range of
the chosen cell parameters (road length, probability of stopping
at a crossing and cell radius) using Monte Carlo simulation.
Then, the values are fitted to the empirical model using genetic
algorithms with grammatical evolution. The model can be used
to calculate sojourn time statistical parameters for any urban
circular cell with square street pattern.


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## Body

## 1. Introduction

Sojourn time (ST) is defined as the time spent by a vehicle
user inside a cell while using a communication channel. It is originally
defined for a handover call, in which the vehicle starts its journey at the
edge of the cell. For a new call, where the vehicle starts from a random
position inside the cell, the channel holding time is called remaining sojourn
time (RST). These two random variables, obviously, depend on characteristics of
the cell area. In case of an urban cell, the important characteristics are the
road pattern, the traffic lights, the crossings, and the city traffic.Sojourn time statistics have been investigated in the
past due to their usability in teletraffic analysis and cell planning. In
[1], it is shown that
sojourn time affects the handover ratio and, therefore, it is an important
characteristic of a cellular network regarding the resource management. In
[1], also a simple
analytical model is presented, where movement in a straight line is considered.
The results in [1]
show an exponential-like behavior. In [2], a more realistic approach is chosen where the
vehicles move in a series of vectors and perform random stops in between. In
this case, the result is a statistical model that describes a specific cell
environment. An extension to [2] is found in [3], where the same approach is enhanced with more complex
stochastic models to better describe velocity, street length, and probabilities
of turning. An important addition of [3] is that average step velocity is measured and the data
are fit in a weighted sum of Rice and Gauss distribution. An analytical
approach of the statistical model of [2, 3] is found in [4]. Other previous works, relative to the present, can be
found in [5–7].However, applying all the above models in a real-life
cell remains challenging since all of them demand extra effort for each new
cell scenario, that is, because they fit measurements or simulations of
specific cells to their models, but they do not provide any means of predicting
the behavior of ST and RST in an arbitrary cell. Moreover, measurements of ST
and RST are particularly time consuming and tedious. Channel holding time
measurements are easier to obtain from a mobile operator as in [8], but in this case ST and RST
information is mixed and the cell is assumed to be operating (information not
available in design state). In this paper, we propose an empirical model that
can be directly applied to any city cell with respect to some assumptions for
ST and RST prediction.The proposed model is a statistical model which can be
seen as an extension of  [9]. ST and RST are assumed to follow a gamma distribution
as in [2–4, 9]. However, this model connects a set of cell parameters
with the gamma distribution parameters a and b.
A road map can be used to extract the cell parameters for any cell that
fulfills certain requirements, and then to use them
as inputs to the model. A measurements case study carried through in a real
cell shows a satisfactory model performance.Apart from the empirical formulas for computing ST and
RST statistics, this paper provides some novel ideas on this topic. The use of
fixed velocity distribution is proposed, implying that the driver behavior is
always the same in urban environments. Issues related to traffic are explained
further. In comparison to [9], the cell radius parameter is added to provide
solution to any cell size ranging from 100 m to 1 km.
In addition, the derivation of turning probability is elaborated to incorporate
normal driver behavior. The fitting approach, that
is also improved and combined with the above improvements, yields
better approximation.The mobility model is described in Section2, and the
empirical formulas are given in Section 3. A real cell study is found in
Section 4, where the model is compared to measurements, and finally the paper
is concluded in Section 5.


## 2. The Proposed Deterministic Simulation Model

In this section, the proposed model is described. The
design process is analyzed and the necessary assumptions are explained.


### 2.1. Model Design

The model design procedure is described in Figure1.
An intuitive set of three cell parameters is chosen. These three parameters are
the ones used to characterize an arbitrary cell, and they constitute the domain
of the empirical functions produced by the model. For all possible values of
these parameters, a Monte Carlo simulation is performed and the results are
fitted to the gamma distribution. The gamma parameters a and b are then extracted with a median regression
method. After the simulation loop, aST, aRST, bST,
and bRST data are available and they are used as inputs
to a graph fitting procedure by means of a genetic algorithm with grammatical
evolution. The final model is an expression of a and b in terms of the chosen parameters. Then, the
distribution of ST and RST is obtained as(1)f(x)=1baΓ(a)xa−1e−x/b.Figure 1
Logical diagram of the model design procedure.The chosen cell parameters are the probability of
trafficptr,
the normalized average block area Ab,
and the cell radius R.
The first parameter ptr is the probability that a vehicle user stops
for a pause time in an arbitrary crossing. For a set of crossings X with NX elements which NtrX of them have traffic lights, we
write(2)ptrX=NtrX+0.5(NX−NtrX)NX.The vehicle user faces three possibilities, making an
average pause time at a traffic light crossing, making a pause time at a
nontraffic light crossing coming from the minor road and making no pause time
when coming from a major road. Every traffic light crossing is associated with
an average delay and half of the priority crossings are associated with a
deterministic delay. By using all the crossings in a cell(X=C), ptr=ptrC describes a cell where the movement is
uniformly distributed among all roads. Another approach is to use the set M that contains all the crossings of major roads
that sustain the most important traffic load. This corresponds to transitional
traffic in the cell. Assuming that the probability that an arbitrary user is in
transitional state (P(intransit)) is known, we get (3)ptr=P(intransit)ptrM+(1−P(intransit))ptrC.The normalized average area of block,Ab,
can be used to extract the length of the roads. In case of square
blocks(4)d=Ab+rw≈Ab, where d is the edge, and rw is the road width. The value of Ab,
in case of nonidentical blocks, is computed by averaging which results in a
possible error. The greater the square diversity, the greater the error is
expected.The third parameter is simply the cell radius,R.
However, this parameter is necessary for the application of the model in an
arbitrary cell. The inclusion of this parameter is an extension to [9]. Average values for NX, NtrX, rw,
and R can be found in a road map.


### 2.2. Deterministic Laws of Mobility

The random walk on a lattice used in the model is
showcased in Figure2. The position vector r¯ for step i will be(5)ri¯=r0¯+∑k=1imk¯, where r0¯ is the starting position, and |mk¯| is the distance that a vehicle makes in step k. |mk¯| is fixed and equal to d in the context of square pattern. In every
crossing, the vehicle makes a turn φi=arccos(mi−1¯⋅mi¯/|mi−1¯||mi¯|) which is a discrete random variable with [0,π/2,−π/2] possible values. The vehicle exits the cell of
radius R for the first time at the nth step when |ri¯|>R is true for i=n but not true for all i<n.
Assuming ui to be a positive Gaussian random variable that
describes the average velocity of the vehicle in every step, the sojourn time
for a random walk on a lattice exiting the cell of radius R with n steps will be(6)st=∑i=1n−1(|mi¯|ui+τ⋅tri)+λ⋅(|mn¯|un+τ⋅trn), where tri is a binomial random variable with p=ptr, τ is an average delay in a traffic light or a
stop sign, and λ is the percentage of the last step which is
inside the cell area. Using the law of cosines on the triangle, defined by rn¯ and the intersection of the circle and mn¯,
we can calculate λ from(7)|mn¯|2λ2+(|rn¯|2−|rn−1¯|2−|mn¯|2)λ=R2−|rn¯|2.Figure 2
A random walk on a lattice
example.Equation (7) has only one positive solution
because (|rn¯|2−|rn−1¯|2−|mn¯|2)>0.
The above analysis is used in the simulation model and it corresponds to ST for |r0¯|=R and RST for |r0¯|<R.


### 2.3. Assumptions

#### 2.3.1. Velocity

Measurements in [3] have shown that average velocity in cities can
be approximated by a weighted Gaussian and Rician
distribution. In [9],
however, it is proposed that this behavior is the result of the distortion of a
Gaussian velocity caused by the pause times. This means that modeling the
velocity as Gaussian and including probabilistic pause times at crossings
produce a more realistic and intuitive model that
can fit the measurements of [3].In this paper, average velocity in every step is
modeled as positive Gaussian with fixed mean and deviation (u¯i=40 kmh, σui=10 kmh). This implies that velocity is not a main
characteristic of sojourn time in urban cells as opposed to other previous
models referring to an arbitrary cell. The vehicle velocity in urban
environment depends on the following factors:(i)
the velocity at which the driver would like to
move,(ii)
the city regulations,(iii)
the crossings and the traffic lights,(iv)
the traffic.Assuming that the driver behavior and the regulations
are qualitatively the same in every city, the average velocity depends mostly
on the cell shape and the expected traffic. The effect of traffic jams is not
considered in this paper and, therefore, velocity is considered a positive
Gaussian random variable with fixed parameters which is distorted by the delays
in the crossings. This means that the model is valid for all city cells without
extreme traffic. In case of extreme traffic, sojourn times are expected to be
comparable to call duration times and the number of handovers is expected to be
very small.


#### 2.3.2. Crossing Delay

The crossing delay isτ with probability ptr and 0 with probability 1−ptr as described by the binomial random variable tri.
The value τ=10 seconds is typically used as an average waiting time
in a traffic light or a stop sign. Delay is 0 in case of a priority road crossing. It is
expected that summing up many crossings with traffic lights and/or stop signs,
the sum of waiting time divided by the number of such crossings, converges to
the mean waiting time per crossing.


#### 2.3.3. Turning in the Crossings

The formulation of turns has an important effect on
the realism of the model. In bibliography [3, 9], it is proposed that the probable cases listed in descending
order are 0, −π/2 (right) and π/2 (left). Making a π turn is considered highly improbable in a city
environment. Moreover, conditional probabilities of turning are used to make
the model more realistic. Turning left or right becomes more improbable after
every turn in the same direction. This makes sure that a driver with a chosen
destination is simulated much more times than one that roams around the cell in
random directions. If conditional probabilities are neglected, the resulting
model yields larger values on the average, like in [9]. The values used in our
model are motivated by simulations and [3] pπ/20=0.135, p−π/20=0.165, p0=1−pπ/2k−p−π/2l,
and pπ/2k=0.5pπ/2k−1, p−π/2l=0.5p−π/2l−1,
where k and l count the π/2 and −π/2 turns, respectively.


### 2.4. Complexity versus Accuracy

Summarizing the previous subsections, the proposed
model assumes vehicle movement in a square pattern urban cell with nonrush hour
traffic. Moreover, it assumes that an arbitrary driver behaves in a standard
way with respect to velocity and probability of turning in a crossing. Finally,
the waiting time in a crossing is modeled the same in case of a traffic light
(as an average) and in case of a nonpriority road (with a stop sign). It is
expected that the accuracy of the model can be compromised when applied in a
real life cell with many irregularities. The main advantage of the model is the
simplicity with which it can be used. It should be used in the same way that
radio propagation models for signal strength predictions are used.


### 2.5. Genetic Algorithm with Grammatical Evolution

The results of Monte Carlo simulations are fitted to a
function using a genetic algorithm with grammatical evolution. Grammatical
evolution is an evolutionary process that can create programs in an arbitrary
language [10]. The
production is performed using a mapping process governed by a grammar expressed
in Backus Naur Form. The problem of data fitting can be formulated as given N points and associated values (xi,yi), i∈[1,N],
with xi∈ℝn,
estimate a function f:ℝn→ℝ that minimizes the least squares
error:(8)ET=∑i=1M(f(xi)−yi)2.The specific method takes as input the points(xi,yi) and creates a functional form that minimizes
the quantity in (8) through the procedure of grammatical evolution.


## 2.1. Model Design

The model design procedure is described in Figure1.
An intuitive set of three cell parameters is chosen. These three parameters are
the ones used to characterize an arbitrary cell, and they constitute the domain
of the empirical functions produced by the model. For all possible values of
these parameters, a Monte Carlo simulation is performed and the results are
fitted to the gamma distribution. The gamma parameters a and b are then extracted with a median regression
method. After the simulation loop, aST, aRST, bST,
and bRST data are available and they are used as inputs
to a graph fitting procedure by means of a genetic algorithm with grammatical
evolution. The final model is an expression of a and b in terms of the chosen parameters. Then, the
distribution of ST and RST is obtained as(1)f(x)=1baΓ(a)xa−1e−x/b.Figure 1
Logical diagram of the model design procedure.The chosen cell parameters are the probability of
trafficptr,
the normalized average block area Ab,
and the cell radius R.
The first parameter ptr is the probability that a vehicle user stops
for a pause time in an arbitrary crossing. For a set of crossings X with NX elements which NtrX of them have traffic lights, we
write(2)ptrX=NtrX+0.5(NX−NtrX)NX.The vehicle user faces three possibilities, making an
average pause time at a traffic light crossing, making a pause time at a
nontraffic light crossing coming from the minor road and making no pause time
when coming from a major road. Every traffic light crossing is associated with
an average delay and half of the priority crossings are associated with a
deterministic delay. By using all the crossings in a cell(X=C), ptr=ptrC describes a cell where the movement is
uniformly distributed among all roads. Another approach is to use the set M that contains all the crossings of major roads
that sustain the most important traffic load. This corresponds to transitional
traffic in the cell. Assuming that the probability that an arbitrary user is in
transitional state (P(intransit)) is known, we get (3)ptr=P(intransit)ptrM+(1−P(intransit))ptrC.The normalized average area of block,Ab,
can be used to extract the length of the roads. In case of square
blocks(4)d=Ab+rw≈Ab, where d is the edge, and rw is the road width. The value of Ab,
in case of nonidentical blocks, is computed by averaging which results in a
possible error. The greater the square diversity, the greater the error is
expected.The third parameter is simply the cell radius,R.
However, this parameter is necessary for the application of the model in an
arbitrary cell. The inclusion of this parameter is an extension to [9]. Average values for NX, NtrX, rw,
and R can be found in a road map.


## 2.2. Deterministic Laws of Mobility

The random walk on a lattice used in the model is
showcased in Figure2. The position vector r¯ for step i will be(5)ri¯=r0¯+∑k=1imk¯, where r0¯ is the starting position, and |mk¯| is the distance that a vehicle makes in step k. |mk¯| is fixed and equal to d in the context of square pattern. In every
crossing, the vehicle makes a turn φi=arccos(mi−1¯⋅mi¯/|mi−1¯||mi¯|) which is a discrete random variable with [0,π/2,−π/2] possible values. The vehicle exits the cell of
radius R for the first time at the nth step when |ri¯|>R is true for i=n but not true for all i<n.
Assuming ui to be a positive Gaussian random variable that
describes the average velocity of the vehicle in every step, the sojourn time
for a random walk on a lattice exiting the cell of radius R with n steps will be(6)st=∑i=1n−1(|mi¯|ui+τ⋅tri)+λ⋅(|mn¯|un+τ⋅trn), where tri is a binomial random variable with p=ptr, τ is an average delay in a traffic light or a
stop sign, and λ is the percentage of the last step which is
inside the cell area. Using the law of cosines on the triangle, defined by rn¯ and the intersection of the circle and mn¯,
we can calculate λ from(7)|mn¯|2λ2+(|rn¯|2−|rn−1¯|2−|mn¯|2)λ=R2−|rn¯|2.Figure 2
A random walk on a lattice
example.Equation (7) has only one positive solution
because (|rn¯|2−|rn−1¯|2−|mn¯|2)>0.
The above analysis is used in the simulation model and it corresponds to ST for |r0¯|=R and RST for |r0¯|<R.


## 2.3. Assumptions

### 2.3.1. Velocity

Measurements in [3] have shown that average velocity in cities can
be approximated by a weighted Gaussian and Rician
distribution. In [9],
however, it is proposed that this behavior is the result of the distortion of a
Gaussian velocity caused by the pause times. This means that modeling the
velocity as Gaussian and including probabilistic pause times at crossings
produce a more realistic and intuitive model that
can fit the measurements of [3].In this paper, average velocity in every step is
modeled as positive Gaussian with fixed mean and deviation (u¯i=40 kmh, σui=10 kmh). This implies that velocity is not a main
characteristic of sojourn time in urban cells as opposed to other previous
models referring to an arbitrary cell. The vehicle velocity in urban
environment depends on the following factors:(i)
the velocity at which the driver would like to
move,(ii)
the city regulations,(iii)
the crossings and the traffic lights,(iv)
the traffic.Assuming that the driver behavior and the regulations
are qualitatively the same in every city, the average velocity depends mostly
on the cell shape and the expected traffic. The effect of traffic jams is not
considered in this paper and, therefore, velocity is considered a positive
Gaussian random variable with fixed parameters which is distorted by the delays
in the crossings. This means that the model is valid for all city cells without
extreme traffic. In case of extreme traffic, sojourn times are expected to be
comparable to call duration times and the number of handovers is expected to be
very small.


### 2.3.2. Crossing Delay

The crossing delay isτ with probability ptr and 0 with probability 1−ptr as described by the binomial random variable tri.
The value τ=10 seconds is typically used as an average waiting time
in a traffic light or a stop sign. Delay is 0 in case of a priority road crossing. It is
expected that summing up many crossings with traffic lights and/or stop signs,
the sum of waiting time divided by the number of such crossings, converges to
the mean waiting time per crossing.


### 2.3.3. Turning in the Crossings

The formulation of turns has an important effect on
the realism of the model. In bibliography [3, 9], it is proposed that the probable cases listed in descending
order are 0, −π/2 (right) and π/2 (left). Making a π turn is considered highly improbable in a city
environment. Moreover, conditional probabilities of turning are used to make
the model more realistic. Turning left or right becomes more improbable after
every turn in the same direction. This makes sure that a driver with a chosen
destination is simulated much more times than one that roams around the cell in
random directions. If conditional probabilities are neglected, the resulting
model yields larger values on the average, like in [9]. The values used in our
model are motivated by simulations and [3] pπ/20=0.135, p−π/20=0.165, p0=1−pπ/2k−p−π/2l,
and pπ/2k=0.5pπ/2k−1, p−π/2l=0.5p−π/2l−1,
where k and l count the π/2 and −π/2 turns, respectively.


## 2.3.1. Velocity

Measurements in [3] have shown that average velocity in cities can
be approximated by a weighted Gaussian and Rician
distribution. In [9],
however, it is proposed that this behavior is the result of the distortion of a
Gaussian velocity caused by the pause times. This means that modeling the
velocity as Gaussian and including probabilistic pause times at crossings
produce a more realistic and intuitive model that
can fit the measurements of [3].In this paper, average velocity in every step is
modeled as positive Gaussian with fixed mean and deviation (u¯i=40 kmh, σui=10 kmh). This implies that velocity is not a main
characteristic of sojourn time in urban cells as opposed to other previous
models referring to an arbitrary cell. The vehicle velocity in urban
environment depends on the following factors:(i)
the velocity at which the driver would like to
move,(ii)
the city regulations,(iii)
the crossings and the traffic lights,(iv)
the traffic.Assuming that the driver behavior and the regulations
are qualitatively the same in every city, the average velocity depends mostly
on the cell shape and the expected traffic. The effect of traffic jams is not
considered in this paper and, therefore, velocity is considered a positive
Gaussian random variable with fixed parameters which is distorted by the delays
in the crossings. This means that the model is valid for all city cells without
extreme traffic. In case of extreme traffic, sojourn times are expected to be
comparable to call duration times and the number of handovers is expected to be
very small.


## 2.3.2. Crossing Delay

The crossing delay isτ with probability ptr and 0 with probability 1−ptr as described by the binomial random variable tri.
The value τ=10 seconds is typically used as an average waiting time
in a traffic light or a stop sign. Delay is 0 in case of a priority road crossing. It is
expected that summing up many crossings with traffic lights and/or stop signs,
the sum of waiting time divided by the number of such crossings, converges to
the mean waiting time per crossing.


## 2.3.3. Turning in the Crossings

The formulation of turns has an important effect on
the realism of the model. In bibliography [3, 9], it is proposed that the probable cases listed in descending
order are 0, −π/2 (right) and π/2 (left). Making a π turn is considered highly improbable in a city
environment. Moreover, conditional probabilities of turning are used to make
the model more realistic. Turning left or right becomes more improbable after
every turn in the same direction. This makes sure that a driver with a chosen
destination is simulated much more times than one that roams around the cell in
random directions. If conditional probabilities are neglected, the resulting
model yields larger values on the average, like in [9]. The values used in our
model are motivated by simulations and [3] pπ/20=0.135, p−π/20=0.165, p0=1−pπ/2k−p−π/2l,
and pπ/2k=0.5pπ/2k−1, p−π/2l=0.5p−π/2l−1,
where k and l count the π/2 and −π/2 turns, respectively.


## 2.4. Complexity versus Accuracy

Summarizing the previous subsections, the proposed
model assumes vehicle movement in a square pattern urban cell with nonrush hour
traffic. Moreover, it assumes that an arbitrary driver behaves in a standard
way with respect to velocity and probability of turning in a crossing. Finally,
the waiting time in a crossing is modeled the same in case of a traffic light
(as an average) and in case of a nonpriority road (with a stop sign). It is
expected that the accuracy of the model can be compromised when applied in a
real life cell with many irregularities. The main advantage of the model is the
simplicity with which it can be used. It should be used in the same way that
radio propagation models for signal strength predictions are used.


## 2.5. Genetic Algorithm with Grammatical Evolution

The results of Monte Carlo simulations are fitted to a
function using a genetic algorithm with grammatical evolution. Grammatical
evolution is an evolutionary process that can create programs in an arbitrary
language [10]. The
production is performed using a mapping process governed by a grammar expressed
in Backus Naur Form. The problem of data fitting can be formulated as given N points and associated values (xi,yi), i∈[1,N],
with xi∈ℝn,
estimate a function f:ℝn→ℝ that minimizes the least squares
error:(8)ET=∑i=1M(f(xi)−yi)2.The specific method takes as input the points(xi,yi) and creates a functional form that minimizes
the quantity in (8) through the procedure of grammatical evolution.


## 3. Results

The results are presented in Figures3–6 for R=100 and R=250.
Rate parameter b has monotonous behavior, while the scale
parameter a is fluctuating in some cases. This fluctuation
is justified in cases the actual distribution of sojourn time is changing
shape. In such transition phase, the shape parameter is expected to be
unstable.Figure 3
The
functionaST(ptr,Ab,R) for R=100 and R=250.Figure 4
The
functionbST(ptr,Ab,R) for R=100 and R=250.Figure 5
The
functionaRST(ptr,Ab,R) for R=100 and R=250.Figure 6
The
functionbRST(ptr,Ab,R) for R=100 and R=250.From the figures, it is observed that greater street
lengthsAb,
smaller probability ptr,
and cell radius R yield smaller b and consequently smaller average sojourn
times, which supports common reasoning.The functionsaST, aRST, bST, bRST are produced by using the data from the
deterministic simulation model described in Section 2 and applying the method
of fitting described in Section 2.5 [10]. For the chosen range of parameters values, there is
a clear dependence of a and b,
on each of the parameters. This shows small information redundancy and supports
the choice of parameters:(9)aST(ptr,Ab,R)=2.49⋅2.37sin(log(Ab0.682707/R))+log(log(log6.99cosptr+ptr)),bST(ptr,Ab,R)=R3Ab/R⋅ptr+1.41772/ptr+cos(2.36ptr),aRST(ptr,Ab,R)=log(log[R2logR/433.77−Ab2logptr/7814+R])2,bRST(ptr,Ab,R)=3.68638⋅RptrAb0.31−ptr(logAb+1), where log is the natural logarithm. The mean least
square error of fitting was found to be e(aST)=0.0315, e(bST)=0.0019, e(aRST)=0.4244,
and e(bRST)=0.3794.
2000 generations that were used in the fit process.
The products aST⋅bST and aRST⋅bRST yield the expectation of ST and RST,
respectively, while it is possible to calculate variance by aST⋅[bST]2 and aRST⋅[bRST]2.


## 4. Case Study in a Real Cell

A case study is presented for the cell of Figure7.
Using the road map, the values Ab=9070m2(95m×95m) and R=300 m are extracted. For ptr,
a traffic assumption is made. The 80% of the vehicles are assumed to use the major
roads (yellow on the map) only and the rest are assumed to move randomly in the
cell. Therefore, ptr=0.8⋅ptrM+0.2⋅ptrC,
where ptrM=0.571 and ptrc=0.7 using (2). This yields the value ptr=0.597.Figure 7
The cell in the city of Patras under
test. Measurements took place in this cell.Using (9), aST=3.88094, bST=24.5184, aRST=1.08972, and bRST=44.6443 are calculated. These results are compared to
measurements taken in the same area in nonrush hour. The measurements density
is estimated using Epanechnichov kernel-based estimator with half the smoothing
optimal bandwidth (hos/2), see [11]. In Figure 8, it is evident that the model
conformance to measurements is improved, in comparison to [9]. This is due to the use of
conditional probabilities of turning, the use of R parameter, and better fitting method.Figure 8
Comparison of the proposed model and measurements for the cell of Figure7. The
values used for the model are aST=3.88094, bST=24.5184, aRST=1.08972,
and bRST=44.6443.


## 5. Conclusion

A deterministic simulation model is proposed. The
model can be used along with a road map in order to provide sojourn time and
remaining sojourn time statistics for an arbitrary city cell with low vehicle
traffic. The calculation of the parameterptr is described in detail. The model has been
tested in a real cell, and the results show a satisfactory measurement
conformance. This model can be used for cell planning and handover ratio
estimation in any urban circular cell that has a rectangular block shape.



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*Source: 721386-2008-05-20.xml*