Modeling Call Holding Times Of Public Safety Network

International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013
DOI:10.5121/ijcsa.2013.3301 1
MODELING CALL HOLDING TIMES OF PUBLIC
SAFETY NETWORK
Tuyatsetseg Badarch*, Otgonbayar Bataa†
*Department of Electrical and Computer Engineering,
Northeastern University, Boston, USA
b_tuyatsteseg@yahoo.com
†
Wireless Communication and Broadcasting Technology, School of Information and
Communications Technology (SICT), Mongolian University of Science and Technology
(MUST), Mongolia.
otgonbayar@sict.edu.mn
ABSTRACT
This paper presents parametric probability density models for call holding time (CHT)based on the actual
data collected for over a week from the IP based public Emergency Information Network (EIN) in
Mongolia. When the set of chosen candidates of Gamma distribution family is fitted to the call holding
time data, it is observed that the whole area in the CHT empirical histogram is underestimated due to
spikes of higher probability and long tails of lower probability in the histogram. Therefore, we provide the
Gaussian parametric model of a mixture of lognormal distributions with explicit analytical expressions for
the modeling of CHTs of PSN. Finally, we show that the CHT for the IP based PSN is fitted reasonably by
a mixture of lognormal distributions via the simulation of expectation maximization algorithm. This result
is significant as it expresses a useful mathematical tool in an explicit manner of a mixture of lognormal
distributions.
KEYWORDS
A Mixture of Lognormal Distributions, EM algorithm, Modeling Call Holding Times, Public Safety
Networks.
1. INTRODUCTION
Emergency communication networks that serve various safety personnel, including medical
responders, police, hazard and fire fighters, play a critical role in responding to emergency calls
and managing voice traffic. In such time-conscious networks, the CHT of incoming voice calls
from the affected population contribute to the maximum volume of traffic, which may not be
supported by existing infrastructure because of logistical constraints of system resources.
Studying call holding time provides the ability to estimate the probability of an ongoing call to
hang up the phone during the next t seconds [1]. Therefore, the call holding time distribution
plays a prominent role that is used to analyze the traffic and accurate design of the system
resources, simulation, performance, and resource allocation strategy.
The call duration over fixed, cellular, and trunked radio networks is traditionally assumed to be
negative exponentially distributed. However, this distribution approximation is not valid for
communication networks because many empirical approaches have proved that generalized
gamma, lognormal distributions fit empirical data much better [2]-[9]. For instance, the
exponential distribution in cellular networks is quite inaccurate in capturing the empirical data

2
compared to the mixture of lognormal distributions. It is observed that the probability of very
short occurrences is overestimated, while the area with the highest probability in the empirical
histogram is underestimated [3], [10].
Whereas there exists a literature such as call holding time distribution modeling in fixed PSTN
network [4], in PCS network [6], in cellular networks [7], in private mobile radio PAMR [8], and
public safety network [11]. Studying call holding time of a public safety network provides
insight into the operation of the special purposed network [11].
In this paper, we present the study of more flexible distribution of CHTs with its mathematical
tools in PSNs, which has not has been presented before. It is proved that the parametric mixture
model has been implemented to show that average CHTs of PSNs may be approximated
accurately by a mixture of lognormal distributions compared to the fitting of the statistical
probability distributions of Lognormal, shifted Lognormal, and Weibull distributions, which
provides the example of visual view of non-superimposed distribution graphs (Fig.2).
This article is structured as follows: Section 2 presents the IP based EIN, its system structure and
components. Section 3 describes the statistical methods of deriving CHT distributions. Section 4
describes the proposed modeling of a mixture of lognormal distributions. Section 5 presents the
EM algorithm performance for the proposed model of a mixture of lognormal distributions.
Section 6 presents a data exploration of the existing PSN for the performance of the proposed
model. Section 7 shows the results of the standard probability models. In section 8, we
demonstrate the numerical results based on the proposed mixture models for the PSN compared
to the statistical models such as Lognormal model, and Weilbull model. Finally, we reach a
conclusion on the research.
2. IP BASED EMERGENCY INFORMATION NETWORK
The ability to access emergency services by dialing fixed numbers is a vital component of public
safety and emergency preparedness. The Federal Communications Commission (FCC) defines
Voice over Internet Protocol (VoIP) as a technology that supports some IP based services to
allow a user to call anyone who has a telephone number, including mobile and fixed network
numbers. Gradually, the Emergency telephonic services are migrating to interconnected VoIP
.
The IP based EIN is the national level, integrated PSN with state-of-the-art ICT solutons
compared with previous emergency services which were handled by the national medical, police,
and emergency management agencies in Mongolia separately. Four emergency telephone
numbers are used throughout the country. They are:"103" (ambulance), "102" (police), "101"
(fire), and "105" (hazards, disaster). When dialing any one of these numbers from
telecommunications networks (PSTN, GSM, CDMA) through PSTN, the emergency call is
pushed/forwarded to an emergency call center agency desk in the EIN.
Figure 1 presents the EIN communication architecture that provides emergency call services
through wired (PSTN) and wireless (GSM, CDMA) system services for the main capital
Ulaanbaatar city as well as provinces within 1100 km, and high speed data, voice, and image
transfer services through its main networks. The EIN system is composed of four main networks
including a fiber optical (F/O) backbone network, IP based telecommunications and computer
networks (IP PABX/VoIP/LAN/WAN/WiMax/WiFi), digital and analog Trunk Radio System
(TRS), and CCTV( Digital Camera TV) networks for voice, data, video, and image transfers in
urban area coverage.

3
The EIN automatically associates a location with the origination of the Emergency call though a
GPS system with Geographic Information System (GIS) using a Digital Map of shape type
(**.shp). Additionally, a variety of Emergency data information, such as data and image
Figure 1. Communication Architecture of EIN
transfers, are being handled by the EIN within the network as real-time applications without a
peak load for the system performance.
However, the life-threatening emergency voice calls which are transferred through the bulk of
the existing optical E1 trunks across long haul telecommunication networks are the main part of
the system used for the peak period network performance and traffic analysis because of the
dedicated limited number of inbound and outbound E1 trunks under the government regulation.
If the number of calls addressed to the system is too large during the peak period, the incoming
trunks will be overloaded and operators cannot handle the volume of emergency calls. Hence,
one aspect of particular interest is the peak period network analysis to model the call holding
times and to estimate the call holding time parameters for the system performance.
3. GENERAL METHODS ON DERIVING CALL HOLDING TIME
DISTRIBUTIONS
In this section, as the preliminary study, the set considered is of probability density functions
(pdf) as main statistical candidates to fit the CHT empirical distribution for ambulance, police,
fire, and hazards calls.
For the fitting of these candidate distributions, we use three tests: K-S tends to be more sensitive
near the center of the distribution than at the tails. Due to this limitation above, we may prefer to
use the Anderson-Darling goodness-of-fit test which gives more weight to the tails than does the
K-S test. The Chi- Square(Chi-Sq) test presents a statistic via the observed frequency and the
expected frequency for bins, which can be calculated based on the sample size. Test results show
that these candidate distributions are not best probability models of average CHTs for PSNs.

4
1. The pdf of Exponential distribution:
( ; ) exp( )f x x  = − (1)
where > 0 is the rate parameter of distribution.
2. The pdf of Weibull distribution:
Weibull distribution is a continuous probability distribution with wide applicability, primarily
due to its relation to the Gamma distribution family.
The pdf of Weibull distribution has the form:
( )( ; , ) exp xF x

  = (2)
where  and  are the scale and shape parameters of distribution, respectively.
3. The pdf of Shifted Lognormal distribution:
The pdf of a shifted lognormal distribution has the form:
2
2
1 ( ( ) )( ) exp
2( ) 2
x
Ln xx
x
 
  
 − −= − 
−  
(3)
where >0, We use the likelihood function :
11
( ( )) ( ) ( )
n n
x x i x i
ii
L x x Ln x  
==
= = ∑∏ (4)
where > 0,
4. The pdf of Lognormal Distribution:
Lognormal distribution is a probability distribution of any random variable whose logarithm is
normally distributed. The pdf of a lognormal distribution has the form:
2
2
1 ( )( ) exp
22
x
Lnxx
x

 
 −= − 
 
(5)
The log-likelihood function is given by
2
2
1
2
2
1 1
1 ( )
( ( )) exp
22
1
2 ( )
2 2
n
i
x
ii
n n
i i
i i
Lnx
Ln x Ln
x
n
nLn Ln Lnx Lnx

 
  

=
= =
  −
= −  
   
= − − − − −
∑
∑ ∑
(6)
The parameter values are computed through Maximum Likelihood Estimate (MLE) for a
lognormal distribution [18], [19].
Based on the assumption of smooth histograms of an empirical data, the standard distributions
for the empirical data, or workloads fit quite accurate. However, in practice, a jagged probability
histogram with random spikes and long tails frequently occurred. Therefore, the spikes may
cause the call holding time to have a jagged histogram. The most comprehensive method of a

5
mixture of lognormal distributions with two or more parameters can be used to approximate the
whole call duration, including the spikes and tails, detailed in the following section.
4. PROPOSED MODEL OF MIXED LOGNORMAL DISTRIBUTIONS BASED ON
EXPECTATION MAXIMIZATION ALGORITHM
In the EM approach, when CHTs , are observed, we may consider the
component indicators 1( ,..., )ny y y= as missing like as in the usual mixture situation, so that
becomes the complete data [20]. In literature it might happen that the CHTs in
networks are not explicitly described by a mixture of parametric multivariate distributions [14].
When each mixture component is mapped to a lognormal call holding time distribution using
priority probabilities , we notice that the n independent and identically distributed ( . . )i i d call
holding time observations come from a ﬁnite mixture of k lognormal CHT
components as follows:
1 1
( ) ( ) ( | , )j
k k
j j k k
j j
x x N x      
= =
= =∑ ∑ (7)
where are the component lognormal densities,
are the parameters, and are the component
weights satisfying . Therefore, the probability density function of the average CHTs
may be modelled by a mixture of k random variables of Gaussian densities on a logarithmic time
scale:
2
1
1 2
1 1
2
2
( )1 1
( ) [ exp ...
2 2
( )1
exp ]
2
k
k
k k
Lnx
x
x
Lnx


 
 


 
 −
= − +  
 
 −
−  
 
(8)
It is often assumed that the parametric MLE is to estimate the set of parameters θ for the density
of the samples that maximizes the likelihood function [15].
Hence the likelihood of the lognormal CHT distribution becomes:
2
1
1 2
1 11
2
2
( )1 1
[ exp ...
2 2
( )1
exp
2
n
ii
k
k
k k
Lnx
L
x
Lnx


 


 
=
 −
= − +  
 
 −
−  
 
∏
(9)
The complete CHT data set (observed CHTs plus unobserved CHTs datasets ) exists
with density where . When a many- to-one
mapping from z to is the unobserved component of the
origin of the n call holding times. Hence, it is clear that the indicator implies
Bernoulli random variable indicating that the CHT observation comes from
the exact lognormal distribution with parameter .

6
The parametric MLE problem is to estimate the set of Gaussian parameters for the lognormal
distribution that maximizes the likelihood function [15]. In this multivariate model case, the
likelihood function of this complete density for one data observation becomes
11 1
2
2
11
( ) ( , ) ( )
( )
exp .
2 2
ij j
ij
n n k
i i j y i
ji i
y
n k
j i j
i j jji
L z L x y I x
Lnx
x
     
 
  
== =
==
= = =
  −
  −
    
∑∏ ∏
∑∏
(10)
where the indicator function for individual i that comes from the lognormal component
with j and is zero elsewhere.
Since the logarithm is a convex increasing function, maximizing the likelihood is equivalent to
maximizing the log-likelihood, thus the likelihood (10) is updated in the logarithm form as log-
likelihood function for the complete CHT data:
1
2
2
1 1
1 1
2
2
( ) [ 2
2
1
( ) ]
2
1
[ ( ) 2
2
1
( ) ]
2
ij
ij
k
y j j
j
n n
i i j
ji i
n k
y j j i
i j
i j
j
n
Ln L z I nLn nLn Ln
Lnx Lnx
I Ln Ln Lnx Ln
Lnx
   


  


=
= =
= =
= − −  
− − −
= − + −
− −
∑
∑ ∑
∑∑
(11)
where n ( . . )i i d and
The second component of the expected value of the complete CHT data log-likelihood is the
marginal distribution of the unobserved CHT data on both the observed CHT
data and on the current estimates ( . The
weight is often assumed that .Therefore, we use Bayesian theorem to
compute the expression for the distribution of the unobserved CHT:
( ) ( 1| ; , )
( ) ( )
( )
( )
t
t t
j j
t
t
j
t t
ij j i ij i
t t
j i j i
k
i
ij
j
r x p y x
x x
x
x

 


  
   

 
= = =
= =
∑ 


(12)
where is simply the CHT lognormal distribution evaluated at and likelihood function
given which is similar to the expression in the right side of (9). Given , a mixture of
lognormal distribution is computed for each and .
The expected value of the ”complete data ” log- likelihood would be the
iterative process. It is formed as a function of the estimate θ from (11) and (12) where is the
current value at iteration

7
1 1
1 1 1 1
2
2
( , ) ( ) ( )
( ) ( ) [
1 1
2 ( ) ] ( )
2 2
j
k n
t t
j i j i
j i
k n k n
t
j j i j i j
j i j i
t
i j j i
j
q Ln x x
Ln x Ln Lnx Ln
Ln Lnx x
    
   
  

= =
= = = =
=
= + − −
− − −
∑∑
∑∑ ∑∑ (13)
Therefore, taking derivatives with respect to these and equating them to zero, we deﬁne
the new estimates respectively as follows:
1. The weights of CHT lognormal components:
To derive the expression for , we need the Lagrange multiplier λ with the constraint that
, then we can iteratively get the result:
1
( )
n
t
j i
t i
j
x
n

 =
=
∑
(14)
2. The parameter of CHT lognormal components:
1 1
2
2
[ (
1 1
2 ( ) )] ( ) 0
2 2
k n
j i j
jj i
t
i j j i
j
d
Ln Lnx Ln
d
Ln Lnx x
 

  

= =
− −
− − − =
∑∑
(15)
, it is clear that the parameter can be written in the form:
1
1
( )
( )
n
t
j i i
t i
j n
t
j i
i
x Lnx
x



=
=
=
∑
∑
(16)
3. The parameter of CHT lognormal components:
1 1
2
2
[ (
1 1
2 ( ) )] ( ) 0
2 2
k n
j i j
jj i
t
i j j i
j
d
Ln Lnx Ln
d
Ln Lnx x
 

  

= =
− −
− − − =
∑∑
(17)
we get the result:
1
1
( )( )( )
( )
n
t t t T
j i i j i j
t i
j n
t
j i
i
x Lnx Lnx
x
  


=
=
− −
=
∑
∑
(18)

8
5. IMPLEMENTATION OF EM ALGORITHM FOR THE PROPOSED MODEL OF
MIXED LOGNORMAL DISTRIBUTIONS
The EM algorithm provides computational techniques for distributions that are almost
completely unspecified 14], [16].
The algorithm performs the process with respect to the parametric part of this expected value of
the completed CHT data log-likelihood by assuming the existence of the hidden variables and
making a guess at the initial parameters of the distribution [15]. The EM algorithm iteratively
maximizes by the following two steps.
1. E-step: compute
E- step calculates the expected value of the "complete data" log likelihood from Equation (13)
with respect to the unobserved call holding times for all and .It
means we calculate the expected value of the unobserved CHT data using the observed
incomplete CHT data. Computing this expectation requires the posterior probability as in
equation (12) and for the parameter value (14).
2. M-step: set
This step maximizes the expectation we performed in E-step. More specifically, M step
performs the first part containing and the second part containing iteratively from (13).
We note that the parts are not related; we can maximize independently.
These two steps are iterated as necessary until the saturation value of the expected complete
log-likelihood. At the saturation value, the M step performs the set of parameters , otherwise
we repeat the E-step for the next iteration.
Although the iteration increases the marginal log-likelihood function, the EM algorithm uses a
random restart approach to avoiding a local maximum of the observed data log-likelihood
function.
6. DATA EXPLORATION OF THE EXISTING SYSTEM
We explore the real data from a number of incoming bundled digital trunks that connect into the
main edge port of IP PBX of EIN, the existing national public safety network .
The statistics indicate that the mean of CHT of Emergency ambulance, police, fire, and hazards
incoming calls are 61.16s ( 0.14cv = ), 42.56s ( cv = 0.22), 27.17s ( cv = 0.42), and 27.76s ( cv =
0.58), respectively. The mean CHT data for all emergency incoming call samples measured is
39.66s. This is a shorter call duration and less value of coefficient of variance cv compared to
201s with cv = 1.23 in non-VoIP call center [13], 110s with cv = 2.7 in Taiwan-mobile [5], 113s
withcv = 1.4 in public telephone system [4], 63.3s with cv = 2.91 in PAMR-PCS [10], 40.6s
with cv = 1.7 in non VoIP cellular network [3].
From this comparison, emergency VoIP calls are considered more efficient time-wise than those
of commercial networks. The ambulance customers keep longer holding call behavior than the
other emergency customers, such as police, fire and hazards; 35s ambulance's mean CHT is
longer than the fire's mean CHT.

9
5. RESULTS ON GENERAL STANDARD DISTRIBUTION MODELS
Before the proposed model performance of a mixture of lognormals using the set of empirical
data obtained from the EIN, first, we proceed to compare the statistical fitting of the candidate
standard distributions (Exponential, Weibull, Lognormal, and Shifted Lognormal) on the basis of
this data set of the existing system.
For determination of the best fit distribution function, a typical the significance resulting from the
Kolmogorov- Smirnov (K-S) goodness-of-fit test which is described by the modified K-S
distance , where for the maximum difference between the fitting
distribution and the empirical cdf and the level of significance [12],
[17]. After getting the set of significance resulting from the K-S goodness-of-fit test for all
empirical data, the easiest way to make a decision validating which candidate is best is to plot the
pdf of the set of statistics. The pdf gives us a rough but quick visualized result about the best-
fitted distribution function, which fits best at the fixed significance level.
As a result of such graphs, the statistical fitting results of police, fire, and hazard's CHTs are
similar to the ambulance's CHT pdf which is depicted in Fig.2. The pdf graphs except that of the
-10 0 10 20 30 40 50 60 70 80 90
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Average Holding Time(secs)
ProbabilityDensity
Weibull model
Lognormal model
Exp model
Lognormal model
Figure 2. PDF of CHT of Emergency Ambulance Calls
ambulance CHT, along with their fitting function curves are not necessarily shown in detail in
this paper due to the proposed expected result with the model of a mixture of lognormal
distributions. Looking at the graphs and goodness-of-fit test results, contradicting what we
expected, all of these four distributions do not show the expected best fit for the empirical data.
The reason is explained to be that the spikes and long-tails in the set of data histograms are not
estimated by these standard distribution models (Figure 2).
However, the lognormal distribution among these four candidates is assumed to be the better
fitted distribution with D max value and with p value at the  = 0.02 significance level under
K-S, Chi-Sq, and A-D tests (see Table 1.). In this table, we summarized statistics for the four
distributions for the emergency ambulance ("103") CHTs. The fitting statistics of the other
emergency CHTs are not shown in the paper.
In addition to the spikes, the relative tail frequencies of the real data are much larger than the
values of the Exponential, Weibull, Lognormal and Shifted Lognormal models. Therefore, we
propose a mixture of lognormal distributions instead of a single distribution next.
However MLE does not simplify a solution of a mixture of lognormal distributions for call
holding time. Therefore, the EM approach [14], a specialized approach designed for MLE
problems, is described for a mixture of lognormal distributions in this paper.

10
Table 1. “103” CHTs: Parameters of the candidate standard
distributions at significance level  = 0.2.
8. RESULTS ON THE PROPOSED PARAMETRIC MIXTURE MODELS AND
MODEL VALIDATION
This part presents the illustrative examples to show how analytical computational results
obtained in section III via the EM algorithm may be used to perform modeling CHTs of IP based
PSN.
When simulations of a mixture model are carried out as a result of the analytical approach in
section III, we observe that a mixture of lognormal distributions represents significant
improvements to hold the spikes and as well to estimate tails of frequencies.
We believe that many other similar modeling problems in a network area can be approximated
by the proposed parametric approach in this paper.
In the performance, we performed 50-321 iterations to find reasonable fits of a mixture of
lognormals. The EM algorithm fits a mixture of lognormal distributions to a given distribution,
aiming to accurately capture spikes and tails over a finite interval between possible start CHTs
and possible large CHTs of the empirical data.
As a model validation, pdf, cdf, and ccdf plots are depicted. Whereas pdf plots the probability of
occurrence of the random variable under study, the cdf plots the probability that the random
variable will not exceed specific values. More specifically, small values of the cdf reveal the
accuracy in tracking the head portion of the pdf, while small values of the ccdf reveal the
accuracy in tracking the tail portion of the pdf. These percentile-percentile plots (P-P plots) show
how well the rank statistics of a sample match a model distribution. We show that the CHTs of
PSN are modeled by a mixture of lognormal distributions, as well as that the CHTs have long-
tailed behavior.
Model Validation Results on the Body Fitting
We start to look at how long a call conversation time into emergency ambulance incoming call
services in the PSN.
m=61.16
cν=0.14
“D. max”
K-S
“D. max”
Chi-sq
p value
K-S
p value
Chi-sq
Shifted
Lognormal
Log normal
Weibull
Shifted
Lognormal
Log normal
Weibull
0.037
0.038
0.067
p val
(A-D)
0.1936
0.1959
2.156
2.232
2.23
14.7
“scale”
0.14
σ=0.15
β=64.45
0.9631
0.96
0.4
“shape”
µ=4.1
γ=4.1
α=8.64
0.9458
0.9459
0.0388
“loc”
-1.63
0
0

11
Figure 3 shows the probability results of the distribution fittings, with the Lognormal-5 as the
best fitted distribution. The values of maxD = 0.0274377, 4
2.3 10 −
= × , and component
parameters of the best fitted lognormal-5 are its main parameter estimates (see Table 2).
In a practical matter of models, it is always very difficult to fit the spikes and tails of frequencies.
We demonstrate the mixture models of other emergency call types similar as the mixture of five
lognormal distributions of "103". We observe clearly that the ambulance CHT pdf has extreme
spikes around 56 and 66 seconds and long tails around more than 93 seconds, which shows why
the CHT is not accurately modeled by Lognormal, shifted Lognormal, Exponential, and Weibull
distributions, while a mixture of lognormal distributions can be used to approximate accurately
the empirical data with the random spikes and long tails as a best fitted model (Fig. 3a).
The CHT distributions for emergency police ("102"), fire ("105"), and hazards ("101") are fitted
quite well by five, five, and four lognormal distributions, respectively as a pdf fitting (Fig. 3).
We described the MLE parameters of a mixture of lognormal distributions in each case. Table II
has shown the parameters (all  , ) of lognormal components with their maxD values through
the K-S test and  of the fitting of a mixture of lognormal distributions.
(a)
(b)
Figure 3. Comparison of candidates and the proposed model for CHTs : (a). Ambulance (“103”) and
Police (“102”) CHTs: fitted with the fitting lognormal-5 curves (b). Fire (“101”) and Hazard (“105”)
CHTs: fitted with the fitting lognormal-5 and lognormal-4 curves
To hold the spikes and tails, the second fit corresponds to the Lognormal-3, which is not like the
best fitted Lognormal-5.
From equation (8), we express lognormal-5 model of the call holding time of "103" using its
component parameters, which fits better than the other candidate distributions based on its
maxD value and  as follows:

12
This empirical data is fitted by lognormal-5 model accurately compared to lognormal-3 model
(Figure 3(a)). As a detailed explanation, the best fitted lognormal-5 distribution is composed of
the parameters of its component lognormals.(see Table 2).
Figure 3(b) shows the pdf of the most fitted lognormal-4 model for the emergency hazard CHTs.
Its parameters are detailed in Table 2. The model validation P-P results show quite good fitting
of the lognormal-4 model with maxD = 0.025939 compared to lognormal model with maxD = 0.06.
The Lognormal distribution is fitted with the scale parameter  =93.25, shape parameter  =4.18,
mean 29.3, standard deviation 19.8, and coefficient of variation cv =0.67.
Model Validation Results on the Tail Fitting
Emergency ambulance CHT pdf, which is derivative of the cdf; the highest probability intervals
are easily recognized as the peaks in the pdf in 3(a), while cdf value is compared to the
difference between empirical data and fitting models; the steeper the slope of the cdf, the higher
the probability of values.
Emergency Ambulance CHTs
Figure 4(a), 4(b) show the model validation P-P results with the best fitting of the lognormal-5
model with maxD = 0.0274377 compared to lognormal model with maxD = 0.038. The Lognormal
distribution is fitted with the scale parameter  =0.14, shape parameter  =4.1, mean 61.2,
standard deviation 8.8, and coefficient of variation cv =0.14.
30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Call Holding Time (seconds)
Cumulativedistributionfunction
Lognormal model
Empirical data
Lognormal-5 model
30 40 50 60 70 80 90 100
-7
-6
-5
-4
-3
-2
-1
0
Holding time in seconds
CCDF
LogNormal model
Empirical data
LogNormal-5 model
(b) (b)
Figure 4. Comparison of candidates and the proposed model for emergency ambulance CHTs: (a) The cdf
values with the small differences between empirical data and fitting models (b). The skewed- right
empirical histogram of emergency ambulance CHTs and its best fitting lognormal-5 curve
The top 40 seconds of empirical data, along with the best fitted lognormal-5 and fitted lognormal
distributions is the start time of the total ambulance CHTs, by average. The top 80 seconds of the
empirical data, along with the fitted distributions above named account for more than 95 % of the
total CHTs. Although the cdf fitting looks perfect, the discrepancy in the set of distributions is
clear, while the tail values are not seen clearly well.

13
In a practical matter of difficulties of tails to distinguish for a model, to illustrate the tail fitting
more clearly, we test the ccdf that the large variability of the most fitted model is inherited by the
approximating lognormal-5 model, so that the best fit is done. Specifically, the fitting of the tail
part of frequencies presents that the empirical data fits accurately lognormal-5 distribution.(Fig.
4(b)).
Result on Emergency Police CHTs
We present the best model with the tail fitting based on how long a call conversation time of
emergency policy incoming call stays in the PSN.
20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CDF
LogNormal model
Empirical data
LogNormal-5 model
(a) (b)
Figure 5. Comparison of candidates and the proposed model for emergency police CHTs :(a). The cdf
We claim that the fitted lognormal-5 model is long-tailed model based on the observation of cdf
and ccdf (Figure 5(a), 5(b)). From Figure 5(b), the top 25 seconds of empirical data, along with
the best fitted lognormal-5 and fitted lognormal distributions, is the start time of the total
ambulance CHTs, by average. The top 65 seconds of the empirical data, along with the fitted
distributions, account for more than 95 % of the total CHTs. From the cdf, there is some clear
view for the Lognormal distribution models of CHTs, but the extreme tail behavior is
distinguished in ccdf of long-tailed distributions.
Except the long tails and spikes, the lognormal model is a good match for the empirical data.
Therefore, the empirical data fits a lognormal-5 accurately compared to the candidate Lognormal
model. From the result, it can be seen that the advantage of modeling long tail patterns as
lognormal is so that the multiplicative property holds, i.e. the product of independent
lognormally distributed call holding times is itself lognormally distributed. If components of the
call holding times are lognormally distributed, then call holding time is also lognormally
distributed.
However, in this paper, the lognormal distribution does not adequately predict peak period call
holding times of policy calls, while a mixture of lognormal distributions do adequate
approximation for the jagged distribution with random spikes and long tails for the peak period
call holding time modeling.
Therefore, the empirical data fits a lognormal-5 accurately compared to the candidate lognormal
model. To examine tail behavior, we use the ccdf on log axis that shows that the divergence of
the tail from the model is more accurately distributed by the lognormal-5 model (Figure 5(b)).
20 30 40 50 60 70 80 90
-7
-6
-5
-4
-3
-2
-1
0
Holding time in
CCDF
LogNormal
Empirical data
LogNormal-5 model

14
Result on Emergency Fire CHTs
Figure 3 (a) shows how the probability density of the emergency fire incoming conversation
holding time from the existing network distributes over a one week continuous measurement
period and how its distribution model fits with best model, while Figure 6(a), Figure 6(b)
presents the associated P-P plots for them. Lognormal-5 model provides good approximation to
the cdf in the range of probabilities from 0 to 1 (Figure 6(a)). The accuracy in tails is observed;
long holding time probabilities correspond to small values of the pdf, while small values of the
complementary cdf (ccdf) reveal the accuracy in tracking the tail portion of the pdf.
0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CDF
Weibull model
Empirical data
LogNormal-5 model
LogNormal model
0 10 20 30 40 50 60
-6
-5
-4
-3
-2
-1
0
CCDF
Weibull model
Empirical data
LogNormal-5 model
LogNormal model
(a) (b)
Figure 6. Comparison of candidates and the proposed model for Emergency Fire CHTs (a). The cdf
The less than 3 seconds of empirical average CHT data is the start time of the total fire CHTs,
while the top 50 seconds of the empirical data, along with the fitted distributions, account for
more than 85 % of the total CHTs.
The Weibull model is considered as a good matched model for the empirical data, except the
long tail and spikes. The Weibull distribution is fitted with the scale parameter  =30.46, shape
parameter =2.4,  the mean 27, standard deviation 11.9, and maxD =0.089. Due to the
comparably large scale parameter, it does not have a long-tailed behavior. It clearly cannot fit
accurately (Figure 6(b)).
Therefore, we present that the CHT of emergency incoming fire calls can be described by
lognormal-5 distribution, which ultimately leads to the long tailed model of a emergency call
duration.
Results on Emergency Hazard CHTs
Lognormal-4 model provides good approximation to the cdf in the range of probabilities from 0
to 1 (Figure 7(a)). The accuracy in tails is observed; long holding time probabilities correspond
to small values of the pdf, while small values of the complementary cdf (ccdf) reveal the
accuracy in tracking the tail portion of the pdf.

15
0 20 40 60 80 100 120 140
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Holdingtimeinseconds
CDF
Weibullmodel
Empiricaldata
LogNormal-4model
LogNormalmodel
0 20 40 60 80 100 120 140
-14
-12
-10
-8
-6
-4
-2
0
Holdingtimeinseconds
CCDF
Weibullmodel
Empiricaldata
LogNormal-4model
LogNormalmodel
(b) (b)
Figure 7. Comparison of candidates and the proposed model for Emergency Hazard CHTs: (a). The cdf
The less than 10 seconds of empirical data, along with the best fitted lognormal-5 and fitted
Lognormal and Weibull distributions is the start time of the total fire CHTs, by average, while
the top 50 seconds of the empirical data, along with the fitted distributions, account for more
than 91 % of the total CHTs.
Except the long tail and spikes, the Weibull model is a good match for the empirical data, while
the lognormal model is far away the empirical hazard CHT data. The Weibull distribution is
fitted with the scale parameter  =32.41, shape parameter  =1.95, the mean 28.7, standard
deviation 15.3, and maxD =0.118. The tail portion does not fit accurately by Lognormal model
compared to the models of Lognormal-4 and Weibull (Figure 7(b)).

16
Table 2. Parameters of fitted mixture of lognormal distributions
Result Comparison
As a general workload computation, the analysis shows that among the total number of
emergency calls during the peak period, there is a big gap of the ambulance calls (49.26 %),
police calls (45.02 %), fire calls (4.14 %), and hazard calls (1.59 %) in terms of the loads to the
system performance.
The daily analysis shows the arrival process of emergency ambulance and police calls have a
periodic behavior with daily spikes which show the repeating frequency of everyday use.
While emergency ambulance and police calls increase between 20PM and 24PM, fire calls
increase around morning 10AM and 11AM, and hazards calls have a flat rate. The ambulance
calls have a greater number of calls than the police calls during rush hours.
Similar behavior of spikes and long-tailed with other classes of emergency CHTs is observed for
the emergency police CHT.
As a result of the detailed curve fitting computation, the EM based method is computed to be the
best algorithm to express a mixture of lognormal distributions.
By comparing the parameters in Table 1 and Table 2, we show that moving from Gamma family
distributions to a mixture of lognormals leads us to the more the accurate result and less error.
As stated in section III, although the values of distance maxD in the Table 1 represent a
successful outcome of measurements that the empirical CHT data may come from the Gamma
distribution family, with respect to the all types of emergency incoming call holding times.
However, we observe the peak period empirical data that the conversation time histograms have
the spikes and long tails. Thus, the mixture of lognormal distributions can be used to
approximate the empirical CHT data accurately.
CHTs
D.max
“102”
0.2172569
0.00013
“103”
0.0374377
0.00023
“101”
0.0364377
0.00023
“105”
0.025939
0.00016
µ1
µ2
µ3
µ4
µ5
γ1
γ2
γ3
γ4
γ5
π1
π2
π3
π4
π5
3.590257
3.823747
3.701574
3.968549
3.928348
0.184237
0.017876
0.050927
0.089731
0.253997
0.421289
0.115688
0.220727
0.100068
0.142225
4.009660
4.160792
4.156612
4.181588
4.005497
0.011857
0.1788397
0.102826
0.017986
0.130829
0.0747236
0.178327
0.410079
0.0441521
0.292717
2.787204
3.105966
3.508355
3.333218
4.013681
0.342812
0.057409
0.195627
0.239888
0.0414164
0.237278
0.060063
0.315353
0.351031
0.0362751
2.841544
2.840751
3.131199
3.703959
N/A
0.272628
0.029038
0.200754
0.355409
N/A
0.136657
0.042747
0.465886
0.354708
N/A

17
In Table 2, the parameters (all  ,  ) of components with their maxD values are listed through
the K-S test and  of fitting a mixture of lognormal distributions. The values of maxD are 0.0217,
0.0374, 0.0364 and 0.0259 for the police, ambulance, fire, and hazards CHTs, respectively. The
weights of component lognormal distributions are 0.0747236, 0.178327, 0.410079, 0.0441521,
and 0.292717, respectively.
As described in the previous section, we should have a desired equation for the chosen CHT
model by a mixture of $k$ random variables of Gaussian densities on a logarithmic time scale:
2
2
( )
( ) exp
2 2
k i k
k k
Lnx
x
x

 

  
−
= − (19)
According (19), we would be able to obtain mathematical models that can accurately capture the
spikes and heavy tails of call conversation time distribution of modern networks.
9. Conclusion
In summary, the research demonstrates that the mixture of lognormal distributions based on EM
method presents the best statistical CHTs fitting model for the peak period analysis of PSN. A
mixture of lognormal distributions with explicit analytical expressions is described for the
incoming CHTs of PSN due to spikes of higher probability and long tails of lower probability in
the histogram.
The results contribute to the existing literature in two important ways: First, compared to the
candidate standard skewed - right distributions, the finite mixture lognormal modeling presents a
best example of a statistically suitable method for modeling the distribution function when the
long tails and peak spikes randomly and frequently occurred in the frequencies of occurrence.
Second, the result is important as it provides a useful mathematical tool in an explicit manner for
a mixture of lognormal distributions. We emphasize the EM algorithm is a powerful method to
model the CHTs of IP based PSNs.
ACKNOWLEDGEMENTS
Financial support from the International Fulbright Science and Technology PhD program of
USA government is gratefully acknowledged.
REFERENCES
[1] A. M. Law, W. D. Kelton, Simulation Modeling and Analysis, New York: McGraw-Hill, 1991.
[2] Al. Ajarmeh, J. Yu, M. Amezziane, “Framework for Modeling Call Holding Time for VoIP
Tandem Networks: Introducing the Call Cease Rate Function,” in Proc. GLOBECOM 2011,
Chicago, IL, USA, 5-9 Dec. 2011, pp. 1- 6.
[3] F. Barcelo, J. Jordan, “Channel Holding Time Distribution in Cellular Telephony,” Elect. Letter, vol.
34, no. 2, pp.146-147, 1998.
[4] V. A. Bolotin, “Modeling call holding time distributions for CCS network design and performance
analysis,” IEEE J. Select. Areas Commun., vol. 12, no. 3, pp. 433-438, 1994.
[5] W-E Chen, H. Hung, Y. Lin, “Modeling VoIP Call Holding Times for Telecom,” IEEE Network, vol.
21, no. 6, pp. 22-28, 2007
[6] Y. Fang and I. Chlamtic, “Teletraffic Analysis and Mobility Modeling of the PCS networks,” IEEE
Trans. on Communications, vol. 47, no.7, July 1999.

18
[7] C. Jedrzycky, V. C. M. Leung, “Probability distribution of channel holding time in cellular telephony
system,” in Proc. IEEE Vech. Technol. Conf., Atlanta, GA, May. 1996, pp. 247-251.
[8] J. Jordan, F. Barcelo, “Statistical of Channel Occupancy in Trunked PAMR systems,” Teletraffic
Contributions for the Information Age (ITC’15), Elsevier Science, pp. 1169-1178, 1997.
[9] D. Sharp, N. Cackov., and L.Trajkovic, “Analysis of Public Safety Traffic on Trunked Land Mobile
Radio Systems,” IEEE J. Select. Areas Commun., vol. 22, no.7, pp. 1197-1202, 2004.
[10] F. Barcelò, J. Jordan, “Channel Holding Time Distribution in Public Telephony System (PAMR and
PCS),” IEEE Transaction on Vechular Technology, vol. 49, no.5, pp.1615-1625, 2000.
[11] B. Vujicic, H. Chen, and Lj. Trajkovic, "Prediction of traffic in a public safety network," in Proc.
IEEE Int. Symp. Circuits and Systems, Kos, Greece, May 2006, pp. 2637-2640.
[12] E. Chelbus, “Empirical validation of call holding time distribution in cellular communication
systems,” Teletraffic Contributions of the Information Age, Elsevier Science, pp.1179 -1189, 1997.
[13] L. Brown, N. Gans., and L. Zhao, “Statistical Analysis of a Telephone Call Center: A Queueing -
Science Perspective,” Journal of the American Statistical Association,vol. 100, no. 469, March.
2005.
[14] Dempster AP, Laird NM, Rubin DB, “Maximum Likelihood from Incomplete Data Via the EM
Algorithm,” Journal of the Royal Statistical Society, no. 39. vol.1, pp.1-38, 1977.
[15] T. Benaglia, D. Chauveau, DR. Hunter, and DS. Young, “Mixtools: An R package for analyzing
finite mixture models,” Journal of Statistical Software, vol. 32, no. 6, pp.1-29, 2009.
[16] G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions, 2nd ed, John Wiley and Sons,
New York, 2008.
[17] I. Tang, X. Wei, “Existence of maximum likelihood estimation for Three-parameter lognormal
Distribution, reliability and safety,” in Proc. 8th International Conference ICRMS, 20-24 July, 2009.
[18] L. Eckhard, A. Werner, A. Markus, “Log-normal Distributions across the Sciences: Keys and Clues,”
BioScience, vol. 51, no. 5, pp. 341-351, May. 2001.
[19] R. Vernic, S. Teodorescu, E. Pelican, “Two Lognormal Models for Real Data,” Annals of Ovidius
University, Series Mathematics, vol. 17, no. 3, pp. 263-277, 2009.
[20] J.A.Bilmes, “A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for
Gaussian Mixture and Hiden Markov Models,” International Computer Science Institute, Berkeley,
California, April, 1998.
Author
Ms. Tuyatsetseg Badarch received both a Bachelor and Master’s degree from the Mongolian Science
and Technology University in 1996 and 1998, respectively as well as a postgraduate degree from the
Space Science and Technology Center in India in 2005. She received prestigious government scholarships
such as Mongolia, China, India, Taiwan, as well as USA government Fulbright S&T PhD scholarship.
She studied as a researcher Beijing University of Post and Telecommunications in China from 1998 to
2000 and served as researcher at National Tsinhua University in Taiwan from 2007 to 2008. She worked
as lecturer and senior lecturer at the leading universities of Mongolia: Mongolian University of Science
and Technology (MUST) and National University of Mongolia (NUM) from 1996 to now, and she is with
the Department of Electrical and Computer Engineering, Northeastern University, USA, on leave from
the NUM. She is focused on the modeling network traffic, traffic distribution phenomena, network
resource allocation, advanced computational analysis and performance.
Dr. Otgonbayar Bataa is Professor and Head of the Department of Wireless Communications, School
of Information and Communications Technology (SICT), Mongolian University of Science and
Technology (MUST), Mongolia. She is the author of over many journals, as well as numerous proceeding
papers of highly ranked international conferences, technical papers, and projects. Her research focuses on
the performance analysis and design of advanced mobile and wireless systems, as well as network traffic
models.

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