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Modeling Call Holding Times Of Public Safety Network
 

Modeling Call Holding Times Of Public Safety Network

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International Journal on Computational Sciences & Applications (IJCSA)

International Journal on Computational Sciences & Applications (IJCSA)

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    Modeling Call Holding Times Of Public Safety Network Modeling Call Holding Times Of Public Safety Network Document Transcript

    • 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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 un- observed 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 .
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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)
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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:
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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 Holding time in seconds 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 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 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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 Holding time in seconds CDF Weibull model Empirical data LogNormal-5 model LogNormal model 0 10 20 30 40 50 60 -6 -5 -4 -3 -2 -1 0 Holding time in seconds 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 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 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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 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 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)).
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.
    • International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 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.