Abstract
The sum of random variables are of interest in many areas of the sciences. In teletraffic analysis, the sum of Hyperexponential distribution is used as a model for the holding time distribution. Many authors examined this model and discussed its probability density function. In this paper, we consider the sum of independent Hyper-Erlang distributions. We showed that the probability density function of this distribution is related to probability density function of the sum of independent Erlang distributionsthe Hypoexponential distribution. As a consequence, we find an exact closed expressions for the probability density function of both distribution, which are related to the Kummer confluent hypergeometric function. AMS Subject Classification: 62E15, 60E10, 60E05
Highlights
The sum of Random Variable (RV) or the convolution of RV plays an important role in modeling many events, see Ross [20] and Feller [9], as communications, computer science, teletraffic engineering (Trivedi [24], Jasiulewicz and Kordecki [12], Fang and Chlamtac [8]), Markov process (Kadri et al [16], Jasiulewicz and Kordecki [12]), and reliability and performance evaluation (Trivedi [24], Jasiulewicz and Kordecki [12] and Bolsh [4])
Many authors showed that call holding time, cell residence time and channel holding time are no longer exponentially distributed, see
Zeng and Chlamtac [25] and Zeng et al [26], Generalized Gamma distribution used by Zonoozi and Dassanayake [27] and Zonoozi et al [28], Lognormal distribution by Jedrzycki and Leung [13], Erlang distribution by Fang et al [7] and Glisic and Lorenzo ([10], Ch6. ”Teletraffic Modeling and Analysis”), Hyper-Erlang distribution by Fang and Chlamtac [8], Fang et al [7], and Glisic and Lorenzo [10], Hyperexponential distribution by Fang et al [7], Coxian distribution by Soong and Barria [23], the sum of the Hyper-Exponential distribution (SOHYP) by Orlik and Rappaport, [18] and [19]
Summary
The sum of Random Variable (RV) or the convolution of RV plays an important role in modeling many events, see Ross [20] and Feller [9], as communications, computer science, teletraffic engineering (Trivedi [24], Jasiulewicz and Kordecki [12], Fang and Chlamtac [8]), Markov process (Kadri et al [16], Jasiulewicz and Kordecki [12]), and reliability and performance evaluation (Trivedi [24], Jasiulewicz and Kordecki [12] and Bolsh [4]). We may consider a more general appropriate distribution in traffic theory is the sum of independent Hyper-Erlang distribution. This distribution was not examined previously by any author before. Its particular case, the sum of the Hyper-Exponential distribution (SOHYP), was proposed by some authors as the most appropriate distribution for call holding time as Orlik and Rappaport, [18] and Fang, [6]. We present the PDF of this distribution as a linear combination of the PDF of a special case of Hypoexponential random variable, the sum of two independent Erlang distributions.
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