Convolutions of Hyper-Erlang and of Erlang Distributions

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 distributions-the 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.

Performance Analysis of Cooperative Low-Power Wide-Area Network for Energy-Efficient B5G Systems

“…Outage Probability Figure 3 shows the BER performances on the same system as the simulation environment in Figure 2. The analyzed BER performances in this paper, Equation (13) and (15), are verified to match well with the simulation results by Monte-Carlo. The detailed interpretation of the results is omitted because it is the same as the interpretation of the result of outage probability.…”

“…the distribution of the effective channel gain Z is the same as the distribution of the sum of N non-identical and independent random variables, which follows Erlang distribution. It is known as hyper-Erlang distribution [15] which is the mixture of N mutually independent Erlang distribution and parallel N-phase Erlang distribution weighted with probabilities. Therefore, the p.d.f.…”

“…which is used to verify the tendency of BER performance; i.e., the slope of BER curve, in high SNR regime [17]. The diversity order is obtained through Taylor series expansion for (15), of which the first-order coefficient shows how the system parameters affect the performance in high SNR regime; e.g., the distances between ID and GWs, the number of antennas, pass-loss factor, and so on. An asymptotic expression is given by…”

“…The hyper-Erlang distribution is the distribution of the sum of two or more independent Erlang random variables. In [15], the p.d.f. of hyper-Erlang distribution is given as…”

“…This paper aims to exactly address the error performance analysis of the cooperative uplink LPWAN with multiple antennas GWs. The exact closed-form expressions for the performance analysis are derived using the exact channel distribution information (CDI) ( hyper-Erlang distribution) [15]. LPWAN is basically based on the low power transmission.…”