An Embedded Markov Chain Modeling Method for Movement-Based Location Update Scheme

Abstract-In this paper, an embedded Markov chain model is proposed to analyze the signaling cost of the Movement-Based Location Update (MBLU) scheme under which a Location Update (LU) occurs whenever the number of cells crossed reaches a threshold, called movement threshold. Compared with existing literature, this paper has the following advantages. 1) This paper proposes an embedded Markov chain model in which the cell residence time follows Hyper-Erlang distribution. 2) This paper considers the Location Area (LA) architecture. 3) This paper emphasize the dependency between the cell and LA residence times using a fluid flow model. Close-form expressions for the signaling cost produced by LU and paging operations are derived, and their accuracy is validated by simulation. Based on the derived analytical expressions, we conduct numerical studies to investigate the impact of diverse parameters on the signaling cost

Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm

Erlang distribution is a particular case of the gamma distribution and is often used in modeling queues, traffic congestion in wireless sensor networks, cell residence duration and finding the optimal queueing model to reduce the probability of blocking. The application is limited because of the unavailability of closed-form expression for the quantile (inverse cumulative distribution) function of the distribution. The problem is primarily tackled using approximation since the inversion method cannot be applied. This paper extended a six parameter quantile model earlier proposed to the Nakagami distribution to the Erlang distributions. Consequently, the established relationship between the two distributions is now extended to their quantile functions. The quantile model was used to fit the machine (R software) values with their corresponding quartiles in two ways. Firstly, artificial neural network (ANN) was used to establish that a curve fitting can be achieved. Lastly, differential evolution (DE) algorithm was used to minimize the errors obtained from the curve fitting and hence estimate the values of the six parameters of the quantile model that will ensure the best possible fit, for different values of the parameters that characterize Erlang distribution. Hence, the problem is constrained optimization in nature and the DE algorithm was able to find the different values of the parameters of the quantile model. The simulation result corroborates theoretical findings. The work is a welcome result for the quest for a universal quantile model that can be applied to different distributions