A Model to Simulate the Movement and Behavior of Hajjis

Abstract

A mathematical model that describes the behavior of hajjis (pilgrims) while throwing jamarat has been developed. A two-dimensional probabilistic approach of the Monte Carlo method was used to sample from discrete probability distributions or probability density functions (pdfs) that describe the real behavior of hajjis in their approach towards the area, while throwing, and on the way out. Movements of hajjis (random walks) were traced out in a two dimensional (2-D) plane step by step using standard Monte Carlo procedures. The model takes into account personal traits such as gender and abilities as well as architectural features of the area. Although the geometric configuration focuses on a single pillar (target to be stoned) i.e. one-pillar module; the model is general and could be applied contiguously to model all three pillars by changing event probabilities and pdfs. To further illustrate this point, concatenation of the model for three pillars could be accomplished by setting appropriate values for the probabilities of same-direction returns (preturn) and the probabilities of forward-movement orientations (pfrwd/right , pfrwd/left , and pfrwd/straight). Therefore, for the first pillar module, the probability of same-direction return is set to zero (preturn = 0) and the forward movement probabilities are set to pfrwd/right = 0 , pfrwd/left = 0, and pfrwd/straight = 1; followed by a module (could be of different geometrical dimensions) for the second pillar with preturn = 0 and pfrwd/right = 0 , pfrwd/left = 0, pfrwd/straight = 1; concluding by a module for the third pillar with preturn = optional and pfrwd/right = optional , pfrwd/left = optional, pfrwd/straight = optional. These optional probabilities are explained elsewhere in the manuscript. The time distribution of arrival of hajjis at the beginning of the first pillar's area could be estimated or obtained from field observations. The time distribution of arrival at the beginning of the next pillar's area is taken to be the time distribution of crossing a finish line after the preceding pillar and so forth. The finish time distribution is an output within the computer code.