Introduction:
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.
The probabilistic approach in the model:
For practicality, an illustrative example might be useful. A
Monte Carlo history of a random walk is followed from
"beginning" to "end". A hajji begins to perform the ritual of
stoning by proceeding towards a global direction leading to a
pillar's area. A hajji for instance, could be a male, a female,
alone, within a group of two or more, of particular physical
traits and abilities ( body size, walking speed, range of
throwing), or throwing on behalf of other hajjis . Furthermore,
the time of arrival has to be determined. All such parameters
are to be Monte Carlo sampled from relevant pdfs that had been
obtained from field observations or had been estimated
properly. At a point of approach when the pillar is within
sight, a choice is to be made whether to globally approach the
pillar directly with papproach-DIR
probability, from right-side with papproach-RS
probability, or from left-side with papproach-LS
probability. The probabilities of global approach are estimated
from field observations, and are influenced by religious
believes, intentional crowd management regulations, or even
conveniences such as shaded areas. Traversing the distance to
destination (immediate vicinity around the pillar's enclosure)
the hajji might change local directions of movement due to
pedestrian congestions, i.e. traversing in a zigzag path while
maintaining the global direction of approach. Each change of
local direction and steps between such changes are sampled and
tracked in 2-D following standard Monte Carlo techniques. The
zigzag pattern is a function of crowd density (hajji per unit
area) at the given locality. The closer to the vicinity
immediately surrounding the pillar the higher the crowd density
will be. Henceforth, the hajji is subject to being randomly
pushed around (will forcefully resist pushes that result in
deviations from destination orientation but follows along for
favorable pushes). The act persists until a predetermined
personal circle corresponding to a personal throwing range is
arrived at. The hajji commences throwing one stone at a time
(needs to hold still for t
seconds to do so otherwise a push-around result in a change of
the hajji's position and orientation) and continues to throw as
long as the hajji is within the personal circle. The dwell time
within the personal circle depends on how many push-arounds the
hajji is subjected to and on the number of stones to be thrown;
which in turn depends on whether the hajji is throwing on behalf
of somebody else (wakala) besides self. Going out is treated
similarly to in-going with the exception that the hajji's
preferred out-going direction is radial until out of the heavily
crowded area. The history is carried on such that the hajji
either foregoes doaa with a (1-pdoaa)
probability or proceeds to perform doaa with a pdoaa
probability. The possibility of going back in the same
direction has to be sampled with a pbkwd
probability followed by sampling from a discrete pdf of pbkwd/RS
or pbkwd/LS. In
the event that the hajji happens to be moving in the forward
direction wanting to perform doaa; the global direction will be
the direction of the designated area for doaa and the movement
is characterized by a zigzag path until arriving at the doaa
area at which an arbitrary (estimated from field observations)
dwell time is tallied. On the other hand if doaa is to be
foregone but forward direction is sought or forward direction is
intended after doaa; the direction is sampled from a discrete
pdf comprised of pfrwd/right
, pfrwd/left ,
pfrwd/straight
and the movement is followed until crossing a pre-determined
line at the end of the area and the history is considered to be
ended. The foregoing briefing illustrates that a high degree of
physical realism had been retained in the mathematical
formulation of the problem.
When statistically significant number of histories
is followed; information about expected behaviors or averaged
functionals could be deduced and used in further analysis. For
instance, analyses of crowd densities and dwell times as well as
tallied parameters of interest could be used to draw conclusions
about local pedestrian congestions for regions of interest.
Implementation into a computer code:
The mathematics and cumbersome logics of the
preceeding model were implemented into a computer code. Quick
Basicä language was
adopted for its graphics capabilities and ease of debugging.
The input to the code contains deterministic and probabilistic
information. The generality of the code is exemplified in the
input data. The geometry of the surrounding area and enclosures
of pillars are defined by global coordinate points and
dimensions. The event probabilities and the pdfs are either
estimated from field observations or factually known. Thus, for
a given set of input conditions, the output of the code provides
quantitative measures that could be analyzed to draw conclusions
about the behavior of the real situation. Moreover, by
specifying different sets of input decision conditions and
running the model repeatedly, one can predict fairly accurately
the expected response of the real situation to changes in
various model parameters. A demo-version of the code is
demonstrated during the presentation of this manuscript.
Conclusions:
When a model that accurately describes a system is
evaluated numerically by a digital device (a computer); it
constitutes a powerful versatile tool to facilitate meaningful
studies. Thereby, the code could be utilized to predict
localities of over-crowdedness, to estimate capacities
conforming to predetermined conditions that deemed to be
acceptable congestions, and most of all the code could be used
as a simulator to try out suggested modifications and
alterations within the area or on the event probabilities and
distribution functions (e.g. time distribution of arrival). The
results and consequences of such alterations could be predicted
without having to go through the physical in-situ alterations.
Henceforth, ineffective alterations could be avoided without
high prices to pay. Finally, the stochastic modeling scheme
could be utilized to model areas of crowdedness such as ALMATAF
and ALSAFA WA ALMARWA to render improved services to visitors of
the Holly City of Makkah.
References:
Almasoumi, A., Monte Carlo Model of
a Capture Gamma Ray Analyzer for a Seafloor Core Sample, Ph.D.
Dissertation, Oregon State University (1989).
Cashwell, E.D. and C.J. Everett,
A Practical Manual on the Monte Carlo Method for Random Walk
Problems, Pergamon Press, Inc., New York (1959).
نداء القاضي ، دراسة الجمرات ، مركز
أبحاث الحج , جامعة أم القرى
(1410)
Payne, J.A., Introduction to Simulation, Programming
Techniques and Methods of Analysis, McGraw-Hill Book
Company, New York (1982).
