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For the DES, the density of mentees must be cal-
culated from the mentees present in the system over
time and their respective progression through their
training. Since mentees progress through the medium
at a constant rate, the location of a mentee in the
medium at a point in time can be inferred from their
training time (initial mentees) or arrival time into the
system (added mentees). To generate a surface plot
of mentee densities, mentees were binned together in
time and medium intervals. The bin in time was cho-
sen to be an interval of 1/12 time units, potentially
simulating a progression through the medium once
every month. The bin in the medium was chosen to be
1/60, matching the time interval so a mentee would
progress forward by one bin in the medium every time
interval.
4.3.1 Surface Solution of the Mentee Population
Across the Medium
Fig. 3 shows the surface plot of densities for the an-
alytical PDE and DES with the benchmark scenario.
The simulation was run for 1000 experiments. From
the analytical solution, Fig. 3a and Fig. 3b, the den-
sity of mentees through the medium jumps from 25 to
50 after the contour l = bt. After 5 (1/b) time units,
the entire medium is populated at the final density.
From (Novozhilov, 2020; Ko, 2020), the trans-
port PDE models a constant progression through the
medium at a rate of b. This gives straight-line con-
tours with a slope of b, as seen from the top-down
view in Fig. 3b. Therefore, in the transport model
each mentee has the same training time of 1/b. This
is different from the two-level and multi-stage models
given in Section 2 and Section 3, in which the training
time for mentees was exponentially distributed. This
constant training time between the mentees arises be-
cause as the number of stages is increased to infinity
(n → ∞), the average time a mentee spends in each
stage approaches zero. For an average time of zero
in each stage, there must be no variance between the
training times for each mentee in the system, giving
the same training time for each. Finally, the loca-
tion of a mentee along the medium denotes how far
along they have progressed with their training. This
improves the tracking of mentees through their train-
ing as now a continuous medium is used to denote
their progression.
For the simulation output, Fig. 3c and Fig. 3d,
the initial density of mentees across the medium is
even around 25, and the final density is even around
50. The surface plot also demonstrates nearly identi-
cal behaviour to the analytical solution, in which the
spike in density up to 50 occurs along the contour:
l = bt. From the top-down view, mentees progress
through the medium at a constant rate of b, like in
the analytical model. Given the similarities observed
between the analytical and discrete outputs, it is be-
lieved that the two versions of the transport model are
equivalent.
4.3.2 Solution of the Total Mentee and Mentor
Populations
The solutions for the total mentee and mentor pop-
ulations for the benchmark scenario are plotted in
Fig. 4. The analytical solutions given by Eqn. (18)
and Eqn. (19) are plotted as solid lines. The statistics
of the total populations from the DES model are also
plotted. The mean of the two populations is given
by the dashed line, and the shaded region gives two
standard deviations of the results. The mentee pop-
ulation linearly increases from the initial population
of 25 mentees to the final population of 50. After 5
(1/b) time units, all initial mentees have upgraded to
mentors and the mentees now populate the system at
the final density. Therefore, the mentee population
reaches steady state at this time. The mentor popula-
tion still undergoes its exponential decay at a rate of c.
However, the decay only begins once the mentee pop-
ulation reaches steady state and the incoming mentors
are at the final density. The analytical solution and the
mean of the simulation results of the two populations
lie on top of each other, are within two standard de-
viations of the simulated results, and their standard
errors were found to be identical to the values from
the previous models (0.225 and 0.45 for the mentee
and mentor populations respectively). This gives fur-
ther confirmation that the two versions of the transport
model are equivalent and the discrete model was set
up correctly.
5 COMPARISON BETWEEN THE
THREE MODELS
Fig. 5 shows the analytical solutions of the two-level
(one mentee stage), five- and 25- mentee stage, and
transport models plotted together with the benchmark
scenario for comparison.
From Fig. 5a, as the number of mentee stages is
increased, the decay rate of the final population in-
creases. The transport model, shown in blue, is the
limit case, in which there is an infinite decay rate and
simply reaches the final population at 5 (1/b) time
units. From Fig. 5b, the exponential rise into the de-
cay to steady state for the mentor population also be-
comes sharper as the number of stages is increased.
Due to the varying sharpness in the rise, all solutions
ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems
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