Uncertainty in the Demand for Service - Faculty of Industrial ...

4.2 Future Work
• Updating the existing simulation software - our first step will be generalizing the
simulation code developed by Feldman [4] with the features of random arrival rate. We
will first check the results of the basic stationary Erlang-A model, and after that, we will
consider the expansion of the algorithm to the time varying Erlang-A model in the random
arrival rate case.
• Validate and understand theoretically Figure 4 - this step entails two things :
1. Establish the square-root staffing rule by comparing simulation results for random
arrival rate with theoretical results as in Figure 4, which are based on the square-root
staffing rule (3).
2. Understand theoretically the functions shape. Important and interesting things to
understand from the graphs are the intersection points of each function with the Y
axis (the case of β = 0, namely N = R), with the Garnett function and with the other
functions. In additional, we would like to understand the changes of the functions
from convexity to concavity, and the values of β for which α is stabilized.
• Arrival rate as a continuous random variable - we shall generalize our results to
arrival rates with densities. For example, we may let the prior distribution of Λ be Gamma
which is the conjugate distribution of the Poisson distribution. In other words, if we
sample Λ we get a posterior distribution which is another Gamma distribution with a
simple change of the parameters. Another continuous distribution which may be taken
into consideration is the Normal distribution, which is a natural choice when one has no
detailed analysis of the distribution of Λ. In doing so, we should assume that the standard
deviation of Λ is relatively small compared to its mean, so that negative values would occur
with only a negligible probability. However, the complexity of working with the normal
distribution must be taken into consideration.
• Theoretical results - motivated by the work of Halfin - Whitt [7] and Garnett [5], the
next step will involve developing a parallel theorem to those which are described in Section
2, in the case where the arrival rate is a random variable. For the M/G/N model (Poisson
arrivals, general service time distribution and N servers) there is, currently, no Halfin-
Whitt-like theorem. Hence, instead of mathematical analysis we shall use simulation, which
is applied to this model, in order to obtain similar results to the ones of Halfin-Whitt and
Garnett. Moreover, we shall consider approximations of performance measures that yield
insight as to their dependence on the model’s parameters. Part of these approximations are
diffusion approximations for stochastic processes, such as the process Q = {Q(t), t ≥ 0},
where Q(t) is the number of customer present in the system at time t. This kind of
approximation enables one to analyze the processes beyond steady-state.
• Considering other performance measures - we shall try to determine staffing levels
that will ensure time-stability of performance measures other than the delay probability,
for example the expected waiting time and the probability of abandonment.
• Optimization problems - we shall try to find staffing rules that, when applied, provide
minimum costs or maximum profit.
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