some comments. First, one can not constrain con-
currency by sampling start of operations via realiz-
ing a Poisson process and determine their ends by
sampling the appropriate duration distribution (here
time is not expanded, rather operations are overlaid
on the time axis). This approximation can be suitable
(see, for example, (Fee and Caron, 2021) or (Couil-
lard et al., 2015), where this approach is used in a
military context, and the references within) and does
not suffer from frequency reduction, but, in general,
we can have an unbound number of concurrent op-
erations if we use this approach and so peak concur-
rency can be unrealistic. Second, using the ﬁnite in-
terval process approach allows us to simulate a man-
date to have C concurrent operations by considering C
parallel independent realizations, where each realiza-
tion has appropriately scaled frequencies ( f
i
→ f
i
/C ).
The additive property of Poisson processes will led to
the correct overall frequency for each operation type
when we take the C parallel realizations in combina-
tion, and we see that by considering C parallel realiza-
tions we allow up to C concurrent operations. More
generally, we may have lines of operation, each of
which with relevant operation types and concurrency
requirements and we would break the simulated time
line out to determine the overall realization (for exam-
ple, SSE mandates two major sustained deployments,
one major time-limited deployment, two minor sus-
tained deployments, and two minor time-limited de-
ployments, as well as other requirements (Department
of National Defence, 2017)). It is noteworthy that
FSRA+ does account for these lines of operations and
concurrency requirements, as well as other complex-
ities such as ongoing constant demands and various
constraints on Force Elements (Dobias et al., 2019).
FSRA+ speaks to the capacity and capability of
the CAF, and breaks down operations by Force Ele-
ments and stafﬁng of those Force Elements. For this
reason FSRA+ lives on a classiﬁed network. The op-
eration variants and scheduling parameterizations are
not secret, as the underlying information is available
in the public domain, and so schedules themselves are
not classiﬁed. One motivation of this work was to
work towards a light weight operation schedule gener-
ator which could be used off the classiﬁed network, to
facilitate analysis (e.g., in the study (Serr
´
e, 2021) only
schedules output by FSRA+ where used and analysis
was done off the classiﬁed network). Due to suitabil-
ity for development and analysis Python was selected
as a language, which makes time stepping computa-
tionally expensive, which was a pragmatic rationale
for considering continuous time processes (however,
see the Appendix for a discrete time approach that is
viable in Python). On the formal side, making con-
tact with established theory and basic principles was
a key motivator and rationale for considering con-
tinuous time. The results here lay the ground work
for a more mature implementation that can simulate
CAF operation schedules, taking line of operations
and concurrency into account.
While we have focused on scheduling of military
operations, as this was the application in mind, un-
scheduled maintenance and other such applications
can be addressed with the same approach. For ex-
ample, unplanned air craft maintenance schedules can
be generated, given a mixture of maintenance types
(“operation types”), their frequency, and the number
of aircraft (“concurrency”), and with these schedules
resource use can be determined.
Two key aspects have been established here. First,
we provide an independent validation of the FSRA+
approach by making contact with stochastic processes
and, second, in doing so provide a scheme to ensure
input frequencies are recovered. This is important as
it simpliﬁes interpretation and as otherwise input fre-
quencies are biased downwards in an opaque manner,
which likewise reduces demands. As a consequence
demands will be lower than suggested by the input
frequencies which implies we will plan for an artiﬁ-
cially small demand.
6 CONCLUSION
Here we extended Poisson point processes to ﬁnite
interval processes, for simulating military operations
over time. The work was motivated by a prag-
matic tool in use by the CAF, FSRA+, which uses
a discrete time approach. We outlined the contin-
uous time approach, characterized the frequency re-
duction whereby input frequencies are reduced when
points are extended to intervals, provided a correction
scheme to address this, provided a scheme to ensure
ergodicity (i.e., eliminate initial condition artifacts,
without the need for burn-in), and subjected the ap-
proach to essential numerical tests. We have also pro-
vided a qualiﬁed veriﬁcation of FSRA+’s pool pro-
cess to initiate operations, demonstrating there will be
a bias—but of small (negligible) magnitude. We note
that the additive property of Poisson point processes is
fundamental in allowing the development and under-
standing of the interval process described here, for ex-
ample allowing concurrency to be cleanly modelled,
which recommends consideration of this approach to
model mixtures of military operations with possible
concurrency mandates.
ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems
362