Software Complexes, Numerical Techniques, and Mathematical
Modeling
Sunatova Dilfuza Abatovna
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471397
Keywords:
optimization, rescheduling, modeling, intensity, martingale, and just-in-time.
Abstract:
This article examines the models of basic Just-In-Time (JIT) systems using point processes in reverse time. This
method permits certain presumptions regarding the workings of actual systems. We thus formulate and solve a few
very basic optimal control problems for a system with bounded intensity and for a multi-stage just-in-time system.
For the objective functions, results are computed as expected linear or quadratic forms of the trajectories' deviations
from the intended values. The statements' proofs employ the martingale method. In logistics tasks, just-in-time
systems are frequently taken into consideration, and only (or mostly) deterministic methods are used to describe them.
Nonetheless, it is evident that stochastic events are frequently observed in these systems and the related processes.
It is crucial to identify strategies for the best just-in-time process management in these kinds of stochastic situations.
In this paper, we propose to use martingale methods for this description. Here, straightforward methods for stochastic
JIT process optimization are shown.
I. Introduction.
In this article, we consider some stochastic models of
simple just-in-time systems. The well-known principle
of just-in-time system (abbreviated as JIT system) is
used in many areas. Examples include just-in-time
production systems, pedagogical strategies of just-in-
time teaching, and just-in-time compilation methods in
computer programming. It should be noted that at
present mathematical, especially stochastic, models
for JIT systems are not sufficiently developed. Such
models are necessary for solving optimal control
problems, which could allow optimizing the allocation
of system resources and implementing optimal
planning of a stochastic JIT system. The purpose of
this article is to present an approach to the stochastic
description of JIT systems, which would be suitable
for both analytical methods and computer simulation.
Mathematical models of such systems should allow
assuming that the trajectories of processes must take
the given values at a fixed time. Such behavior of
processes is known in stochastic bridges and stochastic
processes in the reverse time. Thus, we should
consider models of systems with the requirement of
JIT in terms of processes with the behavior of
trajectories close to stochastic bridges. Models should
also allow investigating possible violations of this
requirement that are unavoidable for real systems.
Stochastic process time reversals have been the subject
of research for many years. We observe that the study
of these processes is the focus of several works
pertaining to stochastic bridges. Furthermore, some
studies on reversible Markov processes also adjoin
process descriptions in the opposite direction. In this
paper, we examine semi martingale models of
elementary JIT systems for point processes near the
previously mentioned Poisson bridge. Here, we'll
grant some suppositions regarding the workings of
actual systems. Thus, a system with bounded intensity
and simple cases of multi-stage JIT systems are
examined. As demonstrated, it is possible to formulate
and solve basic optimal control problems for these
situations. The semi martingale technique is used in
the results proofs.
II. Materials and Methods
A basic JIT system's time reversal technique.
Think
about a Just-In-Time (JIT) system that can be
explained using point processes, such as counting. We
assume that, starting from the zero moment, a fixed
time
𝑇
> 0 must be met by an integer number
𝐾
of
operations within the system. This indicates that for
every time t
∈
[0,
𝑇
], the number of operations left, tr,
is equal to the number p minus the value p_t of a
counting process, t = (p_t)_t>0: tr = p_t − p_t.
We now give a formal description of the mathematical
model. Let (Ω, F, P) be a probability space populated
with a nondecreasing right-continuous family of
𝜎
-
algebras F = (F
𝑡
)
𝑡
>0, complete with respect to P (i.e.,
the conditions of [13] hold). On the stochastic basis B
= (Ω, F, F = (F
𝑡
)
𝑡
>0, P) the process
𝑋
= (
𝑋𝑡
)
𝑡
>0 is
supposed to be the point process with trajectories in
the Skorokhod space,
𝑋𝑡
∈
N0 = {0, 1, 2, . . . } and
∆
𝑋𝑡
=
𝑋𝑡
−
𝑋𝑡
−
∈
{−1, 0}
The process
𝑋
can be represented as a difference:
𝑋
=
𝑋
0 −
𝑁
=
𝐾
−
𝑁
, where
𝑁
= (
𝑁𝑡
)
𝑡
>0 is the counting
process of the number of negative jumps of
𝑋
, with the
initial value
𝑋
0 =
𝐾
> 0 (i.e.,
𝐾
∈
N = {1, 2, . . . },
𝑁
0
= 0, and
𝑋𝑡
=
𝐾
−
𝑁𝑡
, for all
𝑡
> 0).
We suppose that the submartingale
𝑁
and
supermartingale
𝑋
on B admit the well-known Doob–
Meyer decompositions (see, e.g., [13]):
𝑁𝑡
=
𝑁
˜
𝑡
+
𝑚𝑁
𝑡
,
𝑋𝑡
=
𝑋
˜
𝑡
−
𝑚𝑁
𝑡
(1) with the
compensators
𝑁
˜ = (
𝑁
˜
𝑡
)
𝑡
>0 and
𝑋
˜ = (
𝑋
˜
𝑡
)
𝑡
>0,
and the square-integrable martingale
𝑚𝑁
= (
𝑚𝑁
𝑡
)
𝑡
>0 with the quadratic characteristic
⟨𝑚𝑁
⟩
𝑡
=
𝑁
˜
𝑡
for all
𝑡
> 0.
We also suppose in this article that
𝑁
˜
𝑡
= ∫︁
𝑡
0 (
𝐾
−
𝑁𝑠
) · 1
𝑇
−
𝑠
· I {
𝑠
<
𝑡
}
𝑑𝑠
, (2) where I{·} is an
indicator function (i.e., I{true} = 1, I{false} = 0). From
(1) and (2) it follows that the process
𝑋
has the
decomposition:
𝑋𝑡
=
𝐾
− ∫︁
𝑡
0
𝑋𝑠
· 1
𝑇
−
𝑠
· I{
𝑠
<
𝑡
}
𝑑𝑠
−
𝑚𝑁
𝑡
. (3)
In the general case, for the basic model we assume that
the point process
𝑋
admits the representation:
𝑋𝑡
=
𝐾
− ∫︁
𝑡
0 ℎ
𝑠
𝑑𝑠
+
𝑚𝑋𝑡
.
With the intensity of negative jumps ℎ = ℎ(
𝑋
) =
(ℎ
𝑡
(
𝑋
))
𝑡
>0 and the martingale
𝑚𝑋
= (
𝑚𝑋
𝑡
)
𝑡
>0. In the
particular case (3), the following equality holds:
ℎ
𝑡
= ℎ
𝑡
(
𝑋
) =
𝑋𝑡
· I{
𝑠
<
𝑡
}/(
𝑇
−
𝑠
), (5) and
𝑚𝑋
= −
𝑚𝑁
, i.e.
𝑚𝑋
𝑡
= −
𝑚𝑁
𝑡
for all
𝑡
> 0.
It is well known that the compensator of the point
process defined by formula (2) corresponds to the
bridge of a Poisson process.
Consider a standard Poisson process
𝜋
= (
𝜋𝑠
)
𝑠∈
[0,
𝑇
]
on the stochastic basis B with the initial value
𝜋
0 = 0
and any positive intensity
𝜆
> 0. Let F0
𝑡
=
𝜎
{
𝜋𝑠
:
𝑇
−
𝑡
6
𝑠
6
𝑇
} for
𝑡
∈
[0,
𝑇
], F0
𝑡
= F0
𝑇
for
𝑡
>
𝑇
, and
nondecreasing family
𝜎
-algebras F = (F
𝑡
)
𝑡
>0 be the
right continuous completion of (F0
𝑡
)
𝑡
>0.
Define the reverse time supermartingale
𝑌
= (
𝑌𝑡
)
𝑡
>0
as
𝑌
=
𝜋𝑇
−
𝑡
for
𝑡
∈
[0,
𝑇
] and
𝑌𝑡
=
𝜋
0 = 0 for
𝑡
>
𝑇
.
Then
𝑌
is F-adapted and it has the decomposition (as
it easily follows, from Theorem 2.6 in [8]):
𝑌𝑡
=
𝜋𝑇
−
∫︁
𝑡
0
𝑌𝑠
𝑇
−
𝑠
· I{
𝑠
<
𝑡
}
𝑑𝑠
+
𝑚𝑌
𝑡
, (6) where
𝑚𝑌
=
(
𝑚𝑌
𝑡
)
𝑡
>0 is a square-integrable martingale with the
quadratic characteristic
⟨𝑚𝑌
⟩
𝑡
= ∫︁
𝑡
0
𝑌𝑠
𝑇
−
𝑠
· I{
𝑠
<
𝑡
}
𝑑𝑠
.
The comparison of (3) and (6) illustrates the fact
known for bridge processes: the representation of the
process
𝑋
=
𝐾
−
𝑁
(with the initial value
𝐾
and the
Poisson bridge
𝑁
) coincides with the reverse time
representation
𝑌
of the Poisson process
𝜋
(with any
strictly positive intensity
𝜆
) under the condition for the
initial value
𝑌
0 =
𝜋𝑇
=
𝐾
. Thus, we can consider the
behavior of the trajectories of the process
𝑋
with
𝑋
0 =
𝐾
and
𝑋𝑡
= 0 for
𝑡
>
𝑇
as the embodiment of the just-
in-time requirement.
Therefore, the main idea of the presented description
of JIT systems is the realization of the corresponding
behavior of trajectories by means of proper control of
ℎ = (ℎ
𝑡
)
𝑡
>0, which is the intensity of the negative
jumps of
𝑋
in the base model (4). This intensity can be
regarded as a negative feedback tending to −∞ as
𝑡
→
𝑇
in the case of nonzero
𝑋𝑡
.
Note that in (6) it does not directly depend on the
intensity
𝜆
of the initial process
𝜋
. The distribution of
the main process
𝑋
in (4) is determined by the intensity
of the negative jumps ℎ, which in the particular case of
(5) depends on the values of
𝐾
and
𝑇
> 0. Along with
𝑋
, we define for the base model (4) the auxiliary
functions for E
𝑋𝑡
, E
𝑋
2
𝑡
and E(
𝑋𝑡
−
𝑅𝑡
) 2 =
𝐺𝑡
−
𝑅
2
𝑡
(i.e., for the mean, the second moment, and the
variance of
𝑋
, respectively).
For the functional ℎ = ℎ(
𝑋
) of general form in (4), and
the initial value
𝐾
, it is assumed that
𝑅𝑡
=
𝑅𝑡
(
𝐾
; ℎ) =
E
𝑋𝑡
,
𝐺𝑡
=
𝐺𝑡
(
𝐾
; ℎ) = E
𝑋
2
𝑡
,
𝑉𝑡
=
𝑉𝑡
(
𝐾
; ℎ) = E(
𝑋𝑡
−
𝑅𝑡
) 2 . (7)
In the particular case (5), these functions depend only
on the values of
𝑡
,
𝐾
, and
𝑇
. Therefore, for (5) we use
the notations:
𝑟𝑡
(
𝐾
;
𝑇
) =
𝑅𝑡
= E(
𝑋𝑡
/
𝑋
0 =
𝐾
;
𝑋𝑡
= 0),
(8)
𝑔𝑡
(
𝐾
;
𝑇
) =
𝐺𝑡
= E(
𝑋
2
𝑡
/
𝑋
0 =
𝐾
;
𝑋𝑡
= 0), (9)
𝑣𝑡
(
𝐾
;
𝑇
) =
𝑉𝑡
= E((
𝑋𝑡
−
𝑅𝑡
) 2 |
𝑋
0 =
𝐾
;
𝑋𝑡
= 0) =
𝑔𝑡
(
𝐾
;
𝑇
) −
𝑟𝑡
(
𝐾
;
𝑇
) 2 . (10)
For the functions (8), (9) and (10) defined for
𝑋
in (4)
with the intensity (5), we have
𝑟𝑡
(
𝐾
;
𝑇
) =
𝐾
·
𝑇
−
𝑡
𝑇
· I{
𝑡
6
𝑇
},
𝑔𝑡
(
𝐾
;
𝑇
) = (
𝐾
·
𝑇
−
𝑡
𝑇
)2 · /{
𝑡
6
𝑇
} +
𝐾
· (
𝑇
−
𝑡
) ·
𝑡
𝑇
2 · /{
𝑡
6
𝑇
}, (12)
𝑣𝑡
(
𝐾
;
𝑇
) =
𝐾
· (
𝑇
−
𝑡
) ·
𝑡
𝑇
2 · /{
𝑡
6
𝑇
}.
Problems of optimal planning for a multi-stage JIT
process
. Consider a model of simple multi-stage JIT
systems in terms of the proposed description. We
assume that it is a set of separate processes in reverse
time (or bridges of corresponding processes) with a
single aggregate plan. This section presents a simple
solution to the problem of the optimal times for
changing the stages for the model. In the cases
considered here, the mean-square deviations of the
trajectories from the planned values are minimized. In
addition, we consider the problem of optimal
rescheduling for the case of two stages and for its
multistage generalization. 2.1. Separate processes in
reverse time. Let us consider optimal control problem
for the following scheduling model. Let the execution
of (
𝐾
+ 1) operations in time
𝑇
be subdivided into
𝑛
∈
N stages: every successive
𝐾
(
𝑖
) operations must be
performed in stage
𝑖
, which lasts the time
𝜍
(
𝑖
), for all
𝑖
= 1, 2, . . . ,
𝑛
.
The following conditions for the time and number of
operations must be fulfilled:
∑︁
𝑛
𝑖
=1
𝜍
(
𝑖
) =
𝑇
, (14) ∑︁
𝑛
𝑖
=1
𝐾
(
𝑖
) =
𝐾
. (15)
We also define the condition for the uniformity of the
operations:
𝐾
(
𝑖
) =
𝐾
(
𝑖
) ·
𝜍
(
𝑖
)/
𝑇
for all
𝑖
= 1, 2, . . . ,
𝑛
.
Thus, the model of this JIT system is a set of separate
processes in reverse time (or of proper bridges).
Suppose that we must insure the uniform fulfillment of
the plan
𝜍
= {
𝜍
(1),
𝜍
(2), . . . ,
𝜍
(
𝑛
)} in the sense of ,
minimizing the weighted variance of the deviation
from it.
We consider the problem of finding an optimal plan
𝜍
* = {
𝜍
* (1),
𝜍
* (2), . . . ,
𝜍
* (
𝑛
)} for which Φ(
𝜍
*) = inf
𝜍
Φ(
𝜍
), (17) where the objective function Φ(
𝜍
) is the
sum of weighted variances (10) for the processes in (4)
with initial values
𝐾
(
𝑖
) and times of performance
𝜍
(
𝑖
),
𝑖
= 1, 2, . . . ,
𝑛
: Φ(
𝜍
) = ∑︁
𝑛
𝑖
=1
𝛼
(
𝑖
) · ∫︁
𝜍
(
𝑖
) 0
𝑣𝑡
(
𝐾
(
𝑖
),
𝜍
(
𝑖
))
𝑑𝑠
(18) under conditions (14) and (15), and for
strictly positive weights:
𝛼
(
𝑖
) > 0 for all
𝑖
= 1, 2, . . . ,
𝑛
.
Theorem 1
. For the plan that minimizes the objective
function Φ(
𝜍
),
𝜍
* (
𝑖
) =
𝑇
· {
𝛼
(
𝑖
) ·
𝑛
· ∑︁
𝑛
𝑗
=1 1/
𝛼
(
𝑗
)
}−1/2 for all
𝑖
= 1, 2, . . . ,
𝑛
. (20) Remark 1. Theorem
1 implies the trivial consequence that for equal weights
the equal times are optimal: for
𝛼
(1) =
𝛼
(2) = . . . =
𝛼
(
𝑛
) > 0,
𝜍
* (
𝑖
) =
𝑇
/
𝑛
for all
𝑖
= 1, 2, . . . ,
𝑛
. (21) 2.2.
The problem of optimal rescheduling for a two-stage
JIT process. As it follows from (21), for
𝑛
= 2, in the
case of equal weights, it then holds that
𝜍
* (1) =
𝜍
*
(2) =
𝑇
/2.
But in real systems, rescheduling—a process for
reviewing the plan while it's being implemented—
occurs in addition to a priori stage planning. The JIT
system's operations in this instance are carried out in
line with the process intensity in (3) for the planned
initial value
𝐍
and the planned time
𝑇
for the following
cases:
𝐡
∈
[0,
𝜎
],
𝜎
∈
[0,
𝑇
], where
𝜎
is the
rescheduling time. Thus, the initial plan with the
values of
𝐾
and
𝑇
is executed in the first stage for
𝑡
∈
[0,
𝜎
]. The subsequent re-planning process is put into
action at time
𝜏
. The time interval [
𝜎
,
𝑇
] is when the
second stage is completed. Here, following the
rescheduling, the new execution time (
𝑇
−
𝜎
) and the
starting value of the number of operations (
𝑋𝜎
) are set
in the interval [
𝜎
,
𝑇
] for the new process in reserve
time.
plan in the first stage and the deviation from the new
plan in the second stage. Thus, we consider the
problem of finding an optimal value
𝜎
* for which Ψ(
𝜎
* ) = inf
𝜎
Ψ(
𝜎
), (23) where the objective function
Ψ(
𝜎
) is the integrated variance (7) for the intensity ℎ =
ℎ(
𝑋
) is equal to Ψ(
𝜎
) = ∫︁
𝑇
0
𝑉𝑡
(
𝐾
; ℎ)
𝑑𝑠
. (24) Here the
intensity for the rescheduling is equal to ℎ
𝑡
(
𝑋
) = ℎ (1)
𝑡
(
𝑋
) · I{
𝑡
∈
[0,
𝜎
)} + ℎ (2)
𝑡
(
𝑋
) · I{
𝑡
∈
[
𝜎
,
𝑇
)}, (25)
where ℎ (1)
𝑡
(
𝑋
) =
𝑋𝑡
/(
𝑇
−
𝑡
), ℎ(2)
𝑡
(
𝑋
) =
𝑋𝑡
/(
𝑇
−
𝜎
−
𝑡
). (26) Lemma 2. For the time
𝜎
that minimizes the
objective function Ψ(
𝜎
),
𝜎
* =
𝑇
/3.
The problem of the optimal level of resources of a
simple system with possible violations of the
condition.
In this section, we consider some
assumptions about violations of the JIT condition in
processes inherent in real systems. Thus, we assume
that the intensities of point processes can be bounded.
We note that such a representation of the process
𝑋
in
(4) does not correspond to the time reversal procedure
for a point process with fixed initial value.
Nevertheless, such a representation in terms of point
processes is useful for describing a controlled system
with a violation of the condition of JIT. For such a
model, the task of optimal control arises – to find the
value of the maximum level of intensity of the point
process for each operation under conditions of
payment for the value of this boundary, and payment
for non-compliance with the JIT requirement. We
suppose that the intensity h in (4) can be represented
as ℎ
𝑡
= ℎ
𝑡
(
𝑋
) =
𝑋𝑡
· min{Λ,I{
𝑡
<
𝑇
}/(
𝑇
−
𝑡
)}, (32)
where Λ
∈
[0, ∞) is a finite maximum level of intensity
for each operation. Under this assumption for ℎ, the
JIT-condition
𝑋𝑇
= 0 may not hold, and obviously
P{
𝜔
:
𝑋𝑇
(
𝜔
) > 1} > 0 and E
𝑋𝑇
> 0.
We assume that the payment for this violation of the
JIT condition is proportional to the mean value of the
number of uncompleted operation E
𝑋𝑇
. The
coefficient of proportionality is denoted by
𝛼
. The
greater the upper level Λ, the smaller the value of E
𝑋𝑇
and the closer to the fulfillment of the JIT requirement.
Since the resources of the real system provide the level
Λ, it also has a certain positive cost with a
proportionality factor of
𝛽
. Moreover, Λ can serve as
a control parameter in the system (4). Thus, we
consider the problem of optimal control of the process
𝑋
in (4) for fixed
𝐾
∈
N and
𝑇
> 0, and under the
assumption for ℎ. It is necessary to find an optimal
value Λ * for which the problem is analogous to the
problems (17), (23) and (29): Θ(Λ* ) = inf Λ>0 Θ(Λ),
(33) where the objective function Θ(Λ) is equal to
Θ(Λ) =
𝛼
· E
𝑋𝑇
+
𝛽
· Λ (34) under the conditions:
𝛼
> 0,
𝛽
> 0. (35) Theorem
For the maximum intensity level, which minimizes the
objective function
Θ(Λ), Λ * = √︃
𝛼
·
𝐾
𝛽
·
𝑒
·
𝑇
if
𝛼
·
𝐾
·
𝑇
/
𝛽
∈
[
𝑒
,
+∞), (36) Λ * = [log(
𝛼
·
𝐾
·
𝑇
/
𝛽
)]/
𝑇
if
𝛼
·
𝐾
·
𝑇
/
𝛽
∈
(1,
𝑒
), (37) and Λ * = 0 if
𝛼
·
𝐾
·
𝑇
/
𝛽
∈
(0, 1].
III. Discussion.
The main purpose of this article is to show the
possibilities of using of the time reversal approach in
problems concerning just-in-time. We demonstrate
simple methods for optimizing JIT systems, for the
case of a point (counting) process, represented in
semimartingale terms. We also note that the statements
of Theorem 1, Lemma 2, and Theorem 2 are valid in
the case of a random walk in reverse time (Lemma 1
and Theorem 2 remain true if the coefficients are
properly replaced). In this case, the semimartingale
representation methods and optimal control problems
are close. In the case of nonstationary processes in
direct time, the results are also anticipated. Finally,
note that the method of representing JIT systems
discussed in the article in terms of predictable
semimartingale characteristics creates opportunities
for simple and clear computer modeling. Obviously,
the simulation is easy to implement on the basis of the
infinitesimal relation for
𝑋
: P{∆
𝑋𝑡
=
𝑋𝑡
+Δ −
𝑋𝑡
= −1|F
𝑡
} = ℎ
𝑡
(
𝑋
) · ∆ +
𝑜
(∆)
as ∆ → 0, for all
𝑡
> 0.
Thus, it follows that the discussed approach can serve
as an initial step for the analysis of stochastic JIT
systems.
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