An innovative parameter optimization of Spark Streaming based on D3QN with Gaussian process regression

Hong Zhang; Zhenchao Xu; Yunxiang Wang; Yupeng Shen; Hong Zhang; Zhenchao Xu; Yunxiang Wang; Yupeng Shen

doi:10.3934/mbe.2023647

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 14464-14486. doi: 10.3934/mbe.2023647

Previous Article Next Article

Research article

An innovative parameter optimization of Spark Streaming based on D3QN with Gaussian process regression

1.
School of Cyber Security and Computer, Hebei University, Baoding, China
2.
Bureau of Geophysical Prospecting, Baoding, China

Academic Editor: Yang Kuang

Received: 05 March 2023 Revised: 07 June 2023 Accepted: 20 June 2023 Published: 03 July 2023

Nowadays, Spark Streaming, a computing framework based on Spark, is widely used to process streaming data such as social media data, IoT sensor data or web logs. Due to the extensive utilization of streaming media data analysis, performance optimization for Spark Streaming has gradually developed into a popular research topic. Several methods for enhancing Spark Streaming's performance include task scheduling, resource allocation and data skew optimization, which primarily focus on how to manually tune the parameter configuration. However, it is indeed very challenging and inefficient to adjust more than 200 parameters by means of continuous debugging. In this paper, we propose an improved dueling double deep Q-network (DQN) technique for parameter tuning, which can significantly improve the performance of Spark Streaming. This approach fuses reinforcement learning and Gaussian process regression to cut down on the number of iterations and speed convergence dramatically. The experimental results demonstrate that the performance of the dueling double DQN method with Gaussian process regression can be enhanced by up to 30.24%.

Keywords:

Citation: Hong Zhang, Zhenchao Xu, Yunxiang Wang, Yupeng Shen. An innovative parameter optimization of Spark Streaming based on D3QN with Gaussian process regression[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14464-14486. doi: 10.3934/mbe.2023647

Related Papers:

[1]	Shaimaa A. M. Abdelmohsen, D. Sh. Mohamed, Haifa A. Alyousef, M. R. Gorji, Amr M. S. Mahdy . Mathematical modeling for solving fractional model cancer bosom malignant growth. AIMS Biophysics, 2023, 10(3): 263-280. doi: 10.3934/biophy.2023018
[2]	Mati ur Rahman, Mehmet Yavuz, Muhammad Arfan, Adnan Sami . Theoretical and numerical investigation of a modified ABC fractional operator for the spread of polio under the effect of vaccination. AIMS Biophysics, 2024, 11(1): 97-120. doi: 10.3934/biophy.2024007
[3]	Yasir Nadeem Anjam, Mehmet Yavuz, Mati ur Rahman, Amna Batool . Analysis of a fractional pollution model in a system of three interconnecting lakes. AIMS Biophysics, 2023, 10(2): 220-240. doi: 10.3934/biophy.2023014
[4]	Marco Menale, Bruno Carbonaro . The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities. AIMS Biophysics, 2020, 7(3): 204-218. doi: 10.3934/biophy.2020016
[5]	Mehmet Yavuz, Kübra Akyüz, Naime Büşra Bayraktar, Feyza Nur Özdemir . Hepatitis-B disease modelling of fractional order and parameter calibration using real data from the USA. AIMS Biophysics, 2024, 11(3): 378-402. doi: 10.3934/biophy.2024021
[6]	Larisa A. Krasnobaeva, Ludmila V. Yakushevich . On the dimensionless model of the transcription bubble dynamics. AIMS Biophysics, 2023, 10(2): 205-219. doi: 10.3934/biophy.2023013
[7]	Carlo Bianca . Differential equations frameworks and models for the physics of biological systems. AIMS Biophysics, 2024, 11(2): 234-238. doi: 10.3934/biophy.2024013
[8]	Mohammed Alabedalhadi, Mohammed Shqair, Ibrahim Saleh . Analysis and analytical simulation for a biophysical fractional diffusive cancer model with virotherapy using the Caputo operator. AIMS Biophysics, 2023, 10(4): 503-522. doi: 10.3934/biophy.2023028
[9]	Bertrand R. Caré, Pierre-Emmanuel Emeriau, Ruggero Cortini, Jean-Marc Victor . Chromatin epigenomic domain folding: size matters. AIMS Biophysics, 2015, 2(4): 517-530. doi: 10.3934/biophy.2015.4.517
[10]	David Gosselin, Maxime Huet, Myriam Cubizolles, David Rabaud, Naceur Belgacem, Didier Chaussy, Jean Berthier . Viscoelastic capillary flow: the case of whole blood. AIMS Biophysics, 2016, 3(3): 340-357. doi: 10.3934/biophy.2016.3.340

Abstract

1. Introduction

Numerous mathematical models have been established to predict and study the biological system. In the past four decades, there have been far-reaching research on improving cell mass production in chemical reactors [1]. The chemostat model is used to understand the mechanism of cell mass growth in a chemostat. A chemostat is an apparatus for continuous culture that contains bacterial populations. It can be used to investigate the cell mass production under controlled conditions. This reactor provides a dynamic system for population studies and is suitable to be used in a laboratory. A substrate is continuously added into the reactor containing the cell mass, which grows by consuming the substrate that enters through the inflow chamber. Meanwhile, the mixture of cell mass and substrate is continuously harvested from the reactor through the outflow chamber. The dynamics in the chemostat can be investigated by using the chemostat model [2].

Ordinary differential equations (ODEs) are commonly used for modelling biological systems. However, most biological systems behaviour has memory effects, and ODEs usually neglect such effects. The fractional-order differential equations (FDEs) are taken into account when describing the behaviour of the systems' equations. A FDEs is a generalisation of the ODEs to random nonlinear order [3]. This equation is more effective because of its good memory, among other advantages [4]–[8]. The errors occuring from the disregarded parameters when modelling of phenomena in real-life also can be reduced. FDEs are also used to efficiently replicate the real nature of various systems in the field of engineering and sciences [9]. In the past few decades, FDEs have been used in biological systems for various studies [5], [6], [10]–[17].

Since great strides in the study of FDEs have been developed, the dynamics in the chemostat can be investigated using the mathematical model of the chemostat in the form of FDEs. Moreover, there have been few studies on the expansion of the chemostat model with fractional-order theory. Thus, we deepen and complete the analysis on the integer-order chemostat model with fractional-order theory and discuss the stability of the equilibrium points of the fractional-order chemostat model. Next, the bifurcation analysis for the fractional-order chemostat model is conducted to identify the bifurcation point that can change the stability of the system. The analysis identifies the values of the fractional-order and the system parameters to ensure the operation of the chemostat is well-controlled.

2. Materials and methods

2.1. Definition of Caputo derivative

Recently, there are many approaches to define fractional operators such as by Caputo, Riemann-Liouville, Hadamard, and Grunwald-Letnikov [3]. However, Caputo is often used due to its convenience in various applications [18]. Caputo is also useful to encounter an obstacle where the initial condition is done in the differential of integer-order [19]. In this paper, we applied Caputo derivative to define the system of fractional-order. The Caputo derivative for the left-hand side is defined as

$D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} \frac{f^{(n)} τ}{{(t - τ)}^{α - n + 1}} d τ,$ (2.1)

where Г denotes the function of gamma, n is an integer, where $n - 1 < α < n$ [18].

2.2. Adams-type predictor corrector method

The Adams-type predictor-corrector method is one of the technique that have been proposed for fractional-order differential equations [19]–[21]. The Adams-type predictor-corrector method is a analysis of numerical algorithm that involves two basics steps: predictor and corrector. The predictor formula can be described as

$\begin{array}{l} y_{h}^{P} (t_{n + 1}) = \sum_{k = 0}^{α - 1} \frac{t_{n + 1}^{k}}{k!} y_{0}^{(k)} + \frac{1}{Γ (α)} \sum_{j = 0}^{n} b_{j, n + 1} f (t_{j}, y_{h} (t_{j})), \end{array}$ (2.2)

meanwhile the corrector formula can be determined by

$y_{h} (t_{n + 1}) = \sum_{k = 0}^{α - 1} \frac{t_{n + 1}^{k}}{k!} + \frac{h^{α}}{Γ (α + 2)} f (t_{n + 1}, y_{h}^{P} (t_{n + 1})) + \frac{h^{α}}{Γ (α + 2)} \sum_{j = 0}^{n} f (t_{j}, y_{h} (t_{j})) .$ (2.3)

The predictor-corrector method is also called as the PECE (Predict, Evaluate, Correct, Evaluate) method [22]. The procedure of the predictor-corrector method can be explained as follows

(i) Calculate the predictor step, $y_{h}^{P} (t_{n + 1})$ in Eq. (2.2).

(ii) Evaluate $f (t_{n + 1}, y_{h}^{P} (t_{n + 1}))$ .

(iii) Calculate the corrector step, $y_{h} (t_{n + 1})$ in Eq. (2.3).

(iv) Evaluate $f (t_{n + 1}, y_{h} (t_{n + 1}))$ .

The procedure repeatedly predicts and corrects the value until the corrected value becomes a converged number [21]. This method is able to maintain the stability of the properties and has good accuracy. Moreover, this method also has lower computational cost than other methods [23]. The algorithm of the Adams-type predictor-corrector method proposed by [22] is shown in Appendix.

2.3. Stability analysis of fractional-order system

The conditions of stability for integer-order differential equations and fractional-order differential equations are different. Both systems could have the same steady-state points but different stability conditions [24], [25]. The stability condition for fractional-order differential equations can be stated by Theorem 1 and the Routh-Hurwitz stability condition as described by Proposition 1.

Theorem 1 [5], [6], [26]. The commensurate system of fractional-order where $x \in ℝ$ and $0 < α < 1$ is locally asymptotically stable if the eigenvalues of the Jacobian matrix evaluated at the steady-state point is satisfied by

$| a r g (λ) | > \frac{α π}{2} .$ (2.4)

Proposition 1 [5], [6], [26]. Suppose the characteristic polynomial is $P (λ) = λ^{2} + b λ + c$ of the Jacobian matrix which evaluated at the steady-state. The eigenvalues of the Jacobian matrix will satisfy Eq. (2.4) in Theorem 1 if

$b > 0, c > 0,$ (2.5)

$b < 0, 4 c > b^{2}, | t a n^{- 1} (\frac{\sqrt{4 c - b^{2}}}{b}) | > \frac{α π}{2} .$ (2.6)

The stability theorem on the fractional-order systems and fractional Routh-Hurwitz stability conditions are introduced to analyze the stability of the model. The fractional Routh-Hurwitz stability conditions is specifically introduced for the eigenvalues of the Jacobian matrix that obtained in quadratic form. The proof of this proposition is shown in Appendix.

2.4. Hopf bifurcation analysis of fractional-order system

Bifurcation can be defined as any sudden change that occurs while a parameter value is varied in the differential equation system and it has a significant influence on the solution [27]. An unstable steady-state may becomes stable and vice versa. A slight changes in the parameter value may change the system's stability. Despite the steady-state point and the eigenvalues of the system of fractional-order are similar as the system of integer-order, the discriminant method used for the stability of the steady-state point is different. Accordingly, the Hopf bifurcation condition in the fractional-order system is slightly different as compared with the integer-order system.

2.4.1. Hopf bifurcation analysis of fractional order α

Fractional order α can be selected as the bifurcation parameter in a fractional-order system, but this is not allowed in an integer-order system. The existence of Hopf bifurcation can be stated as in Theorem 2.

Theorem 2 [28]. Assume α* as the critical value of the fractional-order. When bifurcation parameter α passes over critical value α*, which is $α^{*} \in (0,1)$ , Hopf bifurcation occurs at the steady-state point if the following conditions are satisfied

(i) 1. The characteristic equation of chemostat system has a pair of complex conjugate roots, $λ_{1,2} = p \pm i q$ , while the other eigenvalues are negative real roots.

(ii) Critical value $m (α^{*}) = \frac{α^{*} π}{2} - m i n | a r g (λ) | = 0$ .

(iii) $\frac{d m (α)}{d α} |_{α = α^{*}} \neq 0$ (condition of transversality).

Proof. Condition (i) is not easy to obtain due to the selected parameter's value. However, this condition can be managed under some confined conditions. In fact, the washout steady-state solution of the chemostat model has two negative real roots. The remaining two roots depend on the characteristic of the polynomial from the no-washout steady-state solution.

Condition (ii) can be satisfied with the existence of critical value α* and when $a r g (λ)$ is equivalent to $a r c t a n (\frac{q}{p})$ . Thus, the solution of critical value $m (α^{*})$ can be written as

$α^{*} = \frac{α^{*} π}{2} - a r c t a n (\frac{q}{p}) = 0, α^{*} \in (0,1) .$ (2.7)

The integer system required p = 0 for the bifurcation's operating condition. For the fractional-order system, the operating condition of the system will change into $m (α^{*}) = \frac{α^{*} π}{2} - m i n | a r g (x) | = 0$ . For condition (iii), the condition of $m (α)$ changes when bifurcation parameter $α$ passes over critical value α*. For example, the steady-state point is asymptotically stable for $0 < α < α^{*}$ and unstable when $α < α^{*} < 1$ . Thus, Hopf bifurcation exists at $α = α^{*}$ .

2.4.2. Hopf bifurcation analysis of parameters value

In studying the dynamic process of chemostat, the parameters such as $Q, S_{0}, µ, k, γ$ and β are usually used as the bifurcation parameter since these parameters have significant effects on the dynamic process of the system of fractional-order and integer-order. Fractional-order α is considered fixed and the initial substrate concentration S₀ is studied as the control parameter. The existence of the Hopf bifurcation can be stated as in Theorem 3.

Theorem 3 [28]. Assume $S_{0}^{*}$ as the critical value of the fractional-order. When bifurcation parameter S₀ passes over critical value $S_{0}^{*}$ , Hopf bifurcation occurs at the steady-state point if the following conditions are satisfied

(i) The characteristic equation of chemostat system has a pair of complex conjugate roots, $λ_{1,2} = p (S_{0}) \pm i q (S_{0})$ , while the other eigenvalues are negative real roots.

(ii) Critical value $m (S_{0}^{*}) = \frac{α π}{2} - m i n | a r g (λ (S_{0}^{*})) | = 0$ .

(iii) $\frac{d m (S_{0}^{*})}{d (S_{0}^{*})} |_{S_{0} = S_{0}^{*}} \neq 0$ (condition of transversality).

Proof. This theorem can be proved in the same way as Theorem 2. Therefore, condition (i) can be guaranteed. Condition (ii) can be satisfied with the existence of critical value $S_{0}^{*}$ and when $a r g [λ (S_{0}^{*})]$ is equivalent to $a r c t a n [\frac{q (S_{0}^{*})}{p (S_{0}^{*})}]$ . Thus, the solution of critical value $m (S_{0}^{*})$ can be written as

$S_{0}^{*} = \frac{α π}{2} - a r c t a n (\frac{q (S_{0}^{*})}{p (S_{0}^{*})}) = 0.$ (2.8)

For condition (iii), the condition of $m (S_{0}^{*})$ changes when bifurcation parameter S₀ passes over critical value $S_{0}^{*}$ . For example, the steady-state point is asymptotically stable when $0 < S_{0} < S_{0}^{*}$ and unstable when $S_{0} < S_{0}^{*} < 1$ . Thus, Hopf bifurcation exists at $S_{0} = S_{0}^{*}$ [28].

Firstly, determine the steady-states, Jacobian matrix and eigenvalues of the fractional-order chemostat model. The stability properties of the fractional-order chemostat model were estimated by using the stability and bifurcation analyses with FDEs by referring to Theorem 1 and Proposition 1. Then, determine the bifurcation point of fractional-order by referring to Theorem 2 and determine the bifurcation point of parameter values by referring to Theorem 3. Next, plot the phase portrait of fractional-order chemostat model by using Adam-types predictor-corrector method to study the dynamic behaviour of the system. Figure 1 depicts the flowchart of this research. This flowchart can be applied to all problems with suitable parameter values.

Figure 1. Mathematical analysis of fractional-order chemostat model.

DownLoad: Full-Size Img PowerPoint

3. Results and Discussion

3.1. Fractional-order chemostat model

An integer-order chemostat model that considered a variable yield coefficient and the Monod growth model from [1] is studied in this section. The chemostat system can be written as

$\begin{array}{l} \frac{d S}{d t} & = Q (S_{0} - S) - \frac{µ S X}{(k + S) (γ + β S)}, \\ \frac{d X}{d t} & = Q (- X) + \frac{µ S X}{k + S}, \end{array}$ (3.1)

with the initial value of $X_{0} = 0$ , where the sterile feed case was assumed. The integer-order chemostat system of Eq. (3.1) is extended to the fractional-order differential equation

$\begin{array}{l} \frac{d^{α} S}{d t^{α}} & = Q (S_{0} - S) - \frac{µ S X}{(k + S) (γ + β S)}, \\ \frac{d^{α} X}{d t^{α}} & = Q (- X) + \frac{µ S X}{k + S} . \end{array}$ (3.2)

Let Eq. (3.2) equal to zero in order to find the steady-state solutions

$Q (S_{0} - S) - \frac{µ S X}{(k + S) (γ + β S)} = 0,$ (3.3)

$Q (- X) + \frac{µ S X}{k + S} = 0.$ (3.4)

By solving Eq. (3.4), the following solutions are obtained

$S^{*} = \frac{k Q}{µ - Q},$ (3.5)

$X^{*} = 0.$ (3.6)

From Eq. (3.3), if $X^{*} = 0$ , then $S^{*} = S_{0}$ . If $S^{*} = \frac{k Q}{µ - Q}$ , then

$X^{*} = \frac{(k Q + Q S_{0} - S_{0} µ) (- Q γ + k Q β + γ µ)}{{(µ - Q)}^{2}} .$ (3.7)

Hence, the solutions of steady-state for the chemostat model are

(i) Washout:

$(S_{0}^{*}, X_{0}^{*}) = (S_{0},0) .$ (3.8)

(ii) No Washout:

$(S_{1}^{*}, X_{1}^{*}) = (ρ, (S_{0} - ρ) (γ + β ρ)),$ (3.9)

where

$\begin{array}{l} ρ = \frac{k Q}{µ - Q} . \end{array}$

The steady-state solutions are physically meaningful if their components are positive. Therefore, $S_{0} > 0$ for the washout steady-state solution exists by biological meaning. The no-washout steady-state solution will only exist when $0 < ρ < S_{0}$ . The Jacobian matrix of no washout steady-state as in Eq. (3.10) can be used to investigate the stability properties of the fractional-order chemostat model.

$J = [\begin{matrix} - Q + \frac{X (- k γ + S^{2} β) µ}{{(k + S)}^{2} {(γ + S β)}^{2}} & \frac{S µ}{(k + S) (γ + S β)} \\ \frac{k X µ}{{(k + S)}^{2}} & - Q + \frac{S µ}{k + S} \end{matrix}] .$ (3.10)

3.2. Stability of washout and no-washout steady-state solution

The solution of steady-state in Equation (3.8) represents the washout situation, where the cell mass is wholly removed from the reactor and where the substrate concentration is at the same stock as in the beginning. This state must always be unstable in order to ensure that the cell mass is able to grow in the chemostat. This is because the cell mass will be continuously removed from the chemostat if the washout steady-state is stable. The Jacobian matrix for the washout steady-state solution can be written as

$J = [\begin{matrix} - Q & - \frac{S_{0} µ}{(k + S_{0}) (γ + S_{0} β)} \\ 0 & - Q + \frac{S_{0} µ}{k + S_{0}} \end{matrix}] .$ (3.11)

The eigenvalues of this matrix are

$λ_{1} = - Q,$ (3.12)

$λ_{2} = \frac{- k Q - Q S_{0} + S_{0} µ}{k + S_{0}} .$ (3.13)

The eigenvalues in Eq. (3.12) and Eq. (3.13) are real. The washout steady-state solution is stable if Q>0 and $ρ > S_{0}$ where $ρ = \frac{k Q}{µ - Q}$ . The steady-state solution in Eq. (3.9) represents the no-washout situation. No-washout situation is where the cell mass is not removed and stay growth in the chemostat.This state is important. The steady-state solution is substituted into the Jacobian matrix in Eq. (3.10) and can be written as

$J = [\begin{matrix} - Q + \frac{µ (S_{0} - ρ) (β ρ^{2} - k γ)}{{(k + ρ)}^{2} (β ρ - γ)} & - \frac{µ ρ}{(k + ρ) (β ρ + γ)} \\ \frac{k µ (S_{0} - ρ) (β ρ - γ)}{{(k + ρ)}^{2}} & - Q + \frac{µ ρ}{k + ρ} \end{matrix}] .$ (3.14)

The eigenvalues of the Jacobian matrix in terms of the characteristic polynomial are

$P (λ) = λ^{2} + b λ + c,$ (3.15)

where

$b = 2 Q + \frac{µ (S_{0} - 2 ρ)}{(k + ρ)} + \frac{µ ρ (ρ - S_{0})}{{(k + ρ)}^{2}} + \frac{β µ ρ (ρ - S_{0})}{(k + ρ) (γ + β ρ)},$ (3.16)

and

$\begin{array}{l} c = & Q^{2} + \frac{Q µ (S_{0} - 2 ρ)}{(k + ρ)} + \frac{(µ ρ - S_{0} µ) (µ ρ - Q ρ)}{{(k + ρ)}^{2}} + \frac{µ^{2} ρ^{2} (S_{0} - ρ)}{{(k + ρ)}^{3}} \\ - \frac{Q β µ ρ (S_{0} - ρ)}{(k + ρ) (γ + β ρ)} + \frac{(S_{0} µ - µ ρ) (µ ρ γ + 2 µ β ρ^{2})}{{(k + ρ)}^{2} (γ + β ρ)} \\ + \frac{(µ ρ - S_{0} µ) (µ ρ^{2} γ + µ β ρ^{3})}{{(k + ρ)}^{3} (γ + β ρ)} . \end{array}$ (3.17)

The eigenvalues of the no-washout steady-state solution were evaluated with Routh-Hurwitz condition in Proposition 1. Based on the eigenvalues in Eq. (3.15) and by referring to the study by [5], the eigenvalues' condition can be simplified as the following two cases

(i) If b>0 or equivalent to $\frac{γ}{β} > P_{1}$ , the no-washout steady-state solution of the system in Eq. (3.2) is asymptotically stable. P₁ can be written as

$P_{1} = - \frac{µ ρ (ρ - S_{0})}{2 Q (k + ρ)} - \frac{ρ (ρ - S_{0})}{(S_{0} - ρ)} - \frac{(k + ρ)}{µ ρ (ρ - S_{0})} - ρ,$ (3.18)

(ii) If b<0 or equivalent to $\frac{γ}{β} < P_{1}$ and $t a n^{- 1} (\frac{\sqrt{4 c - b^{2}}}{b}) > \frac{α π}{2}$ , the no-washout steady-state solution of the system in Eq. (3.2) is asymptotically stable. The condition of $t a n^{- 1} (\frac{\sqrt{4 c - b^{2}}}{b}) > \frac{α π}{2}$ is also equivalent to $4 c o s^{2} (\frac{α π}{2}) c > b^{2}$ , which can be simplified as $\frac{γ}{β} > P_{2}$ . Then, this case can be concluded and written as $P_{2} < \frac{γ}{β} < P_{1}$ where

$P_{2} = \frac{µ ρ (ρ - S_{0})}{2 c o s (\frac{α π}{2}) \sqrt{c} (k + ρ)} - \frac{µ ρ (ρ - S_{0})}{2 Q (k + ρ)} - \frac{ρ (ρ - S_{0})}{(S_{0} - ρ)} - k - 2 ρ .$ (3.19)

Then, if $P_{2} < \frac{γ}{β} < P_{1}$ , the no-washout steady-state solution of the system in Eq. (3.2) is asymptotically stable.

The parameter values of the fractional-order chemostat model are provided in Table 1. The initial substrate concentration, S₀ and ρ were assumed as non-negative values to ensure that the steady-state solutions were physically meaningful. The stability diagram of the steady-state solutions is plotted in Figure 2.

Table 1. Parameter values.

Parameters	Description	Values	Units
k	Saturation constant	1.75	gl⁻¹
Q	Dilution rate	0.02	l²gr⁻¹
µ	Maximum growth rate	0.3	h⁻¹
γ	Constant in yield coefficient	0.01	–
β	Constant in yield coefficient	5.25	lg⁻¹
S₀	Input concentration of substrate	1	gl⁻¹

| Show Table

DownLoad: CSV

Figure 2. Stability diagram of the steady-state solutions when $α = 1$ .

DownLoad: Full-Size Img PowerPoint

The washout steady-state solution is stable if Q > 0 and $ρ > S_{0}$ . From Figure 2, it shows that the unstable solution of washout steady-state, as the eigenvalues did not fulfil the condition of $ρ > S_{0}$ . By choosing the appropriate parameter values, the unstable washout steady-state solution could ensure that the washout condition does not occur in the chemostat. Meanwhile, the solution of no-washout steady-state is stable.

3.3. Hopf bifurcation analysis of the order of fractional-order system

The steady-state solutions of the fractional-order chemostat model for the parameter values given in Table 1 are

(i) Washout:

$(S_{0}^{*}, X_{0}^{*}) = (1,0),$ (3.20)

(ii) No Washout:

$(S_{1}^{*}, X_{1}^{*}) = (\frac{1}{8}, \frac{3731}{6400}) .$ (3.21)

The eigenvalues obtained from the washout steady-state solution are

$λ_{1} = - \frac{1}{50},$ (3.22)

$λ_{2} = - \frac{49}{550},$ (3.23)

and the eigenvalues from the no-washout steady-state solution are

$λ_{1} = \frac{- 2552 + \sqrt{411098126 i}}{399750},$ (3.24)

$λ_{2} = \frac{- 2552 - \sqrt{411098126 i}}{399750},$ (3.25)

Based on Eq. (3.22) to Eq. (3.25), these satisfied the first condition of Hopf bifurcation in Theorem 2. There exists a pair of complex conjugate roots and the other eigenvalues are negative real roots. The transversality condition as the third condition is also satisfied. The eigenvalues of the washout steady-state solution based on the chemostat system is not imaginary, and so there is no existence of Hopf bifurcation in the washout steady-state solution. According to Theorem 2, the critical value of the fractional-order as stated in the second condition can be obtained as

$m (α^{*}) = \frac{α^{*} π}{2} - m i n | a r g (λ) | = 0,$ (3.26)

$α^{*} = \frac{2}{π} m i n | a r g (λ) |,$ (3.27)

where

$a r g (λ) = a r c t a n (\frac{q}{p}),$ (3.28)

$α^{*} = \frac{2}{π} a r c t a n (\frac{q}{p}) = \frac{2}{π} a r c t a n (\frac{\frac{\sqrt{411098126}}{399750}}{\frac{2552}{399750}}) = 0.9202904711 \approx 0.9.$ (3.29)

Value of p and q are obtained from Eq. (3.24) and Eq. (3.25) by assuming parameter value in Table 1. Hence, when $α^{*} = 0.9$ , the chemostat system in Eq. (3.2) shows Hopf bifurcation, at which the system stability would be altered.

Figure 3 is plotted to determine the dynamic behaviour at the Hopf bifurcation point. The phase portrait diagrams of cell mass concentration against substrate concentration are plotted for values of order of the fractional is $α = 0.9$ .

Figure 3. Phase portrait plot of fractional-order chemostat system with $α = 0.9$ .

DownLoad: Full-Size Img PowerPoint

The running state of the fractional-order chemostat system when fractional order α at the Hopf bifurcation point is shown. The fractional-order chemostat system changed its stability once Hopf bifurcation occurred. Therefore, we conjecture that the system of fractional-order chemostat may be lost or gain its stability when the fractional order α is less than the Hopf bifurcation point, or $α < 0.9$ or otherwise. This shows that increasing or decreasing the value of α may destabilise the stable state of the chemostat system. Therefore, these results show that the running state of the fractional-order chemostat system is affected by the value of α.

3.4. Hopf bifurcation analysis of initial concentration of substrate

The initial concentration of the substrate, S₀, was chosen as the control parameter, while fractional order α was fixed. The solutions of steady-state of the fractional-order chemostat model with S₀ as the control parameter are

(i) Washout:

$(S_{0}^{*}, X_{0}^{*}) = (S_{0},0),$ (3.30)

(ii) No Washout:

$(S_{1}^{*}, X_{1}^{*}) = (\frac{1}{8}, \frac{533 (- 1 + 8 S_{0})}{6400}) .$ (3.31)

The eigenvalues obtained from the washout steady-state solution are

$λ_{1} = - \frac{1}{50},$ (3.32)

$λ_{2} = - \frac{7 (1 - 8 S_{0})}{50 (- 7 - 4 S_{0}},$ (3.33)

and the eigenvalues from the no-washout steady-state solution are

$λ_{1} = - 4204 + 1652 S_{0} + \frac{\sqrt{2} \sqrt{1364552 S_{0}^{2} - 245579768 S_{0} + 38666153}}{399750} i,$ (3.34)

$λ_{2} = - 4204 + 1652 S_{0} - \frac{\sqrt{2} \sqrt{1364552 S_{0}^{2} - 245579768 S_{0} + 38666153}}{399750} i .$ (3.35)

These satisfied the first condition of Hopf bifurcation in Theorem 3. There exist a pair of complex conjugate roots in terms of S₀, and the other eigenvalues were negative real roots in terms of S₀. The transversality condition as the third condition is also satisfied. According to Theorem 3, the critical value of the fractional order as stated in the second condition can be obtained as follows

$m (S_{0}^{*}) = \frac{α^{*} π}{2} - m i n | a r g (λ) | = 0.$ (3.36)

By referring to the study by [18], Eq. (3.37) can also be calculated as

$\frac{q^{'} (S_{0}^{*}) p (S_{0}^{*}) - q (S_{0}^{*}) p^{'} (S_{0}^{*})}{q^{2} (S_{0}^{*}) + p^{2} (S_{0}^{*})} \neq 0.$ (3.37)

From the calculations, the critical value of the initial concentration of the substrate is $S_{0} = 2.54$ . When $S_{0} = 2.54$ , the chemostat system shows Hopf bifurcation, at which the stability of the system would be altered.

Figure 4 presents the phase portrait diagrams of concentration of cell mass against concentration of substrate when $α = 1$ for different values of the initial concentration of the substrate, which are $S_{0} = 2$ , $S_{0} = 2.54$ and $S_{0} = 3.5$ .

Figure 4. Phase portrait plot of chemostat system with $α = 1$ (a) $S_{0} = 2$ (b) $S_{0} = 2.54$ and (c) $S_{0} = 3.5$ .

DownLoad: Full-Size Img PowerPoint

The change in the running state when the value of the initial substrate concentration passes through the Hopf bifurcation point is shown. The stability of the fractional-order chemostat system changed once Hopf bifurcation occurred. In Figure 3(a), the fractional-order chemostat system is in a stable state when the initial substrate concentration value is less than the Hopf bifurcation point, or $S_{0} < 2.54$ . Meanwhile, when the value of the initial substrate concentration passes through the Hopf bifurcation point, or $S_{0} \geq 2.54$ , the fractional-order chemostat system lost its stability. This shows that increasing the value of the initial substrate concentration may destabilise the stable state of the chemostat system. These results show that the running state of the fractional-order chemostat system is affected by the value of the initial substrate concentration. In real-life application, the value of the initial substrate should remain at $S_{0} \geq 2.54$ to ensure that the chemostat system is at the unstable state. This is because the unstable state is suitable for the production of cell mass [1]. Unstable state means the system always move away after small disturbance, so the system must be at the unstable state because there will be a change in amount of cell mass production.

Figure 5 depicts the phase portrait diagrams of cell mass concentration against substrate concentration when $α = 0.9$ for different values of the initial concentration of the substrate, which are $S_{0} = 2$ , $S_{0} = 2.54$ and $S_{0} = 3.5$ .

Figure 5. Phase portrait plot of chemostat system with $α = 0.9$ (a) $S_{0} = 2$ (b) $S_{0} = 2.54$ and (c) $S_{0} = 3.5$ .

DownLoad: Full-Size Img PowerPoint

The Hopf bifurcation points of system of fractional-order chemostat and system of integer-order chemostat are different. Figure 4 shows the fractional-order chemostat system at a stable state for all values of the initial substrate concentration when $α = 0.9$ . The chemostat system destabilised the stable state when the initial substrate concentration value is $S_{0} \geq 2.54$ , as shown in Figure 3(b) and Figure 3(c). This shows that the dynamic behaviour of the fractional-order chemostat system is different compared with the integer-order chemostat system. In actual application, the value of the initial substrate should remain at $S_{0} \geq 2.54$ to ensure that the chemostat system can be well controlled in order to be suitable for cell mass production.

4. Conclusions

The stability analysis of the fractional-order chemostat model was conducted based on the stability theory of FDEs. The integer-order chemostat model was extended to the FDEs. There are two steady-state solutions obtained, which are washout and no-washout steady-state solutions. The Hopf bifurcation of the order of α occured at the solutions of steady-state when the Hopf bifurcation conditions is fulfilled. The results show that the increasing or decreasing the value of α may stabilise the unstable state of the chemostat system. Therefore, the running state of the fractional-order chemostat system is affected by the value of α. The Hopf bifurcation of the initial concentration of the substrate, S₀, also occurred when the Hopf bifurcation condition is fulfilled. As the evidence from the phase portrait plots, increase the value of the initial substrate concentration may destabilise the stable state of the chemostat system. The value of the initial substrate should remain at $S_{0} \geq 2.54$ to ensure that the chemostat system is at the unstable state since the unstable state is suitable for the production of cell mass. These dynamical analyses are important to provide suitable values of the fractional-order and the parameters in order to ensure the controllability and stability of the chemostat to suit the actual chemostat environment.

References

[1]	Apache storm. Available from: https://storm.apache.org/.
[2]	Apache spark streaming. Available from: https://spark.apache.org/docs/latest/streaming-programming-guide.html.
[3]	Apache flink. Available from: https://flink.apache.org/.
[4]	D. Cheng, X. Zhou, Y. Wang, C. Jiang, Adaptive scheduling parallel jobs with dynamic batching in spark streaming, IEEE Trans. Parallel Distrib. Syst., 29 (2018), 2672–2685. https://doi.org/10.1109/TPDS.2018.2846234 doi: 10.1109/TPDS.2018.2846234
[5]	H. Du, P. Han, Q. Xiang, S. Huang, Monkeyking: Adaptive parameter tuning on big data platforms with deep reinforcement learning, Big Data, 8 (2020), 270–290.
[6]	A. Kordelas, T. Spyrou, S. Voulgaris, V. Megalooikonomou, N. Deligiannis, KORD-I: A framework for real-time performance and cost optimization of apache Spark Streaming, in 2023 IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS), (2023), 1–3.
[7]	L. Liu, G. Shen, C. Guo, Y. Cui, C. Jiang, D. Wu, A spark streaming parameter optimization method based on deep reinforcement learning, Comput. Modernization, 2021 (2021), 49–56.
[8]	J. Wang, An intuitive tutorial to Gaussian processes regression, preprint, arXiv: 200910862. https://doi.org/10.48550/arXiv.2009.10862
[9]	Z. Wang, T. Schaul, M. Hessel, H. Hasselt, M. Lanctot, N. Freitas, Dueling network architectures for deep reinforcement learning, in Proceedings of Machine Learning Research, (2016), 1995–2003.
[10]	H. Van Hasselt, A. Guez, D. Silver, Deep reinforcement learning with double q-learning, in Proceedings of the AAAI conference on artificial intelligence, 30 (2016). https://doi.org/10.1609/aaai.v30i1.10295
[11]	E. Schulz, M. Speekenbrink, A. Krause, A tutorial on gaussian process regression: Modelling, exploring, and exploiting functions, J. Math. Psychol., 85 (2018), 1–16. https://doi.org/10.1016/j.jmp.2018.03.001 doi: 10.1016/j.jmp.2018.03.001
[12]	X. Wang, S. Wang, X. Liang, D. Zhao, J. Huang, X. Xu, et al., Deep reinforcement learning: A survey, IEEE Trans. Neural Networks Learn. Syst., 2022 (2022). https://doi.org/10.1109/TNNLS.2022.3207346 doi: 10.1109/TNNLS.2022.3207346
[13]	L. P. Swiler, M. Gulian, A. L. Frankel, C. Safta, J. D. Jakeman, A survey of constrained gaussian process regression: Approaches and implementation challenges, J. Machine Learn. Model. Comput., 1 (2020). https://doi.org/10.1615/JMachLearnModelComput.2020035155 doi: 10.1615/JMachLearnModelComput.2020035155
[14]	R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction, MIT press, 2018.
[15]	Z. H. Zhou, Machine Learning, Springer Nature, 2021.
[16]	J. C. Lin, M. C. Lee, I. C. Yu, E. B. Johnsen, Modeling and simulation of spark streaming, in 2018 IEEE 32nd International Conference on Advanced Information Networking and Applications (AINA), (2018), 407–413. https://doi.org/10.1109/AINA.2018.00068
[17]	S. Venkataraman, A. Panda, K. Ousterhout, M. Armbrust, A. Ghodsi, M. J. Franklin, et al., Drizzle: Fast and adaptable stream processing at scale, in Proceedings of the 26th Symposium on Operating Systems Principles, (2017), 374–389. https://doi.org/10.1145/3132747.3132750
[18]	T. Ajila, S. Majumdar, Data driven priority scheduling on spark based stream processing, in 2018 IEEE/ACM 5th International Conference on Big Data Computin Applications and Technologies (BDCAT), (2018), 208–210. https://doi.org/10.1109/BDCAT.2018.00034
[19]	M. Petrov, N. Butakov, D. Nasonov, M. Melnik, Adaptive performance model for dynamic scaling Apache Spark Streaming, Proc. Comput. Sci., 136 (2018), 109–117.
[20]	W. Li, D. Niu, Y. Liu, S. Liu, B. Li, Wide-area spark streaming: Automated routing and batch sizing, IEEE Trans. Parallel Distrib. Syst., 30 (2018), 1434–1448.
[21]	H. Zhao, L. B. Yao, Z. X. Zeng, D. H. Li, J. L. Xie, W. L. Zhu, et al., An edge streaming data processing framework for autonomous driving, Connect. Sci., 33 (2021), 173–200. https://doi.org/10.1080/09540091.2020.1782840 doi: 10.1080/09540091.2020.1782840
[22]	B. Liu, X. Tan, W. Cao, Dynamic resource allocation strategy in spark streaming, J. Comput. Appl., 37 (2017), 1574. https://doi.org/10.11772/j.issn.1001-9081.2017.06.1574 doi: 10.11772/j.issn.1001-9081.2017.06.1574
[23]	D. Cheng, Y. Chen, X. Zhou, D. Gmach, D. Milojicic, Adaptive scheduling of parallel jobs in spark streaming, in IEEE INFOCOM 2017-IEEE Conference on Computer Communications, (2017), 1–9. https://doi.org/10.1109/INFOCOM.2017.8057206
[24]	B. R. Prasad, S. Agarwal, Performance analysis and optimization of spark streaming applications through effective control parameters tuning, in Progress in Intelligent Computing Techniques: Theory, Practice, and Applications, Springer, (2018), 99–110. https://doi.org/10.1007/978-981-10-3376-6_11
[25]	G. Liu, X. Zhu, J. Wang, D. Guo, W. Bao, H. Guo, SP-Partitioner: A novel partition method to handle intermediate data skew in spark streaming, Future Gener. Comput. Syst., 86 (2018), 1054–1063. https://doi.org/10.1016/j.future.2017.07.014 doi: 10.1016/j.future.2017.07.014
[26]	Z. Fu, Z. Tang, L. Yang, K. Li, K. Li, Imrp: A predictive partition method for data skew alleviation in spark streaming environment, Parallel Comput., 100 (2020), 102699. https://doi.org/10.1016/j.parco.2020.102699 doi: 10.1016/j.parco.2020.102699
[27]	J. Clifton, E. Laber, Q-learning: Theory and applications, Ann. Rev. Stat. Appl., 7 (2020), 279–301. https://doi.org/10.1146/annurev-statistics-031219-041220 doi: 10.1146/annurev-statistics-031219-041220
[28]	V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, et al., Human-level control through deep reinforcement learning, Nature, 518 (2015), 529–533. https://doi.org/10.1038/nature14236 doi: 10.1038/nature14236
[29]	W. Samek, G. Montavon, S. Lapuschkin, C. J. Anders, K. R. Müller, Explaining deep neural networks and beyond: A review of methods and applications, Proc. IEEE, 109 (2021), 247–278. https://doi.org/10.1109/JPROC.2021.3060483 doi: 10.1109/JPROC.2021.3060483
[30]	M. G. Titelbaum, Fundamentals of Bayesian Epistemology 2: Arguments, Challenges, Alternatives, Oxford University Press, 2022.
[31]	B. Gaye, D. Zhang, A. Wulamu, Improvement of support vector machine algorithm in big data background, Math. Prob. Eng., 2021 (2021), 1–9. https://doi.org/10.1155/2021/5594899 doi: 10.1155/2021/5594899
[32]	N. Ihde, P. Marten, A. Eleliemy, G. Poerwawinata, P. Silva, I. Tolovski, et al., A survey of big data, high performance computing, and machine learning benchmarks, in Performance Evaluation and Benchmarking: 13th TPC Technology Conference, (2021), 98–118. https://doi.org/10.1007/978-3-030-94437-7_7
[33]	Datanami, Kafka Tops 1 Trillion Messages Per Day at LinkedIn. Available from: https://goo.gl/cY7VOz.
[34]	Observability at twitter: Technical overview. Available from: https://goo.gl/wAHi2I.
[35]	Q. Ye, W. Liu, C. Q. Wu, Nostop: A novel configuration optimization scheme for Spark Streaming, in 50th International Conference on Parallel Processing, (2021), 1–10.
[36]	Spark tuning guide. Available from: https://spark.apache.org/docs/latest/streaming-programming-guide.html#deploying-applications.

This article has been cited by:

Xiaomeng Ma, Zhanbing Bai, Sujing Sun, Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders, 2022, 20, 1551-0018, 437, 10.3934/mbe.2023020

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)