Improving the Performance of MMPP/M/C Queue by Convex Optimization–A Real-World Application in Iron and Steel Industry

Now the information system of the iron and steel industry is a typical sensor cloud computing system. The system has accumulated a lot of relevant manufacturing data. Using these data can well solve the slab storage problem in the production process for the iron and steel industry. In this paper, we investigate a queueing system where customers arrive according to a Markov Modulated Poisson Process (MMPP). MMPP can describe how the arrival rate changes with the environment, which is more realistic. We develop an MMPP(3)/M/C queueing model to solve the congestion problem in the iron and steel industry. In the actual production process, the slab arrival rates vary with states, therefore MMPP is used to model the arrival process in this paper. Based on explicit performance measures, we develop a nonlinear optimization model of queueing system, and convert the model into a convex optimization problem. Through the convex optimization method, the MMPP(3)/M/C model, resulted from the practical system, can be analyzed by the M/M/C model approximately.


I. INTRODUCTION
The iron and steel industry collects the data of production and transportation through wireless sensors in the machine shop. In addition, the data is uploaded to the cloud database. Therefore, the information system of the iron and steel industry is a typical sensor cloud computing system. Managers can read relevant data from the database through the system and optimize it. This can reduce the company's operating costs and improve production efficiency [1].
This paper focuses on the optimization problem of slab storage in the iron and steel industry of China. In actual production, the slab arriving rate to the slab yard varies with the rate of upstream production. This situation affects the number of crane which needs to serve the arrival slabs.
The associate editor coordinating the review of this manuscript and approving it for publication was Md. Zakirul Alam Bhuiyan . A good arrangement for the arrival rate of slabs and the service rate of cranes can improve the work load of queueing system [2]. After continuous casting, slabs need to be stored in the slab yard for the next production stage. Slab storage is completed by cranes. A simplified diagram of the slab production process is shown in Fig.1. Based on the process, VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ we analyze the storage problem by the queueing theory. Slabs are considered as customers in the queueing system. Cranes are equivalent to servers. We mainly analyze the optimal configuration between arrival rate and service rate in the queueing system. When the service rate is high, it will increase the efficiency of the slab storage, but some servers will be idle. This will result in low utilization of equipment and waste of resources. When the arrival rate is high, it will produce slab backlog. This will generate waiting costs. So there is a tradeoff between the arrival rate of slabs and the service rate of cranes.
Therefore, it is very important to determine a reasonable optimal arrival rate and service rate in slab storage process. This can make the performance of the queueing system optimal.
In the actual production process, the arrival rate of slab varies with steel grade. MMPP can describe how the arrival rate changes with the environment. Therefore, we use MMPP to describe the arrival process, which makes the model more realistic. And for the optimal model of MMPP/M/C queueing system about service load, the solving process is NP hard. So we discuss the approximate method to solve the problem, which can get the optimal solution easily.After a long period of investigation, the arrival rates of slabs have three levels, which can describe the arrival process very appropriately. We assume that the service time of crane obeys exponential distribution. In the production process of slabs, the service guideline is ''first come first service''. During the service process, sometimes there is a situation of ''two slabs by one crane'', i.e., one crane can pick up two slabs at the same time, but this situation is rare in actual production. This article analyses the general situation, i.e., ''one slab by one crane''.

II. RELATED WORK AND MAIN FRAME
About the research of slab storage, Tang et al. [3] modelled and optimized slab storage in slab warehouses, and focused on the slab stack shuffling problem in the warehouses and solved the problem by a genetic algorithm, which reduced the number of slab shuffles. Jia et al. [4] discussed the MMPP/M/C queue with thresholds, which used the matrix geometry method to optimize the problem of slab yard. They also used the matrix geometry method to optimize other problem of the iron and steel industry [5].In this paper, we also optimize the slab storage in the warehouse, but focus on the service load of the queueing system. According to queueing models, the nonlinear optimization model is established for the queueing system. Then the MMPP(3)/M/C model can be optimized approximately by the results of M/M/C. Dohn and Clausen [6] studied the issues involving slab warehouse plans and crane scheduling, and a two-stage heuristic algorithm is used to optimize the calculation. We study the optimization problem of slab stocking through a new method. According to the queueing theory, the queueing model of slab storage is established, and the corresponding optimal service load is solved by convex optimization method.
There are many studies on the MMPP model. Baiocchi and Blefari-Melazzi [7] analysed the MMPP/G/1/K queueing model and designed a corresponding algorithm. Ching et al. [8] discussed the application of the MMPP queueing system, and proposed a hedging strategy for production control. They obtained the optimal hedging point by solving the model with a numerical algorithm to obtain the steady state distribution of the system. Ching [9] investigated the inventory system of multi-location and built a new model for the inventory system of consumable goods. The inventory system of each warehouse and main warehouse were modelled by Markov queueing network. The transhipments were described by MMPP. Shah-Heydari and Le-Ngoc [10] presented a study about Markov modulated Poisson processes, which characterized the multimedia traffic with shortterm and long-term correlation. Alain et al. showed how the analysis of Markov modulated rate processes can be used to address the problem of computing the distribution of W [11]. Wang et al. [12] proposed various buffer management and congestion control mechanisms to support differentiated service-of-quality(QoS) requirements. They developed an analytical model using the non-bursty Poisson process and the bursty MMPP for a finite buffer queueing system. They extensive simulation experiments are employed to validate the accuracy of the analytical model in their work. Lu [13] discussed a MMPP whose arrival time was associated with state-dependent marks, and addressed parameter estimation through an EM algorithm and got the optimal policy. Nasr and Maddah [14] studied a MMPP system, which was a continuous inventory replenishment system. The demands varied with the environment. They proposed a heuristic algorithm to optimize the queueing system. Romano et al [15] gave an approximate method to calculate the MMPP/M/1 queueing model, which also considered the accuracy of the approximate calculation and the calculation cost. The accuracy and calculation cost are evaluated by the super exponential approximation in their work. For a comprehensive analysis of MMPP, see Fischer and Meier-Hellstern [16]. All of the above studies create MMPP models for practical problems and developed the corresponding algorithms to optimize the solutions. In this paper, we aim at the problem of slab storage in the iron and steel industry. First, an MMPP(3)/M/C queueing system is investigated. Since the special structure of MMPP(3)/M/C that the queue length and waiting time of it can just be calculated by matrix geometric method. According to the queueing theory, a nonlinear optimization model about the configuration of queueing system is established. Then we design a transformation method to convert the model into convex optimization model, which makes it easy to solve the optimization problem of multiserver queueing system.
There are few studies on the convex optimization methods to optimize queueing systems. Chiang et al [17] first provided a suite of generalized weighted fair queuing formulations for output link scheduling, where the weights can be dynamically optimized under QoS constraints using the tool of geometric programming. Chiang et al. [18] proposed an effective nonlinear optimization system. They used convex optimization calculation tools and fast polynomial time algorithms to obtain the global optimality. Several convex structures of the queueing system were shown and followed by numerical examples for a single server queue. Neely and Modiano [19] studied the convexity of the G/G/1 queueing system. Bertsimas and Natarajan [20] used a semi-positive optimization method to analyse the steady state of the queueing system. Li and Neely [21] studied the convex optimization problem in a multi-class M/G/1 queue with controllable service rate. Chiang et al. [17] first provided a suite of generalized weighted fair queuing formulations for output link scheduling, where the weights can be dynamically optimized under QoS constraints using the tool of geometric programming. Marques et al. [22] relied on stochastic convex optimization to develop optimal algorithms that used instantaneous fading and queue length information to allocate resources at the transport(flow-control), link, and physical layers. Lardjane and Messaci [23] discussed a new numerical method. It is based on linear programming and convex optimization, is performed for the computation of the steady state solution of the queueing system. Egan and Collings [24] achieved queue stability by employing a stochastic scheduling technique, where the realizations of tuned random variables determine whether each BS transmits, in addition to the minimum SINR target. Ganesh and Anantharam [25] used large deviations estimates of the probabilities of these paths, and solved a constrained convex optimization problem to find the most likely path leading to a large queue size. Casmir and Effanga [26] discussed the use of response surface methodology to search for the optimal conditions for improving grinding process in case of convex situations in paper producing industries. Ziegler [27] considered the problem of minimizing a special convex function subject to one linear constraint, which is applied on production planning. All of the above studies are combined with convex optimization methods to analyse the queueing systems with single server. In this paper, we also use the convex optimization method to optimize the queueing system. However, the queueing system with several servers is discussed, which is more complex and realistic.
The main contributions of this paper are as follows: (1) An MMPP(3)/M/C queueing model is developed according to the slab storage in the iron and steel industry. (2) According to the queueing theory, a nonlinear optimization model is established, and a transformation method is designed to transform the model into a convex optimization model.
(3) The convex optimization model established in this paper can be applied to the warehousing in the slab logistics system. By optimizing the arrival rate of slab and service rate of crane, the performance of the queueing system can be optimized. (4) The convex optimization model established in this paper can be applied to the warehousing in the slab logistics system. By optimizing the arrival rate of slab and service rate of crane, the performance of the queueing system can be optimized.
This paper is organized as follows: section 2 introduces the related work and main work. section 3 sets up an MMPP (3)/M/C model for the slab storage in the iron and steel industry. Section 4 presents the comparison of MMPP(3)/M/C model with M/M/C and does a significance test between them. Section 5 describes the nonlinear optimization model of queueing system and the model is transformed into convex optimization model. Section 6 optimizes the problems of slab stocking in the Iron and Steel Complex. By convex optimization model, we give the policy to optimize the queueing system of slab storage. Finally, section 7 concludes.

III. THE MODEL OF SLAB STORAGE
In the iron and steel industry, after slabs being produced from the upstream production, the slabs need to be stored in warehouses and placed in a designated location by cranes. Different slabs have different positions, so the service time for them varies. In the queueing system, Slabs are considered as customers and cranes are equivalent to servers. The arriving slabs are the input in the queueing model. After the slabs being stored in the warehouse by cranes, the stored slabs are output. We assume that the service time obeys exponential distribution. The flow chart of slab production can be seen from Fig.2.
The number of cranes is C in the slab yard. Because the slabs are produced at different rates from the upstream production, the slab arrival rate varies with the states. We use MMPP to describe the arrival process. Based on the investigation from the industry, and divide the slab arrival rates into the following three categories: low arrival rate λ 0 , medium arrival rate λ 1 , and high arrival rate λ 2 . Each arrival state corresponds to a Poisson process, which has different arrival rate. These Poisson processes can transfer with each another. The tran-  Through the diagram of state transition, the corresponding transition matrix Q can be get, which is The sub-matrices are q 0 , q 1 , q 6 , A, A 0 , A 1 , and A 2 , as shown at the bottom of the next page. When the number of customers in the system exceeds C, the corresponding elements of Q matrix are the same, which are A 0 , A 1 and A 2 respectively.
π i is the steady state distribution of i customers in the system. There is an R matrix that makes π i = π C R i−C , i = C + 1, C + 2, ... ; therefore, we can get: The normalized condition is  (1), (2) and (3).The average queue length of the model can be derived by: In this paper, MMPP(3)/M/C model with three states is considered, which are high, medium and low arrival rates. These three states can describe the process of slab storage more realistic. So we can get the balance equation: r 0 , r 1 , r 2 are probabilities which are respectively for customers in the high, medium and low arrival rate. By balance equation (4), we can get r 0 , r 1 , r 2 , (5), as shown at the bottom of the next page.
Then we can obtain the average arrival rate: Since the average queue length of the MMPP(3)/M/C is formula (3), the waiting time is: When the waiting time of the queueing system is taken, we can use the convex optimal method to optimize the performance of the queueing system. But for the complexity of the formula (3) and (7), the convex optimal process is NP hard. Next we find a method to simplified convex optimization model.

IV. ANALYSIS AND COMPARISON BETWEEN MMPP(3)/M/C AND M/M/C
Due to the special structure of MMPP/M/C, there is no analytical expression to describe the queue length and waiting time for it. Therefore, this phenomenon motivates us to do an analysis between MMPP/M/C and M/M/C model which has specific analytical expression. Next the analysis and comparison are displayed in detail.  (3) and (7), we can get the numerical solutions of the average queue length and waiting time of the MMPP/M/C model, which are calculated by the matrix geometry method. Fig. 4 shows the average queue length for the two models with the same parameters. The number of servers C is 6; the service rate of each server is 0.1; the average arrival rate of MMPP(3)/M/C and M/M/C are same. It can be seen from the graph that the average queue length of the models are almost same.
Finally, we can also get the waiting time of MMPP/M/C and M/M/C. Fig. 5 shows the waiting times for the two models with the same parameters. It can be seen from the    two models vary greatly, although they are calculated by the same parameters. When the number of the state increases, the difference between MMPP/M/C and M/M/C will be more significant.

B. SIGNIFICANCE TEST BETWEEN MMPP(3)/M/C AND M/M/C MODEL
Since the above results are obtained when the average arrival rate and the service rate are same, in this section, we will use the F-test to analyze the significance between the different models (MMPP(3)/M/C and M/M/C).   Table 1 and Table 2.
Before doing the F-test, we need to check the homogeneity of variance for the data sets. In Table 1, we find that P = 0.982 > 0.05, which means the variance has the homogeneity. Then we can do the F-test in the next step.
From Table 2, we see that P = 0.99 > 0.05, which means there is no significant difference for the queue length between the MMPP (3) Table 4, we find that P = 0.988 > 0.05. It means there is no significant difference for the waiting time between the MMPP(3)/M/C model and M/M/C model.
In the M/M/C queueing system, the customer arrival process is Poisson process. When n of binomial distribution is large and p is small, the binomial distribution can be approximated by a Poisson distribution. The Poisson distribution is suitable for describing the random events occurring in a unit of time (or space). The Poisson distribution is one of the most important discrete distributions, and many arrival processes can be approximated by the Poisson process.    In the iron and steel industry, the number of cranes in the slab yard is represented as C; λ is the arrival rate; the service rate of the crane is µ; the time that slabs stay in the queueing system is W ; the time that slabs wait for storage is W q ; the queue length is L q . Based on M/M/C queueing system, an optimal model of queueing system about service load is developed.
The specific expressions in the model are as follows: It can be seen from the model (8) that the objective function and the constraints of it are all non-linear functions with respect to λ and µ. Even the simplest queueing system (M/M/1) is hard-pressed to optimize the performance. Fortunately, we can use the convexity of the queueing system to solve this problem, and the calculation time is polynomial time. Subsequently, we will do some conversion, and get the convex optimization model. First, an intermediate variable t 1 is introduced; the first inequality of model (8) can be VOLUME 8, 2020 transformed to the followings: Let t 1 be a constant number.
Then we can get 1 ≤ 1 Next, we change the second inequality in the constraints of the model (8). The difference from the first inequality is that the left side of this inequality is a polynomial, which needs to be divided before the transformation. In addition to the introduction of variable t 2 , it is also necessary to introduce a variable t 3 to transform the model. Both t 2 and t 3 are positive real numbers. The specific inequality transformation process is as follows: 2 ≤ 1 After a series of transformations, the following set of inequalities can be obtained: 2 ≤ 1 Similarly, the third inequality in the constraint condition can also be transformed to obtain the corresponding set of inequalities. The specific transformation is as follows: Finally, a group of inequalities can be obtained: 4 ≤ 1 While transforming the constraints above, according to the characteristics of the convex optimization objective function, we transform the objective function into min Cµ λ . The similar transformation is done for the other inequalities. The model (8) can be transformed into: The model (10) obtained by transformation is a geometric programming model. At this time, the MATLAB toolkit can be used to optimize the model. In order to more intuitively reflect the characteristics of the convex model from the model, we let λ = eλ, µ = eμ, t 1 = e˜t 1 , t 2 = e˜t 2 , t 3 = e˜t 3 , t 4 = e˜t 4 , Then the final convex optimization model can be obtained: Obviously, the model (10) is one kind of convex optimization model, which can be directly solved by various convex optimization packages. Although the number of variables and constraints of the convex optimization model increase, the convex optimization calculation is still fast. Therefore, the model can still be solved quickly.

VI. ANALYSIS OF NUMERICAL EXAMPLE
The data is from the sensor cloud computing system of the iron and steel industry, which is used to analyze. The production process of slabs is iron making, refining, continuous casting and slab storage. In this paper, we focus on the process of slab storage. As mentioned as before, the arrival rate of slabs is determined by the upstream production. Different steel grades lead to different arrival rate of slabs. We find that the arrival rate varies with the state and the number of states is three. So the MMPP (3)  According to the data of actual production, the total number of cranes in the production line C is 5, which is servers' number of the queueing system. If the number of arrival slabs is too large, the number of cranes may be not enough. Then the slabs need to wait, and there will be waiting costs. If the number of arrival slabs is too small, cranes may not work at full capacity, which will cause the waste of resources. In the model, the unit time is one day. The minimum value of arrival rate (λ min ) is 20; the maximum service rate of each crane (µ max ) is 99; the maximum time that slabs stay in the queueing system (W max ) is 1.8; the maximum time that slabs waited for storage (W q max ) is 1.5; the maximum queue length (L q max ) is 36. Finally, we can get the optimal arrival rate and service rate calculated by MATLAB. The arrival rate is 39.25 and the service rate of each crane is 56.63. This policy optimizes system configuration. The optimal service load is 19.05. In this situation, the performance of the queueing system in the slab storage process is optimal. Through the results of M/M/C queue, we can get the approximate policy to optimize MMPP(3)/M/C queue. Then we can use equation (5) and (6) to assign the arrival rate for the three different levels, while the average arrival rate is 39.25. That is to say, this policy also can make the system configuration of MMPP(3)/M/C queueing system optimal approximately. We applied the proposed convex optimization model to the actual slab storage process of the iron and steel enterprise in China to evaluate its performance. Based on the actual data, we calculate the optimal arrival rate and service rate in slab storage process to optimize the performance of the queueing system. YANHE JIA received the B.Eng. degree in mathematics and applied mathematics from the Shanxi University of Finance and Economics, China, in 2011, the M.Eng. degree in industrial engineering from Southwest Jiaotong University, China, in 2013, and the Ph.D. degree in systems engineering from Northeastern University, China, in 2018. From 2015 to 2017, she was a Visiting Student with Simon Fraser University, Canada. She is currently a Lecturer with the School of Economics and Management, Beijing Information Science & Technology University, China. Her research interests include industrial big data science, queueing systems, random process, data analytics and machine learning, dynamic optimization, convex optimization, logistics planning, production and logistics scheduling and engineering applications in manufacturing, resources industry, logistics systems, and service industry. His research interests include queueing theory and applications, stochastic dynamic programming, probability models in reliability, and supply chain management issues in manufacturing and service organizations. His studies include both theoretical work and a wide range of applications in business, engineering, economics, and applied mathematics. The main theme of his research is to bridge the gap between theory and application, obtaining unobservable and sometimes counter-intuitive but significant/practical management insights via modeling and quantitative analysis. He is one of Editor-in-Chiefs' of Journal of the Operational Research Society (JORS). As one of the premier Operation Research (OR) journals, JORS is the first OR journal in the world. He is also one of the founding Editors-in-Chief of Queueing Models and Service Management. He was an Associate Editor of Information Systems and Operational Research (INFOR). He is on the editorial board of several international journals.
TE XU received the B.Eng. degree in computer science from Northeastern University, China, in 2002, and the M.Eng. and Ph.D. degrees in industrial engineering & systems from Changwon National University, South Korea, in 2006 and 2010, respectively. He is currently a Lecturer with the Key Laboratory of Data Analytics and Optimization for Smart Industry, Northeastern University. His research interests include industrial big data science, data analytics and machine learning, reinforcement learning and dynamic optimization, computational intelligent optimization, plant-wide production and logistics planning, production and logistics batching and scheduling and engineering applications in manufacturing (steel, petroleum-chemical, and nonferrous), energy, resources industry, and logistics systems.