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Journal of Industrial and Management Optimization

October 2017 , Volume 13 , Issue 4

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A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations
Elham Mardaneh, Ryan Loxton, Qun Lin and Phil Schmidli
2017, 13(4): 1601-1623 doi: 10.3934/jimo.2017009 +[Abstract](6258) +[HTML](631) +[PDF](725.38KB)

This paper introduces a non-standard vehicle routing problem (VRP) arising in the oil and gas industry. The problem involves multiple offshore production facilities, each of which requires regular servicing by support vessels to replenish essential commodities such as food, water, fuel, and chemicals. The support vessels are also required to assist with oil off-takes, in which oil stored at a production facility is transported via hose to a waiting tanker. The problem is to schedule a series of round trips for the support vessels so that all servicing and off-take requirements are fulfilled, and total cost is minimized. Other constraints that must be considered include vessel suitability constraints (not every vessel is suitable for every facility), depot opening constraints (base servicing can only occur during specified opening periods), and off-take equipment constraints (the equipment needed for off-take support can only be deployed after certain commodities have been offloaded). Because of these additional constraints, the scheduling problem under consideration is far more difficult than the standard VRP. We formulate a mixed-integer linear programming (MILP) model for determining the optimal vessel schedule. We then verify the model theoretically and show how to compute the vessel utilization ratios for any feasible schedule. Finally, simulation results are reported for a real case study commissioned by Woodside Energy Ltd, Australia's largest dedicated oil and gas company.

On subspace properties of the quadratically constrained quadratic program
Xin Zhao and Jinyan Fan
2017, 13(4): 1625-1640 doi: 10.3934/jimo.2017010 +[Abstract](3925) +[HTML](439) +[PDF](392.3KB)

In this paper, we study subspace properties of the quadratically constrained quadratic program (QCQP). We prove that, if an appropriate subspace is chosen to satisfy subspace properties, then the solution of the QCQP lies in that subspace. So, we can solve the QCQP in that subspace rather than solve it in the original space. The computational cost could be reduced significantly if the dimension of the subspace is much smaller. We also show how to construct such subspaces and investigate their dimensions.

Optimal pension decision under heterogeneous health statuses and bequest motives
Lin He and Zongxia Liang
2017, 13(4): 1641-1659 doi: 10.3934/jimo.2017011 +[Abstract](3184) +[HTML](476) +[PDF](433.65KB)

In this paper, we study the optimal decision between ELA(Equity Linked Annuity) and ELID(Equity Linked Income Drawdown) pension plans under heterogeneous personal health statuses and bequest motives. In the ELA pension plan, the survival member receives the mortality credit, and leaves no bequest at the time of death, while the member receives no mortality credit and receives the fund wealth as bequest at the time of death in the ELID pension plan. The pension member controls the asset allocation and benefit outgo policies to achieve the objectives. We explore the square deviations between the actual benefit outgo and the pre-set target, and the square and negative linear deviations between the actual bequest and the pre-set target as the disutility function. The minimization of the disutility function is the objective of the stochastic optimal control problem. Using HJB (Hamilton-Jacobi-Bellman) equations and variational inequality methods, the closed-form optimal policies of the ELA and ELID pension plans are derived. Furthermore, the optimal decision boundary between the ELA and ELID plans is established. It is the first time to study the impacts of heterogeneous personal health status and bequest motive on the optimal choice between the ELA and ELID pension plans under the original performance criterions. The worse health status and higher bequest motive result in the higher utility of the ELID pension plan, and vice versa. The worse heath status increases the proportion allocated in the risky asset and increases the benefit outgo in both pension plans. The bequest motive has positive impacts on the proportion in the risky asset and negative impacts on the benefit outgo in the ELID pension plan.

Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit
Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang and Yung-Fu Huang
2017, 13(4): 1661-1683 doi: 10.3934/jimo.2017012 +[Abstract](3749) +[HTML](516) +[PDF](470.55KB)

In today's competitive markets, the supplier let the buyer to pay the purchasing cost after receiving the items, this strategy motivates the retailer to buy more items from the supplier and gains some benefit from the money which they did not pay at the time of receiving of the items. However, the retailer will be unable to pay off the debt obligations to the supplier in the future, so this study extends Yen et al. (2012) to consider the above situation and assumes the retailer can either pay off all accounts at the end of the delay period or delay incurring interest charges on the unpaid and overdue balance due to the difference between interest earned and interest charged. We will discuss the explorations of the function behaviors of the objection function to demonstrate the retailer's optimal replenishment cycle time not only exists but also is unique. Finally, numerical examples are given to illustrate the theorems and gained managerial insights.

Continuity of approximate solution maps to vector equilibrium problems
Lam Quoc Anh, Pham Thanh Duoc and Tran Ngoc Tam
2017, 13(4): 1685-1699 doi: 10.3934/jimo.2017013 +[Abstract](3775) +[HTML](383) +[PDF](367.2KB)

This paper considers the parametric primal and dual vector equilibrium problems in locally convex Hausdorff topological vector spaces. Based on linear scalarization technique, we establish sufficient conditions for the continuity of approximate solution maps to these problems. As applications, some new results for vector optimization problem and vector variational inequality are derived. Our results are new and improve the existing ones in the literature.

Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization
Roya Soltani, Seyed Jafar Sadjadi and Mona Rahnama
2017, 13(4): 1701-1721 doi: 10.3934/jimo.2017014 +[Abstract](5603) +[HTML](597) +[PDF](493.35KB)

This study makes use of the artificial intelligence approaches combined with some nonlinear optimization techniques for optimization of a well-known problem in financial engineering called yield curve. Yield curve estimation plays an important role on making strategic investment decisions. In this paper, we use two well-known parsimonious estimation models, Nelson-Siegel and Extended Nelson-Siegel, for the yield curve estimation. The proposed models of this paper are formulated as continuous nonlinear optimization problems. The resulted models are then solved using some nonlinear optimization and meta-heuristic approaches. The optimization techniques include hybrid GPSO parallel trust region-dog leg, Hybrid GPSO parallel trust region-nearly exact, Hybrid GPSO parallel Levenberg-Marquardt and Hybrid genetic electromagnetism like algorithm. The proposed models of this paper are examined using some real-world data from the bank of England and the results are analyzed.

An extension of hybrid method without extrapolation step to equilibrium problems
Van Hieu Dang
2017, 13(4): 1723-1741 doi: 10.3934/jimo.2017015 +[Abstract](4000) +[HTML](465) +[PDF](470.61KB)

In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method and the hybrid (outer approximation) method. In this algorithm, only one optimization program is solved at each iteration without any extra-step dealing with the feasible set like as in the hybrid extragradient method and the hybrid Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in the proposed algorithm. The strongly convergent theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.

Structure analysis on the k-error linear complexity for 2n-periodic binary sequences
Jianqin Zhou, Wanquan Liu and Xifeng Wang
2017, 13(4): 1743-1757 doi: 10.3934/jimo.2017016 +[Abstract](3320) +[HTML](487) +[PDF](400.39KB)

In this paper, in order to characterize the critical error linear complexity spectrum (CELCS) for \begin{document}$2^n$\end{document}-periodic binary sequences, we first propose a decomposition based on the cube theory. Based on the proposed \begin{document}$k$\end{document}-error cube decomposition, and the famous inclusion-exclusion principle, we obtain the complete characterization of the \begin{document}$i$\end{document}th descent point (critical point) of the k-error linear complexity for \begin{document}$i=2,3$\end{document}. In fact, the proposed constructive approach has the potential to be used for constructing \begin{document}$2^n$\end{document}-periodic binary sequences with the given linear complexity and \begin{document}$k$\end{document}-error linear complexity (or CELCS), which is a challenging problem to be deserved for further investigation in future.

Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy
Fanwen Meng, Kiok Liang Teow, Chee Kheong Ooi, Bee Hoon Heng and Seow Yian Tay
2017, 13(4): 1759-1770 doi: 10.3934/jimo.2017017 +[Abstract](4823) +[HTML](493) +[PDF](368.52KB)

Timely access of care has been widely recognized as an important dimension of health care quality. Waiting times can affect patient satisfaction and quality of care in the emergency department (ED). This study analyzes a general patient waiting policy such that ED patients who wait beyond a threshold have their wait shortened. Assuming that the policy is implemented to accelerate the long-waiting cases within a short time interval, we transform the original waiting distribution to a piecewise distribution. The objective of this paper is to examine the reliability of the induced waiting system by minimizing the coefficient of variation (CV) of waiting time. We convert the CV minimization problem to an approximation counterpart using the sampling technique. With patient waiting time data from an emergency department in Singapore, we derive the optimal values of parameters, such as the threshold and the length of the underlying time interval, needed in the policy. Numerical results show that CV and variance of new waiting time will be reduced remarkably by 38% and 58% respectively, in comparison with the original ones.

Minimizing expected time to reach a given capital level before ruin
Xiaoqing Liang and Lihua Bai
2017, 13(4): 1771-1791 doi: 10.3934/jimo.2017018 +[Abstract](3465) +[HTML](406) +[PDF](563.4KB)

In this paper, we consider the optimal investment and reinsurance problem for an insurance company where the claim process follows a Brownian motion with drift. The insurer can purchase proportional reinsurance and invest its surplus in one risky asset and one risk-free asset. The goal of the insurance company is to minimize the expected time to reach a given capital level before ruin. By using the Hamilton-Jacobi-Bellman equation approach, we obtain explicit expressions for the value function and the optimal strategy. We also provide some numerical examples to illustrate the results obtained in this paper, and analyze the sensitivity of the parameters.

A numerical scheme for pricing American options with transaction costs under a jump diffusion process
Donny Citra Lesmana and Song Wang
2017, 13(4): 1793-1813 doi: 10.3934/jimo.2017019 +[Abstract](5354) +[HTML](542) +[PDF](565.47KB)

In this paper we develop a numerical method for a nonlinear partial integro-differential complementarity problem arising from pricing American options with transaction costs when the underlying assets follow a jump diffusion process. We first approximate the complementarity problem by a nonlinear partial integro-differential equation (PIDE) using a penalty approach. The PIDE is then discretized by a combination of a spatial upwind finite differencing and a fully implicit time stepping scheme. We prove that the coefficient matrix of the system from this scheme is an M-matrix and that the approximate solution converges to the viscosity solution to the PIDE by showing that the scheme is consistent, monotone, and unconditionally stable. We also propose a Newton's iterative method coupled with a Fast Fourier Transform for the computation of the discretized integral term for solving the fully discretized system. Numerical results will be presented to demonstrate the convergence rates and usefulness of this method.

An uncertain wage contract model for risk-averse worker under bilateral moral hazard
Xiulan Wang, Yanfei Lan and Wansheng Tang
2017, 13(4): 1815-1840 doi: 10.3934/jimo.2017020 +[Abstract](3293) +[HTML](451) +[PDF](458.14KB)

This paper investigates a wage mechanism design problem faced by a risk neutral firm (he) who employs a risk averse worker (she) to sell products for him. The effort levels of both the firm and the worker are unobservable to each other, which results in bilateral moral hazard. The firm offers a wage contract menu to the worker with the objective of maximizing his expected profit. The results show that the firm will provide the same wage contract to the worker when the worker's effort is observable regardless of the market condition being full or private information. The optimal wage contract is related to the worker's risk averse level when the bilateral moral hazard exists. The information values of the worker's effort and the market condition are studied, respectively. The results show that the firm benefits from the worker's observable effort under full information and only when the sales uncertainty is sufficiently low, can the firm profit from that under private information. Moreover, only if the cost coefficient of the firm's effort is sufficiently high, the firm can benefit from full information in the scenario when the worker's effort is unobservable.

Incremental gradient-free method for nonsmooth distributed optimization
Jueyou Li, Guoquan Li, Zhiyou Wu, Changzhi Wu, Xiangyu Wang, Jae-Myung Lee and Kwang-Hyo Jung
2017, 13(4): 1841-1857 doi: 10.3934/jimo.2017021 +[Abstract](3978) +[HTML](469) +[PDF](457.57KB)

In this paper we consider the minimization of the sum of local convex component functions distributed over a multi-agent network. We first extend the Nesterov's random gradient-free method to the incremental setting. Then we propose the incremental gradient-free methods, including a cyclic order and a randomized order in the selection of component function. We provide the convergence and iteration complexity analysis of the proposed methods under some suitable stepsize rules. To illustrate our proposed methods, extensive numerical results on a distributed $l_1$-regression problem are presented. Compared with existing incremental subgradient-based methods, our methods only require the evaluation of the function values rather than subgradients, which may be preferred by practical engineers.

Non-convex semi-infinite min-max optimization with noncompact sets
Meixia Li, Changyu Wang and Biao Qu
2017, 13(4): 1859-1881 doi: 10.3934/jimo.2017022 +[Abstract](3555) +[HTML](439) +[PDF](468.2KB)

In this paper, first we study the non-convex sup-type functions with noncompact sets. Under quite mild conditions, the expressions of its derivative and subderivative along arbitrary direction are given. Furthermore, the structure of its subdifferential is characterized completely. Then, using these results, we establish first-order optimality conditions for semi-infinite min-max optimization problems. These results generalize and improve the corresponding results in the relevant literatures.

A new analytical model for optimized cognitive radio networks based on stochastic geometry
Seunghee Lee and Ganguk Hwang
2017, 13(4): 1883-1899 doi: 10.3934/jimo.2017023 +[Abstract](3779) +[HTML](441) +[PDF](440.84KB)

In this paper, we consider an underlay type cognitive radio network with multiple secondary users who contend to access multiple heterogeneous licensed channels. With the help of stochastic geometry we develop a new analytical model to analyze a random channel access protocol where each secondary user determines whether to access a licensed channel based on a given access probability. In our analysis we introduce the so-called interference-free region to derive the coverage probability for an arbitrary secondary user. With the help of the interference-free region we approximate the interferences at an arbitrary secondary user from primary users as well as from secondary users in a simple way. Based on our analytical model we obtain the optimal access probabilities that maximize the throughput. Numerical examples are provided to validate our analysis.

Analysis of a discrete-time queue with general service demands and phase-type service capacities
Michiel De Muynck, Herwig Bruneel and Sabine Wittevrongel
2017, 13(4): 1901-1926 doi: 10.3934/jimo.2017024 +[Abstract](4481) +[HTML](439) +[PDF](540.38KB)

In this paper, we analyze a non-classical discrete-time queueing model where customers demand variable amounts of work from a server that is able to perform this work at a varying rate. The service demands of the customers are integer numbers of work units. They are assumed to be independent and identically distributed (i.i.d.) random variables. The service capacities, i.e., the numbers of work units that the server can process in the consecutive slots, are also assumed to be i.i.d. and their common probability generating function (pgf) is assumed to be rational. New customers arrive in the queueing system according to a general independent arrival process. For this queueing model we present an analysis method, which is based on complex contour integration. Expressions are obtained for the pgfs, the mean values and the tail probabilities of the customer delay and the system content in steady state. The analysis is illustrated by means of some numerical examples.

Single server retrial queues with speed scaling: Analysis and performance evaluation
Tuan Phung-Duc, Wouter Rogiest and Sabine Wittevrongel
2017, 13(4): 1927-1943 doi: 10.3934/jimo.2017025 +[Abstract](3825) +[HTML](423) +[PDF](449.82KB)

Recently, queues with speed scaling have received considerable attention due to their applicability to data centers, enabling a better balance between performance and energy consumption. This paper proposes a new model where blocked customers must leave the service area and retry after a random time, with retrial rate either varying proportionally to the number of retrying customers (linear retrial rate) or non-varying (constant retrial rate). For both, we first study a basic case and then subsequently incorporate the concepts of a setup time and a deactivation time in extended versions of the model. In all cases, we obtain a full characterization of the stationary queue length distribution. This allows us to evaluate the performance in terms of the mentioned balance between performance and energy, using an existing cost function as well as a newly proposed variant thereof. This paper presents the derivation of the stationary distribution as well as several numerical examples of the cost-based performance evaluation.

Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue
Hideaki Takagi
2017, 13(4): 1945-1973 doi: 10.3934/jimo.2017026 +[Abstract](3832) +[HTML](472) +[PDF](339.9KB)

We present a unified and refined analysis of the response time and waiting time in the M/M/$ m $ FCFS preemptive-resume priority queueing system in the steady state by scrutinizing and extending the previous studies such as Brosh (1969), Segal (1970), Buzen and Bondi (1983), Tatashev (1984), and Zeltyn et al. (2009). In particular, we analyze the durations of interleaving waiting times and service times during the response time of a tagged customer of each priority class that is preempted by the arrivals of higher-priority class customers. Our new contribution includes the explicit formulas for the second and third moments of the response time and the third moment of the waiting time. We also clarify the dependence between the waiting time and the total service time. Numerical examples are shown in order to demonstrate the computation of theoretical formulas.

An integrated inventory model with variable transportation cost, two-stage inspection, and defective items
Biswajit Sarkar, Bijoy Kumar Shaw, Taebok Kim, Mitali Sarkar and Dongmin Shin
2017, 13(4): 1975-1990 doi: 10.3934/jimo.2017027 +[Abstract](5933) +[HTML](548) +[PDF](1170.2KB)

The paper deals with an integrated inventory model with make-to-order policy from buyer to vendor. A variable transportation cost is used as a power function of the delivery quantity for either tapering or considering proportional rate data to maintain single-setup multi-delivery (SSMD) policy with reduced transportation cost. A two-stage inspection process is introduced by the vendor to ensure the perfect quality of product even though the first inspection process indicates a constant defective rate of imperfect production is present during the long-run production system and all defective items are reworked with some fixed cost. The aim is to minimize the total cost of the integrated inventory model by using classical optimization technique. Two numerical examples, sensitivity analysis, and graphical representations are given to illustrate the model.

Prox-dual regularization algorithm for generalized fractional programs
Mostafa El Haffari and Ahmed Roubi
2017, 13(4): 1991-2013 doi: 10.3934/jimo.2017028 +[Abstract](3161) +[HTML](473) +[PDF](429.8KB)

Prox-regularization algorithms for solving generalized fractional programs (GFP) were already considered by several authors. Since the standard dual of a generalized fractional program has not generally the form of GFP, these approaches can not apply directly to the dual problem. In this paper, we propose a primal-dual algorithm for solving convex generalized fractional programs. That is, we use a prox-regularization method to the dual problem that generates a sequence of auxiliary dual problems with unique solutions. So we can avoid the numerical difficulties that can occur if the fractional program does not have a unique solution. Our algorithm is based on Dinkelbach-type algorithms for generalized fractional programming, but uses a regularized parametric auxiliary problem. We establish then the convergence and rate of convergence of this new algorithm.

An efficient Tabu Search neighborhood based on reconstruction strategy to solve the blocking job shop scheduling problem
Adel Dabah, Ahcene Bendjoudi and Abdelhakim AitZai
2017, 13(4): 2015-2031 doi: 10.3934/jimo.2017029 +[Abstract](5287) +[HTML](547) +[PDF](484.69KB)

The blocking job shop scheduling problem (BJSS) is a version of the classical job shop scheduling with no intermediate buffer between machines. The BJSS is known to be NP-hard in the strong sense. A known way to solve such a problem is to use the Tabu Search algorithm (TS) which is a higher level heuristic procedure for solving optimization problems, designed to guide other methods to escape the trap of local optimality. However, the use of the classical TS neighborhood on BJSS problem produces infeasible solutions in most cases (98% of cases). This leads to waste valuable time in exploring infeasible solutions. To overcome this drawback, we propose a new tabu search neighborhood based on reconstruction strategy. This neighborhood consists to remove arcs causing the infeasibility and rebuild the neighbor solutions by using heuristics. Experiments on the reference benchmark instances show that the TS algorithm using the proposed neighborhood improves most of the known results in the literature and gives new upper bounds for more than 52 benchmarks in both BJSS cases (BJSS with Swap and BJSS no-Swap). Moreover, the proposed approach reaches much faster the optimal solution for most of the optimally solved benchmarks.

Stability strategies of manufacturing-inventory systems with unknown time-varying demand
Lizhao Yan, Fei Xu, Yongzeng Lai and Mingyong Lai
2017, 13(4): 2033-2047 doi: 10.3934/jimo.2017030 +[Abstract](4610) +[HTML](403) +[PDF](529.4KB)

For a manufacturing-inventory system, its stability and robustness are of particular important. In the literature, most manufacturing-inventory models are constructed based on deterministic demand assumption. However, demands for many real-world manufacturing-inventory systems are non-deterministic. To minimize the gap between theory and practice, we construct two models for the inventory control problem involving multi-machine and multi-product manufacturing-inventory systems with uncertain time-varying demand, where physical decay rate and shelf life are accounted for in the models. We then design state feedback control strategies to stabilize such systems. Based on the Lyapunov stability theory, sufficient conditions for robust stability, stabilization and control are derived in the form of linear matrix inequalities. Numerical examples are presented to show the potential applications of the proposed models.

A cooperative game with envy
Jiahua Zhang, Shu-Cherng Fang, Yifan Xu and Ziteng Wang
2017, 13(4): 2049-2066 doi: 10.3934/jimo.2017031 +[Abstract](4312) +[HTML](495) +[PDF](413.15KB)

This paper proposes an envy-incorporated cooperative game model and investigates the envy effects on players' coalition-forming and payoff-allocating decisions. A player's utility depends on her/his own payoff and the envy toward other players, inside and outside the same coalition, with higher payoffs. Envy core is defined to characterize stable coalition structures and payoff allocations of this new game. Conditions for the envy core to be nonempty are provided. The relative significance of in-coalition envy and out-coalition envy is shown to be a key factor to the form of the envy core. Application to a simple game shows that envy may significantly change players' decisions.

Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller
Valeria Artale, Cristina L. R. Milazzo, Calogero Orlando and Angela Ricciardello
2017, 13(4): 2067-2091 doi: 10.3934/jimo.2017032 +[Abstract](5067) +[HTML](478) +[PDF](2891.44KB)

In the present paper different approaches for flight control stabilization tuning of a prototype of Unmanned Aerial Vehicle (UAV) are investigated. The mathematical model of a multirotor airframe is presented and its dynamical system is described by means of Newton-Euler equations for a rigid body. The proposed controller systems for the stabilization of the altitude and the attitude of the multirotor around its hovering configuration are based on Proportional Derivative (PD) regulator tuned by optimization techniques such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) or, for the linearized system, by means of Linear Quadratic Regulator (LQR). In this case, PSO and GA are applied to set entries of LQR scheme. Control flight simulation results have been compared in terms of minimization of work carried out by the control actions. Performed simulations pointed out that all approaches lead to zero the error of the position along gravity acceleration direction, stop the rotation of UAV around body axes and stabilize the multirotor. The direct PSO tuning of the PD parameters (PD-PSO) shows better results in terms of errors, performance and speed of convergence.

Light-tailed asymptotics of GI/G/1-type Markov chains
Tatsuaki Kimura, Hiroyuki Masuyama and Yutaka Takahashi
2017, 13(4): 2093-2146 doi: 10.3934/jimo.2017033 +[Abstract](3526) +[HTML](396) +[PDF](831.2KB)

This paper studies the light-tailed asymptotics of the stationary distribution of the GI/G/1-type Markov chain. We consider three cases:(ⅰ) the tail decay rate is determined by a certain parameter \begin{document}$\theta$\end{document} associated with the transition block matrices \begin{document}$\{\boldsymbol{A}(k);k=0,\pm1,\pm2,\dots\}$\end{document} in the non-boundary levels; (ⅱ) by the convergence radius of the generating function of the transition block matrices \begin{document}$\{\boldsymbol{B}(k);k=1,2,\dots\}$\end{document} in the boundary level; and (ⅲ) by the convergence radius of \begin{document}$\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$\end{document}. In the case (ⅰ), we extend the existing asymptotic formula for the M/G/1-type Markov chain to the GI/G/1-type one. In the case (ⅱ), we present general asymptotic formulas that include, as special cases, the existing results in the literature. In the case (ⅲ), we derive new asymptotic formulas. As far as we know, such formulas have not been reported in the literature.

2021 Impact Factor: 1.411
5 Year Impact Factor: 1.441
2021 CiteScore: 2.1




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