# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

January 2014 , Volume 10 , Issue 1

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2014, 10(1): 1-19 doi: 10.3934/jimo.2014.10.1 +[Abstract](1219) +[PDF](612.4KB)
Abstract:
In this paper, we consider a decode-and-forward wireless relay network consisting of a source node, a destination node and multiple relay nodes under Rayleigh fading. Cooperative diversity of the network is achieved by using an opportunistic relay selection. To maximize the benefit of cooperative diversity, we propose a novel cross-layer single relay selection scheme, called the First-Channel-and-Second-Buffer (FCSB) scheme, that considers the performances of both the physical and data link layers. To examine the performance of the proposed FCSB scheme, we develop an analytical model that describes the queueing process of the source node, and derive the average packet delay of the FCSB scheme from our analytical model. Our analytical model is verified through simulation. We discuss how to optimally design the proposed FCSB scheme to minimize the average packet delay. We also show that the optimized FCSB scheme outperforms other relay selection schemes from the viewpoint of average packet delay.
2014, 10(1): 21-40 doi: 10.3934/jimo.2014.10.21 +[Abstract](1395) +[PDF](412.5KB)
Abstract:
In cognitive radio networks, secondary spectrum users detect available frequency channels by spectrum sensing. In general, the sensing time is communication overhead, and affects system's performance. In this paper, we theoretically consider the effect of sensing overhead on the system performance for cognitive radio networks with channel bonding. Specifically, we model the system with a multidimensional continuous-time Markov chain whose state is defined by the numbers of primary users, secondary users, and sensing users. The blocking probability, the forced termination probability and the throughput are derived. The analysis is validated by Monte Carlo simulation. Numerical examples show that the forced termination probability is not affected by sensing overhead, while the blocking probability and the throughput degrade with the increase in the sensing time. It is also shown that the optimal number of bonded sub-channels for the throughput performance significantly depends on the offered load from primary users.
2014, 10(1): 41-55 doi: 10.3934/jimo.2014.10.41 +[Abstract](2343) +[PDF](214.3KB)
Abstract:
This paper develops a catastrophe equity put (CatEPut) option model under realistic assumptions. To reflect the phenomena of real data, we adopt the following assumptions. First, following the reasoning in Lin and Wang [12], we assume that the loss index follows a compound Poisson process with jumps of a mixture of Erlangs. Second, the volatility of stock return is assumed to be stochastic as in Heston [8]. Under the assumptions, we derives a pricing formula for CatEPut options. Numerical examples are given to insist that the pricing formula can be easily implemented numerically. We also confirm the validity and accuracy of implementation of the pricing formula by comparing the numerical results obtained by the pricing formula with those obtained by the Monte Carlo simulation.
2014, 10(1): 57-87 doi: 10.3934/jimo.2014.10.57 +[Abstract](1481) +[PDF](492.8KB)
Abstract:
We consider a FIFO single-server queue with disasters and multiple Markovian arrival streams. When disasters occur, all customers are removed instantaneously and the system becomes empty. Both the customer arrival and disaster occurrence processes are assumed to be Markovian arrival processes (MAPs), and they are governed by a common underlying Markov chain with finite states. There are $K$ classes of customers, and the amounts of service requirements brought by arriving customers follow general distributions, which depend on the customer class and the states of the underlying Markov chain immediately before and after arrivals. For this queue, we first analyze the first passage time to the idle state and the busy cycle. We then obtain two different representations of the Laplace-Stieltjes transform of the stationary distribution of work in system, and discuss the relation between those. Furthermore, using the result on the workload distribution, we analyze the waiting time and sojourn time distributions, and derive the joint queue length distribution.
2014, 10(1): 89-112 doi: 10.3934/jimo.2014.10.89 +[Abstract](1863) +[PDF](459.4KB)
Abstract:
In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.
2014, 10(1): 113-129 doi: 10.3934/jimo.2014.10.113 +[Abstract](1638) +[PDF](410.4KB)
Abstract:
2014, 10(1): 131-149 doi: 10.3934/jimo.2014.10.131 +[Abstract](1437) +[PDF](594.2KB)
Abstract:
In this paper, we analyse the behaviour of a discrete-time single-server queueing system with general service times, equipped with the $NT$-policy. This is a threshold policy designed to reduce the number of service unit activation/deactivation cycles, whilst ensuring an acceptable delay trade-off. Once the server is deactivated, reactivation will be postponed until either $N$ customers have accumulated in the queue or the first customer has been in the queue for $T$ slots, whichever happens first. Due to this modus operandi, the system circulates between three phases: empty, accumulating and serving.
We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.
2014, 10(1): 151-166 doi: 10.3934/jimo.2014.10.151 +[Abstract](1698) +[PDF](300.8KB)
Abstract:
Peer-to-Peer (P2P) storage systems are a prevalent and important mode for implementing cost-efficient, large-scale distributed storage. Considering the random departure feature of the peers and the diverse popularity of the data objects, a proper number of replicas needs to be maintained, and a reasonable trigger threshold of replica repair needs to be set for high data availability and low system overhead. In this paper, based on the working principle of the lazy replica repair policy in a P2P storage system, a three-dimensional Markov chain model is constructed, and the model is analyzed in steady-state by using a matrix-geometric method. Then, the performance measures in terms of the availability of one data object, the average access latency, and the replication rate are given. Moreover, numerical results with analysis are provided to demonstrate how system parameters such as the replica number and the replica repair instant influence the system performance. Finally, we develop benefit functions to optimize the replica number and the repair trigger threshold.
2014, 10(1): 167-192 doi: 10.3934/jimo.2014.10.167 +[Abstract](1753) +[PDF](500.6KB)
Abstract:
In this paper we consider the analysis of a tandem queueing model $M/G/1 -> ./M/1$. In contrast to the vast majority of the previous literature on tandem queuing models we consider the case with GI service time at the first queue and with infinite buffers. The system can be described by an M/G/1-type Markov process at the departure epochs of the first queue. The main result of the paper is the steady-state vector generating function at the embedded epochs, which characterizes the joint distribution of the number of customers at both queues. The steady-state Laplace-Stieljes transform and the mean of the sojourn time of the customers in the system are also obtained.
We provide numerical examples and discuss the dependency of the steady-state mean of the sojourn time of the customers on several basic system parameters. Utilizing the structural characteristics of the model we discuss the interpretation of the results. This gives an insight into the behavior of this tandem queuing model and can be a base for developing approximations for it.
2014, 10(1): 193-206 doi: 10.3934/jimo.2014.10.193 +[Abstract](2021) +[PDF](585.1KB)
Abstract:
In this paper the focus is on class clustering" in a continuous-time queueing model with two classes and dedicated servers. Class clustering" means that customers of any given type may (or may not) have a tendency to arrive back-to-back". We believe this is a concept that is often neglected in literature and we want to show that it can have a considerable impact on multiclass queueing systems, especially on the system considered in this paper. This system adopts a global FCFS" service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. The major aim of our paper is to quantify the intuitively expected (due to the service discipline) negative impact of class clustering" on the performance measures of our system. The motivation of our work are systems where this kind of inherent blocking is encountered, such as input-queueing network switches, road splits or security checks at airports.
2014, 10(1): 207-218 doi: 10.3934/jimo.2014.10.207 +[Abstract](1572) +[PDF](371.6KB)
Abstract:
In this paper, we consider a variance-optimal hedge for target volatility options, under exponential Lévy dynamics. Since the payoff of target volatility options is related with realized volatility of some underlying asset, which is path-dependent, it is difficult to price this instrument. Here we will derive an explicit Föllmer-Schweizer decomposition of the contingent claim of target volatility options and then give the explicit expressions of hedging strategies in both discrete time and continuous time.
2014, 10(1): 219-241 doi: 10.3934/jimo.2014.10.219 +[Abstract](2065) +[PDF](466.3KB)
Abstract:
This paper presents a review of single machine scheduling to minimize the weighted number of tardy jobs. The problem involves processing n jobs on a single machine, each having processing time $p_j$ and due date $d_j$. The aim is to schedule the jobs to meet their due date. A job is tardy if the completion time of job $j$ is $C_j>d_j$ and on-time otherwise. This paper assesses works done to minimize the weighted number of tardy jobs by providing an extensive review of authors, methods and techniques used. Finally, the possible direction for future research is presented.
2014, 10(1): 243-258 doi: 10.3934/jimo.2014.10.243 +[Abstract](1994) +[PDF](304.0KB)
Abstract:
In this paper, we introduce a new approach based on DC (Difference of Convex functions) Programming and DCA (DC Algorithm) for minimizing the maintenance cost involving flow-time and tardiness penalties by optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. The main idea is to divide the horizon considered into $H$ intervals. The problem is first formulated as a mixed integer linear problem (MILP) and then reformulated as a DC program. A solution method based on DCA is used to solve the resulting problem. The efficiency of DCA is compared with the algorithm based on the new flow-time and tardiness rule (FTR) given in [1]. The computational results on several test problems show that the solutions provided by DCA are better.
2014, 10(1): 259-273 doi: 10.3934/jimo.2014.10.259 +[Abstract](1594) +[PDF](366.4KB)
Abstract:
We consider the parallel machine scheduling problem in which the finished jobs need to be delivered to a customer in batches by a single vehicle. The goal is to minimize the makespan, i.e., the time by which the vehicle has delivered all jobs and returned to its initial location. We distinguish two types of batching strategies. The strategy of Type 1 permits the jobs processed on different machines to compose a delivery batch, and the strategy of Type 2 assumes that only the jobs processed on the same machine can compose a batch. For both types of the $m$-machine case, we propose $(2-\frac{1}{m})$-approximation algorithms respectively. For both types of the two-machine case, we obtain two improved $\frac{4}{3}$-approximation algorithms.
2014, 10(1): 275-309 doi: 10.3934/jimo.2014.10.275 +[Abstract](4052) +[PDF](598.1KB)
Abstract:
The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.
2014, 10(1): 311-336 doi: 10.3934/jimo.2014.10.311 +[Abstract](2125) +[PDF](630.1KB)
Abstract:
We study convergence properties of Euler discretization of optimal control problems with ordinary differential equations and mixed control-state constraints. Under suitable consistency and stability assumptions a convergence rate of order $1/p$ of the discretized control to the continuous control is established in the $L^p$-norm. Throughout it is assumed that the optimal control is of bounded variation. The convergence proof exploits the reformulation of first order necessary optimality conditions as nonsmooth equations.
2014, 10(1): 337-362 doi: 10.3934/jimo.2014.10.337 +[Abstract](1406) +[PDF](10159.8KB)
Abstract:
We use concepts and techniques of network optimization theory to gain a better understanding of force transmission in dense granular materials. Specifically, we represent a deforming granular material over the different stages of a quasi-static biaxial compression test as a series of representative flow networks, and analyze force transmission through these networks. The forces in such a material are transmitted through the contacts between the constituent grains. As the sample deforms during the various stages of the biaxial test, these grains rearrange: while many contacts are preserved in this rearrangement process, some new contacts form and some old contacts break. We consider the maximum flow problem and the minimum cost maximum flow (MCMF) problem for the flow networks constructed from this evolving network of grain contacts. We identify the flow network bottleneck and establish the sufficient and necessary conditions for a minimum cut of the maximum flow problem to be unique. We also develop an algorithm to determine the MCMF pathway, i.e. a set of edges that always transmit non-zero flow in every solution of the MCMF problem. The bottlenecks of the flow networks develop in the locality of the persistent shear band, an intensively-studied phenomenon that has long been regarded as the signature failure microstructure for dense granular materials. The cooperative evolution of the most important structural building blocks for force transmission, i.e. the force chains and 3-cycles, is examined with respect to the MCMF pathways. We find that the majority of the particles in the major load-bearing columnar force chains and 3-cycles consistently participate in the MCMF pathways.

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