JIMO
Optimal execution strategy with an endogenously determined sales period
Guibin Lu Qiying Hu Youying Zhou Wuyi Yue
We discuss the problem of the optimal liquidation of a financial product in which both the market risk of the asset and the market impact of the investor's own dealings are considered, and where the asset is liquidated over several sales periods with a constant sales interval. The investor chooses the sales volume in each period as well as the volume over the entire sales period in order to minimize the expected execution costs under a certain level of risk. We obtain an explicit solution for the optimal execution strategies and present four numerical examples to show that the proportion between market risk and liquidity risk exerts a major influence over the optimal execution strategy. We also show that to obtain the optimal result the investor should liquidate his holdings over a short sales period.
keywords: optimal execution strategy market impact dynamic programming. sales period market risk
JIMO
A dual tandem queueing system with GI service time at the first queue
Zsolt Saffer Wuyi Yue
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.
keywords: Queueing theory M/G/1-type Markov process tandem queueing model G/M/1-type Markov process.
JIMO
Optimal control for resource allocation in discrete event systems
Qiying Hu Wuyi Yue
Supervisory control for discrete event systems (DESs) belongs essentially to the logic level for control problems in DESs. Its corresponding control task is hard. In this paper, we study a new optimal control problem in DESs. The performance measure is to maximize the maximal discounted total reward among all possible strings (i.e., paths) of the controlled system. The condition we need for this is only that the performance measure is well defined. We then divide the problem into three sub-cases where the optimal values are respectively finite, positive infinite and negative infinite. We then show the optimality equation in the case with a finite optimal value. Also, we characterize the optimality equation together with its solutions and characterize the structure of the set of all optimal policies. All the results are still true when the performance measure is to maximize the minimal discounted total reward among all possible strings of the controlled system. Finally, we apply these equations and solutions to a resource allocation system. The system may be deadlocked and in order to avoid the deadlock we can either prohibit occurrence of some events or resolve the deadlock. It is shown that from the view of the maximal discounted total cost, it is better to resolve the deadlock if and only if the cost for resolving the deadlock is less than the threshold value.
keywords: resource allocation system. discrete event systems Optimal control
JIMO
Performance analysis of a P2P storage system with a lazy replica repair policy
Shunfu Jin Yuan Zhao Wuyi Yue Lingling Chen
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.
keywords: three-dimensional Markov chain lazy replica repair benefit function. matrix-geometric method P2P storage system
DCDS-B
Optimal control for discrete event systems with arbitrary control pattern
Qiying Hu Wuyi Yue
In this paper, we present a new model for optimal control of discrete event systems (DESs) with an arbitrary control pattern. Here, a discrete event system is defined as a collection of event sets that depend on strings. When the system generates a string, the next event that may occur should be in the corresponding event set. In the optimal control model, there are rewards for choosing control inputs at strings and the sets of available control inputs also depend on strings. The performance measure is to find a policy under the condition where the discounted total reward among strings from the initial state is maximized. By applying ideas from Markov decision processes, we divide the problem into three sub-cases where the optimal value is respectively finite, positive infinite and negative infinite. For the case with finite optimal values, the optimality equation is shown and further characterized with its solutions. We also characterize the structure of the set of all optimal policies. Moreover, we discuss invariance and closeness of several languages. We present a new supervisory control problem of DESs with the control pattern being dependent on strings. We study the problem in both the event feedback control and the state feedback control by generalizing concepts of invariant and closed languages/predicates. Finally, we apply the above model and results to a job-matching problem.
keywords: Markov decision processes. supervisory control problem Optimal control arbitrary control patterns optimality equation
JIMO
Two new optimal models for controlling discrete event systems
Qiying Hu Wuyi Yue
Supervisory control belongs essentially to the logic level for control problems in discrete event systems (DESs) and its corresponding control task is hard. This is unlike many practical optimal control problems which belong to the performance level and whose control tasks are soft. In this paper, we present two new optimal control problems of DESs: one with cost functions for choosing control inputs, and the other for occurring events. Their performance measures are to minimize the maximal discounted total cost among all possible strings that the system generates. Since this is a nonlinear optimization problem, we model such systems by using Markov decision processes. We then present the optimality equations for both control problems and obtain their optimal solutions. When the cost functions are stationary, we show that both the optimality equations and their solutions are also stationary. We then use these equations and solutions to describe and solve uniformly the basic synthesizing problems in the two branches of the supervisory control area: those being the event feedback control and the state feedback control. Moreover, we show that the control invariant languages and the control invariant predicates with their permissive supervisors and state feedbacks not only have meanings in supervisory control of DESs, but are also the optimal solutions for some optimal control problems. This shows a link existing between the logic level and the performance level for the control of discrete event systems. Finally, a numerical example is given to illustrate some results for supervisory control of a DES.
keywords: Optimal control discrete event systems state feedback. supervisor
NACO
Preface
Wuyi Yue Herwig Bruneel Bong Dae Choi Shoji Kasahara
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Yutaka Takahashi on the occasion of his 60th birthday and in recognition of his fundamental contributions in queueing theory and network applications. It is a great honor and a pleasure for the Guest Editors to have this privilege to edit this Special Issue.

For more information please click the “Full Text” above.
keywords:
JIMO
M/M/c multiple synchronous vacation model with gated discipline
Zsolt Saffer Wuyi Yue
In this paper we present the analysis of an M/M/c multiple synchronous vacation model. In contrast to the previous works on synchronous vacation model we consider the model with gated service discipline and with independent and identically distributed vacation periods. The analysis of this model requires different methodology compared to those ones used for synchronous vacation model so far. We provide the probability-generating function and the mean of the stationary number of customers at an arbitrary epoch as well as the Laplace-Stieljes transform and the mean of the stationary waiting time. The stationary distribution of the number of busy servers and the stability of the system are also considered. In the final part of the paper numerical examples illustrate the computational procedure.
    This vacation queue is suitable to model a single operator controlled system consisting of more machines. Hence the provided analysis can be applied to study and optimize such systems.
keywords: Queueing theory multi-server queue service discipline. synchronous vacation model
JIMO
A heterogeneous two-server network system with balking and a Bernoulli vacation schedule
Dequan Yue Wuyi Yue
In this paper, we study a two-server Markovian network system with balking and a Bernoulli schedule under a single vacation policy, where servers have different service rates. After every service, only one server may take a vacation or continue to stay in the system. The vacation time follows an exponential distribution. An arriving customer finding both servers free will choose the faster server. If the customer finds only one server is free, this customer chooses this free server. If the customer finds both servers are not free, then this customer may join the system or balk. For this system, we obtain the steady state condition, the stationary distribution of the number of customers in the system, and the mean system size by using a matrix-geometric method. Some special cases are deduced, which match with earlier exiting results. Extensive numerical illustrations are provided. Motivation for this system model also comes from some computer communication networks with different types of traffic such as real-time traffic and non-real-time traffic, where messages can be processed by two channels (servers) with different transmission rates. The behavior of abandoning messages can be equated with the balking of customers in this system model.
keywords: heterogeneous servers single vacation Network systems Bernoulli schedule matrix-geometric solution. balking
JIMO
Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns
Dequan Yue Wuyi Yue
In this paper, we present analysis for an M/M/R/N queueing system with balking, reneging and server breakdowns. The server is subject to breakdowns with different Poisson breakdown rates $\alpha_0 $ and $\alpha$ for the empty period of the system and the nonempty period of the system, respectively. When the server breaks down, it will be repaired immediately by a repair facility attended by $R$ repairmen. The repair times of the servers are assumed to follow a negative exponential distribution with different repair rates $\beta_0$ and $\beta$ corresponding to whether the server breaks down in the empty period of the system and the nonempty period of the system. We study not only some queueing problems of the system, but also some reliability problems of the servers. By using the partitioned block matrix method, we solved the steady-state probability equations iteratively and derived the steady-state probabilities in a matrix form. Some performance measures of queueing and reliability are obtained. A cost model is developed to determine the optimum number of servers while the system availability is maintained at a certain level. The cost analysis is also investigated by numerical results.
keywords: Queueing system availability server breakdowns. balking reneging

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