Zaiming Liu

An M/G/1 retrial G-queue with preemptive resume and feedback under N-policy subject to the server breakdowns and repairs

An M/G/1 retrial G-queue with preemptive resume and feedback under N-policy vacation subject to the server breakdowns and repairs, is investigated in this paper. Arrivals of both positive customers and negative customers are two independent Poisson processes, and positive customers receive service immediately if the server is free upon their arrivals. Otherwise, they may enter a retrial orbit and try their luck after a random time interval. All positive customers must receive preliminary first phase of service and primary second phase of service. While at the preliminary service, the server may push out the customer undergoing such service to the orbit, to commence preliminary service of an arriving positive customer. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for an N-policy vacation as soon as the system empties. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities. The effects of various parameters on the system performance are analyzed numerically.

Some properties of two subclasses of k-fold symmetric functions associated with Srivastava–Attiya operator☆

Abstract In the present paper, by using the Srivastava–Attiya operator, we define two new subclasses of k -fold symmetric analytic functions. Some interesting properties of these subclasses such as integral representations, extreme points, close convex hulls and subordinations are obtained, which generalize and refine some previous results.

M/M/1 retrial queue with collisions and working vacation interruption under N-policy

Consider an M/M/1 retrial queue with collisions and working vacation interruption under N-policy. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.

Some properties of α-convex and α-quasiconvex functions with respect to n-symmetric points☆

The main object of the present paper is to investigate α-convex and α-quasiconvex functions with respect to n-symmetric points. Some interesting properties such as inclusion relationships, integral representations, convolution conditions are obtained.

The MX/M/1 queue with working breakdown

In this paper, we consider a batch arrival M X /M /1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical M X /M / 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.

Equilibrium strategies of the unobservable M/M/1 queue with balking and delayed repairs

The balking strategies for a queue with breakdowns and delayed repairs is studied.We provide supplement to Wang and Zhang (2011) by studying the unobservable cases.The model is viewed as an M/M/1 queue that operates in a random environment.From a methodological point of view, the almost unobservable case is interesting. Wang and Zhang (2011) studied the equilibrium threshold balking strategies for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Customers decide whether to join or balk the system based on observation of the queue length and status of the server at their arrival instants. The present paper aims to study the corresponding unobservable cases in which the queue length is unknown to arriving customers. The model under consideration can be viewed as an M/M/1 queue in a random environment. Equilibrium mixed strategies are derived for the almost unobservable and fully unobservable queues. Finally, we illustrate the effect of several system parameters on the equilibrium behavior via numerical examples.

Heavy-traffic asymptotics of a priority polling system with threshold service policy

In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system with three queues under threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues, which possess the same distributions as a two-class priority queue with N-policy vacation. Further, we provide an approximation of the tail queue-length distribution of the stable queues, which shows that it has the same prefactors and decay rates as the classical two-class preemptive priority queue. Stochastic simulations are taken to support the results. HighlightsThe heavy-traffic behavior of a polling system with threshold policy is discussed.The scaled queue-length of the critically loaded queue is exponentially distributed.Approximations of queue lengths of the stable queues are provided.The decay rates of the stable queues are the same decay rate as the classical preemptive priority queues.

Maximum principle for mean-field zero-sum stochastic differential game with partial information and its application to finance

Abstract In this paper, we investigate the stochastic optimal control problem for the zero-sum stochastic differential game of mean-field type with partial information. We derive a necessary and sufficient maximum principle for that problem by virtue of the duality method and the mean-field backward stochastic differential equations. As an application, we apply the result to the mean-field stochastic differential portfolio game problem, and obtain an equilibrium point of such game.

Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.

Fluid queue driven by a multi-server queue with multiple vacations and vacation interruption

This paper studies a fluid queue driven by a multi-server queue with multiple working vacations and vacation interruption. The stationary distribution of the background environment is obtained after some manipulation. A system of differential equations satisfied by the fluid queue is presented, by which we gain the matrix-geometric structure of the Laplace transform of the stationary buffer content. Furthermore, we derive the explicit expression of the mean buffer content. Finally, the numerical example is employed to illustrate our results.