Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.     

Flow control using the theory of zero sum Markov games

Considers the problem of dynamic flow control of arriving packets into an infinite buffer. The service rate may depend on the state of the system, may change in time, and is unknown to the controller. The goal of the controller is to design an efficient policy which guarantees the best performance under the worst service conditions. The cost is composed of a holding cost, a cost of rejecting customers (packets), and a cost that depends on the quality of the service. The problem is studied in the framework of zero-sum Markov games, and a value iteration algorithm is used to solve it. It is shown that there exists an optimal stationary policy (such that the decisions depend only on the actual number of customers in the queue); it is of a threshold type, and it uses randomization in at most one state     

Optimal control of parallel queues with impatient customers

We consider a queueing system with a number of identical exponential servers. Each server has its own queue with unlimited capacity. The service discipline in each queue is first-come-first-served (FCFS). Customers arrive according to a state-dependent Poisson process with an arrival rate which is a non-increasing function of the number of customers in the system. Upon arrival, a customer must join a server’s queue according to a stationary state-dependent policy, where the state is taken to be the number of customers in servers’ queues. No jockeying among queues is allowed. Each arriving customer is limited to a generally distributed patience time after which it must depart the system and is considered lost. Two models of customer behavior are considered: deadlines until the beginning of service and deadlines until the end of service. We seek an optimal policy to assign an arriving customer to a server’s queue. We show that, when the distribution of customer impatience satisfies certain property, the policy of joining shortest queue (SQ) stochastically minimizes the number of lost customers during any finite interval in the long run. This property is shown to always hold for the case of deterministic customer impatience.     

Fuzzy control of arrivals to tandem queues with two stations

We consider two queues in tandem, each of which has its own input of arriving customers, which, in turn, may either be accepted or rejected. Suppose that the system receives a fixed reward for each accepted customer and pays a holding cost per customer per unit time in the system. The objective is to dynamically determine the optimal admission policies based on the state of the system so as to maximize the average profit (reward minus cost) over an infinite horizon. This model finds applications in flow control in communication systems, industrial job shops, and traffic-flow systems. In this paper, a novel approach is presented using fuzzy control. This approach explicitly finds the optimal policy, which up until now has defied analytical solutions     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.     

串联排队网络顾客准入的模糊控制

研究一个由两个服务器串联而成的排队网络 .每个服务器拥有其自己的到达顾客群 ,到达顾客可以被接收或被拒绝进入系统 .假设这个系统每接收一个顾客收到一个固定的收益 ,同时为系统中的每个顾客每单位时间付一笔占位费 .控制目标是根据系统的状态动态地确定最优准入策略以保证系统平均利润最大 .这个控制模型可以应用在通讯系统的流量控制、Job shop及交通控制中 .本文提出一模糊控制新方法解决这个问题 .     

串联排队网络顾客准入的模糊控制

研究一个由两个服务器串联而成的排队网络.每个服务器拥有其自己的到达顾客 群,到达顾客可以被接收或被拒绝进入系统.假设这个系统每接收一个顾客收到一个固定的 收益,同时为系统中的每个顾客每单位时间付一笔占位费.控制目标是根据系统的状态动态 地确定最优准入策略以保证系统平均利润最大.这个控制模型可以应用在通讯系统的流量控 制、Job shop及交通控制中.本文提出一模糊控制新方法解决这个问题.     

Equilibrium balking strategies in renewal input batch arrival queues with multiple and single working vacation

This paper investigates equilibrium threshold balking strategies of customers in a renewal input batch arrival queue with multiple and single working vacation of the server. The vacation period, service period during normal service and vacation period are considered to be independent and exponentially distributed. Upon arriving, the customers decide whether to join or balk the queue based on observation of the system-length and status of the server. The waiting time in the system is associated with a linear cost–reward structure for estimating the net benefit if a customer wishes to participate in the system. Equilibrium customer strategy is studied under four cases: fully observable, almost observable, almost unobservable and fully unobservable. Using embedded Markov chain approach and system cost analysis, we obtain the equilibrium threshold. The analysis of unobservable cases is based on the roots of the characteristics equations formed using the probability generating function of embedded pre-arrival epoch probabilities. Equilibrium balking strategy may be useful in quality of service for EPON (ethernet passive optical network) as a multiple working vacation model and accounting through gatekeeper based H.323 protocols as a single working vacation model.     

Optimal scheduling in some multiqueue single-server systems

The authors address the problem of optimal scheduling in a multiqueue single-server (MQSS) model. An arriving customer joins queue j with probability 1/N, j=1,. . ., N. The server visits each queue for a random period of time whose duration is independent of the queue length. At the end of a visit period, either all customers of the attended queue leave the system (variant I), or only customers that were present in the queue upon the arrival of the server leave the system (variant II). A scheduling policy is a rule that selects the next queue to be visited by the server. When the controller has no information on the state of the system, it is shown that a cyclic policy minimizes the expected number of customers in the system. When the controller knows the number of customers in each queue, it is shown that the so-called most customers first (MCF) policy minimizes, in the sense of strong stochastic ordering, the vector of the number of customers in each queue whose components are arranged in decreasing order     

Decomposition of Queueing Networks with Dependent Service and Negative Customers

Queueing networks with negative customers (G-networks) and dependent service at different nodes are studied. Every customer arriving at the network is defined by a set of random parameters: his route over the network (a sequence of nodes visited by the customers), route length, and volume and service length of the customer at every stage of the route. For G-networks, which are the analogs of BCMP-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form.     

On waiting for simultaneous access to two resources: Deterministic service distribution

Suppose that a test customer in anM/D/1queueing system can get service only if he has access to the server and a separate eventEhas occurred. All other customers only require access to the server. The time until the eventEoccurs is assumed to be an exponentially distributed random variable, if the test customer reaches the server beforeEoccurs, he must then return to the back of the queue. At any time, however, the test customer is allowed to give up his place in the queue and join the back of the queue. The test customer represents a computational task that depends upon the results of an associated task. The test customer's mean delay until service is derived assuming that he always maintains his position in the queue until he reaches the server. Conditions are given for which this "move-along" policy is optimal, i.e., minimizes the test customer's mean delay until service. A condition is also given for which the move-along policy is not optimal.     

Optimal control of a queueing system with two heterogeneous servers

The problem considered is that of optimally controlling a queueing system which consists of a common buffer or queue served by two servers. The arrivals to the buffer are Poisson and the servers are both exponential, but with different mean service times. It is shown that the optimal policy which minimizes the mean sojourn time of customers in the system is of threshold type. The faster server should be fed a customer from the buffer whenever it becomes available for service, but the slower server should be utilized if and only if the queue length exceeds a readily computed threshold value.     

面向在线客服系统的调度算法研究

Different from traditional customer service systems, online customer service systems offer business services for multiple customers simultaneously, which makes the adaptation and scheduling between service providers and customers a big challenge. Based on the characteristics of online customer service, this paper proposes a scheduling model for online customer service systems. The scheduling model is composed of three constituents: a multi-priority customer queue, the states of the scheduling system and the transition relations between them, and the correspondence between scheduling strategies and states of the system. Its scheduling algorithm is designed. Experiments verify the rationality of the scheduling model and the effectiveness of the scheduling algorithm. In comparison to the operating customer service system, the algorithm can not only considerably reduce the average waiting time of customers, but also achieve load balancing among service providers, when guaranteeing high quality of services.     

Optimal routing and buffer allocation for a class of finitecapacity queueing systems

The problem of routing jobs to K parallel queues with identical exponential servers and unequal finite buffer capacities is considered. Routing decisions are taken by a controller which has buffering space available to it and may delay routing of a customer to a queue. Using ideas from weak majorization, it is shown that the shorter nonfull queue delayed (SNQD) policy minimizes both the total number of customers in the system at any time and the number of customers that are rejected by that time. The SNQD policy always delays routing decisions as long as all servers are busy. Only when all the buffers at the controller are occupied is a customer routed to the queue with the shortest queue length that is not at capacity. Moreover, it is shown that, if a fixed number of buffers is to be distributed among the K queues, then the optimal allocation scheme is the one in which the difference between the maximum and minimum queue capacities is minimized, i.e. becomes either 0 or 1     

Equilibrium joining strategies in the single server queues with negative customers

We consider an M/M/1 queue with negative customers. An arriving negative customer will break the server down and the positive customer being served (if any) is forced to leave the system. Once a breakdown occurs, the server is sent immediately for repair while positive customers are not allowed to join the system during the repair process. When the server is available, positive arrivals decide whether to join or balk the system based on a common reward-cost structure. We consider an observable case that the positive arrivals are informed about the number of customers in the system and an unobservable case without any information. The corresponding Nash equilibrium strategies and the socially optimal joining strategies are explored. We get a socially optimal threshold in the observable case and a mixed joining strategy in the unobservable case. The profit maximization issue is studied, and we derive optimal strategies in two information cases. Finally, numerical examples are provided to show the influence of different parameters on the strategies and social benefit.     

Parameter estimation for product-form distributions of queueing networks

Basic parameters of a queueing network are its routing matrix, arrival flow rate, and service rates at network nodes. To estimate these parameters, one has to solve a system of balance equations. In turn, a product-form limiting distribution of the number of customers at the network nodes is defined through loading factors. Therefore, in the paper we propose to estimate loading factors through estimates of the limiting distribution based on observations of the number of customers at the nodes. This makes it possible to avoid solving a system of balance equations. This algorithm is realized for Jackson networks: classical, in a random environment, with blocked transitions.     

The stationary distribution invariance of states in a closed queueing network with temporarily non-active customers

A stationary functioning of a closed queueing network with temporarily non-active customers is analyzed. Non-active customers are located at network nodes in queues, being not serviced. For a customer, the feasibility of passing from its ordinary state to the temporarily non-active state (and backwards) is provided. Service times of customers at different nodes possess arbitrary distributions. Finally, the stationary distribution invariance of network states is established with respect to the functional form of customer service time distributions under fixed first-order moments.     

Optimal control of service for facilities holding inventory

This paper addresses the problem of optimally controlling service rates for an inventory system of service facilities. We consider a finite capacity system with Poisson arrivals and exponentially distributed leadtimes and service times. For given values of maximum inventory and reorder levels, we determine the service rates to be employed at each instant of time so that the long-run expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. We establish the existence of a stationary optimal policy and we solve it by employing linear programming. Several instances of a numerical example, which provide insight into the behaviour of the system, are presented.Scope and purposeIn this article we discuss the problem of inventory control of service parts at a service facility where there is only a limited waiting space for customers. If a customer enters the service facility and sees all the waiting spaces occupied he/she will leave the facility, which results in both intangible losses (loss of goodwill) and tangible losses (loss in profit). Hence, the service provider aims at obtaining an optimal rate at which service is to be provided by balancing costs due to waiting time and limited waiting spaces against costs due to ordering and overheads due to storing items. We develop an algorithm that controls the service rate as a function of the number of customers waiting for service.     

Allocation of distinguishable servers

The model considered is a variation of the Hypercube model in which there are R distinguishable servers and N types of customers. We consider two interesting special cases—light and heavy traffic intensities—and use asymptotic techniques to derive policies for assigning servers to arriving customers which are optimal under these conditions. For light traffic, we show that it is optimal to assign the server with the smallest assignment cost while, for heavy traffic, we derive an efficient algorithm for finding the optimal policy.