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 queueing systems with heterogeneous servers

We consider the problem of optimal control of queueing systems with heterogeneous servers in parallel. The system objective is to assign customers dynamically to idle servers based on the state of the system so as to minimize the average cost of holding customers. Three cases, either known in the literature or new, are studied in detail: queueing systems with server heterogeneity in-service rates, in-service functions, and both in-service rates and in-service functions. An approach is presented using fuzzy control to solve these problems. Simulation shows that this approach is efficient and promising, especially in cases where analytical solutions do not exist     

Analysis of a finite capacity non preemptive priority queue

In this paper a finite capacity priority queue with multiple servers and non preemptive service discipline is analysed. The arrival and service processes were assumed Markovian and only two priority classes were considered. An analytical solution for blocking probabilities for the customers in the two classes is obtained under the assumption that a high priority customer upon arrival can displace a low priority customer from the waiting line if the queue is full.     

Customer routing to parallel servers with different rates

We consider the problem of routing customers to parallel servers having different rates. There are no buffers in the system. Each customer must be rooted to a server immediately upon its arrival and if the server to which it is routed is occupied, then the customer is aborted. The aim is to maximize throughput (the proportion of customers which are successfully routed to a free server), when the routing must be done without knowing which servers are occupied and which are free. An upper bound on the throughput is found for a general renewal arrival process and geometric service times. Furthermore, a new routing policy, the golden ratio policy, is suggested and shown to approach a limit which is within at least 98.4 percent of the upper bound. The golden ratio policy is a generalization of the round robin policy, when the service rates of the servers are different.     

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     

Performance prediction of loss and delay Markovian queueing model with nopassing and removable additional servers

In this investigation, the loss and delay Markovian queueing system with nopassing is proposed. The customers may balk or renege with certain probability, on finding all servers busy on their arrival. To cope up with the balking and reneging behaviour of the customers, there is provision of removable additional servers apart from permanent servers so as to provide the better grade of service at optimal cost operating conditions. The customers are classified into two classes depending upon whether they can wait or lost when all servers are busy. The customers can also be categorized into two classes from service point of view. Type A customers have zero service time whereas type B customers have exponential service time. The explicit expressions for the average number of customers in the system, the expected waiting time for both types of customers, etc., are derived by using steady-state queue size distribution. Some earlier results are deduced by setting appropriate system parameters. The system behaviour is examined with the help of numerical illustrations by varying different parameters.Scope and purposeThe performance prediction of various systems in communication switching network, remote border security check post, jobs processing in computers, etc., are influenced by the customers behaviour, in particular, when nopassing constraints are prevalent. The incorporation of loss and delay phenomena is likely to bring about understanding whether the customers would like to wait in the queue or would be lost in case when all servers are busy. The provision of additional removable servers will be helpful in upgrading the service and to reduce the discouragement behaviour of the customers in such congestion situations.     

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.     

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     

Machine scheduling with job class setup and delivery considerations

We study machine scheduling problems in which the jobs belong to different job classes and they need to be delivered to customers after processing. A setup time is required for a job if it is the first job to be processed on a machine or its processing on a machine follows a job that belongs to another class. Processed jobs are delivered in batches to their respective customers. The batch size is limited by the capacity of the delivery vehicles and each shipment incurs a transport cost and takes a fixed amount of time. The objective is to minimize the weighted sum of the last arrival time of jobs to customers and the delivery (transportation) cost. For the problem of processing jobs on a single machine and delivering them to multiple customers, we develop a dynamic programming algorithm to solve the problem optimally. For the problem of processing jobs on parallel machines and delivering them to a single customer, we propose a heuristic and analyze its performance bound.     

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     

Optimal scheduling of two competing queues with blocking

Consideration is given to the problem of optimal scheduling for two queues for service at a single station, where the queues have finite capacities and the service rate is class-dependent. The cost structure is linear in the number of holding customers in the queues, combined with blocking costs incurred whenever arrivals encounter a full queue. The authors derive the optimal policy minimizing this criterion when the blocking cost is larger than the holding cost for each queue. Then they show that the optimal policy is of the switching type. Numerical results are included to support the analytical findings     

A tandem queue with general bulk service and servers' “vacation”

This paper analyses two queueing models consisting of two units I and II connected in series, separated by a finite buffer of size N. In both models, unit I has only one exponential server capable of serving customers one at a time and unit II consist of c parallel exponential servers, each of them serving customers in groups according to general bulk service rules. When the queue length in front of unit II is less than the minimum of batch size, the free servers take a vacation. On return from vacation, if the queue length is less than the minimum, they leave for another vacation in the first model, whereas in the second model they wait in the system until they get the minimum number of customers and then start servicing. The steady-state probability vector of the number of customers waiting and receiving service in unit I and waiting in the buffer is obtained for both the models, using the modified matrix geometric method. Numerical results are also presented.     

Routing and scheduling in heterogeneous systems: a sample pathapproach

Consider the problem of routing customers to a set of K parallel servers that have different rates. Each server has a buffer with infinite capacity. The arrival process is general and the service times are assumed to be i.i.d. exponential random variables. Using sample path arguments, we show that, given any Bernoulli policy π, there exists another policy ρ which outperforms π by partially using a randomized version of a round-robin policy. Moreover, ρ is easily specified and implemented. We present extensions of our results to systems with finite capacities and service times that have an increasing hazard rate. Finally, a similar result is shown to hold in the context of scheduling customers from a set of K parallel queues     

Solving fuzzy capacitated location routing problem using hybrid variable neighborhood search and evolutionary local search

A fuzzy capacitated location routing problem (FCLRP) is solved by using a heuristic method that combines variable neighborhood search (VNS) and evolutionary local search (ELS). Demands of the customer and travel times between customers and depots are considered as fuzzy and deterministic variables, respectively in FCLRP. Heterogeneous and homogeneous fleet sizes are performed together to reach the least multi-objective cost in a case study. The multi-objective cost consists of transportation cost, additional cost, vehicle waiting cost and delay cost. A fuzzy chance constrained programming model is added by using credibility theory. The proposed method reaches the solution by performing four stages. In the first stage, initial solutions are obtained by using a greedy heuristic method, and then VNS heuristic, which consists of seven different neighborhood structures, is performed to improve the solution quality in the second stage. In the third stage, a perturbation procedure is applied to the improved solution using ELS algorithm, and then VNS heuristic is applied again in the last stage. The combination of VNS and ELS is called VNSxELS algorithm and applied to a case study, which has fifty-seven customers and five distributing points, effectively in a reasonable time.     

Priority tandem queueing model with admission control

A two-stage multi-server tandem queue with two types of processed customers is analyzed. The input is described by the Marked Markovian Arrival Process (MMAP). The first stage has an infinite number of servers while the second stage has a finite number of servers. The service time at the both stages has an exponential distribution. Priority customers are always admitted to the system. Non-priority customers are admitted to the system only if the number of busy servers at the second stage does not exceed some pre-assigned threshold. Queueing system’s behavior is described in terms of the multi-dimensional asymptotically quasi-Toeplitz continuous time Markov chain. It allows to exploit a numerically stable algorithm for calculation of the stationary distribution of the queueing system. The loss probability at the both stages of the tandem is computed. An economic criterion of the system operation is optimized with respect to the threshold. The effect of control on the main performance measures of the system is numerically demonstrated.     

Approximate analysis of multiqueue systems with multiple cyclic servers

In this paper we consider an approximate model of a system of multiple queues served cyclically by a number of identical servers, which is an extension of the Kuehn model for the single server multiqueueing system. The arrival process of customers is Poissonian, walking and service times are general, servers work in the repeated mode, and they serve at most one customer per visit at a queue (node) in a cycle. Applications in the performance evaluation of the existent network of processors of the Brazilian switching system TRÓPICO are considered. The approximation is validated by means of computer simulations.     

Analysis of an unobservable queue using arrival and departure times

Suppose one wants to assess the efficiency of a queueing system but is unable to observe directly its internal operations. This situation might arise if one works with a restricted set of historical data or because secrecy restricts access to the queueing facility. One might still be able to observe, from outside the system, the exact arrival and departure times of each customer. Using observations of arrival and departure times and knowledge of whether the service discipline is either first-come-first-served or last-come-first-served, one can exactly reconstruct the unobserved queue delays and service times of any sequence of arrivals during busy periods when the number of customers is greater than the number of servers. If the number of servers is also unknown, it too can be estimated. In this paper, we propose optimization models which determine the unknown number of servers.     

Analysis of the busy period in threshold control system

Consideration was given to the controllable Markov queuing system with nonhomogeneous servers and threshold policy of server activation depending on the queue length. By the busy period is meant the time interval between the instant of customer arrival to the empty system and the instant of service completion when the system again becomes free. The system busy period and the number of serviced customers were analyzed. Recurrent relations for the distribution density in terms of the Laplace transform and generating function, as well as formulas for calculation of the corresponding arbitrary-order moments, were obtained.