Abstract: Queueing problems in which a customer having received a unit of service, returns to the waiting line, under some decision rule, to receive another unit of service occur often in applications. Inspection procedures provide such a framework for units that must be reworked. A large class of such problems appears in computer modelling under the name of round-robin models and foreground-background models. In the present paper, such a system is referred to as a queue with instantaneous feedback. Throughout the text it is assumed that there are two independent Poisson arrival processes giving two types of customers. Each type of customer has its own service time distribution. The decision to feedback to receive another unit of service or not is based on the type of customer completing service. Conditions for the existence of a steady state queue length are found and some joint and marginal distribution of these queue lengths are given. Moreover it is shown that several earlier results in queueing and computer modelling can be obtained simply from the results given here. A particular case of the foreground-background model in computer systems analysis serves as an example.

Abstract: We consider a two-server queuing system in which the service time distributions of the two servers are different. Customer arrivals are Poisson distributed. Customers renege if their wait in queue exceeds T (a random time). Customers arriving when the system is empty can be served by either of the two- servers. Under policy i, they are served by server i, i = 1, 2. Our objective is to find the policy that maximizes the number of customers served. Let server 1's service time be stochastically less than server 2's service time. Intuitively, the optimal policy is policy 1; a counter-example shows that this is not always true. A TWO-SERVER heterogeneous queuing system is considered. Customer arrivals are Poisson distributed. Customers renege if their wait in queue exceeds T (a random time). When an arriving customer finds both servers free, a policy determines which server serves him. The optimal policy maximizes the fraction of customers served. We consider two cases. Case 1 has two servers, each having exponentially distributed service times with different rates. Customers renege (leave the

Abstract: We consider a two-server queuing system in which the service time distributions of the two servers are different. Customer arrivals are Poisson distributed. Customers renege if their wait in queue exceeds T a random time. Customers arriving when the system is empty can be served by either of the two servers. Under policy i, they are served by server i, i = 1, 2. Our objective is to find the policy that maximizes the number of customers served. Let server 1's service time be stochastically less than server 2's service time. Intuitively, the optimal policy is policy 1; a counter-example shows that this is not always true.

Abstract: We consider a discrete time queuing system composed of a set of servers in parallel in which both customers and servers may be of several types. Customers in service may be switched from one server to another at the beginning of each period. We consider dynamic assignment rules and derive conditions that ensure that it is optimal (in a very strong sense) always to assign customers with longer service times to faster servers.

Abstract: This paper describes the behavior of communication on a land mobile radio channel and suggests a mathematical model to be used in analyzing such communication. Mobile radio is gaining wide recognition as an important business tool, but its growing use brings the problem of congestion. Although the communication on one radio channel by twelve different companies has the structure of a complex queuing system not all of which can be observed, the paper shows that such communication has some of the behavior of a simple queue. This facilitates the analysis of the radio channel and provides an illustration that models of simple queues may offer a helpful means of analyzing those which are more complicated. Once the situation can be analyzed it becomes possible to try to manage it as effectively as possible. In the case of mobile radio the analysis provides a means of measuring the cost of congestion under alternate circumstances.

Abstract: We consider a queuing system in which both customers and servers may be of several types. The distribution of a customer's service time is assumed to depend on both the customer's type and the type of server to which he is assigned. For a model with two servers and two customer types, conditions are presented which ensure that the discounted number of service completions is maximized by assigning customers with longer service times to faster servers. Generalizations to more complex models are discussed.

Abstract: In order to provide for the safe and expeditious passage of maritime traffic in congested waters, the U.S. Coast Guard is authorized by the Ports and Waterways Safety Act of 1972 to establish, operate, and maintain Vessel Traffic Services (VTS) where needed. In larger areas, a VTS will generally require a communications system to enable the vessel traffic center and the participating vessels to exchange information. In designing such a system, it is necessary to assess the expected communications loading in order to determine frequency requirements and evaluate alternative configurations for the system. Here, VTS communications are viewed as a queuing system. The customers (messages) arrive at the service facility (communications channel) according to some probabilistic process, and are then serviced (transmitted) according to some other probabilistic process. Queues or waiting lines form as arriving messages wait to be transmitted, because the communications channels are busy. Three classes of messages are considered in the arrival process: check in/check out (basic VTS) messages; Vessel Movement Reporting System (VMRS) messages; and bridge-to-bridge messages. Each class is characterized by an independent Poisson distribution, and the resultant arrival process is a well-defined nonhomogeneous Poisson process. The service time is characterized by a general distribution with a known mean and variance. The queuing results, which are developed, include the utilization factor, the expected waiting time, and the expected number of messages waiting to be transmitted. The arrival process and the queuing results vary according to the time of the day, reflecting the varying traffic load throughout the day. A detailed example is given for a preliminary analysis of New York Harbor VTS communications.

Abstract: We consider a finite-capacity single-server queuing system in which the problems of capacity design and service quality control are integrated. The amount of time spent on servicing a customer is used as a measure of the quality of service provided. The expected reward from servicing is assumed to be a nondecreasing function of the service duration, while a linear holding cost is assumed for customers waiting in the system. We maximize the long-run average return per unit time by optimally controlling the quality of service to be provided as a function of the workload facing the server for different waiting-room capacities. We show that the optimal control policy for a given capacity is monotone and also examine the effect of varying the system capacity on the optimal control policy. Furthermore, we analyze the design problem of selecting an optimal capacity by assuming that an optimal service control policy will be followed for each system capacity.

Abstract: In order to provide for the safe and expeditious passage of maritime traffic in congested waters, the U.S. Coast Guard is authorized by the Ports and Waterways Safety Act of 1972 to establish, operate, and maintain Vessel Traffic Services (VTS) where needed. In larger areas, a VTS will generally require a communications system to enable the vessel traffic center and the participating vessels to exchange information. In designing such a system, it is necessary to assess the expected communications loading in order to determine frequency requirements and evaluate alternative configurations for the system. Here, VTS communications are viewed as a queuing system. The customers (messages) arrive at the service facility (communications channel) according to some probabilistic process, and are then serviced (transmitted) according to some other probabilistic process. Queues or waiting lines form as arriving messages wait to be transmitted, because the communications channels are busy. Three classes of messages are considered in the arrival process: check in/check out (basic VTS) messages; Vessel Movement Reporting System (VMRS) messages; and bridge-to-bridge messages. Each class is characterized by an independent Poisson distribution, and the resultant arrival process is a well-defined nonhomogeneous Poisson process. The service time is characterized by a general distribution with a known mean and variance. The queuing results, which are developed, include the utilization factor, the expected waiting time, and the expected number of messages waiting to be transmitted. The arrival process and the queuing results vary according to the time of the day, reflecting the varying traffic load throughout the day. A detailed example is given for a preliminary analysis of New York Harbor VTS communications.

Abstract: This paper deals with the continuous simulation of complex queuing systems on a digital computer. Con tinuous-system mathematical models of multicompart ment queuing systems based on a birth-death process consist of large sets of differential equations. A dynamic allocation algorithm can substantially de crease the required computer time and storage by con tinuously adjusting the size of the system of differ ential equations. The applicability of the continuous approach to simulation of complex queuing systems is demonstrated for a four-compartment outpatient clinic.

Abstract: The optimization of throughput in locally balanced queueing network models is investigated. A general result, useful in the design of computer system models, shows that throughput is a nondecreasing function of the number of customers contained in any subnetwork. Then processor allocation algorithms that maximize throughput are shown for the case where processing power can be switched between queues, as when several queues are served at a single multiprocessor system. The maximization of throughput is shown first in the case that processing power allocations to a queue depend on the queue state only, and then, in an extension of known locally balanced queue, the case in which processing power is allocated on the basis of an entire subnetwork state. The latter case provides a simple and optimum rule for processor allocations that maximize throughput in networks containing multiprocessor systems.

Abstract: This paper considers tandem queuing system in which the order of performing service at two serial service stations can be changed. Interarrival distribution and service distribution of each station are assumed to be ex­ ponential. For the above system, we derive the mean queue length and the mean availability per station for finite queue case. And some numerical results will be attached.

Abstract: We consider a queuing system consisting of a finite number of identical exponential servers. Each server has its own queue, and upon arrival each customer must be assigned to some server's queue. Under the assumption that no jockeying between queues is permitted, it is shown that the intuitively satisfying rule of assigning each arrival to the shortest line maximizes, with respect to stochastic order, the discounted number of customers to complete their service in any time t.

Abstract: We consider a queuing network with Poisson arrivals at each node. At each service completion epoch, a reward is received and the serviced customer changes nodes or leaves the system according to specified probabilities. In addition, linear holding costs are incurred. The problem is to schedule the server so as to maximize the expected discounted reward over an infinite planning horizon. This model is equivalent to a single-server, multi-class queuing system with feedback of the customers. We study two cases: general service times with a non-preemptive service discipline and exponential service times with a preemptive service discipline. For each case we show that a modified static policy of priority form is optimal and we provide an algorithm for computing an optimal policy.