Queues served in cyclic order: Waiting times
TL;DR: The Laplace-Stieltjes transforms of the order-of-arrival waiting time distribution functions and, for the exhaustive service model, the mean waiting time for a unit arriving at a queue, are obtained.
Abstract: This paper extends the results of a previous paper in which two models of a system of queues served in cyclic order were studied. One model is an exhaustive service model, in which the server waits on all customers in a queue before proceeding to the next queue in cyclic order. The other is a gating model, in which a gate closes behind the waiting units when the server arrives, and the server waits on only those customers in front of the gate, deferring service of later arrivals until the next cycle. In the present paper, the Laplace-Stieltjes transforms of the order-of-arrival waiting time distribution functions and, for the exhaustive service model, the mean waiting time for a unit arriving at a queue, are obtained.
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TL;DR: This survey gives an overview of some general decomposition results and the methodology used to obtain these results for two vacation models and attempts to provide a methodological overview to illustrate how the seemingly diverse mix of problems is closely related in structure and can be understood in a common framework.
Abstract: Queueing systems in which the server works on primary and secondary (vacation) customers arise in many computer, communication, production and other stochastic systems. These systems can frequently be modeled as queueing systems with vacations. In this survey, we give an overview of some general decomposition results and the methodology used to obtain these results for two vacation models. We also show how other related models can be solved in terms of the results for these basic models. We attempt to provide a methodological overview with the objective of illustrating how the seemingly diverse mix of problems is closely related in structure and can be understood in a common framework.
1,136 citations
Cites methods from "Queues served in cyclic order: Wait..."
...One approach to obtain the waiting time distribution, used by Eisenberg [10], is to embed at all service completion epochs; another, used by Cooper [4], is to use the available generating functions to get the generating function of the queue length in queue i when the server arrives there....
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TL;DR: In this article, the authors considered a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time and showed that the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables.
Abstract: This paper considers a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. As has been noted by other researchers, for several specific models of this type, the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables, one of which is the stationary number of customers present in the standard M/G/1 queue i.e., the server is always available at a random point in time. In this paper we demonstrate that this type of decomposition holds, in fact, for a very general class of M/G/1 queueing models. The arguments employed are both direct and intuitive. In the course of this work, moreover, we obtain two new results that can lead to remarkable simplifications when solving complex M/G/1 queueing models.
664 citations
TL;DR: There are also disclosed a new desoldering tool, a power cylinder, clamping fixtures and suction cups operable with the vacuum pump of the invention.
Abstract: This paper studies an M/G/1 queue where the idle time of the server is utilized for additional work in a secondary system. As usual, the server is busy as long as there are units in the main system...
398 citations
TL;DR: The cyclic polling model, its enhancement by customer routing, and the replacement of a fixed polling order by a random polling order are reviewed.
Abstract: The cyclic polling model, its enhancement by customer routing, and the replacement of a fixed polling order by a random polling order are reviewed. Modeling of polling systems, performance improvement, and system optimization issues are discussed. Examples are given that include token rings, ARQ and time-sharing schemes, random-access protocols, robotics and manufacturing systems. Emphasis is not on the analytical derivations of polling systems but rather on the description of the capabilities and limitations of the different polling models. >
343 citations
IBM1
TL;DR: An overview of the state of the art of polling model analysis is presented, in particular, single- buffer systems and infinite-buffer systems with exhaustive, gated, and limited service disciplines are treated.
Abstract: A polling model is a system of multiple queues accessed by a single server in cyclic order. Polling models provide performance evaluation criteria for a variety of demand-based, multiple-access schemes in computer and communication systems. This paper presents an overview of the state of the art of polling model analysis, as well as an extensive list of references. In particular, single-buffer systems and infinite-buffer systems with exhaustive, gated, and limited service disciplines are treated. There is also some discussion of systems with a noncyclic order of service and systems with priority. Applications to computer networks are illustrated, and future research topics are suggested.
327 citations
Additional excerpts
...P) * (14) ACM Computing Surveys, Vol. 20, No. 1, March 1988 12 l Hideaki Takagi Note that the case in which N = 1 is an M/G/l queuing model with a server that goes on vacations [Cooper 1970; Doshi 1986; Scholl and Kleinrock 1983; Skinner 19671....
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TL;DR: In this paper, the authors study two models of a system of queues served in cyclic order by a single server, and find expressions for the mean number of units in a queue at the instant it starts service, the mean cycle time, and the Laplace-Stieltjes transform of the cycle time distribution function.
Abstract: We study two models of a system of queues served in cyclic order by a single server. In each model, the ith queue is characterized by general service time distribution function H i (·) and Poisson input with parameter Λ i . In the exhaustive service model, the server continues to serve a particular queue until the server becomes idle and there are no units waiting in that queue; at this time the server advances to and immediately starts service on the next nonempty queue in the cyclic order. The gating model differs from the exhaustive service model in that when the server advances to a nonempty queue, a gate closes behind the waiting units. Only those units waiting in front of the gate are served during this cycle, with the service of subsequent arrivals deferred to the next cycle. We find expressions for the mean number of units in a queue at the instant it starts service, the mean cycle time, and the Laplace-Stieltjes transform of the cycle time distribution function.
150 citations
TL;DR: It is supposed that customers of type 1 and type 2 arrive at a service system in accordance with a Poisson process and the service times have a general distribution.
Abstract: It is supposed that customers of type 1 and type 2 arrive at a service system in accordance with a Poisson process. There are two counters in the system. Customers of type 1 receive service at coun...
104 citations
TL;DR: The alternating priority rule is compared to the "head-of-the-line" and the "first-come, first-served" rules with respect to expected queuing times and queue sizes and the results are related to the case where switching from one type of customers to another is not penalized by setup time.
Abstract: For a service facility serving more than one class of customers a problem arises as to the order in which customers should be served i.e., the priority rule. A typical example is a manufacturing facility that produces two products to meet a random stream of incoming orders. In this paper the alternating priority rule also known as the "zero switch rule" is introduced and investigated. Two classes of customers are served by a single service facility. Customers of class ii = 1, 2 have priority over customers of class jj = 1, 2; j â 1 whenever a class i customer is in service. When the service facility is idle, the first arriving customer enters service and acquires the priority right for customers of his class. Within classes the "first-come, first-served" rule is observed. Customers' arrivals are assumed to be Poisson and service times are assumed to be arbitrarily distributed independent random variables. The steady-state densities of queuing times are formulated by the use of a special mathematical procedure. The expectations of queuing times and sizes of queues as well as the first two moments of the busy periods are obtained in terms of the basic parameters of the arrival process and service time distributions. All the results are related to the case where switching from one type of customers to another is not penalized by setup time. The alternating priority rule is compared to the "head-of-the-line" and the "first-come, first-served" rules with respect to expected queuing times and queue sizes. The last part of the paper discusses the possibilities of extending the suggested rule to cases involving more than two classes of customers and nonzero setup times and setup costs.
73 citations