Longest path analysis in networks of queues: Dynamic scheduling problems (original) (raw)
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Distribution function of the shortest path in networks of queues
OR Spectrum, 2005
In this paper, we consider acyclic networks of queues as a model to support the design of a dynamic production system. Each service station in the network represents a manufacturing or assembly operation. Only one type of product is produced by the system, but there exist several distinct production processes for manufacturing this product, each one corresponding with a directed path in the network of queues. In each network node, the number of servers in the corresponding service station is either one or infinity. The service time in each station is either exponentially distributed or belongs to a special class of Coxian distribution. Only in the source node, the service system may be modeled by an M/G/∞ queue. The transport times between every pair of service stations are independent random variables with exponential distributions. In method proposed in this paper, the network of queues is transformed into an equivalent stochastic network. Next, we develop a method for approximating the distribution function of the length of the shortest path of the transformed stochastic network, from the source to the sink node. Hence, the method leads to determining the distribution function of the time required to complete a product in this system (called the manufacturing lead time). This is done through solving a system of linear differential equations with non-constant coefficients, which is obtained from a related continuous-time Markov process. The results are verified by simulation.
Performance Evaluation, 1993
Nelson, R.D. and T.K. Philips, An approximation for the mean response time for shortest queue routing with general interarrival and service times, Performance Evaluation 17 (1993) 123-139. In this paper we derive an approximation for the mean respone time of a multiple queue system in which shortest queue routing is used. We assume there are K identical queues with infinite capacity. Interarrival and service times are generally distributed, and an arriving job is routed to a queue of minimal length. Our approximation is a simple closed form equation that requires only the mean and coefficient of variation of job's interarrival and service times. The approximation is extensively compared to simulated values for values of K ~< 8, and has small relative errors, typically less than 5%, for systems where the coefficient of variation of interarrival and serivce times are both ~< 1. For the system consisting of Poisson arrivals and exponential service times, we extend the approximation so that the error is less than one half of one percent for K~<8.
Queueing networks: customers, signals and product form solutions
1999
Networks of queues (= stochastic networks) have been a field of intensive research over the last three decades. The foundation for this research is classical queueing theory, which dealt mainly with single node queueing systems [4, 5, 8]. There is now a well-developed theory of stochastic networks accompanied by an unbounded set of open problems which originates directly from applications as well as from theoretical considerations. Most of these open problems are easily stated and put into a theoretical framework, but they often require either intricate techniques on an ad hoc basis or deep mathematical methods. Often, both of these approaches have to be combined to tackle successfully the solution of quickly formulated problem. The development of queueing network theory, which provided application areas with solutions, formulas, and algorithms, commenced around 1950 in the area of Operations Research, with special emphasis on production, inventory, and transportation. The first breakthroughs were the works of Jackson [10] and Gordon and Newell [10]; both papers appeared in Operations Research. The second breakthrough in queueing network theory was already connected with Computer Science: the celebrated papers of Baskett, Chandy, Muntz, and Palacios [2], which appeared in the Journal of the Association for Computing Machinery, and of Kelly [11]. The two volumes of Kleinrock's book [13, 14] appeared at the same time as [11]. From that
Networks of queues in discrete time
Zeitschrift für Operations Research, 1983
We study a new class of networks of queues whose nodes operate in round-robin fashion and other ways of interest to computer science. We compute a stationary law of product form for the Markov process describing the state of the net~vork. Moreover, we obtain the conditional expected travel time of a job given the job's requested processing times at particular nodes along its route. Zusammenfassung: Die Arbeit untersucht ein Netzwerk yon Bedienem, die nach der round-robinoder anderen Regein arbeiten, wie sie etwa bei Rechenanlagen benutzt werden. Es wird ein Markovscher Zustandsprozet~ ffir das Netzwerk definiert und dessen invariantes Gesetz angegeben. Ferner wi~d die bedingte mittlere Aufenthaltszeit eines Kunden ira Netzwerk berechnet, gegeben des Kunden Route und seine Bedienungszeitforderuv~en entlang der Route.
Response Time Distributions in Networks of Queues
International Series in Operations Research & Management Science, 2010
The problem of computing the response (sojourn) time distribution in queuing networks has been researched extensively during the past few decades. (For a somewhat dated survey see .) In case of open queuing networks, a considerable amount of work has been done in computing the response time distribution in the domain of Jackson networks. Closed form solutions have been derived for the (Laplace-Stieltjes transform of) response time distributions through a particular path in product-form queuing networks .
An approximation to the response time for shortest queue routing
1989
In this paper we derive an approximation for the mean response time of a multiple queue system in which shortest queue routing is used. We assume there are K identical queues with infinite capacity and service times that are exponentially distributed. Arrivals of jobs to this system are Poisson and are routed to a queue of minimal length. We develop an approximation which is based on both theoretical and experimental considerations and, for K 5 8, has an relative error of less than one half of one percent when compared to simulation. For K = 16, the relative error is still acceptable, being less than 2 percent. An application to a model of parallel processing and a comparison of static and dynamic load balancing schemes are presented.
On the asymptotic optimality of the SPT rule for the flow shop average completion time problem
Operations Research, 2000
Consider a flow shop with M machines in series, through which a set of jobs are to be processed. All jobs have the same routing, and they have to be processed in the same order on each of the machines. The objective is to determine such an order of the jobs, often referred to as a permutation schedule, so as to minimize the total completion time of all jobs on the final machine. We show that when the processing times are statistically exchangeable across machines and independent across jobs, the Shortest Processing Time first (SPT) scheduling rule, based on the total service requirement of each job on all M machines, is asymptotically optimal as the total number of jobs goes to infinity. This extends a recent result of Kaminsky and Simchi-Levi (1996), in which a crucial assumption is that the processing times on all M machines for all jobs must be i.i.d.. Our work provides an alternative proof using martingales, which can also be carried out directly to show the asymptotic optimality of the weighted SPT rule for the Flow Shop Weighted Completion Time Problem. Subject classifications: Asymptotically optimal scheduling: flow shop average completion time problem. Flow shop scheduling: average completion time, asymptotic optimality. Tandem queueing: asymptotic optimality of SPT. Area of review: STOCHASTIC MODELS.
An Approximate Analytical Method for General Queueing Networks
IEEE Transactions on Software Engineering, 2000
In this paper, we present an approximate solution for the asymptotic behavior of relatively general queueing networks. In the particular case of networks with general service time distributions (i.e., fixed routing matrix, one or many servers per station, FIFO discipline), the application of the method gives relatively accurate results in a very short time. The approximate stationary state probabilities are identified with the solution of a nonlinear system. The proposed method is applicable to a larger class of queueing networks (dependent routing matrix, stations with fimite capacity, etc.). In this case, the structure of the network studied must satisfy certain decomposability conditions.
Lecture Notes in Computer Science, 2007
Stochastic models of resource sharing systems computer, communication, traffic, manufacturing systems Customers compete for the resource service => queue QN are powerful and versatile tool for system performance evaluation and prediction Stochastic models based on queueing theory * queuing system models (single service center) represent the system as a unique resource * queueing networks represent the system as a set of interacting resources => model system structure => represent traffic flow among resources System performance analysis * derive performance indices (e.g., resource utilization, system throughput, customer response time) * analytical methods exact, approximate * simulation Queueing Network a system model set of service centers representing the system resources that provide service to a collection of customers that represent the users Outline I) II) III) IV) Queueing systems various hypotheses analysis to evaluate performance indices underlying stochastic Markov process Queueing networks (QN) model definition analysis to evaluate performance indices types of customers: multi-chain, multi-class models types of QN Markovian QN underlying stochastic Markov process Product-form QN have a simple closed form expression of the stationary state distribution BCMP theorem => efficient algorithms to evaluate average performance measures
Title Analysis of Queueing Networks and Models
Depends R (>= 2.11.1) Suggests Description It provides versatile tools for analysis of birth and death based Markovian Queueing Models and Single and Multiclass Product-Form Queueing Networks.