Cyclic Queueing Networks Research Articles

Analysis of a two-node closed queueing cyclic network with one non-exponential node

Consider a two-node closed cyclic queueing network with a total of M customers. The first node consists of N⩽M parallel identical servers having exponential service times. Node 2 consists of a single server having a general service time distribution. Waiting space at each node is sufficient to accommodate all M customers.We derive an exact solution for the steady state system size probabilities. Our approach is based on developing and solving an imbedded Markov chain at Node 2 service completion epochs coupled with a birth process over Node 1 service times. In that sense, our methodology is intuitively appealing and easy to apply. We focus on analytically and numerically investigating the effect of Node 2 service time variability on system performance. We find that closed queueing networks, like the one considered here, are less vulnerable to variability than open networks. However, the effect of variability is still significant to merit considerable attention. Numerical results are presented to illustrate the viability of our method for many service time distributions. As by-products, our work yield new results on birth processes and combinatorial identities, which can be useful in other contexts.A primary application of this model is the well-known machine repair model where a set of identical machines are attended by a single repairman. Other applications include performance evaluation of manufacturing and computer networks, as well as reliability studies where our model can be easily used to compute system availability.

Cyclic Queueing Networks With Subexponential Service Times and Finite Buffers

In this technical note, we consider closed tandem queueing networks with finite buffers and blocking. We assume that at least one station has subexponential service time distribution. We analyze this system under communication blocking and manufacturing blocking rules. Our objective is to derive expressions for the tail asymptotics of transient cycle times and waiting times. Furthermore, we study under which conditions on system parameters these tail asymptotics also hold for their stationary counter parts. Finally, we provide numerical examples to understand the convergence behavior of the tail asymptotics.

Analysis of cyclic queueing networks with parallelism and vacation

The aim of this paper is to improve the machine interference model with vacation to deal with more recent problems of the communication area. To this scope the model is extended to include parallelism in the vacation station. The underlying Markov process is analyzed and a state arrangement is found that yields an efficient matrix-analytic technique that substantially lowers down the time- and space-complexity of standard methods. A numerical example of the method effectiveness is presented, and an example of resource allocation is introduced that finds applications in the QoS management of wireless networks.

Cyclic queueing networks with subexponential service times

For a K-stage cyclic queueing network with N customers and general service times, we provide bounds on the nth departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.

Cyclic queueing networks with subexponential service times

For a K-stage cyclic queueing network with N customers and general service times, we provide bounds on the nth departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.

Sensitivity formulas of performance in two‐server cyclic queueing networks with phase‐type distributed service times

AbstractWith the approach of the infinitesimal generator perturbation, sensitivities of the steady‐state performance are studied in two‐server cyclic queueing networks with phase‐type distributed service times. Sensitivity formulas expressed by the potentials of the networks are given for parameter‐dependent performance functions. An algorithm to compute the potentials and the performance derivatives of the networks is given, and a numerical example is provided.

Sensitivity formulas of performance in two-server cyclic queueing networks with phase-type distributed service times

With the approach of the infinitesimal generator perturbation, sensitivities of the steady-state performance are studied in two-server cyclic queueing networks with phase-type distributed service times. Sensitivity formulas expressed by the potentials of the networks are given for parameter-dependent performance functions. An algorithm to compute the potentials and the performance derivatives of the networks is given, and a numerical example is provided.

Queue length and occupancy in discrete-time cyclic networks with several types of nodes

For a discrete-time, closed, cyclic queueing network, where the nodes have independent, geometric service times, the equilibrium rate of local progress is determined. Faster nodes are shown to have a capacity depending only on the service probabilities. A family of such networks, each with the same number of types of nodes, is analyzed. If the number of nodes approaches infinity, and if the ratio of jobs to nodes has a positive limit and each node type has an asymptotic density, then for a given node type, the limits of the proportion of occupied nodes and the expected queue length are calculated. These values depend on the service parameter and on the asymptotic rate of local progress. The faster nodes can attain their capacity only when the limiting density of nodes of slowest type is zero.

Throughput estimation in cyclic queueing networks with blocking

The paper presents a new approach for estimating the throughput of a closed queueing network with exponential servers, finite buffer capacity and a blocking after service discipline. The problem is tackled by decomposing the network. The population constraint is enforced by requiring that the sum of the expected number of customers in the various subsystems is equal to the population size. Each subsystem is analyzed as an M/M/1/Ci+1 queue with state-dependent arrival and service rates. The rationale behind this last assumption is that the behavior of the system at a given time is affected by the history of blockings and starvations. The results obtained by applying the proposed algorithm to a set of test problems show a good agreement with those obtained with simulation, the difference on the maximum throughput of the network being of the order of 3%. The obtained results also compare favorably with those described in the literature.

MVA for product-form cyclic queueing networks with blocking.

The paper analyzes the class of product-form queueing network models with cyclic topology and blocking. For this class of networks, the concept of a dual network was introduced in [9]. First we derive a set of properties about the primal and dual performance indices. Then on the basis of the dual network, we are able to define an MVA algorithm to evaluate the mean performance indices of a cyclic network with blocking. The computational time complexity of the algorithm is polynomial in the number of nodes and number of jobs of the network. Finally, we define an equivalence relation for the class of product-form queueing networks with blocking, based on the dual network. This equivalence relation can be used to derive some properties of the equivalent networks, related to the performance indices, which simplify the application of the algorithm.

Approximate mean value performance analysis of cyclic queueing networks with production blocking

A simple approximation method is developed for performance analysis of exponential cyclic queues with production blocking. With a modified Arrival Instant Theorem to cater for finite queue capacity, a set of equations involving the mean values of performance measures are established and an efficient algorithm is proposed. Numerical experiments show that the approximation method is very efficient in providing results with good accuracy.

Performance analysis of the parallel cyclic two-stage queueing model

The 2-stage cyclic queueing network model is an analytical tool that has been widely used for the performance analysis of sequential processing systems in computer, manufacturing, logistic and communication modeling contexts. No similar tool exists for the parallel processing case, because of the structural complexity of the underlying stochastic process. Consequently, only numerical approaches can be used. On the other hand, numerical solutions are difficult to deal with for at least three reasons: (1) because of the growth of the process' rate matrix with the number of jobs N; (2) because of the fact that the N + 1, N + 2,…, N + k job solutions cannot be derived from the N-job solution, and (3) because of the recalculations that are needed each time new parameters are introduced. This paper deals with the parallel version of the 2-stage cyclic queueing model. It shows that the parallel cyclic server belongs to the class of Quasi-Birth-and-Death finite-state processes for which an explicit expression can be found for the invariant probability vector, without recourse to the more expensive Matrix-Geometric method. It then provides an analytical solution, in the time domain, based on an efficient recurrence which solves the model without use of the N-dependent rate matrix. It is in symbolic form. The effect of parameter change is therefore easily evaluated, and the N + 1, N + 2, etc., job analysis is easily derived from the N-job solution.

Features of some discrete-time cyclic queueing networks

A class of discrete-time closed cyclic networks is analyzed, where queues at each node have ample waiting room and have independent geometric service times with possibly unequal means. If each node has a single server or if there are sufficiently many parallel servers at each node to accommodate all jobs, equilibrium vectors of product form are obtained. For some other cases, equilibrium vectors of product form need not exist. For the single-server model, a normalization constant is computed and used to determine the queue-length distribution at a node.

A study of regeneration times in cyclic queueing networks

A study of regeneration times in cyclic queueing networks

Max-algebra model of cyclic queueing networks and its explicit eigenvalue formula

Cyclic queueing network with deterministic service times is modeled as 'linear dynamical system', using max-algebra. Its unique eigenvalue λ is given in the 'close form', λ = f(S1,S2, Sn, N), where S1 is the service time of the i-th server, and N is the number of customs in system. This result is applied to optimize the production lines with finite number of pallets.

Analysis of cyclic queueing networks with finite buffers and blocking before service

We consider cyclic queueing networks with exponential servers and finite buffers. The blocking mechanism we consider is blocking-before-service, i.e., a server is allowed to start processing a job only if a space is available in the next queue. We present an approximate technique for analyzing such networks based on a decomposition approach. We first consider the case where at least one buffer has an infinite capacity. The network is decomposed into a set of M/M/1 x/ K queues. An iterative algorithm for calculating the performance parameters is proposed. This technique is then extended to the case where all queues have finite capacity. Numerical experiments show that this technique provides a fairly good estimate of the throughput and the average queue lengths.

Approximate throughput analysis of cyclic queueing networks with finite buffers

An approximation method for obtaining the throughput of cyclic queueing networks with blocking as a function of the number of customers in it is presented. The approximation method was developed for two different blocking mechanisms. It was also extended to the case of the central server model with blocking. Validation tests show that the algorithm is fairly accurate. >

Sensitivity analysis and optimization of throughput in a production line with blocking

Production lines with limited storage capacities can be modeled as cyclic queueing networks with finite buffers and general service times. A new technique, called perturbation analysis of discrete event dynamic systems, is applied to these queueing networks. An estimate of the gradient of the system throughput is obtained by perturbation analysis based on only one sample trajectory of the system. We show that the estimate is strongly consistent. Using this perturbation analysis estimate of gradient, we can apply the Robbins-Monro stochastic procedure in optimizing the system throughput. Compared to the conventional Kiefer-Wolfowitz optimization procedure, this approach saves a large amount of computation. For a real system, the gradient estimate can be obtained without changing any parameters in the system. The results also hold for systems with general routing but in which no server can block more than one other server simultaneously.

M-LiPS規律を含む循環型待ち行列網の滞在時間について

We consider a cyclic queueing network with two service stations, nodes 1 and 2. Node 1 has s servers and the FCFS discipline, Node 2 processes jobs under the m-limited processor sharing (m-LiPS) discipline, which is a generalization of the processor sharing (PS) discipline. We consider the joint distribution of successive sojourn times of a job at the two nodes, and propose a simple method to describe the Laplace-Stieltjes transform (LST) and the moments of these variables by their conditional marginal distributions. Further, we deduce linear equations giving the moments and the LST for an m-LiPS node. For some cases, the correlation coefficients and the coefficients of variation of the total sojourn times are calculated numerically.

Maximum entropy and the G/G/1/N queue

A new “hybrid” analytic framework, based on the principle of maximum entropy, is used to derive a closed form expression for the queue length distribution of a G/G/1 finite capacity queue. It is shown that “Birth-Death” homogeneous recursions for a single resource queue are special cases of maximum entropy “one-step” transitions which can be applied either in an operational or stochastic context. Furthermore, an “equivalence” relationship is used to analyse two-stage cyclic queueing networks with general service times, and favourable comparisons are made with global balance and approximate results. Numerical examples provide useful information on how critically system behaviour is affected by the distributional form of interarrival and service patterns. Comments on the implication of the work to the performance analysis and aggregation of computer systems are included.