Asymptotic Throughput in Discrete‐Time Cyclic Networks with Queue‐Length‐Dependent Service Rates

Abstract

For a discrete‐time closed cyclic network of single server queues whose service rates are non‐decreasing in the queue length, we compute the queue‐length distribution at each node in terms of throughputs of related networks. For the asymptotic analysis, we consider sequences of networks where the number of nodes grows to infinity, service rates are taken only from a fixed finite set of non‐decreasing sequences, the ratio of customers to nodes has a limit, and the proportion of nodes for each possible service‐rate sequence has a limit. Under these assumptions, the asymptotic throughput exists and is calculated explicitly. Furthermore, the asymptotic queue‐length distribution at any node can be obtained in terms of the asymptotic throughput. The asymptotic throughput, regarded as a function of the limiting customer‐to‐node ratio, is strictly increasing for ratios up to a threshold value (possibly infinite) and is constant thereafter. For ratios less than the threshold, the asymptotic queue‐length distribution at each node has finite moments of all orders. However, at or above the threshold, bottlenecks (nodes with asymptotically‐infinite mean queue length) do occur, and we completely characterize such nodes.     

A MAP/G/1 Queue with an Underlying Birth–Death Process

Abstract

In this paper, we define a birth–death‐modulated Markovian arrival process (BDMMAP) as a Markovian arrival process (MAP) with an underlying birth–death process. It is proved that the zeros of det(zI ? A(z)) in the unit disk are real and simple. In order to analyze a BDMMAP/G/1 queue, two spectral methods are proposed. The first one is a bisection method for calculation of the zeros from which the boundary vector is derived. The second one is the Fourier inversion transform of the probability generating function for the calculation of the stationary probability distribution of the queue length. Eigenvalues required in this calculation are obtained by the Duran–Kerner–Aberth (DKA) method. For numerical examples, the stationary probability distribution of the queue length is calculated by using the spectral methods. Comparisons of the spectral methods with the currently best methods available are discussed.     

An Exact Analysis of the Multi‐class M/M/k Priority Queue with Partial Blocking

Abstract

We consider a multi‐server queuing model with two priority classes that consist of multiple customer types. The customers belonging to one priority class customers are lost if they cannot be served immediately upon arrival. Each customer type has its own Poisson arrival and exponential service rate. We derive an exact method to calculate the steady state probabilities for both preemptive and nonpreemptive priority disciplines. Based on these probabilities, we can derive exact expressions for a wide range of relevant performance characteristics for each customer type, such as the moments of the number of customers in the queue and in the system, the expected postponement time and the blocking probability. We illustrate our method with some numerical examples.     

Best Spatial Two‐Stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances

Abstract

Estimation of a cross‐sectional spatial model containing both a spatial lag of the dependent variable and spatially autoregressive disturbances are considered. [Kelejian and Prucha (1998)] Kelejian, H. H. and Prucha, I. R. 1998. A generalized spatial two‐stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. J. Real Estate Financ. and Economics, 17: 99–121. [Crossref], [Web of Science ®] , [Google Scholar]described a generalized two‐stage least squares procedure for estimating such a spatial model. Their estimator is, however, not asymptotically optimal. We propose best spatial 2SLS estimators that are asymptotically optimal instrumental variable (IV) estimators. An associated goodness‐of‐fit (or over identification) test is available. We suggest computationally simple and tractable numerical procedures for constructing the optimal instruments.