On the overflow processes from the PH1 + PH2/M/S/K queue with two independent PH-renewal inputs

This paper presents an analysis of overflow processes from a PH₁ + PH₂/M/S/K queue having two independent phase type renewal input streams. Both the superposed overflow process and individual overflow processes for the PH₁- and PH₂-streams are analyzed using first passage time distributions for the number of customers in the system. Each overflow process is characterized as a Markov renewal process. The nth moment of the number of customers in an infinite server group to which these overflows have been offered is derived using a theory for the MR/M/∞ queue with a Markov renewal input. The numerical examples for means and variance-to-mean ratios (peakednesses) of the individual overflow streams are given for an H₂ + H₂/M/S/S queue with interrupted Poisson inputs, which is a vital model for telephone network planning. In addition, overflow traffic characteristics are discussed by using these examples.     

Asymptotic analysis and simple approximation of the loss probability of the GI/M/c/K queue

We consider the GI^X/M/c/K queues with partial rejection or total rejection, and find an asymptotic behavior of loss probability of the GI^X/M/c/K queue as K tends to infinity. The asymptotic loss probability is expressed only in terms of the roots of the characteristic equation and the boundary probabilities of the corresponding GI^X/M/c queue. Numerical examples show that the asymptotic loss probability is a quite accurate approximation for the loss probability of the GI^X/M/c/K queue even when the system capacity K is moderate.     

On the M/Ck/1 queue

We obtain the exact analytic expression of the probability distribution of the number of units in a single server queue with Poisson arrivals and Coxian service time distribution (notated as M/C_k/1). A recursive procedure for calculating this probability distribution is given. The well-known queues M/E_k/1 and M/D/1 are re-derived as special cases of the M/C_k/1 queue. Finally, the cases of M/C₂/1 and M/C₃/1 are fully worked out.     

Efficient implementation of Johnson''s rule for the n/2/F/Fmax scheduling problem

Several efficient algorithms of O(n log n) computational complexity, for the Johnson's rule to schedule a set of simultaneously available jobs on two machines in a flowship to minimize the maximum job flowtime have appeared in the literature. A modified version of one of these algorithms is presented which not only simplifies the programming effort for implementation but is also able to generate all possible optimal sequences obtainable from Johnson's rule.     

The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV). During the SWV, the server works at a different rate rather than completely stopping working. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrary customer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it with the GI/M/1/SWV queue.     

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

Summary 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.Some of the material included in this paper has been presented to the Performance '86 and ACM Sigmetrics 1986 Joint Conference on Computer Modelling, Measurement and Evaluation, May 28–30, 1986, University of North Carolina, USA     

Sojourn time distribution in the M/M/1 queue with discriminatory processor-sharing

In this paper, we consider a queue with multiple K job classes, Poisson arrivals, exponentially distributed required service times in which a single processor serves according to the DPS discipline. More precisely, if there are n_i class i jobs in the system, i=1,…,K, each class j job receives a fraction _j/∑_i=1^K_in_i of the processor capacity. For this queue, we obtain a system of equations for joint transforms of the sojourn time and the number of jobs. Using this system of equations we find the moments of the sojourn time as a solution of linear simultaneous equations, which solves an open problem.     

Remarks on the L/sup p/-input converging-state property

Let X /spl sub/ /spl Ropf//sup N/ and consider a system x/spl dot/ = f(x,u), f : X /spl times/ /spl Ropf//sup M/ /spl rarr/ /spl Ropf//sup N/, with the property that the associated autonomous system x/spl dot/ = f (x,0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,/spl infin/), corresponding to an input function u. Assume further that, for each compact K /spl sub/ X, there exists k > 0 such that |f(z,v) - f(z,0)| /spl les/ k|v| for all (z,v) /spl isin/ /spl times/ /spl Ropf//sup M/. A simple proof is given of the following L/sup p/-input converging-state property: if u /spl isin/ L/sup p/ for some p /spl isin/ [1,/spl infin/) and x has an /spl omega/-limit point in A, then x approaches C.     

Approximations for the waiting time distribution of the M/G/c queue

For the M/G/c queue we present an approximate analysis of the waiting time distribution. The result is given in the form of the defective renewal equation. This integral equation can be numerically solved by a simple recursion procedure. Also, asymptotic results for the waiting times are presented. Numerical results indicate that the approximations are sufficiently accurate for practical purposes.     

On the dead-time compensation from L/sup 1/ perspectives

It is known that both H/sup 2/ and H/sup /spl infin// optimization problems for dead-time systems are solved by controllers having the so-called modified Smith predictor (dead-time compensator) structure. This note shows that this is also true for the L/sup 1/ control problem. More precisely, it is demonstrated that the use of the modified Smith predictor enables one to reduce the standard L/sup 1/ problem for systems with a single loop delay to an equivalent delay-free problem. The (sub)optimal solution therefore always contains the modified Smith predictor.     

On approximating the generalized occupancy of the G/M/K queueing system

The G/M/K is one of very few multiserver queueing systems for which analytical results exist. In 1951 Kendall [4] showed how to compute the steady-state probabilities of a G/M/K queueing system. Later, Takacs [6] suggested an iterative procedure for the evaluation of a needed component in Kendall's scheme; namely, the generalized occupancy ω^*. However, Takàcs' algorithm requires the computation of a general integral for each of its interations.In this paper we propose a simple and explicit approximation for the generalized occupancy of the G/M/K system. Several numerical results are also included.     

A functional approximation for the M/G/1/N queue

This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value.     

Thermodynamic study of the KCl-K2SO4-K2Cr2O7-H2O system at the temperature 298.15 K

The Pitzerion-interaction model has been used for thermodynamic simulation of the ternary solutions KCl-K₂Cr₂O₇-H₂O, KBr-K₂Cr₂O₇-H₂O and K₂SO₄-K₂Cr₂O₇-H₂O and the quaternary system KCl-K₂SO₄-K₂Cr₂O₇-H₂O at T=298.15 K. The necessary thermodynamic functions (binary and ternary parameters of interionic interaction and thermodynamic solubility products) have been calculated and the theoretical solubility isotherms have been plotted. Good agreement between experimentally determined and calculated solubilities has been found.     

Linear dependency between /spl epsi/ and the input noise in /spl epsi/-support vector regression

In using the /spl epsi/-support vector regression (/spl epsi/-SVR) algorithm, one has to decide a suitable value for the insensitivity parameter /spl epsi/. Smola et al. considered its "optimal" choice by studying the statistical efficiency in a location parameter estimation problem. While they successfully predicted a linear scaling between the optimal /spl epsi/ and the noise in the data, their theoretically optimal value does not have a close match with its experimentally observed counterpart in the case of Gaussian noise. In this paper, we attempt to better explain their experimental results by studying the regression problem itself. Our resultant predicted choice of /spl epsi/ is much closer to the experimentally observed optimal value, while again demonstrating a linear trend with the input noise.     

M/G/c/K blocking probability models and system performance

An exact solution for the M/G/c/K model is only possible for special cases, such as exponential service, a single server, or no waiting room at all. Instead of basing the approximation on an infinite capacity queue as is often the case, an approximation based on a closed-form expression derivable from the finite capacity exponential queue is presented. Properties of the closed-form expression along with its use in approximating the blocking probability of M/G/c/K systems are discussed. Extensive experiments are provided to test and verify the efficacy of our approximate results.     

The complete analysis of the discrete time finite DBMAP/G/1/N queue

A comprehensive analysis of the DBMAP/G/1/N queueing system for both the “arrival first” and “departure first” policies is given including detailed proofs and latest results on an algorithmic recipe to compute the z-transform of the waiting time probability function as well as the probability function itself. Additionally, expressions for the probability of simultaneous arrivals and departures and an algorithm to compute them are presented for the first time. The framework presented allows dealing with a superposition DMAP+DBMAP as an input process. For both streams in the superposition, per stream waiting time probability functions are given for the case of a deterministic service time, which is relevant for the modeling of ATM networking (both wired and wireless). Together with findings for the continuous time case, this paper completes the insights into finite queueing systems of the M/G/1-type. In comparison with the method of the unfinished work, which can be used to analyze finite queueing systems with deterministic service-time only and, besides other semi-Markovian input processes, with the DBMAP as an input, the M/G/1-paradigm provides much faster algorithms to compute loss probabilities and waiting time moments, due to the smaller system matrix. Note that the D[B]MAP has proved a versatile stochastic process, which can also be tuned to represent periodic correlation functions and not only geometrically decaying ones.     

The Brownian approximation for rate-control throttles and the G/G/1/C queue

This paper studies approximations to describe the performance of a rate-control throttle based on a token bank, which is closely related to the standard G/G/1/C queue and the two-node cyclic network of ·/G/1/∞ queues. Several different approximations for the throttle are considered, but most attention is given to a Brownian or diffusion approximation. The Brownian approximation is supported by a heavy-traffic limit theorem (as the traffic intensity approaches the upper limit for stability) for which an upper bound on the rate of convergence is established. Means and squared coefficients of variation associated with renewal-process approximations for the overflow processes are also obtained from the Brownian approximation. The accuracy of the Brownian approximation is investigated by making numerical comparisons with exact values. The relatively simple Brownian approximation for the job overflow rate is not very accurate for small overflow rates, but it nevertheless provides important insights into the way the throttle design parameters should depend on the arrival-process characteristics in order to achieve a specified overflow rate. This simple approximation also provides estimates of the sensitivity of the overflow rates to the model parameters.     

Thermodynamics and phase diagram of the binary system Bi2O3-B2O3

We analyzed the thermodynamic data of the binary system Bi₂O₃-B₂O₃. The phase diagram contains five compounds, of which four melt congruently, and a miscibility gap, close to pure B₂O₃. The three sets of transformation data of Bi₂O₃, the enthalpy of mixing data in the liquid, and the critical point of the miscibility gap gave five equations for the excess Gibbs energy of mixing and five sets of values for the enthalpies of formation of the binary compounds. The critical point data yielded another set of transformation data for Bi₂O₃. Using the monotectic composition and the enthalpy of mixing data also yielded a set of transformation data for Bi₂O₃. The values obtained using the enthalpy of mixing data are the most reliable values.     

Distribution of busy period for the bulk-service queueing system Ek/Ma,b/1

Using the Erlangian technique the busy-period equations for the single-server bulkservice system E_k/M^a,b/1. are solved to obtain the Laplace transform of the probability density function (pdf) of the busy period. The Laplace transform is expressed in terms of an easily computable real root of the characteristic equation. Expressions for the mean and variance of the busy-period distribution are given. Explicit results for the pdf of the busy period are obtained for the special systems M/M^a,b/1 and E_k/M^a,a/1.     

Inducing NNC-trees with the R/sup 4/-rule

An NNC-Tree is a decision tree (DT) with each nonterminal node containing a nearest neighbor classifier (NNC). Compared with the conventional axis-parallel DTs (APDTs), the NNC-Trees can be more efficient, because the decision boundary made by an NNC is more complex than an axis-parallel hyperplane. Compared with single-layer NNCs, the NNC-Trees can classify given data in a hierarchical structure that is often useful for many applications. This paper proposes an algorithm for inducing NNC-Trees based on the R/sup 4/-rule, which was proposed by the author for finding the smallest nearest neighbor based multilayer perceptrons (NN-MLPs). There are mainly two contributions here. 1) A heuristic but effective method is given to define the teacher signals (group labels) for the data assigned to each nonterminal node. 2) The R/sup 4/-rule is modified so that an NNC with proper size can be designed automatically in each nonterminal node. Experiments with several public databases show that the proposed algorithm can produce NNC-Trees effectively and efficiently.