Computation Of Steady-state Probabilities Research Articles

Optimizing stock levels for service-differentiated demand classes with inventory rationing and demand lead times

In this paper, we study a service parts inventory management system for a single product at a parts distribution center serving two priority-demand classes: critical and non-critical. Distribution center keeps a common inventory pool to serve the two demand classes, and provides differentiated levels of service by means of inventory rationing. We assume a continuous review one-for-one ordering policy with backorders and Poisson demand arrivals. Only one demand class provides advance demand information whose orders are due after a deterministic demand lead time, whereas the orders of the other demand class need to be satisfied immediately. The problem has been studied before, but remained a challenging problem. The quality of the existing heuristic for estimating the critical class service levels can diminish significantly in some settings and the search routine for the service level optimization model relies on a brute force approach. Our contribution to the literature is twofold. For the given class of inventory replenishment and allocation policies, first we determine the form of the optimal solution to the service level optimization model, and then we derive an exact optimization routine to determine the optimal policy parameters provided the steady-state distribution is available. The computation of steady-state probabilities is needed only once. Second, we propose an alternative approach to estimate steady-state probabilities. By analyzing the limiting behavior of transition probabilities during infinitesimal time intervals, we are able to characterize the relationships between the steady-state probabilities, which satisfy nicely formed balance equations under the so-called Independence Assumption. In the numerical study section, we show that our approach provides superior performance in estimating service levels than the existing heuristic for all the examples considered. We also compare the performance of using the critical class service levels computed according to our method against the service levels computed by the existing heuristic, and show that our method can provide inventory savings up to 16.67%.

ASSA-PBN: A Toolbox for Probabilistic Boolean Networks.

As awell-established computational framework, probabilistic Boolean networks (PBNs) are widely used for modelling, simulation, and analysis of biological systems. To analyze the steady-state dynamics of PBNs is of crucial importance to explore the characteristics of biological systems. However, the analysis of large PBNs, which often arise in systems biology, is prone to the infamous state-space explosion problem. Therefore, the employment of statistical methods often remains the only feasible solution. We present ${\mathsf{ASSA-PBN}}$ , asoftware toolbox for modelling, simulation, and analysis of PBNs. ${\mathsf{ASSA-PBN}}$ provides efficient statistical methods with three parallel techniques to speed up the computation of steady-state probabilities. Moreover, particle swarm optimisation (PSO) and differential evolution (DE) are implemented for the estimation of PBN parameters. Additionally, we implement in-depth analyses of PBNs, including long-run influence analysis, long-run sensitivity analysis, computation of one-parameter profile likelihoods, and the visualization of one-parameter profile likelihoods. APBN model of apoptosis is used as acase study to illustrate the main functionalities of ${\mathsf{ASSA-PBN}}$ and to demonstrate the capabilities of ${\mathsf{ASSA-PBN}}$ to effectively analyse biological systems modelled as PBNs.

Markov Processes on a Semi-Infinite Strip and the Geometric Tail Algorithm

Many queuing processes have a semi-infinite strip in the two-dimensional plane as state space. As an alternative to well-established algorithms for the computation of steady-state probabilities, such as the matrix-geometric method and the spectral method, this article discusses a simple and easy-to-implement algorithm which is based on the geometric tail behavior of the steady-state probabilities. Some numerical comparisons between this algorithm and the spectral method are presented.

A more accurate formulation for a class of models for the economic design of control charts

Some of the published models for the economic design of control charts use the expected cost per unit of output as the objective function to be minimized. In these models, the computation of steady-state probabilities that the process is in each possible state does not take into account the effect of the corrective action that may follow a signal from the control chart. As a result, the expected cost per unit is overestimated and the selection of the chart parameters is not optimal. The purpose of this paper is to eliminate this inaccuracy by proposing the exact formulation and to estimate the magnitude of errors resulting from the inaccurate formulation of the objective function. By solving numerical examples of joint design of X and R charts, it is shown that these errors are typically very large and consequently it is imperative to use the exact formulation, proposed in this paper, to avoid inefficient and costly control chart designs. Finally, an additional opportunity for further cost improvements in process control is identified and discussed.

Recursive computation of steady-state probabilities in priority queues

Exact recursive formulas are derived for the state probabilities in priority queueing systems (preemptive and non-preemptive). The derivation is based only on the general structure of the generating function involved, and thus is simpler and more general than previous methods. Furthermore, applications of the method to other queueing systems are discussed.

Recursive computation of steady-state probabilities of nonproduct form queueing networks associated with computer network models

Consideration is given to the exact computation of the equilibrium state probabilities of a class of non-product-form queueing networks that is of interest for communication systems. It is shown that a specific structure in the state transition lattice leads to a sequential method of solution. The method is sequential either in terms of individual states or in terms of subsets of states. In the former case, algebraic recursions for the state probabilities can be developed. In both cases, substantial computational savings are possible. >

Analysis of an Important Class of Non-Markov Systems

Probabilistic modeling of many types of systems generally assumes Markov behavior. However, some important practical systems exhibit memory. For example, in digital computer systems, the probability of occurrence of a transient failure is related to the time period the system has been operating correctly. Analytic methods do not yet exist that allow accurate modeling of such systems for the purpose of reliability analysis and fault-tolerant design. Methods are presented here to analyze an important class of non-Markov systems. In this class, the transition-probability-rate of an out-ward transition from a state is related to the duration the system has continuously been in that state. To analyze such systems, concept of memory profile has been introduced. Methods are first presented which enable computation of steady-state probabilities for both discrete-time and continuous-time processes with two states. These are then extended for general non-steady-state cases and also for systems with more than two states.

A finite waiting room queueing model with multiple servers having Markovian interruptions and its application to computer communications

A queueing model with finite waiting room, Poisson arrivals, multiple synchronous outputs and the number of active servers varying through the birth and death process is studied. Variation of active servers is obtained by providing a switch in each server. The switches are controlled through the birth and death process having finite population. The relationships between overflow probabilities, buffer size and expected queueing delay due to buffering are obtained. An efficient algorithm for computation of steady state probabilities is developed. Two digital voice-data systems; Data multiplexing in speech with synchronous Time Division Multiplexing (STDM) and Time Assignment Speech Interpolation (TASI) systems are considered for the application of the queueing model studied. The results of this study are portrayed on graphs and may be used as guidelines for designing a buffer in digital systems. The two voice-data systems using the developed model are simulated on the EC-1030 computer to check the validity of the analytical results. Although this problem arose in the study of data interpolation in STDM and TASI systems, the queueing model developed is quite general and may be useful for other industrial applications.