State-dependent Rates Research Articles

The exponentialization approach to flexible manufacturing system models with general processing times

Recently flexible manufacturing systems (FMSs) have been modelled as closed networks of queues. In this paper we develop an exponentialization approach to the modeling of FMS networks with general processing times. The idea of the approach is to transform the network into an (approximately) equivalent exponential network, where each station has exponential processing times with state-dependent rates. The approach is formulated as a fixed-point problem. Numerical examples have indicated excellent accuracies of the approach. This approach can also be readily adapted to accommodate limited local buffers and dynamic parts routing.

On the busy-period distributions ofM/G/1/Kqueues by state-dependent arrivals and FCFS/LCFS-P service disciplines

The busy-period distributions ofM/G/1/Kqueues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinaryM/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines

The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

A Class of Diffusion Processes with Killing Arising in Population Genetics

An interesting class of diffusion stochastic processes is studied. These processes arise from discrete models of gene formation and detection in finite populations. The diffusion processes are governed not only by the usual infinitesimal drift and diffusion terms, but also by a state dependent killing rate, which corresponds to formation of certain types of individuals. For one case, the spectral decomposition of the transition function is available, which allows a complete study of the process. Its behavior is compared with other variants of the detection problem.

A diffusion process with killing: the time to formation of recurrent deleterious mutant genes

We analyze the role of recurrent mutation on the time to formation or detection of particular genotypes in finite populations. The traditional method of approximating Markov chain models by diffusion processes is used. However, the diffusion approximations that arise in this context are governed not only by infinitesimal drift and diffusion coefficients, but also by a state-dependent killing rate that arises from the formation events. The resulting processes are particularly tractable, and allow a comprehensive analysis of the role of mutation in this problem.

Moment Formulae for a Class of Mixed Multi-Job-Type Queueing Networks

Queueing network models play an important role during each stage of a computer system's life cycle (from initial conception to system maturity), where in each stage broadly applicable performance analysis tools are needed. This paper presents new results which contribute to the foundations of a tool to support performance analysis and modeling activities. In dealing with some performance issues, it is important to be able to quantify distribution or moment information, because these quantities can influence system capacity and service and performance measures. It is also important that models include the effect of congestion adaptive I/O devices, in a stable and efficient manner, for this inclusion can significantly affect the outcome of studying certain performance issues. We address the problem of direct, recursive computation of moments of the queue size distributions at a class of service centers embedded in a mixed network of queues. The parameterized class includes state-dependent processing rates useful in modeling congestion adaptive I/O devices. We also present results for calculating moments of both the waiting time and virtual delay (work backlog) distributions at a class of service centers. In addition, we obtain a Little's Law type of relation between delay moments and queue size factorial moments. For a class of networks, an algorithm is given for the direct, recursive computation of the tail of the node delay distribution.

Factors affecting the identifiability of compartmental models

The identifiability of compartmental models is analysed through a series of examples which have been used to describe physiological or pharmacokinetic processes. Emphasis is placed on aspects of experimental identifiability which have hitherto received little attention in identifiability analysis. It is shown that where a single-input, single-output experiment results in non-identifiability or local identifiability, it is often possible to improve the situation by measuring more responses or simultaneously perturbing more inputs. Identifiability is then shown usually to depend on whether the observation gains are known and on the shape of the inputs, when more than one is applied. The relative merits of the Laplace transform and normal mode methods of analysing identifiability are discussed and illustrated with a substantial example. The identifiability analysis of a nonlinear compartmental model, with state-dependent rate coefficients, is presented. It is shown that inclusion of a neglected (nonlinear) relationship can make a previously non-identifiable model uniquely identifiable.

Self-similar solution of the spherical detonation problem

Under polytropic equation of state and strong shock assumptions, group-theoretic methods are used to obtain a class of self-similar solutions to a one-dimensional problem in reactive shock hydrodynamics with spherical symmetry and to characterize the class of all state-dependent chemical reaction rates under which such solutions exist. In the special case that the reaction rate is proportional to the internal energy, the system of ordinary differential equations describing the self-similar solution can be reduced to a single, first-order differential equation for which a phase-plane analysis yields the nature of the solution near the equation's critical points. Further, a modeling parameter which characterizes the flow is identified.

Inactivation of voltage-gated delayed potassium current in molluscan neurons. A kinetic model

Voltage-gated delayed potassium current in molluscan neurons is characterized by a marked inactivation. Inactivation can accumulate between repetitive pulses, giving rise to current patterns in which the maximum current during a second voltage pulse is less than the current at the end of the preceding pulse (cumulative inactivation). Other features of inactivation of this current include an onset time-course that can be characterized by the sum of two exponential processes and an early minimum in the recovery-vs.-time curve. A simple four-state model is developed that can, when supplied with rate constants derived from voltage-clamp experiments, reproduce these features of inactivation. The model incorporates state-dependent inactivation rates. Upon depolarization, both open and closed channels can be inactivated, although inactivation of closed channels is much faster. Upon repolarization, recovery from inactivated states is sufficiently slow that little recovery occurs during a short interpulse interval. Cumulative inactivation comes about as a result of fast inactivation during the second pulse, further limiting the peak current from the level at the end of the previous pulse.

A sufficient criterion for particles performing a diffusive motion with state-dependent death rate to die with probability one

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.

A sufficient criterion for particles performing a diffusive motion with state-dependent death rate to die with probability one

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.

The Identifiability of Nonlinear Compartmental Models

A structural identifiability test due to Pohjanpalo is applied to application examples (from pharmacokinetics and chemical reaction kinetics) of compartmental models having state-dependent rate coefficients. In particular, it is demonstrated that allowance in a non-identifiable linear model for nonlinearity of the physical system can result in unique identifiability without necessarily measuring more outputs or perturbing more inputs. Structural identifiability is by its nature a concept of limited utility, and this is exemplified by pharmacokinetic analysis of experimental data on the repeated dosing of human subjects with erythromycin : the analysis shows that satisfaction of the condition for unique identifiability is only one factor in validating a model structure. Since one approach to testing the identifiability of nonlinear compartmental models is to test the equations resulting from linearisation about a steady operating condition, careful distinction is made in the paper between the respective models for perturbation and tracer experiments, using a four-compartment closed model of the continuous pyrolysis of benzene as an example.

Technical Note—Optimality of Monotonic Policies for Multiple-Server Exponential Queuing Systems with State-Dependent Arrival Rates

We consider an exponential queuing system consisting of several removable servers in which the arrival rate depends on the current system state. Costs are incurred at a rate that depends on both the number of customers present and the number of servers in operation. We present conditions that ensure that the number of servers in operation is a non-decreasing function of the number of customers in the system.

Optimal replacement under a general failure model

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

Optimal replacement under a general failure model

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

The Steady State Behaviour Of TheM/Ek/1Queue, With State Dependent Arrival Rates

The steady state behaviour of the M/Ek/1 queue, with state dependent arrival rates, is investigated. The method employed is to divide the service time into k phases and derive the probability distribution for the number of phases recursively from its steady equations; then, the distribution of the number of phases recursively from its steady state equations; and finally, the distribution of the number of phases at the time of an arrival is found. Once these two distributions are determined, one can obtain the distribution of the number of elements in the system, the distribution of the actual and virtual waiting time and the sojourn time distribution. A number of interesting relationships between the different distributions are established. A numerical example completes the article.

Feedback regulators for jump parameter systems with state and control dependent transition rates

A class of linear, stochastic, jump parameter systems is studied in which the probability law of the parameter processes depends upon the control policy used. An optimal controller is given when the loss function is quadratic. Because of difficulties inherent in the design equations, a more easily evaluated, near optimal controller is provided for problems in which the parameter processes are weakly dependent on the control policy.

On the limiting behaviour of a basic stochastic process

Determination of the limiting distributions for a class of mixed-type stochastic processes with state-dependent rates of decline is reduced to the solution of a class of integral equations. For the case where the rate of decline is proportional to the state, some results are obtained by solving the integral equation of the process through Fuchs' method.

On the limiting behaviour of a basic stochastic process

Determination of the limiting distributions for a class of mixed-type stochastic processes with state-dependent rates of decline is reduced to the solution of a class of integral equations. For the case where the rate of decline is proportional to the state, some results are obtained by solving the integral equation of the process through Fuchs' method.

Correction to “Queues with State-Dependent Stochastic Service Rates”

Correction to the article Harris, Carl M. “Queues with State-Dependent Stochastic Service Rates,” Operations Research, Vol. 15, 117–130 (1967).