Time-dependent State Probabilities Research Articles

A Simple Formula for the Transient State Probabilities of an M/M/1/∞Queue

In this paper a series formula is developed for the time-dependent state probabilities for the M/M/1 queue and it is shown that the coefficients in this series satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Furthermore, a closed form solution to these recurrence equations is obtained which leads to a simple formula for the transient state probabilities from which the steady state solution is readily isolated

The final outcome and temporal solution of a carrier-borne epidemic model

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

The final outcome and temporal solution of a carrier-borne epidemic model

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

On a new formula for the transient state probabilities for M/M/1 queues and computational implications

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.

On a new formula for the transient state probabilities for M/M/1 queues and computational implications

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Development and utilization of a generalized three-state model

The basic component representation used in many studies of reliability and availability is the two-state model in which the component is designated as being in the up or down state. In some cases, however, the component can reside in an inbetween state, and a three-state representation is required. This representation is often used to model large generating units in power system adequacy or security evaluation. The inclusion of the third or derated state in this case can have an appreciable influence on the calculated system risk. The basic three-state model can be utilized in a wide range of applications using both limiting and time dependent state probabilities. This paper illustrates the development of a generalized three-state model and presents a basic set of equations which can be utilized in a wide range of applications. The utility of the model is illustrated by three specific applications taken from the power systems area.

Reliability and availability equations for multi-state components

Fundamental relations are derived for the reliability, availability and other failure and repair characteristics of components with multiple failed states. These equations allow general-failure and repair-time distributions, initial conditions, and replacement components to be either old or new, and also allow the repair crew to be old or renewed for each failure. These new equations for multi-state components also provide new results for binary components and a systematic classification structure based on the repair policies. Solutions and solution techniques are introduced for the time-dependent and asymptotic state probabilities and failure and repair intensities. Applications to probabilistic risk assessments and reliability analyses of complex components are discussed.

Short-term reliability of power-system generation

New mathematical models for calculation of time-dependent state probabilities required for evaluation of a shortterm level of reliability to meet a forecasted load demand, under forced outage of generating units, are presented. The calculation of the time-dependent state probabilities of normally operating and standby generators is based on the conventional assumption that the time of units spent in the fully available state is exponentially distributed. Since the time of generator units spent in outage condition is arbitrarily distributed, special methods have been developed that are relatively less complex than the present techniques, more accurate than the conventional assumption of exponential distribution and computationally efficient for practical commercial application

A Semi-Markov Model of a Three-State Generating Unit

This paper shows how the theory of semi-Markov processes can be used to obtain some important figures of merit from non-exponential failure models of generating units. The approach is used to obtain the time dependent state probabilities, the distribution of the time to transit from one state to another and the distribution of the number of visits to a given state in a fixed amount of time for a three-state model in which the occupancy times of two of the states is not exponential.

The Continuous Time Solution of the Equations of the Single Channel Queue with a General Class of Service-Time Distributions by the Method of Generating Functions

SUMMARY Equations describing a single channel queue with random arrivals and a general class of service-times are solved for time-dependent state probabilities. Solutions for a Pearson type III service-time and a “double” Pearson type III service-time are given in terms of the Ik n function. Corresponding results for the distribution of busy periods are also obtained.