An Algorithm for Estimating the Expected Number of Customers for a Class of Markovian Queueing Systems

An algorithm is presented for estimating the expected number of customers for a class of Markovian queueing systems. The class is characterized by those systems whose transition matrix for the underlying customer arrival and departure process is finite, irreducible, and aperiodic. The algorithm does not depend on a closed-form solution for the limiting behavior of the queue. The expected number of customers is frequently used as a measure of effectiveness to describe the behavior of the system or to optimize its design or control. To calculate such a quantity one must usually obtain a closed-form expression for the steady-state probabilities. Unfortunately, of the myriad of Markovian queueing systems, only a few have known closed-form expressions for their steady-state probabilities. The most well-known, using Kendall’s notation, are the M/M/1 and the M/M/c system. The algorithm described below estimates the expected number in the system under steady-state without a need for closed form steady-state probabilities. All that is needed is the transition matrix for the underlying Markov chain.

Some Explicit Results for the Distribution Problem of Stochastic Linear Programming

A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain;indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.

Stochastic Reservoir Systems with Different Assumptions for Storage Losses

Moran considered a dam whose inflow in a given interval of time is a continuous random variable. He then developed integral equations for the probabilities of emptiness and overflow. These equations are difficult to solve numerically;thus, approximations have been proposed that discretize the input. In this paper, extensions are considered for storage systems with different assumptions for storage losses. We also develop discrete approximations for the probabilities of emptiness and overflow.

Convexity Properties for a Function of Two Integer Variables

Sufficient conditions are given for any local minimum of a function of two integer variables to be a global minimum. An example is given to show that a function of two integer variables need not be discrete convex for this condition to hold.