Michael Rubinovitch

The stochastic behavior of a buffer with non-identical input lines

A system consisting of a buffer, N input lines leading to it and one line leading out is considered. Successive active and idle periods on the input lines constitute an alternating renewal process of a special kind. While in previous work the case of identical input lines was considered, the present paper gives a solution to the general case of non-identical input lines. This provides a tool for the analysis of arbitrarily complicated networks of buffers. The paper contains results regarding the traffic pattern on the output lineas well as the content of the buffer and the maximum content of the buffer during intervals of non-emptiness.

Number of cycles distribution in circulating systems

Abstract The characterization of certain circulating systems in terms of the number of cycles distribution (NCD) of the fluid during operation is considered. The theory of stochastic processes is used to obtain the NCD from the cycle time distribution (CTD) for the system. The case of the gamma distributed CTD is solved explicitly and the limiting distribution of the NCD as the operating time becomes large is considered. It is shown that for long operating times, the NCD converges to a normal distribution whose mean and variance depend only on the mean and the variance of the CTD.

Further results for ladder processes in continuous time

The distribution of ladder variables is obtained for the class of processes with stationary independent increments for which they are almost surely positive. This result is used to derive some related distributions. It is proved that a Wiener-Hopf factorization completely analogous to the one for random walks holds if and only if the process is compound Poisson with zero drift.

A Probabilistic Model for Optimal Project Scheduling

This paper formulates a model for the scheduling and control of a multiphase project that consists of a given sequence of jobs to be completed by a specified due date. The duration of each job is the sum of a known fixed time and a random delay; and, while the former may be shortened within given limits, there is no control over the latter. The decision variables are the level of activity in each job and the starting time of the first in the sequence. An optimal decision policy for this model is obtained assuming a simple cost structure. It is shown that, if this policy is followed, a minimum a priori probability of meeting the due date is guaranteed.