An important class of queueing networks is characterized by the following feature: in contrast with ordinary units, a disaster may remove all work from the network. Applications of such networks include computer networks with virus infection, migration processes with mass exodus and serial production lines with catastrophes. In this paper, we deal with a two-stage tandem queue with blocking operating under the presence of a secondary flow of disasters. The arrival flows of units and disasters are general Markovian arrival processes. Using spectral analysis, we determine the stationary distribution at departure epochs. That distribution enables us to derive the distribution of the number of units which leave the network at a disaster epoch. We calculate the stationary distribution at an arbitrary time and, finally, we give numerical results and graphs for certain probabilistic descriptors of the network.

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (Xn,Sn), n ≥0, in which Xn takes values in a general state space and Sn takes values in ℝd. In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when Sn exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for Sn. For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), where T = inf { n : Sn,1 > b } and Sn,1 denotes the first component of Sn.

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

We consider a system that gradually deteriorates. At random times Tn the system is inspected and may or may not be replaced, but it must be if its state is too bad. The process of deterioration is described by a Markovian renewal process. Under some monotonicity conditions we derive a policy for the finite and the infinite horizon which minimizes the total discounted costs. This paper is a generalization of a result of Feldman (1977).

In this paper we treat the general version of the semi-Markov storage model, introduced first by Senturia and Puri: transitions in the state of the system occur at a discrete sequence of time points, described by a two-state semi-Markov process. An input occurs at an instant of transition to state 1 and a demand for release occurs at an instant of transition to state 2.

Assuming general distributions for all the variables involved, we show that the dam contents just after the nth input converges properly in distribution as n →∞ under conditions of stability; likewise that after the nth demand. We also show that the demand lost due to shortage of stock, accumulated over instants of demand as well as over time, obeys a strong law and a central limit theorem.

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused from all previous shocks.

The purpose of this paper is to derive the optimal replacement rule for such a system whose cumulative damage process is a semi-Markov process. This allows for both the time between shocks and the damage due to the next shock to be dependent on the present cumulative damage level.

Only policies within the class of control-limit policies will be considered; namely, policies with which no action is taken if the damage is below a fixed level, and a replacement is made if the damage is above that.

An example will be given illustrating the use of the optimal replacement rule.

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.