This note gives a new strong stationary time (SST) for reversible finite Markov chains. A modification of the initial distribution is represented as a mixture of distributions which have eigenvector interpretations, and for which good simple SSTs exist. This provides some insight into the relationship between SSTs and eigenvalues. Connections to duality and the threshold phenomenon are discussed.

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. Failures can be of two types. With probability p(q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous-parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that both and grow linearly with n. These limit theorems give very useful approximations, in terms of the six parameters characterizing the model, to a distribution that would be difficult to use in practice.