3 Acknowledgments 4 List of Figures 9

We study the stationary distribution of the number of busy servers in a GI/GI/∞ system in which the service-time distribution is identical to the interarrival-time distribution, and obtain several representations for the variance. As a result we can verify an expression for the variance, conjectured by Rajaratnam and Takawira (IEEE Trans. Vehicular Technol. 50 (2001) 954-970), when the common d...

The feature map represented by the set of weight vectors of the basic SOM (Self-Organizing Map) provides a good approximation to the input space from which the sample vectors come. But the timedecreasing learning rate and neighborhood function of the basic SOM algorithm reduce its capability to adapt weights for a varied environment. In dealing with non-stationary input distributions and changi...

In this paper we consider a queuing system in which the service times of customers are independent and identically distributed random variables, the arrival process is stationary and has the property of orderliness, and the queue discipline is arbitrary. For this queuing system we obtain the steady state second moment of the queue size in terms of the stationary waiting time distribution of a s...

A random walk on a directed graph gives a markov chain on the vertices on the graph. An important question that arises often in the context of markov chain is whether the uniform distribution on the vertices of the graph a stationary distribution for the markov chain. Testing whether a distribution is uniform or “far” from being uniform is a well studied problem in property testing and statisti...

Computing the stationary distribution of a large finite or countably infinite state space Markov Chain has become central to many problems such as statistical inference and network analysis. Standard methods involve large matrix multiplications as in power iteration, or simulations of long random walks, as in Markov Chain Monte Carlo (MCMC). For both methods, the convergence rate is is difficul...

In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (Xt) on Ω is a Markov chain if the sequence is Markovian in the following sense, for all t, all x0, . . . , xt, y ∈ Ω, we require Pr(Xt+1 = y|X0 = x0, X1 = x1, . . . , Xt = xt) = Pr(Xt+1 ...