On the continued Erlang loss function

We prove two fundamental results in teletraffic theory. The first is the frequently conjectured convexity of the analytic continuation B(x, a) of the classical Erlang loss function as a function of x, x 0. The second is the uniqueness of the solution of the basic set of equations associated with the ‘equivalent random method’

On a conjecture of Tutte concerning minimal tree numbers

A counterexample is given to a conjecture by Tutte on the minimum number of spanning trees that a 3-connected planar graph with a prescribed number of edges may have

Random walks on graphs

In this paper the following Markov chains are considered: the state space is the set of vertices of a connected graph, and for each vertex the transition is always to an adjacent vertex, such that each of the adjacent vertices has the same probability. Detailed results are given on the expectation of recurrence times, of first-entrance times, and of symmetrized first-entrance times (called commuting times). The problem of characterizing all connected graphs for which the commuting time is constant over all pairs of adjacent vertices is solved almost completely

A Note on the GI/GI/∞ System with Identical Service and Interarrival-Time Distributions

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 distribution of interarrival and service times is a gamma distribution

A note on the GI/GI/∞GI/GI/\infty system with identical service and interarrival-time distributions

We study the stationary distribution of the number of busy servers in a $GI/GI/\infty$ 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 ({\it IEEE Trans. Vehicular Technol.} {\bf 50} (2001) 817-834), when the common distribution of interarrival and service times is a gamma distribution. \u