Connectivity of circulant digraphs

An explicit expression is derived for the connectivity of circulant digraphs

On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

A sufficient condition is obtained for a discrete-time birth-death process to possess the strong ratio limit property, directly in terms of the one-step transition probabilities of the process. The condition encompasses all previously known sufficient conditions

Asymptotic period of an aperiodic Markov chain

We introduce the concept of asymptotic period for an irreducible and aperiodic, discrete-time Markov chain X on a countable state space, and develop the theory leading to its formal definition. The asymptotic period of X equals one - its period - if X is recurrent, but may be larger than one if X is transient; X is asymptotically aperiodic if its asymptotic period equals one. Some sufficient conditions for asymptotic aperiodicity are presented. The asymptotic period of a birth-death process on the nonnegative integers is studied in detail and shown to be equal to 1, 2 or infinity. Criteria for the occurrence of each value in terms of the 1-step transition probabilities are established.Comment: 19 page

The indeterminate rate problem for birth-death processes

A birth-death process is completely determined by its set of rates if and only if this set satisfies a certain condition C, say. If for a set of rates R the condition C is not fulfilled, then the problem arises of characterizing all birth-death processes which have rate set R (the indeterminate rate problem associated with R). We show that the characterization may be effected by means of the decay parameter, and we determine the set of possible values for the decay parameter in terms of JR. A fundamental role in our analysis is played by a duality concept for rate sets, which, if the pertinent rate sets satisfy C, obviously leads to a duality concept for birth-death processes. The latter can be stated in a form which suggests the possibility of extension in the context of indeterminate rate problems. This, however, is shown to be only partially true

On the α-classification of birth-death and quasi-birth-death processes

In several recent papers criteria for the α-classification of birth-death and quasi-birth-death processes have been proposed. In this paper the relations between the various criteria are brought to light

Weighted sums of orthogonal polynomials related to birth-death processes with killing

We consider sequences of orthogonal polynomials arising in the analysis of birth-death processes with killing. Motivated by problems in this stochastic setting we discuss criteria for convergence of certain weighted sums of the polynomials

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’

Weighted sums of orthogonal polynomials with positive zeros

We study the two sequences of polynomials which arise as denominators of the approximants of even and odd order, respectively, of a Stieltjes fraction, and which may be defined alternatively as a sequence of orthogonal polynomials with positive zeros and the associated sequence of kernel polynomials. Motivated by problems in the setting of birth-death processes, where these sequences play a major role, we focus on the asymptotic behaviour of the sequences and establish convergence of certain weighted sums of the polynomials at hand

Analysis of a queuing model for slotted ring networks

We study a multi-server multi-queue system which is intended to model a local area network with slotted ring protocol. Two special cases of the model are analysed and the results are used to motivate an approach to approximate mean queue lengths in the general model