First-passage times and related moments for continuous-time birth–death chains

Spectral analysis of bilateral birth–death processes: some new explicit examples

“…This function gives a discrete version for a similar situation applied to the Kolmogorov forward equation (or Fokker-Planck equation) for diffusion processes (see [12, p. 121]), replacing the partial derivative by an integer-valued analogue. The function j,n (t) has been recently considered in [13,14] to obtain functional relations between the transition probabilities of a birth-death process on N 0 and the transition probabilities of a bilateral birth-death process (see for instance [13,Proposition 5]). Using the Karlin-McGregor formula (3) and the symmetry property…”

“…Now a diagram of the transitions between states is These processes are also known as bilateral birth-death processes (following [29]), doubleended systems, or unrestricted birth-death processes (see [6,7,8,13,14]). Again, we will assume that the set of rates {λ n , μ n } uniquely determines the bilateral birth-death process.…”

“…3. We identify the corresponding orthogonal polynomials (Q α n ) n∈Z , α = 1, 2, by looking at the three-term recurrence relation (13). All the examples we study in this paper are related somehow to the Chebyshev polynomials of the first and second kind.…”

Upper bound on the rate of convergence and truncation bound for nonhomogeneous birth and death processes on $\mathbb{Z}$

“…If the intensities depend on the state of X(t), it implies that the admission of passengers/taxis to the queueing-point is dynamically controlled. Since the seminal paper [31] such queues and similar to X(t) processes have been the subject of extensive research and now they are usually referred to as doublesided or double-ended queues (see [24,33]), unrestricted random walks on lattice and bilateral BDPs [34,35,36]. Another intuitive but otherwise artificial example (which will also be revisited in the numerical section) is the system size/queue length in common queueing systems 1 at epoch t. If one removes the impenetrable barrier at the origin, which means that the departures are also allowed, when the system size is zero or negative, one arrives at another instance of X(t) (see [28,38]).…”

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

“…Very often, in growth models, immigration's effects may occur, due to the circumstance that the population is not isolated. In these cases, it is necessary take into account birth-death processes with a reflecting condition in the zero state (see, for instance, Di Crescenzo et al [8], Crawford and Suchard [9], Giorno and Nobile [10], Lenin et al [11] and Tavaré [12]). These processes provide interesting applications in queuing models in which a reflecting boundary must be imposed to describe the number of customers in the system (cf., for instance, Crawford et al [13], Di Crescenzo et al [14] and Giorno et al [15]).…”