The Classification of Birth and Death Processes

Karlin, Samuel; McGregor, James A.

doi:10.2307/1993021

Cited by 77 publications

(114 citation statements)

References 5 publications

Supporting

Mentioning

112

Contrasting

Order By: Relevance

“…The same result holds when n = r. In conclusion, both the expressions given in (26) and (27) satisfy the system of forward equations for the probabilities of N (t) with initial condition δ n,k , and both vanish when n = s and n = r. Hence, they constitute the {r, s}-avoiding transition probabilities of N (t). Figure 4 shows some plots of probabilities (27). We finally stress that the role of the symmetry properties of the transition probabilities p k,n (t) has been essential to obtain the results on absorption problems given in this section.…”

Section: Two-boundaries Casesupporting

confidence: 51%

See 1 more Smart Citation

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Crescenzo¹,

Martinucci²

2009

Symmetry

View full text Add to dashboard Cite

We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series.

show abstract

Section: Two-boundaries Casesupporting

confidence: 51%

“…The identity between (26) and (27) can be immediately obtained by means of (28), with two appropriate choices of N . Moreover, from (12) one has…”

Section: Two-boundaries Casementioning

confidence: 99%

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Crescenzo¹,

Martinucci²

2009

Symmetry

View full text Add to dashboard Cite

show abstract

“…The adjoint matrix P * t is the fundamental solution for the forward equationż t = A * z t = Kz t given in (2.9). The representation formula of Karlin and McGregor [10,11], see (1.2) and (2.18) in [3], yields…”

Section: A Related Birth and Death Process And The Spectral Decomposmentioning

confidence: 99%

Functional central limit theorems for a large network in which customers join the shortest of several queues

Graham

2004

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Abstract. We consider N single server infinite buffer queues with service rate β. Customers arrive at rate N α, choose L queues uniformly, and join the shortest. We study the processes t ∈ R + → R N t = (R N t (k)) k∈N for large N , where R N t (k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R N 0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R N in equilibrium under the stability condition α < β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim N →∞ lim t→∞ = lim t→∞ lim N →∞ by a compactnessuniqueness method. We deduce a posteriori the CLT for R N 0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit. Key-words:Mean-field interaction, load balancing, resource pooling, ergodicity, non-equilibrium fluctuations, equilibrium fluctuations, birth and death processes, spectral gap, global exponential stability MSC2000: Primary: 60K35. Secondary: 60K25, 60B12, 60F05, 37C75, 37A30. Introduction PreliminariesWe consider a Markovian network constituted of N ≥ L ≥ 1 infinite buffer single server queues.Customers arrive at rate N α, are each allocated L distinct queues uniformly at random, and join the shortest, ties being resolved uniformly. Servers work at rate β. Arrivals, allocations, and services are independent. For L = 1 we have i.i.d. M α /M β /1/∞ queues. For L ≥ 2 the interaction structure depends only on sampling from the empirical measure of L-tuples of queue states: in statistical mechanics terminology, the system is in L-body mean-field interaction. We continue the large N study introduced by Vvedenskaya et al.

show abstract

“…It is well-known that for the (1, 1)-RW, the criteria for the (positive) recurrence and the expression for the stationary distribution have been given explicitly, for example see [8] and [9]). However, for L > 1, no such explicit expressions are found to our best knowledge.…”

Section: The Background and Motivationmentioning

confidence: 99%

Explicit stationary distribution of the (L, 1)-reflecting random walk on the half line

Hong

Zhou

Qiang

et al. 2014

Acta. Math. Sin.-English Ser.

View full text Add to dashboard Cite

In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.

show abstract

The Classification of Birth and Death Processes

Cited by 77 publications

References 5 publications

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Functional central limit theorems for a large network in which customers join the shortest of several queues

Explicit stationary distribution of the (L, 1)-reflecting random walk on the half line

Contact Info

Product

Resources

About