Asymptotic analysis of the Krawtchouk polynomials by the WKB method

Dominici, Diego

doi:10.1007/s11139-007-9078-9

Cited by 27 publications

(14 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The same approach was employed by Carlitz in [2]. From (6), it follows easily that for a fixed value of n P n (x) ∼ n! x n , x → ∞.…”

Section: Introductionmentioning

confidence: 90%

“…The paper is organized as follows: in Section 2 we approach the problem using a singularity analysis of the generating function [14] of the polynomials P n (x). In Section 3 we apply the WKB method to the differential-difference equation (6). In [15], we used this approach in the asymptotic analysis of computer science problems and in [6] to study the Krawtchouk polynomials.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic analysis of a family of polynomials associated with the inverse error function

Dominici¹,

Knessl²

2012

Rocky Mountain J. Math.

Self Cite

View full text Add to dashboard Cite

We analyze the sequence of polynomials defined by the differentialdifference equation P n+1 (x) = P ′ n (x) + x(n + 1)P n (x) asymptotically as n → ∞. The polynomials P n (x) arise in the computation of higher derivatives of the inverse error function inverf(x). We use singularity analysis and discrete versions of the WKB and ray methods and give numerical results showing the accuracy of our formulas.MSC-Class: 33B20 (Primary) 34E20, 33E30 (Secondary).

show abstract

“…The same approach was employed by Carlitz in [2]. From (6), it follows easily that for a fixed value of n P n (x) ∼ n! x n , x → ∞.…”

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of a family of polynomials associated with the inverse error function

Dominici¹,

Knessl²

2012

Rocky Mountain J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The main complication in working with Krawtchouk polynomials is that they have several different forms of asymptotic behavior depending on whether x and k are in the lower, middle or upper part of their range; indeed, [9] breaks the asymptotic properties of κ k (x) into 12 different cases. However, for our purpose, we need only two different upper bounds on the Krawtchouk polynomials, based on whether k/n is greater than or less than 0.14; a somewhat arbitrary threshold that we will see the justification for below.…”

Section: Fourier Analysis and Krawtchouk Polynomialsmentioning

confidence: 99%

Untitled

Harrow¹,

Kolla²,

Schulman³

2014

Theory of Comput.

View full text Add to dashboard Cite

“…The parabolic cylinder functions arise in several contexts associated with the limits of queueing models, such as the Ornstein-Uhlenbeck limit of the appropriately scaled infinite-server queue [20] and various limits associated with the Erlang loss model [40]. We note that the parabolic cylinder functions have been studied as the limits of certain polynomials under the HW scaling, using tools from the theory of differential equations [2,[9][10][11]].…”

Section: 2mentioning

confidence: 99%

On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime

Gamarnik¹,

Goldberg²

2013

Ann. Appl. Probab.

View full text Add to dashboard Cite

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant B * ≈ 1.85772 s.t. when a certain excess parameter B ∈ (0, B * ], the error in the steady-state approximation converges exponentially fast to zero at rate B 2 4 . For B > B * , the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed n by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer].We also prove explicit bounds on the distance to stationarity for the M/M/n queue in the Halfin-Whitt regime, when B < B * . Our bounds scale independently of n in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.and Whitt [18], who studied the GI/M/n system for large n when the traffic intensity scales like 1− Bn −1/2 for some strictly positive excess parameter B. They proved that, under minor technical assumptions on the inter-arrival distribution, this sequence of GI/M/n queueing models has the following properties:(i) the steady-state probability that an arriving job has to wait for service has a nontrivial limit;(ii) the sequence of queueing processes, normalized by n 1/2 , converges weakly to a nontrivial positive recurrent diffusion, a.k.a. the HW diffusion;(iii) the sequence of steady-state queue length distributions, normalized by n 1/2 , is tight and converges distributionally to the mixture of a point mass at zero and an exponential distribution.Since the steady-state behavior of the M/M/n queue in the HW regime is quite simple [18], while the transient dynamics are more complicated [18], it is common to use the steady-state approximation to the transient distribution [16]. Thus it is important to understand the quality of the steady-state approximation. The only work along these lines seems to be the recent papers [38,39], in which the authors study the Laplace transform of the HW and related diffusions, and prove several results analogous to our own for these diffusions. The key difference is that in this paper we study the prelimit diffusion-scaled M/M/n queue, not the limiting diffusion. We note that the relevant transform functions were also studied in [1], although in a different context. Also, similar questions were studied for the associated sequence of fluid-scaled queues in [23].The question of how quickly the positive recurrent M/M/n queue approaches stationarity has a rich history in the queueing literature. In [28], Morse derives an explicit solution for the transient M/M/1 queue, and discusses implications for the exponential rate of convergence to stationarity...

show abstract

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

Cited by 27 publications

References 35 publications

Asymptotic analysis of a family of polynomials associated with the inverse error function

Asymptotic analysis of a family of polynomials associated with the inverse error function

Untitled

On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime

Contact Info

Product

Resources

About