Parallel fluid queues with constant inflows and simultaneous random reductions

We consider I fluid queues in parallel. Each fluid queue has a deterministic inflow with a constant rate. At a random instant subject to a Poisson process, random amounts of fluids are simultaneously reduced. The requested amounts for the reduction are subject to a general I-dimensional distribution. The queues with inventories that are smaller than the requests are emptied. Stochastic upper bounds are considered for the stationary distribution of the joint buffer contents. Our major interest is in finding exponential product-form bounds, which turn out to have the appropriate decay rates with respect to certain linear combinations of buffer contents.

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On the rate of Poisson process approximation to a Bernoulli process

Journal of Applied Probability ◽

10.2307/3215005 ◽

1997 ◽

Vol 34 (4) ◽

pp. 898-907 ◽

Cited By ~ 4

Author(s):

Aihua Xia

Keyword(s):

Poisson Process ◽

Point Process ◽

Discrete Time ◽

Wasserstein Distance ◽

Stein’S Method ◽

Upper Bounds ◽

Stein's Method ◽

Bernoulli Process ◽

Logarithmic Factor ◽

Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

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Prime splitting in abelian number fields and linear combinations of Dirichlet characters

International Journal of Number Theory ◽

10.1142/s1793042114500055 ◽

2014 ◽

Vol 10 (04) ◽

pp. 885-903 ◽

Cited By ~ 3

Author(s):

Paul Pollack

Keyword(s):

Number Field ◽

Nonzero Element ◽

Number Fields ◽

Upper Bounds ◽

Prime Ideals ◽

Linear Combinations ◽

Dirichlet Characters ◽

Abelian Number Field ◽

Splitting Type ◽

A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

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Series representing transcendental numbers that are not U-numbers

International Journal of Number Theory ◽

10.1142/s1793042115500487 ◽

2015 ◽

Vol 11 (03) ◽

pp. 869-892

Author(s):

Emre Alkan

Keyword(s):

Rational Functions ◽

Rational Numbers ◽

Upper Bounds ◽

Integral Representations ◽

Binomial Coefficients ◽

Elementary Functions ◽

Linear Combinations ◽

Type Series ◽

Transcendental Numbers ◽

Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

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Distributional Properties of Fluid Queues Busy Period and First Passage Times

Mathematics ◽

10.3390/math8111988 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1988

Author(s):

Zbigniew Palmowski

Keyword(s):

Markov Process ◽

Busy Period ◽

First Passage Time ◽

Passage Time ◽

Fluid Queue ◽

First Passage ◽

Fluid Queues ◽

Distributional Properties ◽

Finite State ◽

Passage Times

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution.

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Fluid queues driven by anM/M/1/Nqueue

Mathematical Problems in Engineering ◽

10.1155/s1024123x00001423 ◽

2000 ◽

Vol 6 (5) ◽

pp. 439-460 ◽

Cited By ~ 9

Author(s):

R. B. Lenin ◽

P. R. Parthasarathy

Keyword(s):

Distribution Function ◽

Buffer Capacity ◽

Input Rate ◽

Fluid Queue ◽

Buffer Occupancy ◽

Fluid Queues ◽

Markovian Queue ◽

Service Rates ◽

Queue Model ◽

Number Of Customers

In this paper, we consider fluid queue models with infinite buffer capacity which receives and releases fluid at variable rates in such a way that the net input rate of fluid into the buffer (which is negative when fluid is flowing out of the buffer) is uniquely determined by the number of customers in anM/M/1/Nqueue model (that is, the fluid queue is driven by this Markovian queue) with constant arrival and service rates. We use some interesting identities of tridiagonal determinants to find analytically the eigenvalues of the underlying tridiagonal matrix and hence the distribution function of the buffer occupancy. For specific cases, we verify the results available in the literature.

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Markov-renewal fluid queues

Journal of Applied Probability ◽

10.1239/jap/1091543423 ◽

2004 ◽

Vol 41 (3) ◽

pp. 746-757 ◽

Cited By ~ 6

Author(s):

Guy Latouche ◽

Tetsuya Takine

Keyword(s):

Markov Process ◽

Exponential Distribution ◽

Stationary Distribution ◽

Markov Processes ◽

Input Rate ◽

Fluid Queue ◽

Birth And Death Processes ◽

Fluid Queues ◽

Semi Markov Process ◽

Semi Markov Processes

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

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Fluid queues driven by a birth and death process with alternating flow rates

Mathematical Problems in Engineering ◽

10.1155/s1024123x0420103x ◽

2004 ◽

Vol 2004 (5) ◽

pp. 469-489

Author(s):

P. R. Parthasarathy ◽

K. V. Vijayashree ◽

R. B. Lenin

Keyword(s):

State Space ◽

Death Process ◽

Birth And Death Process ◽

Fluid Queue ◽

Flow Rates ◽

Buffer Occupancy ◽

Fluid Queues ◽

Finite State ◽

Service Rates ◽

Finite State Space

Fluid queue driven by a birth and death process (BDP) with only one negative effective input rate has been considered in the literature. As an alternative, here we consider a fluid queue in which the input is characterized by a BDP with alternating positive and negative flow rates on a finite state space. Also, the BDP has two alternating arrival rates and two alternating service rates. Explicit expression for the distribution function of the buffer occupancy is obtained. The case where the state space is infinite is also discussed. Graphs are presented to visualize the buffer content distribution.

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Busy periods in fluid queues with multiple emptying input states

Journal of Applied Probability ◽

10.1239/jap/1276784904 ◽

2010 ◽

Vol 47 (2) ◽

pp. 474-497 ◽

Cited By ~ 2

Author(s):

A. J. Field ◽

P. G. Harrison

Keyword(s):

Busy Period ◽

Arrival Rate ◽

Random Variable ◽

Input Rate ◽

Output Rate ◽

Fluid Queue ◽

Fluid Queues ◽

The Laplace Transform ◽

Constant Output ◽

Positive Input

A semi-numerical method is derived to compute the Laplace transform of the equilibrium busy period probability density function in a fluid queue with constant output rate when the buffer is nonempty. The input process is controlled by a continuous-time semi-Markov chain (CTSMC) with n states such that in each state the input rate is constant. The holding time in states with net positive output rate - so-called emptying states - is assumed to be an exponentially distributed random variable, whereas in states with net positive input rate - so-called filling states - it may have an arbitrary probability distribution. The result is demonstrated by applying it to various systems, including fluid queues with two on-off input sources. The latter exercise in part shows consistency with prior results but also solves the problem in the case where there are two emptying states. Numerical results are presented for selected examples which expose discontinuities in the busy period distribution when the number of emptying states changes, e.g. as a result of increasing the fluid arrival rate in one or more states of the controlling CTSMC.

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Sign changes in linear combinations of derivatives and convolutions of Polya frequency functions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171279000090 ◽

1979 ◽

Vol 2 (1) ◽

pp. 91-101

Author(s):

Steven Nahmias ◽

Frank Proschan

Keyword(s):

Upper Bounds ◽

Linear Combinations ◽

Sign Changes ◽

Variation Diminishing ◽

Totally Positive ◽

Positive Functions ◽

Frequency Functions

We obtain upper bounds on the number of sign changes of linear combinations of derivatives and convolutions of Polya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan.

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