Uniform bounds for the complementary incomplete Gamma function

Borwein, Jonathan M.; Chan, O-Yeat

doi:10.7153/mia-12-10

Cited by 22 publications

(23 citation statements)

References 9 publications

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“…Let δ > 0 and p ≥ 0; then ∞ T e −δt t p dt ≤ 2δ −2 T p−1 e −δT ,provided that δT ≥ 2p.Proof. The statement is an immediate consequence of Corollary 2.5 in[6].…”

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confidence: 65%

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Goldenshluger

Koops

2019

Stochastic Systems

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This paper provides a mathematical framework for estimation of the service time distribution and the expected service time of an infinite-server queueing system with a non-homogeneous Poisson arrival process, in the case of partial information, where only the number of busy servers are observed over time. The problem is reduced to a statistical deconvolution problem, which is solved by using Laplace transform techniques and kernels for regularization. Upper bounds on the mean squared error of the proposed estimators are derived. Some concrete simulation experiments are performed to illustrate how the method can be applied and to provide some insight in the practical performance.Infinite-server queue First we provide some background on the M/G/∞ queueing model, which is well studied and could be considered as a standard model. In such queues there are arrivals according to a homogeneous Poisson process, each customer is served independently of all other customers and customers do not have to queue for service, because there is an infinite number of servers. The model has a wide variety of applications in e.g. telecommunication, road traffic and hospital modeling. It can be used as an approximation to M/G/n systems, where n is relatively large with respect to the arrival rate, but the model is also interesting in its own right. For example, it can be interpreted outside queueing theory as a model for the size of a population. In this paper we will use queueing terminology (servers, customers, etc.), but these terms could be adjusted according to the application. For example, 'customers' could be cars travelling between two locations and their 'service time' could be the travel time. In many applications it is seen that the arrival rate is not constant, but it varies over time. We will provide some examples below. This observation motivates studying the M t /G/∞ queue, where the arrival rate is assumed to be a nonhomogeneous Poisson process. This model is still particularly tractable (cf. [12]) and amenable for statistical analysis, as shown in this paper. Statistical queueing problemsQueueing theory studies probabilistic properties of random processes in service systems on the basis of a complete specification of system parameters. In this paper we are interested in inverse problems when unknown characteristics of a system should be inferred from observations of the associated random processes. Typically such observations are incomplete in the sense that individual customers cannot be tracked as they go through the service system. The importance of such statistical inverse problems with incomplete observations was emphasized in [3].The service time distribution and its expected value are important performance metrics of the infinite-server queue (note that waiting times are identical to service times in infinite-server queues). Our goal is to estimate these characteristics of the M t /G/∞ system from observations of the queue-length process. Specifically, let {τ j , j ∈ Z} be arrival epochs constituting a realizati...

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“…Let δ > 0 and p ≥ 0; then ∞ T e −δt t p dt ≤ 2δ −2 T p−1 e −δT ,provided that δT ≥ 2p.Proof. The statement is an immediate consequence of Corollary 2.5 in[6].…”

mentioning

confidence: 65%

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Goldenshluger

Koops

2019

Stochastic Systems

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“…and (s, x) is an upper incomplete gamma function defined as (s, x) = ∞ x t s−1 e −t dt. Proof: See [4] for the first inequality and [11] for the second inequality.…”

Section: Analysis Of Secure User-scaling Lawmentioning

confidence: 99%

Effects of Multiple Antennas and Imperfect Channel Knowledge on Secrecy Multiuser Diversity

2015

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In this letter, we investigate the effects of multiple antennas at a legitimate receiver and imperfect channel knowledge on secure user-scaling law in uplink multiuser wiretap networks. We generalize the conventional analysis of the secure user-scaling law for a single antenna system with perfect channel knowledge by considering multiple antennas at the receiver and the channel estimation errors.Index Terms-Physical-layer security, secrecy capacity, userscaling law, multiuser diversity, imperfect channel knowledge.

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“…Using a nice bound on Γ due to [11] that can be found in [5] we obtain a bound for B(G) < n. Think of a direct reduction for proving Corollary 14 as implicitly guessing C k+1 via guessing N (C k+1 ) by picking up to B(G) vertices in G and guessing one of at most B(G) connected subgraphs that have the guessed neighborhood. Note that Corollary 14 cannot provide a tighter bound on the loss than Theorem 8.…”

Section: A Bound Using the Maximum Bordermentioning

confidence: 99%

Standard Security Does Imply Security Against Selective Opening for Markov Distributions

Fuchsbauer

Heuer

Kiltz

et al. 2015

Theory of Cryptography

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About three decades ago it was realized that implementing private channels between parties which can be adaptively corrupted requires an encryption scheme that is secure against selective opening attacks. Whether standard (IND-CPA) security implies security against selective opening attacks has been a major open question since. The only known reduction from selective opening to IND-CPA security loses an exponential factor. A polynomial reduction is only known for the very special case where the distribution considered in the selective opening security experiment is a product distribution, i.e., the messages are sampled independently from each other.In this paper we give a reduction whose loss is quantified via the dependence graph (where message dependencies correspond to edges) of the underlying message distribution. In particular, for some concrete distributions including Markov distributions, our reduction is polynomial.

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Uniform bounds for the complementary incomplete Gamma function

Cited by 22 publications

References 9 publications

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Effects of Multiple Antennas and Imperfect Channel Knowledge on Secrecy Multiuser Diversity

Standard Security Does Imply Security Against Selective Opening for Markov Distributions

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