arXiv subjects

Yoni Nazarathy

Publications and source records attributed to Yoni Nazarathy.

18 recordsLinked to original sources

arXiv2026

Second Order Properties of Thinned Counts in Finite Birth--Death Processes

The paper studies the counting process arising as a subset of births and deaths in a birth--death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a state-dependent function of the birth and death and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested.

math.PR↗

arXiv2024

Parameter and State Estimation in Queues and Related Stochastic Models: A Bibliography

This is an annotated bibliography on estimation and inference results for queues and related stochastic models. The purpose of this document is to collect and categorise works in the field, allowing for researchers and practitioners to explore the various types of results that exist. Our focus is on papers that deal with mathematical queueing models as well as related stochastic models motivated by queues. We attempted to make this bibliography exhaustive, yet there are possibly some papers that we have missed. As it is updated continuously, additions and comments are welcomed. Note that this bibliography is also a companion to our survey of parameter and state estimation in queues [20].

math.ST↗

arXiv2022

Nonparametric testing via partial sorting

In this paper we introduce the idea of partially sorting data to design nonparametric tests. This approach gives rise to tests that are sensitive to both the order and the underlying distribution of the data. We focus in particular on a test that uses the bubble sort algorithm to partially sort the data. We show that a function of the data, referred to as the empirical bubble sort curve, converges uniformly to a limiting curve. We define a goodness-of-fit test based on the distance between the empirical curve and its limit. The asymptotic distribution of the test statistic is a generalization of the Kolmogorov distribution. We apply the test in several examples and observe that it outperforms classical nonparametric tests for appropriately chosen sorting levels.

math.ST↗

arXiv2022

On Busy Periods of the Critical GI/G/1 Queue and BRAVO

We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) for the work-output process (namely the busy-time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy-period. In the current paper we strengthen the previous results by reducing to assumptions to existence of $2+ε$ moments.

math.PR↗

arXiv2021

Diffusion Parameters of Flows in Stable Multi-class Queueing Networks

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Towards that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.

math.PR↗

arXiv2021

The Value of Information and Efficient Switching in Channel Selection

We consider a collection of statistically identical two-state continuous time Markov chains (channels). A controller continuously selects a channel with the view of maximizing infinite horizon average reward. A switching cost is paid upon channel changes. We consider two cases: full observation (all channels observed simultaneously) and partial observation (only the current channel observed). We analyze the difference in performance between these cases for various policies. For the partial observation case with two channels, or an infinite number of channels, we explicitly characterize an optimal threshold for two sensible policies which we name "call-gapping" and "cool-off". Our results present a qualitative view on the interaction of the number of channels, the available information, and the switching costs.

cs.IT↗

arXiv2021

Markovian Transition Counting Processes: An Alternative to Markov Modulated Poisson Processes

Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the \textit{Markov Modulated Poisson Process} (MMPP), however it is not the only option. Here, we introduce an alternative that we call \textit{Markovian transition counting process} (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We illustrate the use of such equivalence in a simple queueing example, showing that the MTCP is a comparable and competitive model for performance analysis.

math.PR↗

arXiv2020

Estimation of Semi-Markov Multi-state Models: A Comparison of the Sojourn Times and Transition Intensities Approaches

Semi-Markov models are widely used for survival analysis and reliability analysis. In general, there are two competing parameterizations and each entails its own interpretation and inference properties. On the one hand, a semi-Markov process can be defined based on the distribution of sojourn times, often via hazard rates, together with transition probabilities of an embedded Markov chain. On the other hand, intensity transition functions may be used, often referred to as the hazard rates of the semi-Markov process. We summarize and contrast these two parameterizations both from a probabilistic and an inference perspective, and we highlight relationships between the two approaches. In general, the intensity transition based approach allows the likelihood to be split into likelihoods of two-state models having fewer parameters, allowing efficient computation and usage of many survival analysis tools. {Nevertheless, in certain cases the sojourn time based approach is natural and has been exploited extensively in applications.} In contrasting the two approaches and contemporary relevant R packages used for inference, we use two real datasets highlighting the probabilistic and inference properties of each approach. This analysis is accompanied by an R vignette.

stat.ME↗

arXiv2020

A Survey of Parameter and State Estimation in Queues

We present a broad literature survey of parameter and state estimation for queueing systems. Our approach is based on various inference activities, queueing models, observations schemes, and statistical methods. We categorize these into branches of research that we call estimation paradigms. These include: the classical sampling approach, inverse problems, inference for non-interacting systems, inference with discrete sampling, inference with queueing fundamentals, queue inference engine problems, Bayesian approaches, online prediction, implicit models, and control, design, and uncertainty quantification. For each of these estimation paradigms, we outline the principles and ideas, while surveying key references. We also present various simple numerical experiments. In addition to some key references mentioned here, a periodically-updated comprehensive list of references dealing with parameter and state estimation of queues will be kept in an accompanying annotated bibliography.

stat.ME↗

arXiv2019

The Squared Coefficient of Variation for MMPP is Greater than Unity

Folklore often treats the Markov Modulated Poisson Process as bursty because the variance divided by the expectation of counts is greater than unity. When viewed through the lens of the inter-event process, this ideally corresponds to a squared coefficient of variation greater than unity. As this has not been proved to date, we provide a proof together with an associated stochastic order relation.

math.PR↗

arXiv2019

Stationary Markovian Arrival Processes, Results and Open Problems

We consider two classes of irreducible Markovian arrival processes specified by the matrices $C$ and $D$. The Markov Modulated Poison Process (MMPP) and the Markovian Switched Poison Process (MSPP). The former exhibits a diagonal $D$ while the latter exhibits a diagonal $C$. For these two classes, we consider the following statements: (I) Overdispersion of the counts process. (II) A non-increasing hazard rate of the stationary inter-event time. (III) The squared coefficient of variation of the event stationary process is greater or equal to unity. (IV) A stochastic order showing that the time stationary inter-arrival time dominates the event-stationary time. For general MSPPs and two-state MMPPs, we show that (I)-(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is false due to a counter-example of Miklos Telek and Illes Horvath. For general simple point processes, (III) follows from (IV). For MMPPs we conjecture and numerically test that (IV) and thus (III) hold. Importantly, modeling folklore has often treated MMPPs as ``bursty'' and implicitly assumed that (III) holds. However, this is still an open question.

math.PR↗

arXiv2017

Scheduling for a Processor Sharing System with Linear Slowdown

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a "local search", where we use the problem structure in a novel manner. The second is a one-coordinate "global search", used in coordinate pivot iteration. We then merge these two heuristics into a unified "local-global" heuristic, and numerically illustrate the effectiveness of this heuristic.

math.OC↗

arXiv2016

The Role of Information in System Stability with Partially Observable Servers

We consider a simple discrete-time controlled queueing system, where the controller has a choice of which server to use at each time slot and server performance varies according to a Markov modulated random environment. We explore the role of information in the system stability region. At the extreme cases of information availability, that is when there is either full information or no information, stability regions and maximally stabilizing policies are trivial. But in the more realistic cases where only the environment state of the selected server is observed, only the service successes are observed or only queue length is observed, finding throughput maximizing control laws is a challenge. To handle these situations, we devise a Partially Observable Markov Decision Process (POMDP) formulation of the problem and illustrate properties of its solution. We further model the system under given decision rules, using Quasi-Birth-and-Death (QBD) structure to find a matrix analytic expression for the stability bound. We use this formulation to illustrate how the stability region grows as the number of controller belief states increases. Our focus in this paper is on the simple case of two servers where the environment of each is modulated according to a two-state Markov chain. As simple as this case seems, there appear to be no closed form descriptions of the stability region under the various regimes considered. Our numerical approximations to the POMDP Bellman equations and the numerical solutions of the QBDs hint at a variety of structural results.

math.OC↗

arXiv2014

A Correction Term for the Covariance of Renewal-Reward Processes with Multivariate Rewards

We consider a renewal-reward process with multivariate rewards. Such a process is constructed from an i.i.d.\ sequence of time periods, to each of which there is associated a multivariate reward vector. The rewards in each time period may depend on each other and on the period length, but not on the other time periods. Rewards are accumulated to form a vector valued process that exhibits jumps in all coordinates simultaneously, only at renewal epochs. We derive an asymptotically exact expression for the covariance function (over time) of the rewards, which is used to refine a central limit theorem for the vector of rewards. As illustrated by a numerical example, this refinement can yield improved accuracy, especially for moderate time-horizons.

math.PR↗

arXiv2013

The Second Order Terms of the Variance Curves for Some Queueing Output Processes

We consider queueing output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve, indicating the variance of the number of served customers over a time interval. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to the results by finding explicit expressions for the second order approximation of the variance curve, namely the y-intercept of the linear asymptote. For M/M/1/K queues our results are based on the Drazin inverse of the generator. It turns out that by viewing output processes as MAPs (Markovian Arrival Processes) and considering the Drazin inverse, one can obtain explicit expressions for the y-intercept, together with some further insight regarding the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1 queues our results are based on a classic transform of D.J. Daley. In this case we represent the y-intercept of the variance curve in terms of the first three moments of the service time distribution. A further performance measure that we are able to calculate for both models, is the asymptotic covariance between the queue length and the number of arrivals or departures. In addition we shed light on a classic conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

math.PR↗

arXiv2013

The age of information in gossip networks

We introduce models of gossip based communication networks in which each node is simultaneously a sensor, a relay and a user of information. We model the status of ages of information between nodes as a discrete time Markov chain. In this setting a gossip transmission policy is a decision made at each node regarding what type of information to relay at any given time (if any). When transmission policies are based on random decisions, we are able to analyze the age of information in certain illustrative structured examples either by means of an explicit analysis, an algorithm or asymptotic approximations. Our key contribution is presenting this class of models.

math.PR↗

arXiv2013

BRAVO for many-server QED systems with finite buffers

This paper demonstrates the occurrence of the feature called BRAVO (Balancing Reduces Asymptotic Variance of Output) for the departure process of a finite-buffer Markovian many-server system in the QED (Quality and Efficiency-Driven) heavy-traffic regime. The results are based on evaluating the limit of a formula for the asymptotic variance of death counts in finite birth--death processes.

math.PR↗

arXiv2013

Non-Existence of Stabilizing Policies for the Critical Push-Pull Network and Generalizations

The push-pull queueing network is a simple example in which servers either serve jobs or generate new arrivals. It was previously conjectured that there is no policy that makes the network positive recurrent (stable) in the critical case. We settle this conjecture and devise a general sufficient condition for non-stabilizability of queueing networks which is based on a linear martingale and further applies to generalizations of the push-pull network.

math.PR↗