Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed... more Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution F supported on [−∞, 1] with essential supremum equal to 1 (a charge of −∞ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by C(F). Even in the simplest case where F = pδ 1 + (1 − p)δ −∞ , corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant C(F). Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.
We consider a sample path of a random walk on the integers with bounded local times, conditioned ... more We consider a sample path of a random walk on the integers with bounded local times, conditioned on the event that it hits a high level. Under an auxiliary assumption, we obtain representations for its distribution in terms of the corresponding limiting sequence. Then we prove limiting results as the high level grows. In particular, we generalize results for a simple symmetric random walk obtained earlier by Benjamini and Berectycki (2010).
This text studies heavy-tailed distributions in probability theory, and especially convolutions o... more This text studies heavy-tailed distributions in probability theory, and especially convolutions of such distributions. The mail goal is to provide a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions which includes many novel elements and, in particular, is based on the regular use of the principle of a single big jump.
We introduce a model for the classical synchronised multiple access system with a single transmis... more We introduce a model for the classical synchronised multiple access system with a single transmission channel and a randomised transmission protocol (ALOHA). We assume in addition that there is an energy harvesting mechanism, and any message transmission requires a unit of energy. Units of energy arrive randomly and independently of anything else. We analyze stability and instability conditions for this model.
We consider a discrete-time Markov chain (X t , Y t), t = 0, 1, 2,. . ., where the X-component fo... more We consider a discrete-time Markov chain (X t , Y t), t = 0, 1, 2,. . ., where the X-component forms a Markov chain itself. Assume that (X t) is Harris-ergodic and consider an auxiliary Markov chain { Y t } whose transition probabilities are the averages of transition probabilities of the Y-component of the (X, Y)-chain, where the averaging is weighted by the stationary distribution of the X-component. We first provide natural conditions in terms of test functions ensuring that the Y-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain (X t , Y t). The we prove a "multi-dimensional" extension of the result obtained. In the second part of the paper, we apply our results to two versions of a multi-access wireless model governed by two randomised protocols.
We consider spatial marked Poisson arrivals in a Polish space. These arrivals are accepted or los... more We consider spatial marked Poisson arrivals in a Polish space. These arrivals are accepted or lost in a general state dependent manner. The accepted arrivals remain in the system for a random amount of time, where the individual sojourn times are i.i.d. For such systems, we develop semi-closed form expressions for the steady state probabilities that can be seen to be insensitive to the sojourn time distribution, and that rely essentially on the static probabilities of marked Poisson objects meeting the state acceptance criteria. The latter observation is then exploited to yield straightforward exact simulation algorithms to sample from the steady state distribution. In addition, for the special case where the arrivals are spheres in a Euclidean space that are lost whenever they overlap with an existing sphere, we develop large deviations asymptotics for the probability of observing a large number of spheres in the system in steady state, under diverse asymptotic regimes. Application...
IEEE INFOCOM 2008 - The 27th Conference on Computer Communications, 2008
The split of a multihop, point-to-point TCP connection consists in replacing a plain, end-to-end ... more The split of a multihop, point-to-point TCP connection consists in replacing a plain, end-to-end TCP connection by a cascade of TCP connections. In such a cascade, connection n feeds connection n + 1 through some proxy node n. This technique is used in a variety of
Изучаются нижние пределы отношений F* r (x)/F(x) хвостов распре делений, где F* T-распределение с... more Изучаются нижние пределы отношений F* r (x)/F(x) хвостов распре делений, где F* T-распределение суммы случайного числа т независи мых одинаково распределенных случайных величин с общим распреде лением F, при этом г не зависит от слагаемых. Ключевые слова и фразы: хвост свертки распределений, случайные суммы случайных величин, нижний предел, распределения с тяжелыми и легкими хвостами.
Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an ar... more Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an arbitrary boundary. If the sequence increments satisfy a so-called condition of asymptotic homogeneity and the boundary is asymptotically "smooth," then the occurrence of the weak convergence to a limit shape (as the boundary is sent away) is established for the distribution of the overshoot value. As an application, we obtain a uniform (over the class of distributions) basic renewal theorem and determine the asymptotics of the average time of crossing a curvilinear border by the trajectories of asymptotically homogeneous Markov chains.
Consider a queueing system in which arriving customers are placed on a circle and wait for servic... more Consider a queueing system in which arriving customers are placed on a circle and wait for service. A traveling server moves at constant speed on the circle, stopping at the location of the customers until service completion. The server is greedy: always moving in the direction of the nearest customer. Coffman and Gilbert conjectured that this system is stable if the traffic intensity is smaller than 1; however, a proof or counterexample remains unknown. In this review, we present a picture of the current state of this conjecture and suggest new related open problems.
Random multiple-access protocols of type ALOHA are used to regulate networks with a star configur... more Random multiple-access protocols of type ALOHA are used to regulate networks with a star configuration where client nodes talk to the hub node at the same frequency (finding a wide range of applications among telecommunication systems, including mobile telephone networks and WiFi networks). Such protocols control who talks at what time sharing the common idea "try to send your data and, if your message collides with another transmission, try resending later". In the present paper, we consider a time-slotted ALOHA model where users are allowed to renege before transmission completion. We focus on the scenario that leads to overload in the absence of impatience. Under mild assumptions, we show that the fluid (or law-of-large-numbers) limit of the system workload coincides a.s. with the unique solution to a certain integral equation. We also demonstrate that the fluid limits for distinct initial conditions converge to the same value as time tends to infinity. Keywords ALOHA protocol • Queues with impatience • Queues in overload • Fluid limits Mathematics Subject Classification (2000) 60K25 • 60F17 • 90B15 • 90B22 The research of M. Frolkova and B. Zwart is supported by an NWO VIDI grant.
This note introduces a greedy walk on Poisson and Binomial processes, which is a close relative t... more This note introduces a greedy walk on Poisson and Binomial processes, which is a close relative to the well-known greedy server model. Some open problems are presented.
Regenerative events for different queueing models are considered. The aim of this paper is to con... more Regenerative events for different queueing models are considered. The aim of this paper is to construct these events for continuous-time processes if they are given for the corresponding discrete-time model. The construction uses so-called renovative events revealing the property of the state at time n of the discrete-time model to be independent (in an algebraic sense) of the states referring to epochs not later than n-L (where L is some constant) given that there are some restrictions on the "governing sequence". Different types of multi-server and multi-phase queues are considered.
Probability in the Engineering and Informational Sciences, 1998
We develop an algorithm for simulating approximate random samples from the invariant measure of a... more We develop an algorithm for simulating approximate random samples from the invariant measure of a Markov chain using backward coupling of embedded regeneration times. Related methods have been used effectively for finite chains and for stochastically monotone chains: here we propose a method of implementation which avoids these restrictions by using a “cycle-length” truncation. We show that the coupling times have good theoretical properties and describe benefits and difficulties of implementing the methods in practice.
We consider a generalized vacation or polling system, modeled as an input-output process operatin... more We consider a generalized vacation or polling system, modeled as an input-output process operating over successive “cycles,” in which the service mechanism can be in an “up” mode (processing) or “down” mode (e.g., vacation, walking). Our primary motivation is polling systems, in which there are several queues and the server moves cyclically between them providing some service in each. Our basic assumption is that the amount of work that leaves the system in a “cycle” is no less than the amount present at the beginning of the cycle. This includes the standard gated and exhaustive policies for polling systems in which a cycle begins whenever the server arrives at some prespecified queue. The input and output processes satisfy model-dependent conditions: pathwise bounds on the average rate and the burstiness (Cruz bounds); existence of long-run average rates; a pathwise generalized Law of the Iterated Logarithm; or exponentially or polynomially bounded tail probabilities of burstiness....
In this paper, we give evolution equations for free choice Petri net which generalize the max; +]... more In this paper, we give evolution equations for free choice Petri net which generalize the max; +]algebraic setting already known for event graphs. These evolution equations can be seen as a coupling of two linear systems, a (min; +)-linear system, and a quasi (+;)-linear one. This leads to new methods and algorithms to: In the untimed case: check liveness and several other basic logical properties; In the timed case: establish various conservation and monotonicity properties; In the stochastic case: check stability, i.e. the fact that the marking remains bounded in probability, and construct minimal stationary regimes. The main tools for proving these properties are graph theory, idempotent algebras and ergodic theory.
We consider a single-server cyclic polling system with three queues where the server follows an a... more We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.
We consider two independent homogeneous Poisson processes Π0and Π1in the plane with intensities λ... more We consider two independent homogeneous Poisson processes Π0and Π1in the plane with intensities λ0and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.
Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed... more Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution F supported on [−∞, 1] with essential supremum equal to 1 (a charge of −∞ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by C(F). Even in the simplest case where F = pδ 1 + (1 − p)δ −∞ , corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant C(F). Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.
We consider a sample path of a random walk on the integers with bounded local times, conditioned ... more We consider a sample path of a random walk on the integers with bounded local times, conditioned on the event that it hits a high level. Under an auxiliary assumption, we obtain representations for its distribution in terms of the corresponding limiting sequence. Then we prove limiting results as the high level grows. In particular, we generalize results for a simple symmetric random walk obtained earlier by Benjamini and Berectycki (2010).
This text studies heavy-tailed distributions in probability theory, and especially convolutions o... more This text studies heavy-tailed distributions in probability theory, and especially convolutions of such distributions. The mail goal is to provide a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions which includes many novel elements and, in particular, is based on the regular use of the principle of a single big jump.
We introduce a model for the classical synchronised multiple access system with a single transmis... more We introduce a model for the classical synchronised multiple access system with a single transmission channel and a randomised transmission protocol (ALOHA). We assume in addition that there is an energy harvesting mechanism, and any message transmission requires a unit of energy. Units of energy arrive randomly and independently of anything else. We analyze stability and instability conditions for this model.
We consider a discrete-time Markov chain (X t , Y t), t = 0, 1, 2,. . ., where the X-component fo... more We consider a discrete-time Markov chain (X t , Y t), t = 0, 1, 2,. . ., where the X-component forms a Markov chain itself. Assume that (X t) is Harris-ergodic and consider an auxiliary Markov chain { Y t } whose transition probabilities are the averages of transition probabilities of the Y-component of the (X, Y)-chain, where the averaging is weighted by the stationary distribution of the X-component. We first provide natural conditions in terms of test functions ensuring that the Y-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain (X t , Y t). The we prove a "multi-dimensional" extension of the result obtained. In the second part of the paper, we apply our results to two versions of a multi-access wireless model governed by two randomised protocols.
We consider spatial marked Poisson arrivals in a Polish space. These arrivals are accepted or los... more We consider spatial marked Poisson arrivals in a Polish space. These arrivals are accepted or lost in a general state dependent manner. The accepted arrivals remain in the system for a random amount of time, where the individual sojourn times are i.i.d. For such systems, we develop semi-closed form expressions for the steady state probabilities that can be seen to be insensitive to the sojourn time distribution, and that rely essentially on the static probabilities of marked Poisson objects meeting the state acceptance criteria. The latter observation is then exploited to yield straightforward exact simulation algorithms to sample from the steady state distribution. In addition, for the special case where the arrivals are spheres in a Euclidean space that are lost whenever they overlap with an existing sphere, we develop large deviations asymptotics for the probability of observing a large number of spheres in the system in steady state, under diverse asymptotic regimes. Application...
IEEE INFOCOM 2008 - The 27th Conference on Computer Communications, 2008
The split of a multihop, point-to-point TCP connection consists in replacing a plain, end-to-end ... more The split of a multihop, point-to-point TCP connection consists in replacing a plain, end-to-end TCP connection by a cascade of TCP connections. In such a cascade, connection n feeds connection n + 1 through some proxy node n. This technique is used in a variety of
Изучаются нижние пределы отношений F* r (x)/F(x) хвостов распре делений, где F* T-распределение с... more Изучаются нижние пределы отношений F* r (x)/F(x) хвостов распре делений, где F* T-распределение суммы случайного числа т независи мых одинаково распределенных случайных величин с общим распреде лением F, при этом г не зависит от слагаемых. Ключевые слова и фразы: хвост свертки распределений, случайные суммы случайных величин, нижний предел, распределения с тяжелыми и легкими хвостами.
Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an ar... more Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an arbitrary boundary. If the sequence increments satisfy a so-called condition of asymptotic homogeneity and the boundary is asymptotically "smooth," then the occurrence of the weak convergence to a limit shape (as the boundary is sent away) is established for the distribution of the overshoot value. As an application, we obtain a uniform (over the class of distributions) basic renewal theorem and determine the asymptotics of the average time of crossing a curvilinear border by the trajectories of asymptotically homogeneous Markov chains.
Consider a queueing system in which arriving customers are placed on a circle and wait for servic... more Consider a queueing system in which arriving customers are placed on a circle and wait for service. A traveling server moves at constant speed on the circle, stopping at the location of the customers until service completion. The server is greedy: always moving in the direction of the nearest customer. Coffman and Gilbert conjectured that this system is stable if the traffic intensity is smaller than 1; however, a proof or counterexample remains unknown. In this review, we present a picture of the current state of this conjecture and suggest new related open problems.
Random multiple-access protocols of type ALOHA are used to regulate networks with a star configur... more Random multiple-access protocols of type ALOHA are used to regulate networks with a star configuration where client nodes talk to the hub node at the same frequency (finding a wide range of applications among telecommunication systems, including mobile telephone networks and WiFi networks). Such protocols control who talks at what time sharing the common idea "try to send your data and, if your message collides with another transmission, try resending later". In the present paper, we consider a time-slotted ALOHA model where users are allowed to renege before transmission completion. We focus on the scenario that leads to overload in the absence of impatience. Under mild assumptions, we show that the fluid (or law-of-large-numbers) limit of the system workload coincides a.s. with the unique solution to a certain integral equation. We also demonstrate that the fluid limits for distinct initial conditions converge to the same value as time tends to infinity. Keywords ALOHA protocol • Queues with impatience • Queues in overload • Fluid limits Mathematics Subject Classification (2000) 60K25 • 60F17 • 90B15 • 90B22 The research of M. Frolkova and B. Zwart is supported by an NWO VIDI grant.
This note introduces a greedy walk on Poisson and Binomial processes, which is a close relative t... more This note introduces a greedy walk on Poisson and Binomial processes, which is a close relative to the well-known greedy server model. Some open problems are presented.
Regenerative events for different queueing models are considered. The aim of this paper is to con... more Regenerative events for different queueing models are considered. The aim of this paper is to construct these events for continuous-time processes if they are given for the corresponding discrete-time model. The construction uses so-called renovative events revealing the property of the state at time n of the discrete-time model to be independent (in an algebraic sense) of the states referring to epochs not later than n-L (where L is some constant) given that there are some restrictions on the "governing sequence". Different types of multi-server and multi-phase queues are considered.
Probability in the Engineering and Informational Sciences, 1998
We develop an algorithm for simulating approximate random samples from the invariant measure of a... more We develop an algorithm for simulating approximate random samples from the invariant measure of a Markov chain using backward coupling of embedded regeneration times. Related methods have been used effectively for finite chains and for stochastically monotone chains: here we propose a method of implementation which avoids these restrictions by using a “cycle-length” truncation. We show that the coupling times have good theoretical properties and describe benefits and difficulties of implementing the methods in practice.
We consider a generalized vacation or polling system, modeled as an input-output process operatin... more We consider a generalized vacation or polling system, modeled as an input-output process operating over successive “cycles,” in which the service mechanism can be in an “up” mode (processing) or “down” mode (e.g., vacation, walking). Our primary motivation is polling systems, in which there are several queues and the server moves cyclically between them providing some service in each. Our basic assumption is that the amount of work that leaves the system in a “cycle” is no less than the amount present at the beginning of the cycle. This includes the standard gated and exhaustive policies for polling systems in which a cycle begins whenever the server arrives at some prespecified queue. The input and output processes satisfy model-dependent conditions: pathwise bounds on the average rate and the burstiness (Cruz bounds); existence of long-run average rates; a pathwise generalized Law of the Iterated Logarithm; or exponentially or polynomially bounded tail probabilities of burstiness....
In this paper, we give evolution equations for free choice Petri net which generalize the max; +]... more In this paper, we give evolution equations for free choice Petri net which generalize the max; +]algebraic setting already known for event graphs. These evolution equations can be seen as a coupling of two linear systems, a (min; +)-linear system, and a quasi (+;)-linear one. This leads to new methods and algorithms to: In the untimed case: check liveness and several other basic logical properties; In the timed case: establish various conservation and monotonicity properties; In the stochastic case: check stability, i.e. the fact that the marking remains bounded in probability, and construct minimal stationary regimes. The main tools for proving these properties are graph theory, idempotent algebras and ergodic theory.
We consider a single-server cyclic polling system with three queues where the server follows an a... more We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.
We consider two independent homogeneous Poisson processes Π0and Π1in the plane with intensities λ... more We consider two independent homogeneous Poisson processes Π0and Π1in the plane with intensities λ0and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.
Uploads
Papers by Sergey Foss