Fundamentals of Random and Traffic Processes

2020 ◽  
pp. 77-102
Author(s):  
Marco Guerrieri ◽  
Raffaele Mauro
Keyword(s):  
Traffic Processes

scholarly journals Correlations in Output and Overflow Traffic Processes in Simple Queues

2007 ◽  
Vol 2007 ◽  
pp. 1-13
Author(s):  
Don McNickle
Keyword(s):  
Cross Correlation ◽  
Counting Processes ◽  
Storage Space ◽  
Output Process ◽  
Overflow Traffic ◽  
The Cross ◽  
Departure Processes ◽  
Traffic Processes ◽  
Overflow Process

We consider some simple Markov and Erlang queues with limited storage space. Although the departure processes from some such systems are known to be Poisson, they actually consist of the superposition of two complex correlated processes, the overflow process and the output process. We measure the cross-correlation between the counting processes for these two processes. It turns out that this can be positive, negative, or even zero (without implying independence). The models suggest some general principles on how big these correlations are, and when they are important. This may suggest when renewal or moment approximations to similar processes will be successful, and when they will not.

Characterizations of Poisson traffic streams in Jackson queueing networks

1979 ◽  
Vol 11 (02) ◽  
pp. 422-438 ◽  
Author(s):  
Benjamin Melamed
Keyword(s):  
Queueing Networks ◽  
Simple Graph ◽  
Single Server ◽  
Jackson Network ◽  
Jackson Networks ◽  
Equilibrium Behavior ◽  
Poisson Arrivals ◽  
Stochastic Properties ◽  
Networks Of Queues ◽  
Traffic Processes

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues. A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

Characterizations of Poisson traffic streams in Jackson queueing networks

1979 ◽  
Vol 11 (2) ◽  
pp. 422-438 ◽  
Author(s):  
Benjamin Melamed
Keyword(s):  
Queueing Networks ◽  
Simple Graph ◽  
Single Server ◽  
Jackson Network ◽  
Jackson Networks ◽  
Equilibrium Behavior ◽  
Poisson Arrivals ◽  
Stochastic Properties ◽  
Networks Of Queues ◽  
Traffic Processes

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues.A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

Explosive synchronization induced by traffic processes in complex networks

2020 ◽  
Vol 2020 (8) ◽  
pp. 083404
Author(s):  
Jun-Jie Chen ◽  
Xiang Ling ◽  
Ning Guo
Keyword(s):  
Complex Networks ◽  
Traffic Processes ◽  
Explosive Synchronization

scholarly journals On Markovian traffic with applications to TES processes

1994 ◽  
Vol 7 (3) ◽  
pp. 373-396
Author(s):  
David L. Jagerman ◽  
Benjamin Melamed
Keyword(s):  
Integral Equations ◽  
Queueing Systems ◽  
Renewal Theory ◽  
Transform Domain ◽  
Arrival Times ◽  
Second Order Statistics ◽  
Two Measures ◽  
The Times ◽  
The Relationship ◽  
Traffic Processes

Markov processes are an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose inter-arrival times (separating the time points of discrete arrivals) form a real-valued Markov chain. As such this paper aims to extend the classical results of renewal traffic, where interarrival times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These are then specialized to a subclass, TES+, of a versatile class of random sequences, called TES (Transform-Expand-Sample), consisting of marginally uniform autoregressive schemes with modulo-1 reduction, followed by various transformations. TES models are designed to simultaneously capture both first-order and second-order statistics of empirical records, and consequently can produce high-fidelity models. Two theoretical solutions for TES+ traffic functions are derived: an operator-based solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closed-form representations for two measures of traffic burstiness: peakedness and index of dispersion, elucidating the relationship between them.

Traffic processes in queueing networks

1988 ◽  
Vol 36 (1) ◽  
pp. 135
Author(s):  
János Sztrik
Keyword(s):  
Queueing Networks ◽  
Traffic Processes

scholarly journals Mathematical Problems of Collaborating Robotic Vehicle Traffic

2021 ◽  
pp. 67-74
Author(s):  
В. А. Галкин
Keyword(s):  
Traffic Flows ◽  
Vehicle Traffic ◽  
Vlasov Equations ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Mathematical Problems ◽  
Robotic Vehicle ◽  
Self Consistent Field Theory ◽  
Traffic Processes

Предложен подход для моделирования динамики транспортных потоков для взаимодействующих аппаратов на основе теории самосогласованного поля, основанного на уравнениях А.А. Власова. Сформулированы проблемы применимости таких моделей для описания коллективных явлений трафика в связи с задачами поведения «стаи» роботизированных однородных взаимодействующих аппаратов в фазовом пространстве на основе кинетического подхода. An approach to the simulation of time-dependent collaborating vehicle traffic flows based on the self-consistent field theory and A. Vlasov equations are proposed. The problems of the simulation model applicability to collaborative traffic processes such as the behavior of a swarm of identical collaborating vehicles in phase space using the kinetic approach are stated.

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