Invariance principles in queueing theory

1989 ◽  
Vol 26 (4) ◽  
pp. 845-857 ◽  
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Invariance principles in queueing theory

1989 ◽  
Vol 26 (04) ◽  
pp. 845-857
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk }and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Weak convergence in queueing theory

1973 ◽  
Vol 5 (3) ◽  
pp. 570-594 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Weak Convergence ◽  
Queueing Theory ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Probability Measures ◽  
Queueing Models ◽  
Applied Probability ◽  
Light Traffic ◽  
Survey Paper ◽  
Convergence Of Probability Measures

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (3) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Weak convergence in queueing theory

1973 ◽  
Vol 5 (03) ◽  
pp. 570-594 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Weak Convergence ◽  
Queueing Theory ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Probability Measures ◽  
Queueing Models ◽  
Applied Probability ◽  
Light Traffic ◽  
Survey Paper ◽  
Convergence Of Probability Measures

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (03) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

scholarly journals Heavy Traffic Feasible Hybrid Intracycle and Cyclic Sleep for Power Saving in 10G-EPON

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Xintian Hu ◽  
Liqian Wang ◽  
Zhiguo Zhang ◽  
Xue Chen
Keyword(s):  
Energy Consumption ◽  
Heavy Traffic ◽  
Optical Network ◽  
Power Saving ◽  
Passive Optical Network ◽  
Traffic Load ◽  
Contributing Factors ◽  
Light Traffic ◽  
Gigabit Ethernet ◽  
Qos Guarantee

Energy consumption in optical access networks costs carriers substantial operational expense (OPEX) every year and is one of contributing factors for the global warming. To reduce energy consumption in the 10-gigabit Ethernet passive optical network (10G-EPON), a hybrid intracycle and cyclic sleep mechanism is proposed in this paper. Under heavy traffic load, optical network units (ONUs) can utilize short idle slots within each scheduling cycle to enter intracycle sleep without postponing data transmission. In this way, energy conservation is achieved even under heavy traffic load with quality of service (QoS) guarantee. Under light traffic load, ONUs perform long cyclic sleep for several scheduling cycles. The adoption of cyclic sleep instead of intracycle sleep under light traffic load can reduce unnecessary frequent transitions between sleep and full active work caused by using intracycle sleep. Further, the Markov chain of the proposed mechanism is established. The performances of the proposed mechanism and existing approaches are analyzed quantitatively based on the chain. For the proposed mechanism, power saving ability with QoS guarantee even under heavy traffic and better power saving performance than existing approaches are verified by the quantitative analysis. Moreover, simulations validate the above conclusions based on the chain.

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