scholarly journals Tails in generalized Jackson networks with subexponential service-time distributions

2005 ◽  
Vol 42 (02) ◽  
pp. 513-530 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss ◽  
Marc Lelarge
Keyword(s):  
Closed Form ◽  
Service Time ◽  
Large Deviation ◽  
Exact Asymptotics ◽  
Fluid Limits ◽  
Jackson Networks ◽  
Single Station ◽  
Service Times

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

scholarly journals Tails in generalized Jackson networks with subexponential service-time distributions

2005 ◽  
Vol 42 (2) ◽  
pp. 513-530 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss ◽  
Marc Lelarge
Keyword(s):  
Closed Form ◽  
Service Time ◽  
Large Deviation ◽  
Exact Asymptotics ◽  
Fluid Limits ◽  
Jackson Networks ◽  
Single Station ◽  
Service Times

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme

2007 ◽  
Vol 44 (02) ◽  
pp. 379-392
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Closed Form ◽  
Long Range ◽  
Covariance Function ◽  
Queueing System ◽  
Regenerative Process ◽  
Long Range Dependence ◽  
Exact Asymptotics ◽  
Regenerative Processes ◽  
Service Times ◽  
Heavy Tailed

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

scholarly journals On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme

2007 ◽  
Vol 44 (02) ◽  
pp. 379-392
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Closed Form ◽  
Long Range ◽  
Covariance Function ◽  
Queueing System ◽  
Regenerative Process ◽  
Long Range Dependence ◽  
Exact Asymptotics ◽  
Regenerative Processes ◽  
Service Times ◽  
Heavy Tailed

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

scholarly journals On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme

2007 ◽  
Vol 44 (2) ◽  
pp. 379-392 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Closed Form ◽  
Long Range ◽  
Covariance Function ◽  
Queueing System ◽  
Regenerative Process ◽  
Long Range Dependence ◽  
Exact Asymptotics ◽  
Regenerative Processes ◽  
Service Times ◽  
Heavy Tailed

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

scholarly journals The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

2019 ◽  
Vol 51 (03) ◽  
pp. 835-864 ◽  
Author(s):  
Yuqing Pan ◽  
Konstantin A. Borovkov
Keyword(s):  
Large Deviation ◽  
Risk Process ◽  
Boundary Crossing ◽  
Moment Condition ◽  
Two Dimensional ◽  
Exact Asymptotics ◽  
Negative Component ◽  
Positive Orthant ◽  
Mean Vector ◽  
Target Sets

AbstractFor a multivariate random walk with independent and identically distributed jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results of Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by Borovkov and Mogulskii from around 2000 with new auxiliary constructions, enabling us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a ‘corner’ at the ‘most probable hitting point’. We also discuss how our results can be extended to the case of more general target sets.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

scholarly journals THE TIME-TO-EMPTY FOR TANDEM JACKSON NETWORKS

2004 ◽  
Vol 18 (4) ◽  
pp. 493-502
Author(s):  
Michael A. Zazanis
Keyword(s):  
Jackson Network ◽  
Jackson Networks ◽  
Service Times

We derive the distribution of the time-to-empty for an open tandem Jackson network assuming that while in equilibrium at time 0, the arrival stream is suddenly shut off. The analysis is based on analogous results regarding the distribution of the time-to-empty for the corresponding closed tandem Jackson network. The results obtained are used in the analysis of a two-class tandem Jackson network with FIFO discipline in which customers of the second class have negligible service times.

scholarly journals DELAY MODELS BASED ON SYSTEMS WITH USUAL AND SHIFTED HYPEREXPONENTIAL AND HYPERERLANGIAN INPUT DISTRIBUTIONS

2021 ◽  
pp. 56-64
Author(s):  
V. N. Tarasov ◽  
N. F. Bakhareva
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Waiting Time ◽  
Service Time ◽  
Spectral Decomposition ◽  
Flow Distribution ◽  
Input Flow ◽  
Average Delay ◽  
Spectral Expansions ◽  
Wide Range

Context. In the queueing theory, the study of systems with arbitrary laws of the input flow distribution and service time is relevant because it is impossible to obtain solutions for the waiting time in the final form for the general case. Therefore, the study of such systems for particular cases of input distributions is important. Objective. Getting a solution for the average delay in the queue in a closed form for queuing systems with ordinary and with shifted to the right from the zero point hyperexponential and hypererlangian distributions in stationary mode. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average delay for two systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used. Results. For the first time, a spectral decomposition of the solution of the Lindley integral equation for systems with ordinary and with shifted hyperexponential and hyperelangian distributions is obtained, which is used to derive a formula for the average delay in a queue in closed form. Conclusions. It is proved that the spectral expansions of the solution of the Lindley integral equation for the systems under consideration coincide; therefore, the formulas for the mean delay will also coincide. It is shown that in systems with a delay, the average delay is less than in conventional systems. The obtained expression for the waiting time expands and complements the wellknown incomplete formula of queuing theory for the average delay for systems with arbitrary laws of the input flow distribution and service time. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average waiting time, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time.

A note on the queueing system GI/G/∞

1968 ◽  
Vol 64 (2) ◽  
pp. 477-479 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Queueing System ◽  
Expected Value ◽  
Arrival Times ◽  
Service Times

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

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