Regenerative processes in the theory of queues, with applications to the alternating-priority queue

1972 ◽  
Vol 4 (03) ◽  
pp. 542-577 ◽  
Author(s):  
Shaler Stidham
Keyword(s):  
Stochastic Process ◽  
Steady State ◽  
Wide Class ◽  
Asymptotic Properties ◽  
Arbitrary Point ◽  
Priority Queue ◽  
Regenerative Processes ◽  
Simple Derivation ◽  
State Delay ◽  
Poisson Arrivals

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.

Regenerative processes in the theory of queues, with applications to the alternating-priority queue

1972 ◽  
Vol 4 (3) ◽  
pp. 542-577 ◽  
Author(s):  
Shaler Stidham
Keyword(s):  
Stochastic Process ◽  
Steady State ◽  
Wide Class ◽  
Asymptotic Properties ◽  
Arbitrary Point ◽  
Priority Queue ◽  
Regenerative Processes ◽  
Simple Derivation ◽  
State Delay ◽  
Poisson Arrivals

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.

Polling systems with periodic server routeing in heavy traffic: distribution of the delay

2003 ◽  
Vol 40 (02) ◽  
pp. 305-326 ◽  
Author(s):  
Tava Lennon Olsen ◽  
R. D. van der Mei
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Heavy Traffic ◽  
Linear Equations ◽  
Asymptotic Properties ◽  
Polling Systems ◽  
Parameter Setting ◽  
State Delay ◽  
Traffic Distribution ◽  
General Parameter

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of only M—N linear equations, where M is the length of the polling table and N is the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

Polling systems with periodic server routeing in heavy traffic: distribution of the delay

2003 ◽  
Vol 40 (2) ◽  
pp. 305-326 ◽  
Author(s):  
Tava Lennon Olsen ◽  
R. D. van der Mei
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Heavy Traffic ◽  
Linear Equations ◽  
Asymptotic Properties ◽  
Polling Systems ◽  
Parameter Setting ◽  
State Delay ◽  
Traffic Distribution ◽  
General Parameter

We consider polling systems with mixtures of exhaustive and gated service in which the server visits the queues periodically according to a general polling table. We derive exact expressions for the steady-state delay incurred at each of the queues under standard heavy-traffic scalings. The expressions require the solution of a set of onlyM—Nlinear equations, whereMis the length of the polling table andNis the number of queues, but are otherwise explicit. The equations can even be expressed in closed form for several routeing schemes commonly used in practice, such as the star and elevator visit order, in a general parameter setting. The results reveal a number of asymptotic properties of the behavior of polling systems. In addition, the results lead to simple and fast approximations for the distributions and the moments of the delay in stable polling systems with periodic server routeing. Numerical results demonstrate that the approximations are highly accurate for medium and heavily loaded systems.

scholarly journals Stability and moment bounds under utility-maximising service allocations: Finite and infinite networks

2020 ◽  
Vol 52 (2) ◽  
pp. 463-490
Author(s):  
Seva Shneer ◽  
Alexander Stolyar
Keyword(s):  
Steady State ◽  
Wide Class ◽  
Continuous Time ◽  
Service Rate ◽  
Fluid Limit ◽  
Natural Conditions ◽  
Infinite Systems ◽  
Moment Bounds ◽  
Infinite Networks ◽  
State Moment

AbstractWe study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For finite networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov–Foster-type criteria, and apply to a wide class of systems, including those for which fluid-limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of infinite networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

scholarly journals Geometry and dynamics of admissible metrics in measure spaces

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Anatoly Vershik ◽  
Pavel Zatitskiy ◽  
Fedor Petrov
Keyword(s):  
Invariant Measure ◽  
Wide Class ◽  
Lebesgue Space ◽  
Measure Space ◽  
Asymptotic Properties ◽  
Ergodic Transformation ◽  
Measure Spaces ◽  
Admissible Metric ◽  
Discreteness Criterion ◽  
Uniformly Bounded

AbstractWe study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

The basic equations for a supplemented GSMP and its applications to queues

1988 ◽  
Vol 25 (03) ◽  
pp. 565-578 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Stochastic Process ◽  
Steady State ◽  
Queueing Networks ◽  
General Setting ◽  
Main Concern ◽  
Stationary Condition ◽  
State Equations ◽  
Semi Markov Process ◽  
Basic Equations ◽  
Stieltjes Transforms

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

GRAPHICAL METHODS FOR DETERMINING/ESTIMATING OPTIMAL REPAIR-LIMIT REPLACEMENT POLICIES

2000 ◽  
Vol 07 (01) ◽  
pp. 43-60 ◽  
Author(s):  
TADASHI DOHI ◽  
KENTARO TAKEITA ◽  
SHUNJI OSAKI
Keyword(s):  
Steady State ◽  
Asymptotic Properties ◽  
Repair Time ◽  
Graphical Methods ◽  
Test Statistics ◽  
Numerical Examples ◽  
Time Limits ◽  
Nonparametric Estimators

In this paper, we consider two kinds of repair-limit replacement models and develop the corresponding graphical methods to estimate the optimal repair-time limits which minimize the expected costs per unit time in the steady state. Then, both the total time on test statistics and the Lorenz statistics play important roles to develop nonparametric estimators of the optimal repair-time limits. Numerical examples are devoted to illustrate asymptotic properties of nonparametric estimators for the optimal repair-limit policies.

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