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 Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

2020 ◽  
Vol 45 (3) ◽  
pp. 1069-1103
Author(s):  
Anton Braverman
Keyword(s):  
Steady State ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Time Intervals ◽  
Dimensional Diffusion ◽  
Queue Model ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Process Level ◽  
General Tool

This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models

1992 ◽  
Vol 24 (01) ◽  
pp. 172-201 ◽  
Author(s):  
Søren Asmussen ◽  
Reuven Y. Rubinstein
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Score Function ◽  
Service Rate ◽  
Underlying Distribution ◽  
Control Variate ◽  
Heavy Traffic Limit ◽  
Standard Output

This paper studies computer simulation methods for estimating the sensitivities (gradient, Hessian etc.) of the expected steady-state performance of a queueing model with respect to the vector of parameters of the underlying distribution (an example is the gradient of the expected steady-state waiting time of a customer at a particular node in a queueing network with respect to its service rate). It is shown that such a sensitivity can be represented as the covariance between two processes, the standard output process (say the waiting time process) and what we call the score function process which is based on the score function. Simulation procedures based upon such representations are discussed, and in particular a control variate method is presented. The estimators and the score function process are then studied under heavy traffic conditions. The score function process, when properly normalized, is shown to have a heavy traffic limit involving a certain variant of two-dimensional Brownian motion for which we describe the stationary distribution. From this, heavy traffic (diffusion) approximations for the variance constants in the large sample theory can be computed and are used as a basis for comparing different simulation estimators. Finally, the theory is supported by numerical results.

Markov chains and generalized continued fractions

1992 ◽  
Vol 29 (04) ◽  
pp. 838-849 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Wide Class ◽  
Stochastic Models ◽  
Continuous Time ◽  
Continued Fractions ◽  
Invariant Measures ◽  
Continuous Time Markov Chains ◽  
Generalized Continued Fractions

This paper deals with a class of discrete-time Markov chains for which the invariant measures can be expressed in terms of generalized continued fractions. The representation covers a wide class of stochastic models and is well suited for numerical applications. The results obtained can easily be extended to continuous-time Markov chains.

On the GIX/G/∞ system

1990 ◽  
Vol 27 (3) ◽  
pp. 671-683 ◽  
Author(s):  
L. Liu ◽  
B. R. K. Kashyap ◽  
J. G. C. Templeton
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Continuous Time ◽  
Transient State ◽  
System Size ◽  
Noise Process ◽  
Shot Noise Process ◽  
Special Cases

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

On the GIX/G/∞ system

1990 ◽  
Vol 27 (03) ◽  
pp. 671-683 ◽  
Author(s):  
L. Liu ◽  
B. R. K. Kashyap ◽  
J. G. C. Templeton
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Continuous Time ◽  
Transient State ◽  
System Size ◽  
Noise Process ◽  
Shot Noise Process ◽  
Special Cases

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

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.

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