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.

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.

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 TWO-PHASE GATED SERVICE

2008 ◽  
Vol 22 (4) ◽  
pp. 623-651 ◽  
Author(s):  
R. D. van der Mei ◽  
J. A. C. Resing
Keyword(s):  
Waiting Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Polling Systems ◽  
Polling System ◽  
Two Phase ◽  
General Parameter ◽  
Multitype Branching Processes ◽  
Poisson Arrivals ◽  
Gated Service

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

scholarly journals Customer-Server Population Dynamics in Heavy Traffic

2021 ◽  
Author(s):  
Rami Atar ◽  
Prasenjit Karmakar ◽  
David Lipshutz
Keyword(s):  
Population Dynamics ◽  
Steady State ◽  
Closed Form ◽  
Population Size ◽  
Time Scales ◽  
Performance Measures ◽  
Heavy Traffic ◽  
The Other ◽  
Queueing Model ◽  
Server Vacations

We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin–Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate near Halfin–Whitt regime, whereas the dynamics asymptotically decouple in the other heavy-traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b)—namely, the probability of wait and the slowdown. Although closed-form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed-form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.

scholarly journals Polling Systems with Zero Switchover Times: A Heavy-Traffic Averaging Principle

1995 ◽  
Vol 5 (3) ◽  
pp. 681-719 ◽  
Author(s):  
E. G. Coffman ◽  
A. A. Puhalskii ◽  
M. I. Reiman
Keyword(s):  
Heavy Traffic ◽  
Polling Systems ◽  
Averaging Principle ◽  
Switchover Times

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.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

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