scholarly journals Validity of heavy traffic steady-state approximations in generalized Jackson networks

2006 ◽  
Vol 16 (1) ◽  
pp. 56-90 ◽  
Author(s):  
David Gamarnik ◽  
Assaf Zeevi
Keyword(s):  
Steady State ◽  
Heavy Traffic ◽  
Jackson Networks

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.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (04) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

A heavy traffic result for the finite dam

1973 ◽  
Vol 10 (01) ◽  
pp. 223-228 ◽  
Author(s):  
Nils Blomqvist
Keyword(s):  
Steady State ◽  
Exponential Distribution ◽  
Discrete Time ◽  
Queueing Theory ◽  
Heavy Traffic ◽  
Previous Knowledge ◽  
Finite Dam ◽  
State Content ◽  
General Conditions

The steady state content Z of a finite dam in discrete time is investigated for small absolute values of the expected net input and correspondingly large values of the dam capacity. It is shown that under general conditions Z has, asymptotically, a truncated exponential distribution, a result that supplements previous knowledge in queueing theory.

A new heavy traffic limit for the asymmetric shortest queue problem

1999 ◽  
Vol 10 (5) ◽  
pp. 497-509 ◽  
Author(s):  
CHARLES KNESSL
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Parabolic Partial Differential Equation ◽  
Service Rate ◽  
Heavy Traffic Limit ◽  
Dimensional Approximation ◽  
Shortest Queue ◽  
Shortest Queue Problem

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

scholarly journals HEAVY-TRAFFIC ANALYSIS OF A NON-PREEMPTIVE MULTI-CLASS QUEUE WITH RELATIVE PRIORITIES

2015 ◽  
Vol 29 (2) ◽  
pp. 153-180 ◽  
Author(s):  
A. Izagirre ◽  
I.M. Verloop ◽  
U. Ayesta
Keyword(s):  
Steady State ◽  
State Space ◽  
Waiting Time ◽  
Queue Length ◽  
Heavy Traffic ◽  
Random Variables ◽  
Random Variable ◽  
Traffic Analysis ◽  
Service Requirement ◽  
Heavy Traffic Analysis

We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue-length vector in the heavy-traffic regime, that is, in the limit the scaled queue-length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. We observe that the scaled queue length reduces as classes with smaller mean service requirement obtain relatively larger weights. We finally show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables.

On queues in which customers are served in random order

1962 ◽  
Vol 58 (1) ◽  
pp. 79-91 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Steady State ◽  
Exponential Decay ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Asymptotic Form ◽  
Heavy Traffic ◽  
Formal Solution ◽  
Random Order ◽  
Waiting Time Distribution

ABSTRACTThe queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.

Heavy traffic analysis for continuous polling models

1997 ◽  
Vol 34 (03) ◽  
pp. 720-732 ◽  
Author(s):  
Dirk P. Kroese
Keyword(s):  
Steady State ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Traffic Analysis ◽  
Polling System ◽  
Polling Models ◽  
Heavy Traffic Analysis ◽  
Age Dependent ◽  
The Relationship

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

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.

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