A join the shorter queue model in heavy traffic

2000 ◽  
Vol 37 (01) ◽  
pp. 212-223 ◽  
Author(s):  
Stephen R. E. Turner
Keyword(s):  
State Space ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Resource Pooling ◽  
Heavy Traffic Limit ◽  
Queue Model ◽  
Limit Result

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

A join the shorter queue model in heavy traffic

2000 ◽  
Vol 37 (1) ◽  
pp. 212-223 ◽  
Author(s):  
Stephen R. E. Turner
Keyword(s):  
State Space ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Resource Pooling ◽  
Heavy Traffic Limit ◽  
Queue Model ◽  
Limit Result

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (04) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system withNusers, and any set of positive numbers {an},n= 1, 2,…,N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each timet, it (asymptotically) minimizes maxnanq̃n(t), whereq̃n(t) is the queue length of usernin the heavy-traffic regime.

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.

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (4) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system with N users, and any set of positive numbers {an}, n = 1, 2,…, N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each time t, it (asymptotically) minimizes maxnanq̃n(t), where q̃n(t) is the queue length of user n in the heavy-traffic regime.

Heavy Traffic Analysis of Approximate Max-Weight Matching Algorithms for Input-Queued Switches

2021 ◽  
Vol 48 (3) ◽  
pp. 109-110
Author(s):  
Yu Huang ◽  
Longbo Huang
Keyword(s):  
Approximation Algorithms ◽  
State Space ◽  
Queue Length ◽  
Heavy Traffic ◽  
Traffic Analysis ◽  
The State ◽  
Heavy Traffic Analysis ◽  
Heavy Traffic Limit

In this paper, we propose a class of approximation algorithms for max-weight matching (MWM) policy for input-queued switches, called expected 1-APRX. We establish the state space collapse (SSC) result for expected 1-APRX, and characterize its queue length behavior in the heavy-traffic limit.

Asymptotic Properties of a Recursive Procedure for Simultaneous Estimation

1990 ◽  
Vol 4 (4) ◽  
pp. 461-475
Author(s):  
Martin I. Reiman
Keyword(s):  
Central Limit ◽  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Queueing Network ◽  
Random Variable ◽  
Simultaneous Estimation ◽  
Recursive Procedure ◽  
Heavy Traffic Limit ◽  
Sojourn Time Distribution ◽  
Pass Through

In this paper we consider a problem that arises in estimating the heavy traffic limit of a sojourn time distribution in a queueing network during the course of a medium traffic simulation. We need to estimate α = E[f(γ, M)], where γ is an unknown constant and M a random variable. More specifically, we are given an iid sequence of random vectors {(Xi, Mi), 1 ≤ i ≤ n}, with γ = E[Xi] and Mi having the same distribution as M.For known γ, we have a standard estimation problem, which we describe here. The standard estimate is unbiased and asymptotically (as n → 8 ) consistent. There is also a central limit theorem for this estimator. For unknown γ, we provide two estimation procedures, one that requires two passes through the data (as well as storage of {Mi, 1 ≤ i ≤ n}), and another one, which is recursive, requiring only one pass through and bounded storage. The estimators obtained from these two procedures are shown to be strongly consistent, and central limit theorems are also proven for them.

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

1992 ◽  
Vol 24 (1) ◽  
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.

Resource pooling in queueing networks with dynamic routing

1992 ◽  
Vol 24 (3) ◽  
pp. 699-726 ◽  
Author(s):  
C. N. Laws
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Optimal Control Problem ◽  
Queueing Networks ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Original System ◽  
Dynamic Routing ◽  
Resource Pooling ◽  
Service Stations

In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.

Resource pooling in queueing networks with dynamic routing

1992 ◽  
Vol 24 (03) ◽  
pp. 699-726 ◽  
Author(s):  
C. N. Laws
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Optimal Control Problem ◽  
Queueing Networks ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Original System ◽  
Dynamic Routing ◽  
Resource Pooling ◽  
Service Stations

In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.

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