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 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.

The Equilibrium Distribution for a Clocked Buffered Switch

1992 ◽  
Vol 6 (4) ◽  
pp. 425-438 ◽  
Author(s):  
Steven Jaffe
Keyword(s):  
Joint Distribution ◽  
Equilibrium Distribution ◽  
Heavy Traffic ◽  
Basic Element ◽  
Asymptotic Results ◽  
Heavy Traffic Limit ◽  
Parallel Data ◽  
Shortest Queue ◽  
Bivariate Generating Function ◽  
Shortest Queue Problem

A 2-by-2 buffered switch is the basic element of certain parallel data-processing networks. For a switch fed by two independent Bernoulli input streams, we find the joint distribution of the number of messages waiting in the two buffers at equilibrium, in the form of a bivariate generating function. The derivation uses complex-variable techniques developed by Kingman and by Flatto and McKean for the “shortest queue problem.” A number of asymptotic results are given, the principal one being the variance of the total number of waiting messages in the heavy-traffic limit.

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.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

The steady-state solution of the M/K2/m queue

1980 ◽  
Vol 12 (03) ◽  
pp. 799-823
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Simple Formula ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
Server Queue ◽  
The Mean ◽  
Mean Queue Length ◽  
The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

scholarly journals A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics

1999 ◽  
Vol 12 (4) ◽  
pp. 311-338 ◽  
Author(s):  
Charles Knessl
Keyword(s):  
Diffusion Approximation ◽  
Queue Length ◽  
Heavy Traffic ◽  
Transient Behavior ◽  
Processor Sharing ◽  
Parallel Queues ◽  
Poisson Arrival ◽  
Heavy Traffic Limit ◽  
Service Rates ◽  
Dependent Probability

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate λ and have service rates equal to μ. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-of-the-line processor sharing. We study this model in the heavy traffic limit, where ρ=λ/μ→1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.

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