Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (01) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Exponential approximation of waiting time and queue size for queues in heavy traffic

1990 ◽  
Vol 22 (1) ◽  
pp. 230-240 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Wide Class ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Single Server ◽  
Queue Size ◽  
Exponential Approximation ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

The diffusion approximation for tandem queues in heavy traffic

1978 ◽  
Vol 10 (04) ◽  
pp. 886-905 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Waiting Time ◽  
Limit Distribution ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Boundary Surface ◽  
Sufficient Conditions ◽  
Single Server ◽  
Waiting Time Distribution

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved. It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated. It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z ∗. The generator of Z ∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.

The diffusion approximation for tandem queues in heavy traffic

1978 ◽  
Vol 10 (4) ◽  
pp. 886-905 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Waiting Time ◽  
Limit Distribution ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Boundary Surface ◽  
Sufficient Conditions ◽  
Single Server ◽  
Waiting Time Distribution

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved.It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated.It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z∗. The generator of Z∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (03) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (02) ◽  
pp. 433-445 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

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