scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (2) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (01) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

1971 ◽  
Vol 8 (3) ◽  
pp. 494-507 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Random Variable ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

On up- and downcrossings

1977 ◽  
Vol 14 (2) ◽  
pp. 405-410 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Processes ◽  
Waiting Time ◽  
Release Rate ◽  
Queueing System ◽  
Time Process ◽  
Simple Fact ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Busy Cycle

For the sample functions of the stationary virtual waiting-time process vt of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the vt-process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the vt-process of the GI/G/1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.

On up- and downcrossings

1977 ◽  
Vol 14 (02) ◽  
pp. 405-410 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Processes ◽  
Waiting Time ◽  
Release Rate ◽  
Queueing System ◽  
Time Process ◽  
Simple Fact ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Busy Cycle

For the sample functions of the stationary virtual waiting-time process vt of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the vt -process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the vt -process of the GI/G/1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

1971 ◽  
Vol 8 (03) ◽  
pp. 494-507 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Random Variable ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 &lt; t &lt; ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (1) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}.For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t).These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

The quasi-stationary distributions of queues in heavy traffic

1972 ◽  
Vol 9 (04) ◽  
pp. 821-831 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Stationary Distribution ◽  
Gamma Distribution ◽  
Arrival Time ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Time Process ◽  
Queue Size ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Quasi Stationary Distribution

This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn } of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.

A limit theorem for priority queues in heavy traffic

1973 ◽  
Vol 10 (04) ◽  
pp. 907-912 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Limit Theorem ◽  
Queueing System ◽  
Heavy Traffic ◽  
Critical Value ◽  
Single Server ◽  
Traffic Condition ◽  
Transient Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Priority Queueing

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

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