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.

A limit theorem for priority queues in heavy traffic

1973 ◽  
Vol 10 (4) ◽  
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.

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.

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.

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 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.

On the M/G/1 queue by additional inputs

1984 ◽  
Vol 21 (1) ◽  
pp. 129-142 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Busy Period ◽  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Input Process ◽  
Stochastic Monotonicity ◽  
General Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Monotonicity Results

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system.The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.

On the M/G/1 queue by additional inputs

1984 ◽  
Vol 21 (01) ◽  
pp. 129-142
Author(s):  
Teunis J. Ott
Keyword(s):  
Busy Period ◽  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Input Process ◽  
Stochastic Monotonicity ◽  
General Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Monotonicity Results

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system. The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

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.

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, ∞).

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