The quasi-stationary distributions of queues in heavy traffic

1972 ◽  
Vol 9 (4) ◽  
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 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.

On the quasi-stationary distributions of the GI/M/1 queue

1972 ◽  
Vol 9 (01) ◽  
pp. 117-128 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Gamma Distribution ◽  
Waiting Time ◽  
Limit Distribution ◽  
Weighted Sum ◽  
Traffic Intensity ◽  
Time Process ◽  
Queue Size ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Conditional Limit

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).

On the quasi-stationary distributions of the GI/M/1 queue

1972 ◽  
Vol 9 (1) ◽  
pp. 117-128 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Gamma Distribution ◽  
Waiting Time ◽  
Limit Distribution ◽  
Weighted Sum ◽  
Traffic Intensity ◽  
Time Process ◽  
Queue Size ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Conditional Limit

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).

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, ∞).

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.

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, ∞).

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

1999 ◽  
Vol 36 (04) ◽  
pp. 1194-1209 ◽  
Author(s):  
Raúl Fierro ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Initial Point ◽  
Time Process ◽  
Conditional Distributions ◽  
Exponential Law ◽  
Invariant Distributions ◽  
Virtual Waiting Time ◽  
The Family ◽  
Waiting Time Process ◽  
Negative Drift ◽  
Limiting Behaviour

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λ c ], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

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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