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

1999 ◽  
Vol 36 (4) ◽  
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, λc], 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.

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.

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 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, ∞).

Limiting diffusions for the conditioned M/G/1 queue

1974 ◽  
Vol 11 (02) ◽  
pp. 355-362 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Traffic Intensity ◽  
Time Process ◽  
Brownian Excursion ◽  
Virtual Waiting Time ◽  
Waiting Time Process

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

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 expected number of crossings of a level in certain stochastic processes

1970 ◽  
Vol 7 (3) ◽  
pp. 766-770 ◽  
Author(s):  
P. B. M. Roes
Keyword(s):  
Present Method ◽  
Busy Period ◽  
Time Dependent ◽  
Renewal Processes ◽  
Time Process ◽  
Expected Number ◽  
Original Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

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