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 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.

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.

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 waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (1) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts, R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for the M/M/1/1 retrial queue and derive expressions for mean, variance and generating function of R. Limit theorems are stated for heavy- and light-traffic cases.

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.

On the waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (01) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts,R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for theM/M/1/1 retrial queue and derive expressions for mean, variance and generating function ofR.Limit theorems are stated for heavy- and light-traffic cases.

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