Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

For the distribution functions of the stationary actual waiting time and of the stationary virtual waiting time of the GI/G/l queueing system it is shown that the tails vary regularly at infinity if and only if the tail of the service time distribution varies regularly at infinity.For sn the sum of n i.i.d. variables xi, i = 1, …, n it is shown that if E {x1} < 0 then the distribution of sup, s1s2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x1 varies regularly at infinity and conversely, moreover varies regularly at + ∞.In the appendix a lemma and its proof are given providing necessary and sufficient conditions for regular variation of the tail of a compound Poisson distribution.

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Algorithms for waiting-time distributions under various queue disciplines in the M/G/1 queue by a service-time distribution of phase type

Advances in Applied Probability ◽

10.2307/1425901 ◽

1976 ◽

Vol 8 (2) ◽

pp. 252-252 ◽

Cited By ~ 3

Author(s):

Marcel F. Neuts

Keyword(s):

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Service Time Distribution ◽

Waiting Time Distributions ◽

Phase Type

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Rate conservation for stationary processes

Journal of Applied Probability ◽

10.2307/3214747 ◽

1991 ◽

Vol 28 (1) ◽

pp. 146-158 ◽

Cited By ~ 11

Author(s):

Josep M. Ferrandiz ◽

Aurel A. Lazar

Keyword(s):

Service Time ◽

Conservation Law ◽

Time Distribution ◽

Distribution Density ◽

Queueing System ◽

Arrival Process ◽

Stationary Processes ◽

Service Time Distribution ◽

Stochastic Intensity ◽

Residual Service

We derive a rate conservation law for distribution densities which extends a result of Brill and Posner. Based on this conservation law, we obtain a generalized Takács equation for the G/G/m/B queueing system that only requires the existence of a stochastic intensity for the arrival process and the residual service time distribution density for the G/GI/1/B queue. Finally, we solve Takács' equation for the N/GI/1/∞ queueing system.

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On the service time distribution and the waiting time process of a potentially infinite capacity queueing system

Journal of Applied Probability ◽

10.2307/3212105 ◽

1969 ◽

Vol 6 (3) ◽

pp. 594-603 ◽

Cited By ~ 20

Author(s):

Nasser Hadidi

Keyword(s):

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Queueing System ◽

Service Time Distribution ◽

Waiting Time Distribution ◽

Poisson Stream ◽

Waiting Time Process ◽

The Subject ◽

Effective Service

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.

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