Some results on regular variation for distributions in queueing and fluctuation theory

1973 ◽  
Vol 10 (2) ◽  
pp. 343-353 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Regular Variation ◽  
Time Distribution ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Distribution Functions ◽  
Service Time Distribution ◽  
Virtual Waiting Time ◽  
Necessary And Sufficient

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.

Some results on regular variation for distributions in queueing and fluctuation theory

1973 ◽  
Vol 10 (02) ◽  
pp. 343-353 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Regular Variation ◽  
Time Distribution ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Distribution Functions ◽  
Service Time Distribution ◽  
Virtual Waiting Time ◽  
Necessary And Sufficient

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 {x 1} &lt; 0 then the distribution of sup, s 1 s 2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x 1 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.

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

1969 ◽  
Vol 6 (3) ◽  
pp. 594-603 ◽  
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.

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

1969 ◽  
Vol 6 (03) ◽  
pp. 594-603 ◽  
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.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (3) ◽  
pp. 642-649 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

scholarly journals Note sur un modele de file GI/G/1 a service autonome (avec vacances du serveur)

1987 ◽  
Vol 19 (1) ◽  
pp. 289-291 ◽  
Author(s):  
Christine Fricker
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Conditional Distribution ◽  
Stochastic Decomposition ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Vacation Model ◽  
The Law

Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].

Single-server queues with impatient customers

1984 ◽  
Vol 16 (4) ◽  
pp. 887-905 ◽  
Author(s):  
F. Baccelli ◽  
P. Boyer ◽  
G. Hebuterne
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Distribution Functions ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Input Process ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
Random Threshold ◽  
The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

Insensitivity in queueing systems

1981 ◽  
Vol 13 (04) ◽  
pp. 846-859 ◽  
Author(s):  
David Y. Burman
Keyword(s):  
Stationary Distribution ◽  
Service Time ◽  
Stochastic Models ◽  
Time Distribution ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Service Time Distribution ◽  
Flow Equations

It is well known that the stationary distribution of the number of busy servers in the Erlang blocking system (M/G/c/c) depends on the service-time distribution only through its mean. This insensitivity property is shared by several other queueing systems. In this paper, we give simple sufficient conditions for determining if this insensitivity property holds for general queueing systems and related stochastic models. The conditions involve determining whether the solution of the stationary Markovian flow equations also solves certain restricted flow equations. The proof that these conditions are sufficient is direct and elementary.

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