Approximating the stationary waiting time distribution function of GI/GI/1-queues with arithmetic interarrival time and service time distribution function

OR Spectrum ◽  
1982 ◽  
Vol 4 (3) ◽  
pp. 135-148 ◽  
Author(s):  
D. Wolf
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Interarrival Time ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time

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].

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

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

1987 ◽  
Vol 19 (01) ◽  
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].

On the concavity of the waiting-time distribution in some GI/G/1 queues

1986 ◽  
Vol 23 (02) ◽  
pp. 555-561 ◽  
Author(s):  
R. Szekli
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Service Time Distribution ◽  
Random Sum ◽  
Waiting Time Distribution ◽  
Idle Period ◽  
Period Distribution ◽  
Geometric Compound

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

On exponential bounds for the waiting-time distribution function in GI/G/1

1976 ◽  
Vol 13 (2) ◽  
pp. 411-417 ◽  
Author(s):  
R. Bergmann ◽  
D. Stoyan
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Time ◽  
Distribution Functions ◽  
Integral Inequalities ◽  
Arrival Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Bounds ◽  
Stationary Waiting Time

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

scholarly journals A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution

1998 ◽  
Vol 11 (3) ◽  
pp. 247-254 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Asymptotic Series ◽  
Traffic Load ◽  
Interarrival Time ◽  
Service Time Distribution ◽  
Tail Probabilities ◽  
S Transform ◽  
Stationary Waiting Time

For the GI/G/1 queueing model with traffic load a<1, service time distribution B(t) and interarrival time distribution A(t), whenever for t→∞1−B(t)∼c(t/β)ν+O(e−δt),c>0,1<ν<2,δ>0, and ∫0∞tμdA(t)<∞ for μ>ν, (1−a)1ν−1w converges in distribution for a↑1. Here w is distributed as the stationary waiting time distribution. The L.-S. transform of the limiting distribution is derived and an asymptotic series for its tail probabilities is obtained. The theorem actually proved in the text concerns a slightly more general asymptotic behavior of 1−B(t), t→∞, than mentioned above.

On the concavity of the waiting-time distribution in some GI/G/1 queues

1986 ◽  
Vol 23 (2) ◽  
pp. 555-561 ◽  
Author(s):  
R. Szekli
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Service Time Distribution ◽  
Random Sum ◽  
Waiting Time Distribution ◽  
Idle Period ◽  
Period Distribution ◽  
Geometric Compound

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

1998 ◽  
Vol 35 (02) ◽  
pp. 325-347 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Autocorrelation Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Matrix Measure ◽  
Markov Modulated

Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n &lt; 0, at least one W ij &gt; 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup n≥0 S n &gt; x] → (−𝔼X n )−1 ∫ x ∞ ℙ[X n &gt; u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

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.

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