COMPUTATIONAL ANALYSIS OF STATIONARY WAITING-TIME DISTRIBUTIONS OF GIX/R/1 AND GIX/D/1 QUEUES

2005 ◽  
Vol 19 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Mohan L. Chaudhry ◽  
Dae W. Choi ◽  
Kyung C. Chae
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Computational Analysis ◽  
Rational Functions ◽  
Waiting Time Distribution ◽  
Generalized Distributions ◽  
Analytic Expression ◽  
Stationary Waiting Time ◽  
Stieltjes Transforms

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

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.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

The stationary work in system of a G/G/1 gradual input queue

1993 ◽  
Vol 30 (1) ◽  
pp. 207-222 ◽  
Author(s):  
Issei Kino ◽  
Masakiyo Miyazawa
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Work Load ◽  
Total Work ◽  
Waiting Time Distribution ◽  
Input Process ◽  
Explicit Formulas ◽  
Input Queue ◽  
Work Input ◽  
Stieltjes Transforms

This paper is devoted to the study of a stationary G/G/1 queue in which work input is gradually injected into the system and the work load is processed at unit rate. First, assuming that the input process is stationary, the key formula for the stationary distribution of work in system is derived by appeal to the waiting time of the associated regular G/G/1 queue. The main contribution of this paper is the derivation of Laplace-Stieltjes transforms (LSTs) for the stationary distributions of work in system and related random variables in terms of integrations with respect to a waiting time distribution of the associated regular queue. The results are exemplified by giving explicit formulas for the LST of total work for M/Ek/1 and M/H2/1. The results generalize the results of Pan et al. (1991) for the M/M/1 gradual input queue.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

1988 ◽  
Vol 25 (3) ◽  
pp. 636-641 ◽  
Author(s):  
V. Ramaswami ◽  
D. M. Lucantoni
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Higher Moments ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Recursive Relations

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

1988 ◽  
Vol 25 (03) ◽  
pp. 636-641
Author(s):  
V. Ramaswami ◽  
D. M. Lucantoni
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Higher Moments ◽  
Waiting Time Distribution ◽  
Stationary Waiting Time ◽  
Recursive Relations

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

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